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wordcount.html
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wordcount.html
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<h1>LaTeX word count</h1>
<div class="stylehelp"><p><h2>Format/colour codes of verbose output:</h2>
<span class='error'>ERROR</span> .... TeXcount error message<br>
<span class='word'>Text which is counted</span> .... counted as text words<br>
<span class='hword'>Header and title text</span> .... counted as header words<br>
<span class='oword'>Caption text and footnotes</span> .... counted as caption words<br>
<span class='altwd'>Words in user specified counters</span> .... counted in separate counters<br>
<span class='ignore'>Ignored text or code</span> .... excluded or ignored<br>
<span class='document'>\documentclass</span> .... document start, beginning of preamble<br>
<span class='special'>Special macros, eg require special handling or have side-effects</span> .... <br>
<span class='cmd'>\macro</span> .... macro not counted, but parameters may be<br>
<span class='exclcmd'>\macro</span> .... macro in excluded region<br>
<span class='option'>[Macro options]</span> .... not counted<br>
<span class='optparm'>[Optional parameter]</span> .... content parsed and styled as counted<br>
<span class='envir'>\begin{name} \end{name}</span> .... environment<br>
<span class='exclenv'>\begin{name} \end{name}</span> .... environment in excluded region<br>
<span class='specarg'>Special argument, eg with side-effects</span> .... <br>
<span class='mathgroup'>$ $</span> .... counted as one equation<br>
<span class='exclmath'>$ $</span> .... equation in excluded region<br>
<span class='math'>2+2=4</span> .... maths (inside $...$ etc.)<br>
<span class='mathcmd'>$\macro$</span> .... macros inside maths<br>
<span class='comment'>% Comments</span> .... not counted<br>
<span class='tc'>%TC:TeXcount instructions</span> .... not counted<br>
<span class='fileinc'>File to include</span> .... not counted but file may be counted later<br>
</p></div>
<div class='filegroup'>
<h2>File: C:/dev/projects/uni/FinalYearProject/LaTeX/tex/chapters/01-introduction.tex</h2>
<div class='parse'><p>
<span class='comment'>% !TEX root = ../main.tex </span><br>
<br>
<span class='cmd'>\chapter</span><span class='ignore'>{</span><span class='hword'>Introduction</span><span class='ignore'>}</span><span class='cmd'>\label</span><span class='ignore'>{</span><span class='ignore'>ch</span><span class='ignore'>:</span><span class='ignore'>intro</span><span class='ignore'>}</span><br>
<span class='word'>In</span> <span class='word'>this</span> <span class='word'>report</span> <span class='word'>we</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>looking</span> <span class='word'>for</span> <span class='word'>best</span> <span class='word'>responses</span> <span class='word'>of</span> <span class='word'>strategies</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>Iterated</span> <span class='word'>Prisoners</span> <span class='word'>Dilemma</span> <span class='ignore'>(</span><span class='word'>IPD</span><span class='ignore'>)</span><span class='ignore'>;</span><br>
<span class='word'>we</span> <span class='word'>will</span> <span class='word'>try</span> <span class='word'>to</span> <span class='word'>identify</span> <span class='word'>the</span> <span class='word'>best</span> <span class='word'>overall</span> <span class='word'>score</span><span class='ignore'>,</span> <span class='word'>and</span> <span class='word'>its</span> <span class='word'>corresponding</span> <span class='word'>set</span> <span class='word'>of</span> <span class='word'>moves</span><span class='ignore'>,</span> <span class='word'>for</span> <span class='word'>a</span> <span class='word'>player</span> <span class='word'>on</span> <span class='word'>the</span> <span class='word'>other</span> <span class='word'>side</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>game</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>strategy</span><span class='ignore'>.</span><br>
<span class='word'>We</span> <span class='word'>will</span> <span class='word'>look</span> <span class='word'>into</span> <span class='word'>explicit</span> <span class='word'>sequences</span> <span class='word'>which</span> <span class='word'>should</span> <span class='word'>be</span> <span class='word'>played</span> <span class='word'>for</span> <span class='mathgroup'>$</span><span class='math'>1</span><span class='mathgroup'>$</span> <span class='word'>vs</span> <span class='mathgroup'>$</span><span class='math'>1</span><span class='mathgroup'>$</span> <span class='word'>games</span> <span class='word'>of</span> <span class='word'>length</span> <span class='word'>200</span> <span class='word'>to</span> <span class='word'>gain</span> <span class='word'>the</span> <span class='word'>highest</span> <span class='word'>score</span> <span class='word'>per</span> <span class='word'>turn</span> <span class='word'>upon</span> <span class='word'>the</span> <span class='word'>games</span> <span class='word'>conclusion</span><span class='ignore'>.</span><br>
<br>
<span class='word'>This</span> <span class='word'>task</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>completed</span> <span class='word'>using</span> <span class='word'>reinforcement</span> <span class='word'>techniques</span><span class='ignore'>,</span> <span class='word'>specifically</span> <span class='word'>genetic</span> <span class='word'>algorithms</span><span class='ignore'>,</span> <span class='word'>to</span> <span class='word'>continuously</span> <span class='word'>improve</span> <span class='word'>our</span> <span class='word'>sequences</span> <span class='word'>during</span> <span class='word'>training</span><span class='ignore'>.</span> <span class='word'>Once</span> <span class='word'>we</span> <span class='word'>have</span> <span class='word'>best</span> <span class='word'>responses</span><span class='ignore'>,</span> <span class='word'>these</span> <span class='word'>sequences</span> <span class='word'>will</span> <span class='word'>then</span> <span class='word'>be</span> <span class='word'>compared</span> <span class='word'>and</span> <span class='word'>contrasted</span> <span class='word'>to</span> <span class='word'>understand</span> <span class='word'>how</span> <span class='word'>opponents</span> <span class='word'>could</span> <span class='word'>be</span> <span class='word'>grouped</span> <span class='word'>together</span> <span class='word'>even</span> <span class='word'>though</span> <span class='word'>they</span> <span class='word'>have</span> <span class='word'>no</span> <span class='word'>structure</span> <span class='word'>in</span> <span class='word'>common</span><span class='ignore'>.</span><br>
<br>
<span class='word'>This</span> <span class='word'>work</span> <span class='word'>assumes</span> <span class='word'>a</span> <span class='word'>basic</span> <span class='word'>knowledge</span> <span class='word'>of</span> <span class='word'>game</span> <span class='word'>theory</span> <span class='word'>and</span> <span class='word'>how</span> <span class='word'>to</span> <span class='word'>model</span> <span class='word'>a</span> <span class='word'>normal</span> <span class='word'>form</span> <span class='word'>game</span><span class='ignore'>.</span><br>
<span class='word'>Knowledge</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>Prisoners</span> <span class='word'>Dilemma</span> <span class='word'>is</span> <span class='word'>helpful</span> <span class='word'>but</span> <span class='word'>not</span> <span class='word'>required</span><span class='ignore'>,</span> <span class='word'>the</span> <span class='word'>IPD</span> <span class='word'>game</span> <span class='word'>itself</span> <span class='word'>is</span> <span class='word'>described</span> <span class='word'>in</span> <span class='word'>Section</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>iteratedPrisonersDilemma</span><span class='ignore'>}</span> <span class='word'>and</span> <span class='word'>literature</span> <span class='word'>is</span> <span class='word'>discussed</span> <span class='word'>in</span> <span class='word'>Chapter</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>ch</span><span class='ignore'>:</span><span class='ignore'>literature</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<span class='word'>There</span> <span class='word'>is</span> <span class='word'>also</span> <span class='word'>the</span> <span class='word'>assumption</span> <span class='word'>the</span> <span class='word'>reader</span> <span class='word'>is</span> <span class='word'>familiar</span> <span class='word'>with</span> <span class='word'>basic</span> <span class='word'>algorithms</span> <span class='word'>and</span> <span class='word'>programming</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\section</span><span class='ignore'>{</span><span class='hword'>The</span> <span class='hword'>Prisoners</span> <span class='hword'>Dilemma</span> <span class='hword'>and</span> <span class='hword'>Its</span> <span class='hword'>Iterated</span> <span class='hword'>Form</span><span class='ignore'>}</span><span class='cmd'>\label</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>iteratedPrisonersDilemma</span><span class='ignore'>}</span><br>
<span class='comment'>% \begin{center}</span><br>
<span class='comment'>% \itshape~Two members of a criminal gang are arrested and imprisoned.</span><br>
<span class='comment'>% Each prisoner is in solitary confinement with no means of communicating with the other.</span><br>
<span class='comment'>% The prosecutors lack sufficient evidence to convict the pair on the principal charge.</span><br>
<span class='comment'>% They hope to get both sentenced to a year in prison on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain.</span><br>
<span class='comment'>% Each prisoner is given the opportunity either to: betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent.</span><br>
<span class='comment'>% \end{center}</span><br>
<br>
<span class='word'>The</span> <span class='word'>Prisoners</span> <span class='word'>Dilemma</span> <span class='ignore'>(</span><span class='word'>PD</span><span class='ignore'>)</span> <span class='word'>is</span> <span class='word'>a</span> <span class='word'>normal</span> <span class='word'>form</span> <span class='word'>game</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>space</span> <span class='mathgroup'>$</span><span class='mathcmd'>\mathbb</span><span class='ignore'>{</span><span class='math'>R</span><span class='ignore'>}</span><span class='math'>^</span><span class='ignore'>{</span><span class='ignore'>{</span><span class='math'>2</span><span class='mathcmd'>\times</span> <span class='math'>2</span><span class='ignore'>}</span><span class='math'>^</span><span class='math'>2</span><span class='ignore'>}</span><span class='mathgroup'>$</span> <span class='word'>with</span> <span class='word'>utility</span> <span class='word'>matrices</span> <span class='word'>and</span> <span class='word'>mixed</span> <span class='word'>strategies</span> <span class='word'>for</span> <span class='word'>the</span> <span class='word'>row</span> <span class='word'>and</span> <span class='word'>column</span> <span class='word'>player</span> <span class='word'>as</span> <span class='word'>follows</span><span class='ignore'>:</span><br>
<span class='mathgroup'>$$</span><br>
<span class='math'>A</span><span class='math'>=</span><span class='exclenv'>\begin</span><span class='exclenv'>{pmatrix}</span><span class='math'>R</span> <span class='math'>&</span> <span class='math'>S</span><span class='mathcmd'>\\</span> <span class='math'>T</span> <span class='math'>&</span> <span class='math'>P</span><span class='exclenv'>\end</span><span class='exclenv'>{pmatrix}</span><span class='mathcmd'>\quad</span><br>
<span class='math'>B</span><span class='math'>=</span><span class='exclenv'>\begin</span><span class='exclenv'>{pmatrix}</span><span class='math'>R</span> <span class='math'>&</span> <span class='math'>T</span><span class='mathcmd'>\\</span> <span class='math'>S</span> <span class='math'>&</span> <span class='math'>P</span><span class='exclenv'>\end</span><span class='exclenv'>{pmatrix}</span><br>
<span class='mathgroup'>$$</span><br>
<span class='mathgroup'>$$</span><br>
<span class='mathcmd'>\sigma</span><span class='math'>_</span><span class='math'>r</span><span class='math'>=</span><span class='exclenv'>\begin</span><span class='exclenv'>{pmatrix}</span><span class='math'>C</span> <span class='math'>&</span> <span class='math'>D</span><span class='mathcmd'>\\</span> <span class='exclenv'>\end</span><span class='exclenv'>{pmatrix}</span><span class='mathcmd'>\quad</span><br>
<span class='mathcmd'>\sigma</span><span class='math'>_</span><span class='math'>c</span><span class='math'>=</span><span class='exclenv'>\begin</span><span class='exclenv'>{pmatrix}</span><span class='math'>C</span> <span class='math'>&</span> <span class='math'>D</span><span class='mathcmd'>\\</span> <span class='exclenv'>\end</span><span class='exclenv'>{pmatrix}</span><br>
<span class='mathgroup'>$$</span><br>
<span class='word'>with</span> <span class='word'>the</span> <span class='word'>following</span> <span class='word'>utility</span> <span class='word'>constraints</span><span class='ignore'>:</span><br>
<span class='mathgroup'>$$</span><span class='math'>T</span><span class='math'>></span><span class='math'>R</span><span class='math'>></span><span class='math'>P</span><span class='math'>></span><span class='math'>S</span><span class='math'>,</span> <span class='mathcmd'>\qquad</span> <span class='math'>2R</span><span class='math'>></span><span class='math'>T</span><span class='math'>+</span><span class='math'>S</span><span class='mathgroup'>$$</span><br>
<span class='word'>These</span> <span class='word'>constraints</span> <span class='word'>mean</span> <span class='word'>that</span> <span class='word'>the</span> <span class='word'>defection</span> <span class='word'>action</span><span class='ignore'>,</span> <span class='mathgroup'>$</span><span class='math'>D</span><span class='mathgroup'>$</span><span class='ignore'>,</span> <span class='word'>dominates</span> <span class='word'>the</span> <span class='word'>cooperation</span> <span class='word'>action</span><span class='ignore'>,</span> <span class='mathgroup'>$</span><span class='math'>C</span><span class='mathgroup'>$</span><span class='ignore'>,</span> <span class='word'>for</span> <span class='word'>both</span> <span class='word'>player</span> <span class='word'>and</span> <span class='word'>that</span> <span class='word'>the</span> <span class='word'>largest</span> <span class='word'>payoff</span> <span class='word'>comes</span> <span class='word'>when</span> <span class='word'>both</span> <span class='word'>players</span> <span class='word'>cooperate</span><span class='ignore'>.</span> <span class='word'>This</span> <span class='word'>payoff</span> <span class='word'>model</span> <span class='word'>has</span> <span class='word'>the</span> <span class='word'>following</span> <span class='word'>interpretation</span><span class='ignore'>:</span><br>
<span class='envir'>\begin</span><span class='envir'>{itemize}</span><br>
<span class='cmd'>\item</span> <span class='mathgroup'>$</span><span class='math'>T</span><span class='mathgroup'>$</span> <span class='ignore'>-</span><span class='ignore'>-</span><span class='ignore'>-</span> <span class='word'>Temptation</span><span class='ignore'>,</span> <span class='word'>the</span> <span class='word'>utility</span> <span class='word'>for</span> <span class='word'>successfully</span> <span class='word'>tricking</span> <span class='word'>your</span> <span class='word'>opponent</span> <span class='word'>to</span> <span class='word'>cooperate</span> <span class='word'>while</span> <span class='word'>you</span> <span class='word'>defect</span><span class='ignore'>.</span><br>
<span class='cmd'>\item</span> <span class='mathgroup'>$</span><span class='math'>R</span><span class='mathgroup'>$</span> <span class='ignore'>-</span><span class='ignore'>-</span><span class='ignore'>-</span> <span class='word'>Reward</span><span class='ignore'>,</span> <span class='word'>by</span> <span class='word'>both</span> <span class='word'>coopering</span> <span class='word'>with</span> <span class='word'>your</span> <span class='word'>respective</span> <span class='word'>opponents</span> <span class='word'>you</span> <span class='word'>both</span> <span class='word'>receive</span> <span class='word'>the</span> <span class='word'>reward</span><span class='ignore'>.</span><br>
<span class='cmd'>\item</span> <span class='mathgroup'>$</span><span class='math'>P</span><span class='mathgroup'>$</span> <span class='ignore'>-</span><span class='ignore'>-</span><span class='ignore'>-</span> <span class='word'>Punishment</span><span class='ignore'>,</span> <span class='word'>if</span> <span class='word'>both</span> <span class='word'>you</span> <span class='word'>and</span> <span class='word'>your</span> <span class='word'>opponent</span> <span class='word'>defect</span> <span class='word'>you</span> <span class='word'>will</span> <span class='word'>both</span> <span class='word'>receive</span> <span class='word'>the</span> <span class='word'>punishment</span><br>
<span class='cmd'>\item</span> <span class='mathgroup'>$</span><span class='math'>S</span><span class='mathgroup'>$</span> <span class='ignore'>-</span><span class='ignore'>-</span><span class='ignore'>-</span> <span class='word'>Sucker</span><span class='ignore'>,</span> <span class='word'>the</span> <span class='word'>utility</span> <span class='word'>for</span> <span class='word'>being</span> <span class='word'>tricked</span> <span class='word'>by</span> <span class='word'>your</span> <span class='word'>opponent</span> <span class='word'>to</span> <span class='word'>cooperate</span> <span class='word'>while</span> <span class='word'>they</span> <span class='word'>defect</span><span class='ignore'>;</span> <span class='word'>they</span> <span class='word'>get</span> <span class='word'>temptation</span> <span class='word'>and</span> <span class='word'>you</span> <span class='word'>get</span> <span class='word'>sucker</span><span class='ignore'>.</span><br>
<span class='envir'>\end</span><span class='envir'>{itemize}</span><br>
<span class='word'>Because</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>way</span> <span class='word'>the</span> <span class='word'>game</span> <span class='word'>is</span> <span class='word'>put</span> <span class='word'>together</span> <span class='word'>we</span> <span class='word'>arrive</span> <span class='word'>at</span> <span class='word'>the</span> <span class='word'>dilemma</span><span class='ignore'>.</span><br>
<span class='word'>Do</span> <span class='word'>you</span> <span class='word'>cooperate</span> <span class='word'>and</span> <span class='word'>risk</span> <span class='word'>being</span> <span class='word'>taken</span> <span class='word'>advantage</span> <span class='word'>of</span> <span class='word'>by</span> <span class='word'>your</span> <span class='word'>opponent</span><span class='ignore'>,</span> <span class='word'>first</span> <span class='word'>row</span> <span class='word'>or</span> <span class='word'>column</span><span class='ignore'>.</span><br>
<span class='word'>Or</span> <span class='word'>do</span> <span class='word'>you</span> <span class='word'>defect</span> <span class='word'>and</span> <span class='word'>hope</span> <span class='word'>your</span> <span class='word'>opponent</span> <span class='word'>cooperates</span> <span class='word'>to</span> <span class='word'>give</span> <span class='word'>you</span> <span class='word'>a</span> <span class='word'>bonus</span><span class='ignore'>,</span> <span class='word'>second</span> <span class='word'>row</span> <span class='word'>or</span> <span class='word'>column</span><span class='ignore'>.</span><br>
<span class='word'>In</span> <span class='word'>this</span> <span class='word'>work</span> <span class='word'>we</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>using</span> <span class='word'>the</span> <span class='word'>T</span><span class='ignore'>,</span> <span class='word'>R</span><span class='ignore'>,</span> <span class='word'>S</span> <span class='cmd'>\&</span> <span class='word'>P</span> <span class='word'>utilities</span> <span class='word'>originally</span> <span class='word'>introduced</span> <span class='word'>in</span> <span class='word'>axelrods</span> <span class='word'>1980s</span> <span class='word'>work</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>axelrod1980effective</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<span class='mathgroup'>$$</span><br>
<span class='math'>A</span><span class='math'>=</span><span class='exclenv'>\begin</span><span class='exclenv'>{pmatrix}</span><span class='math'>3</span> <span class='math'>&</span> <span class='math'>0</span> <span class='mathcmd'>\\</span> <span class='math'>5</span> <span class='math'>&</span> <span class='math'>1</span><span class='exclenv'>\end</span><span class='exclenv'>{pmatrix}</span><span class='mathcmd'>\quad</span><br>
<span class='math'>B</span><span class='math'>=</span><span class='exclenv'>\begin</span><span class='exclenv'>{pmatrix}</span><span class='math'>3</span> <span class='math'>&</span> <span class='math'>5</span> <span class='mathcmd'>\\</span> <span class='math'>0</span> <span class='math'>&</span> <span class='math'>1</span><span class='exclenv'>\end</span><span class='exclenv'>{pmatrix}</span><br>
<span class='mathgroup'>$$</span> <br>
<span class='word'>From</span> <span class='word'>a</span> <span class='word'>game</span> <span class='word'>theoretic</span> <span class='word'>point</span> <span class='word'>of</span> <span class='word'>view</span> <span class='word'>it</span> <span class='word'>can</span> <span class='word'>be</span> <span class='word'>seen</span> <span class='word'>that</span> <span class='word'>there</span> <span class='word'>exists</span> <span class='word'>a</span> <span class='word'>strongly</span> <span class='word'>dominant</span> <span class='word'>strategy</span> <span class='word'>for</span> <span class='word'>both</span> <span class='word'>the</span> <span class='word'>row</span> <span class='word'>and</span> <span class='word'>column</span> <span class='word'>player</span><span class='ignore'>:</span> <span class='word'>defection</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>however</span> <span class='word'>does</span> <span class='word'>not</span> <span class='word'>reflect</span> <span class='word'>on</span> <span class='word'>the</span> <span class='word'>iterated</span> <span class='word'>version</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>game</span> <span class='word'>where</span> <span class='word'>repetition</span> <span class='word'>can</span> <span class='word'>allow</span> <span class='word'>for</span> <span class='word'>higher</span> <span class='word'>overall</span> <span class='word'>scores</span> <span class='word'>and</span> <span class='word'>players</span> <span class='word'>can</span> <span class='word'>take</span> <span class='word'>advantage</span> <span class='word'>of</span> <span class='word'>each</span> <span class='word'>other</span><span class='ignore'>.</span><br>
<br>
<span class='word'>We</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>looking</span> <span class='word'>into</span> <span class='word'>the</span> <span class='word'>iterated</span> <span class='word'>form</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>game</span> <span class='word'>which</span><span class='ignore'>,</span> <span class='word'>simply</span> <span class='word'>put</span><span class='ignore'>,</span> <span class='word'>is</span> <span class='word'>a</span> <span class='word'>series</span> <span class='word'>of</span> <span class='word'>single</span> <span class='word'>games</span> <span class='word'>played</span> <span class='word'>back</span> <span class='word'>to</span> <span class='word'>back</span><span class='ignore'>.</span> <br>
<span class='word'>Players</span> <span class='word'>do</span> <span class='word'>not</span> <span class='word'>know</span> <span class='word'>what</span> <span class='word'>their</span> <span class='word'>opponent</span> <span class='word'>will</span> <span class='word'>play</span> <span class='word'>on</span> <span class='word'>any</span> <span class='word'>given</span> <span class='word'>turn</span><span class='ignore'>,</span> <span class='word'>they</span> <span class='word'>only</span> <span class='word'>know</span> <span class='word'>what</span> <span class='word'>has</span> <span class='word'>been</span> <span class='word'>played</span> <span class='word'>on</span> <span class='word'>previous</span> <span class='word'>turns</span><span class='ignore'>.</span><br>
<span class='word'>Players</span> <span class='word'>can</span> <span class='word'>algorithmically</span><span class='ignore'>,</span> <span class='word'>stochastically</span> <span class='word'>or</span> <span class='word'>using</span> <span class='word'>a</span> <span class='word'>combination</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>two</span> <span class='word'>decide</span> <span class='word'>on</span> <span class='word'>their</span> <span class='word'>next</span> <span class='word'>move</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>number</span> <span class='word'>of</span> <span class='word'>turns</span> <span class='word'>is</span> <span class='word'>often</span> <span class='word'>provided</span> <span class='word'>beforehand</span><span class='ignore'>,</span> <span class='word'>but</span> <span class='word'>does</span> <span class='word'>not</span> <span class='word'>need</span> <span class='word'>to</span> <span class='word'>be</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>axelrod1980more</span><span class='ignore'>}</span><span class='ignore'>,</span> <span class='word'>and</span> <span class='word'>the</span> <span class='word'>overall</span> <span class='word'>score</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>game</span> <span class='word'>is</span> <span class='word'>normalised</span> <span class='word'>to</span> <span class='word'>by</span> <span class='word'>the</span> <span class='word'>number</span> <span class='word'>of</span> <span class='word'>turns</span> <span class='word'>to</span> <span class='word'>allow</span> <span class='word'>for</span> <span class='word'>comparisons</span> <span class='word'>between</span> <span class='word'>games</span> <span class='word'>of</span> <span class='word'>different</span> <span class='word'>length</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\section</span><span class='ignore'>{</span><span class='hword'>Machine</span> <span class='hword'>Learning</span> <span class='cmd'>\&</span> <span class='hword'>Computer</span> <span class='hword'>Intelligence</span><span class='ignore'>}</span><span class='cmd'>\label</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>machineLearningAndcomputerIntelligence</span><span class='ignore'>}</span><br>
<span class='word'>Machine</span> <span class='word'>learning</span> <span class='ignore'>(</span><span class='word'>ML</span><span class='ignore'>)</span> <span class='word'>was</span> <span class='word'>most</span> <span class='word'>famously</span> <span class='word'>introduced</span> <span class='word'>by</span> <span class='word'>questions</span> <span class='word'>posed</span> <span class='word'>by</span> <span class='word'>Alan</span> <span class='word'>Turing</span> <span class='word'>in</span> <span class='word'>1950</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>turing1950computing</span><span class='ignore'>}</span> <span class='word'>asking</span> <span class='ignore'>`</span><span class='word'>can</span> <span class='word'>machines</span> <span class='word'>think</span><span class='ignore'>?</span><span class='ignore'>'</span><span class='ignore'>,</span> <span class='word'>a</span> <span class='word'>question</span> <span class='word'>that</span> <span class='word'>has</span> <span class='word'>been</span> <span class='word'>refined</span> <span class='word'>and</span> <span class='word'>analysed</span> <span class='word'>to</span> <span class='word'>this</span> <span class='word'>day</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>field</span> <span class='word'>of</span> <span class='word'>computer</span> <span class='word'>intelligence</span> <span class='word'>is</span> <span class='word'>rich</span> <span class='word'>in</span> <span class='word'>its</span> <span class='word'>complexities</span> <span class='word'>and</span> <span class='word'>has</span> <span class='word'>recently</span> <span class='word'>been</span> <span class='word'>making</span> <span class='word'>breakthroughs</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>knight2017alphaZeroMIT</span><span class='ignore'>}</span> <span class='word'>on</span> <span class='word'>topics</span> <span class='word'>which</span> <span class='word'>would</span> <span class='word'>traditionally</span> <span class='word'>be</span> <span class='word'>considered</span> <span class='ignore'>`</span><span class='word'>thinking</span><span class='ignore'>'</span><span class='ignore'>.</span><br>
<span class='word'>Along</span> <span class='word'>with</span> <span class='word'>this</span><span class='ignore'>,</span> <span class='word'>recently</span> <span class='word'>there</span> <span class='word'>has</span> <span class='word'>also</span> <span class='word'>been</span> <span class='word'>record</span> <span class='word'>levels</span> <span class='word'>of</span> <span class='word'>funding</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>chui2017artificial</span><span class='ignore'>}</span> <span class='word'>put</span> <span class='word'>in</span> <span class='word'>to</span> <span class='word'>companies</span> <span class='word'>which</span> <span class='word'>operate</span> <span class='word'>in</span> <span class='word'>this</span> <span class='word'>field</span><span class='ignore'>,</span> <span class='word'>producing</span> <span class='word'>results</span> <span class='word'>in</span> <span class='word'>areas</span> <span class='word'>that</span> <span class='word'>would</span> <span class='word'>usually</span> <span class='word'>seem</span> <span class='ignore'>`</span><span class='word'>solved</span><span class='ignore'>'</span><span class='ignore'>.</span> <br>
<br>
<span class='word'>Computer</span> <span class='word'>intelligence</span> <span class='word'>breaks</span> <span class='word'>down</span> <span class='word'>into</span> <span class='word'>2</span> <span class='word'>general</span> <span class='word'>topics</span><span class='ignore'>,</span> <span class='word'>supervised</span> <span class='word'>and</span> <span class='word'>unsupervised</span> <span class='word'>learning</span><span class='ignore'>.</span><br>
<span class='word'>Supervised</span> <span class='word'>learning</span> <span class='word'>is</span> <span class='word'>done</span> <span class='word'>with</span> <span class='word'>help</span> <span class='word'>from</span> <span class='word'>an</span> <span class='word'>external</span> <span class='word'>source</span><span class='ignore'>;</span> <span class='word'>inferring</span> <span class='word'>a</span> <span class='word'>function</span> <span class='word'>from</span> <span class='word'>labelled</span> <span class='word'>examples</span> <span class='word'>that</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>used</span> <span class='word'>to</span> <span class='word'>map</span> <span class='word'>unseen</span> <span class='word'>data</span> <span class='word'>to</span> <span class='word'>their</span> <span class='word'>potential</span> <span class='word'>labels</span><span class='ignore'>.</span><br>
<span class='word'>Unsupervised</span> <span class='word'>learning</span> <span class='word'>is</span> <span class='word'>the</span> <span class='word'>process</span> <span class='word'>of</span> <span class='word'>creating</span> <span class='word'>new</span> <span class='word'>data</span> <span class='word'>about</span> <span class='word'>a</span> <span class='word'>dataset</span> <span class='word'>which</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>used</span> <span class='word'>to</span> <span class='word'>expand</span> <span class='word'>the</span> <span class='word'>understanding</span> <span class='word'>of</span> <span class='word'>existing</span> <span class='word'>data</span><span class='ignore'>;</span> <span class='word'>for</span> <span class='word'>example</span> <span class='word'>finding</span> <span class='word'>a</span> <span class='word'>hidden</span> <span class='word'>structure</span> <span class='word'>or</span> <span class='word'>searching</span> <span class='word'>a</span> <span class='word'>solution</span> <span class='word'>space</span><span class='ignore'>.</span><br>
<span class='word'>Reinforcement</span> <span class='word'>learning</span> <span class='word'>is</span> <span class='word'>a</span> <span class='word'>technique</span> <span class='word'>that</span> <span class='word'>can</span> <span class='word'>be</span> <span class='word'>applied</span> <span class='word'>to</span> <span class='word'>both</span> <span class='word'>supervised</span> <span class='word'>and</span> <span class='word'>unsupervised</span> <span class='word'>learning</span><span class='ignore'>.</span><br>
<span class='word'>What</span> <span class='word'>it</span> <span class='word'>means</span> <span class='word'>is</span> <span class='word'>to</span> <span class='word'>repeat</span> <span class='word'>an</span> <span class='word'>algorithm</span> <span class='word'>using</span> <span class='word'>results</span> <span class='word'>from</span> <span class='word'>a</span> <span class='word'>previous</span> <span class='word'>run</span> <span class='word'>to</span> <span class='word'>improve</span> <span class='word'>parameters</span> <span class='word'>of</span> <span class='word'>its</span> <span class='word'>next</span> <span class='word'>run</span><span class='ignore'>;</span> <span class='word'>genetic</span> <span class='word'>algorithms</span> <span class='word'>are</span> <span class='word'>a</span> <span class='word'>prime</span> <span class='word'>example</span> <span class='word'>of</span> <span class='word'>this</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\subsection</span><span class='ignore'>{</span><span class='hword'>Genetic</span> <span class='hword'>Algorithms</span><span class='ignore'>}</span><span class='cmd'>\label</span><span class='ignore'>{</span><span class='ignore'>subsec</span><span class='ignore'>:</span><span class='ignore'>geneticAlgorithms</span><span class='ignore'>}</span><br>
<span class='word'>This</span> <span class='word'>report</span> <span class='word'>will</span> <span class='word'>cover</span> <span class='word'>a</span> <span class='word'>class</span> <span class='word'>of</span> <span class='word'>ML</span> <span class='word'>called</span> <span class='word'>genetic</span> <span class='word'>algorithms</span> <span class='ignore'>(</span><span class='word'>GAs</span><span class='ignore'>)</span><span class='ignore'>.</span><br>
<span class='word'>GAs</span> <span class='word'>fall</span> <span class='word'>under</span> <span class='word'>a</span> <span class='word'>branch</span> <span class='word'>of</span> <span class='word'>ML</span> <span class='word'>called</span> <span class='word'>evolutionary</span> <span class='word'>algorithms</span><span class='ignore'>.</span><br>
<span class='word'>More</span> <span class='word'>generally</span><span class='ignore'>,</span> <span class='word'>genetic</span> <span class='word'>algorithms</span> <span class='word'>are</span> <span class='word'>put</span> <span class='word'>into</span> <span class='word'>a</span> <span class='word'>class</span> <span class='word'>of</span> <span class='word'>unsupervised</span> <span class='word'>reinforcement</span> <span class='word'>learning</span> <span class='word'>algorithms</span><span class='ignore'>.</span><br>
<span class='word'>These</span> <span class='word'>represent</span> <span class='word'>techniques</span> <span class='word'>of</span> <span class='word'>using</span> <span class='word'>genetic</span> <span class='word'>algorithms</span> <span class='word'>for</span> <span class='word'>generating</span> <span class='word'>solutions</span> <span class='word'>to</span> <span class='word'>problems</span> <span class='word'>that</span> <span class='word'>typically</span> <span class='word'>revolve</span> <span class='word'>around</span> <span class='word'>heuristically</span> <span class='word'>improving</span> <span class='word'>members</span> <span class='word'>of</span> <span class='word'>a</span> <span class='word'>population</span> <span class='word'>who</span> <span class='word'>represent</span> <span class='word'>these</span> <span class='word'>solutions</span><span class='ignore'>,</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>horn1994niched</span><span class='ignore'>,</span><span class='ignore'>rahmat1999electromagnetic</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>concept</span> <span class='word'>of</span> <span class='word'>a</span> <span class='word'>genetic</span> <span class='word'>algorithm</span><span class='ignore'>,</span> <span class='word'>and</span> <span class='word'>more</span> <span class='word'>generally</span> <span class='word'>an</span> <span class='word'>evolutionary</span> <span class='word'>algorithm</span><span class='ignore'>,</span> <span class='word'>comes</span> <span class='word'>from</span> <span class='word'>nature</span><span class='ignore'>;</span><br>
<span class='word'>like</span> <span class='word'>nature</span> <span class='word'>we</span> <span class='word'>create</span> <span class='word'>a</span> <span class='word'>survival</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>fittest</span> <span class='word'>selection</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>darwin2009origin</span><span class='ignore'>}</span> <span class='word'>competition</span> <span class='word'>to</span> <span class='word'>evaluate</span> <span class='word'>a</span> <span class='word'>population</span> <span class='word'>then</span> <span class='word'>kill</span> <span class='word'>off</span> <span class='word'>the</span> <span class='word'>weakest</span> <span class='word'>members</span><span class='ignore'>.</span><br>
<span class='word'>After</span> <span class='word'>this</span> <span class='word'>cull</span> <span class='word'>we</span> <span class='word'>create</span> <span class='word'>offspring</span> <span class='word'>from</span> <span class='word'>the</span> <span class='word'>most</span> <span class='word'>successful</span> <span class='word'>population</span> <span class='word'>or</span> <span class='word'>introduce</span> <span class='word'>new</span> <span class='word'>members</span> <span class='word'>from</span> <span class='word'>a</span> <span class='word'>predefined</span> <span class='word'>source</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>process</span> <span class='word'>is</span> <span class='word'>then</span> <span class='word'>repeated</span> <span class='word'>until</span> <span class='word'>we</span> <span class='word'>stop</span> <span class='word'>it</span><span class='ignore'>,</span> <span class='word'>or</span> <span class='word'>forever</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>case</span> <span class='word'>of</span> <span class='word'>nature</span><span class='ignore'>.</span><br>
<br>
<span class='word'>We</span> <span class='word'>say</span> <span class='word'>a</span> <span class='word'>genetic</span> <span class='word'>algorithm</span> <span class='word'>is</span> <span class='word'>structured</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>following</span> <span class='word'>way</span><span class='ignore'>.</span><br>
<span class='word'>Given</span> <span class='word'>a</span> <span class='word'>population</span><span class='ignore'>,</span> <span class='mathgroup'>\(</span><span class='math'>P</span><span class='mathgroup'>\)</span><span class='ignore'>,</span> <span class='word'>each</span> <span class='word'>with</span> <span class='word'>unique</span> <span class='word'>genes</span> <span class='ignore'>(</span><span class='word'>the</span> <span class='word'>terms</span> <span class='ignore'>`</span><span class='word'>genes</span><span class='ignore'>'</span> <span class='word'>and</span> <span class='ignore'>`</span><span class='word'>member</span> <span class='word'>properties</span><span class='ignore'>'</span> <span class='word'>are</span> <span class='word'>interchangeable</span><span class='ignore'>)</span><span class='ignore'>,</span> <span class='word'>and</span> <span class='word'>a</span> <span class='word'>number</span> <span class='word'>of</span> <span class='word'>generations</span><span class='ignore'>,</span> <span class='mathgroup'>\(</span><span class='math'>G</span><span class='mathcmd'>\in</span> <span class='mathcmd'>\mathbb</span><span class='ignore'>{</span><span class='math'>N</span><span class='ignore'>}</span><span class='mathgroup'>\)</span><span class='ignore'>,</span> <span class='word'>the</span> <span class='word'>algorithm</span> <span class='word'>will</span> <span class='word'>create</span> <span class='mathgroup'>\(</span><span class='math'>G</span><span class='mathgroup'>\)</span> <span class='word'>cycles</span> <span class='word'>of</span> <span class='word'>scoring</span> <span class='word'>and</span> <span class='word'>potentially</span> <span class='word'>removing</span> <span class='word'>each</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>members</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>population</span><span class='ignore'>,</span> <span class='mathgroup'>\(</span><span class='math'>p</span><span class='math'>_</span><span class='math'>i</span> <span class='mathcmd'>\in</span> <span class='math'>P</span><span class='mathgroup'>\)</span><span class='ignore'>.</span><br>
<span class='word'>It</span> <span class='word'>does</span> <span class='word'>this</span> <span class='word'>by</span> <span class='word'>using</span> <span class='word'>a</span> <span class='word'>mapping</span> <span class='word'>from</span> <span class='word'>a</span> <span class='word'>member</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>population</span> <span class='word'>to</span> <span class='word'>an</span> <span class='word'>ordered</span> <span class='word'>set</span><span class='ignore'>,</span> <span class='word'>for</span> <span class='word'>example</span> <span class='mathgroup'>\(</span><span class='math'>f</span><span class='math'>(</span><span class='math'>p</span><span class='math'>_</span><span class='math'>i</span><span class='math'>)</span><span class='mathcmd'>\mapsto</span> <span class='mathcmd'>\mathbb</span><span class='ignore'>{</span><span class='math'>R</span><span class='ignore'>}</span><span class='mathgroup'>\)</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>function</span><span class='ignore'>,</span> <span class='mathgroup'>\(</span><span class='math'>f</span><span class='mathgroup'>\)</span><span class='ignore'>,</span> <span class='word'>is</span> <span class='word'>defined</span> <span class='word'>beforehand</span> <span class='word'>in</span> <span class='word'>a</span> <span class='word'>way</span> <span class='word'>which</span> <span class='word'>describes</span> <span class='word'>the</span> <span class='word'>goal</span> <span class='word'>of</span> <span class='word'>our</span> <span class='word'>investigation</span><span class='ignore'>.</span><br>
<span class='word'>Defining</span> <span class='word'>a</span> <span class='word'>cut-off</span> <span class='word'>or</span> <span class='word'>bottleneck</span> <span class='mathgroup'>\(</span><span class='math'>b</span><span class='math'><</span><span class='math'>|</span><span class='math'>P</span><span class='math'>|</span><span class='mathgroup'>\)</span><span class='ignore'>,</span> <span class='word'>such</span> <span class='word'>that</span> <span class='word'>on</span> <span class='word'>conclusion</span> <span class='word'>of</span> <span class='word'>completing</span> <span class='word'>any</span> <span class='word'>cycle</span><span class='ignore'>,</span> <span class='word'>the</span> <span class='word'>top</span> <span class='mathgroup'>\(</span><span class='math'>b</span><span class='mathgroup'>\)</span> <span class='word'>ranking</span> <span class='word'>members</span> <span class='ignore'>(</span><span class='word'>or</span> <span class='word'>proportion</span> <span class='word'>of</span> <span class='word'>members</span><span class='ignore'>)</span> <span class='word'>by</span> <span class='word'>score</span> <span class='word'>can</span> <span class='word'>be</span> <span class='word'>kept</span> <span class='word'>and</span> <span class='word'>the</span> <span class='word'>rest</span> <span class='word'>discarded</span><span class='ignore'>.</span><br>
<span class='word'>By</span> <span class='word'>doing</span> <span class='word'>this</span> <span class='word'>we</span> <span class='word'>are</span> <span class='word'>saving</span> <span class='word'>the</span> <span class='word'>more</span> <span class='word'>successful</span> <span class='word'>candidates</span> <span class='word'>allowing</span> <span class='word'>us</span> <span class='word'>to</span> <span class='word'>rebuild</span> <span class='word'>the</span> <span class='word'>population</span> <span class='word'>using</span> <span class='word'>a</span> <span class='word'>series</span> <span class='word'>of</span> <span class='word'>crossovers</span> <span class='word'>and</span> <span class='word'>mutations</span> <span class='ignore'>(</span><span class='word'>and</span> <span class='word'>possibly</span> <span class='word'>introducing</span> <span class='word'>new</span> <span class='word'>members</span> <span class='word'>into</span> <span class='word'>the</span> <span class='word'>population</span><span class='ignore'>)</span> <span class='word'>with</span> <span class='word'>the</span> <span class='word'>genes</span> <span class='word'>which</span> <span class='word'>were</span> <span class='word'>successful</span><span class='ignore'>.</span><br>
<br>
<span class='envir'>\begin</span><span class='envir'>{itemize}</span><br>
<span class='cmd'>\item</span> <span class='word'>Crossovers</span> <span class='word'>take</span> <span class='word'>in</span> <span class='word'>2</span> <span class='word'>members</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>population</span> <span class='word'>and</span> <span class='word'>return</span> <span class='word'>a</span> <span class='word'>new</span> <span class='word'>member</span> <span class='word'>based</span> <span class='word'>on</span> <span class='word'>some</span> <span class='word'>parameters</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>2</span> <span class='ignore'>`</span><span class='word'>parents</span><span class='ignore'>'</span><span class='ignore'>.</span><br>
<span class='word'>For</span> <span class='word'>example</span><span class='ignore'>,</span> <span class='word'>our</span> <span class='word'>crossover</span> <span class='word'>takes</span> <span class='word'>the</span> <span class='word'>first</span> <span class='word'>half</span> <span class='word'>of</span> <span class='word'>a</span> <span class='word'>sequence</span> <span class='word'>from</span> <span class='word'>one</span> <span class='word'>and</span> <span class='word'>the</span> <span class='word'>second</span> <span class='word'>half</span> <span class='word'>from</span> <span class='word'>the</span> <span class='word'>other</span><span class='ignore'>,</span> <span class='word'>merging</span> <span class='word'>them</span> <span class='word'>to</span> <span class='word'>form</span> <span class='word'>the</span> <span class='word'>third</span><span class='ignore'>.</span><br>
<span class='cmd'>\item</span> <span class='word'>Mutations</span> <span class='word'>allow</span> <span class='word'>a</span> <span class='ignore'>(</span><span class='word'>possibly</span> <span class='word'>targeted</span><span class='cmd'>\footnote</span><span class='ignore'>{</span><span class='oword'>For</span> <span class='oword'>example</span> <span class='oword'>using</span> <span class='oword'>intuition</span> <span class='oword'>and</span> <span class='oword'>targeting</span> <span class='oword'>specific</span> <span class='oword'>genes</span><span class='ignore'>,</span> <span class='oword'>or</span> <span class='oword'>allowing</span> <span class='oword'>another</span> <span class='oword'>algorithm</span> <span class='oword'>to</span> <span class='oword'>improve</span> <span class='oword'>the</span> <span class='oword'>targeting</span> <span class='oword'>of</span> <span class='oword'>this</span> <span class='oword'>meta</span> <span class='oword'>function</span><span class='ignore'>.</span><span class='ignore'>}</span><span class='ignore'>)</span> <span class='word'>change</span> <span class='word'>in</span> <span class='word'>a</span> <span class='word'>single</span> <span class='word'>member</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>population</span><span class='ignore'>.</span><br>
<span class='word'>A</span> <span class='word'>mutation</span> <span class='word'>has</span> <span class='word'>2</span> <span class='word'>parameters</span><span class='ignore'>,</span> <span class='word'>a</span> <span class='word'>potency</span> <span class='mathgroup'>\(</span><span class='math'>M</span><span class='math'>_</span><span class='math'>p</span><span class='mathcmd'>\in</span> <span class='mathcmd'>\mathbb</span><span class='ignore'>{</span><span class='math'>R</span><span class='ignore'>}</span><span class='math'>></span><span class='math'>0</span><span class='mathgroup'>\)</span> <span class='word'>and</span> <span class='word'>a</span> <span class='word'>frequency</span> <span class='mathgroup'>\(</span><span class='math'>M</span><span class='math'>_</span><span class='math'>f</span><span class='mathcmd'>\in</span> <span class='option'>[0,1]</span><span class='mathgroup'>\)</span><span class='ignore'>.</span><br>
<span class='mathgroup'>\(</span><span class='math'>M</span><span class='math'>_</span><span class='math'>p</span><span class='mathgroup'>\)</span> <span class='word'>describes</span> <span class='word'>how</span> <span class='word'>strong</span> <span class='word'>the</span> <span class='word'>mutation</span> <span class='word'>is</span><span class='ignore'>,</span> <span class='word'>the</span> <span class='word'>higher</span> <span class='word'>it</span> <span class='word'>is</span> <span class='word'>the</span> <span class='word'>larger</span> <span class='word'>change</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>member</span> <span class='word'>occurs</span><span class='ignore'>.</span><br>
<span class='mathgroup'>\(</span><span class='math'>M</span><span class='math'>_</span><span class='math'>f</span><span class='mathgroup'>\)</span> <span class='word'>explains</span> <span class='word'>the</span> <span class='word'>percentage</span> <span class='word'>of</span> <span class='word'>how</span> <span class='word'>many</span> <span class='word'>members</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>population</span> <span class='word'>are</span> <span class='word'>mutated</span><span class='ignore'>.</span><br>
<span class='envir'>\end</span><span class='envir'>{itemize}</span><br>
<br>
<span class='word'>Figure</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>genericGACycle</span><span class='ignore'>}</span> <span class='word'>shows</span> <span class='word'>a</span> <span class='word'>flow</span> <span class='word'>diagram</span> <span class='word'>of</span> <span class='word'>a</span> <span class='word'>generic</span> <span class='word'>genetic</span> <span class='word'>algorithm</span> <span class='word'>cycle</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>specific</span> <span class='word'>algorithm</span> <span class='word'>we</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>using</span> <span class='word'>is</span> <span class='word'>described</span> <span class='word'>in</span> <span class='word'>Subsection</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>buildingTheAlgorithm</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<br>
<span class='envir'>\begin</span><span class='envir'>{figure}</span><span class='option'>[ht]</span><br>
<span class='exclcmd'>\centering</span><br>
<span class='exclcmd'>\includegraphics</span><span class='option'>[width=1.0\textwidth, center]</span><span class='ignore'>{</span><span class='ignore'>.</span><span class='ignore'>/</span><span class='ignore'>img</span><span class='ignore'>/</span><span class='ignore'>flows</span><span class='ignore'>/</span><span class='ignore'>ga</span><span class='ignore'>_</span><span class='ignore'>cycle.pdf</span><span class='ignore'>}</span><br>
<span class='cmd'>\caption</span><span class='ignore'>{</span><span class='oword'>Generic</span> <span class='oword'>genetic</span> <span class='oword'>algorithm</span> <span class='oword'>cycle</span> <span class='oword'>diagram</span><span class='ignore'>}</span><span class='exclcmd'>\label</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>genericGACycle</span><span class='ignore'>}</span> <br>
<span class='envir'>\end</span><span class='envir'>{figure}</span><br>
<br>
<span class='word'>Genetic</span> <span class='word'>algorithms</span> <span class='word'>are</span> <span class='word'>designed</span> <span class='word'>in</span> <span class='word'>way</span> <span class='word'>to</span> <span class='word'>avoid</span> <span class='word'>local</span> <span class='word'>maxima</span><span class='ignore'>/</span><span class='word'>minima</span> <span class='word'>when</span> <span class='word'>searching</span> <span class='word'>a</span> <span class='word'>solution</span> <span class='word'>space</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>property</span> <span class='word'>of</span> <span class='word'>mutating</span> <span class='word'>a</span> <span class='word'>member</span> <span class='word'>of</span> <span class='word'>its</span> <span class='word'>population</span> <span class='word'>allows</span> <span class='word'>alterations</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>members</span> <span class='word'>features</span><span class='ignore'>;</span> <br>
<span class='word'>this</span> <span class='word'>in</span> <span class='word'>turn</span> <span class='word'>will</span> <span class='word'>lead</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>change</span> <span class='word'>in</span> <span class='word'>that</span> <span class='word'>members</span> <span class='word'>fitness</span><span class='ignore'>.</span><br>
<span class='word'>If</span> <span class='word'>a</span> <span class='word'>member</span> <span class='word'>is</span> <span class='ignore'>`</span><span class='word'>stuck</span><span class='ignore'>'</span> <span class='word'>in</span> <span class='word'>a</span> <span class='word'>local</span> <span class='word'>maxima</span><span class='ignore'>/</span><span class='word'>minima</span> <span class='word'>with</span> <span class='word'>its</span> <span class='word'>fitness</span> <span class='word'>score</span><span class='ignore'>,</span> <span class='word'>a</span> <span class='word'>mutation</span> <span class='word'>of</span> <span class='word'>a</span> <span class='word'>certain</span> <span class='word'>feature</span> <span class='word'>could</span> <span class='word'>potentially</span> <span class='word'>move</span> <span class='word'>their</span> <span class='word'>position</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>solution</span> <span class='word'>space</span> <span class='word'>to</span> <span class='word'>better</span> <span class='word'>local</span> <span class='word'>maxima</span><span class='ignore'>/</span><span class='word'>minima</span><span class='ignore'>.</span><br>
<span class='word'>Along</span> <span class='word'>with</span> <span class='word'>mutation</span><span class='ignore'>,</span> <span class='word'>the</span> <span class='word'>crossover</span> <span class='word'>property</span> <span class='word'>explicitly</span> <span class='word'>combines</span> <span class='word'>2</span> <span class='word'>members</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>population</span> <span class='word'>to</span> <span class='word'>create</span> <span class='word'>a</span> <span class='word'>new</span> <span class='word'>member</span><span class='ignore'>.</span> <br>
<span class='word'>Crossover</span> <span class='word'>adds</span> <span class='word'>more</span> <span class='word'>variance</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>population</span> <span class='word'>by</span> <span class='word'>taking</span> <span class='word'>properties</span> <span class='word'>of</span> <span class='word'>both</span> <span class='word'>parent</span> <span class='word'>members</span> <span class='word'>and</span> <span class='word'>introducing</span> <span class='word'>the</span> <span class='word'>evaluation</span> <span class='word'>of</span> <span class='word'>their</span> <span class='word'>combination</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>solution</span> <span class='word'>space</span><span class='ignore'>.</span><br>
<span class='word'>By</span> <span class='word'>doing</span> <span class='word'>this</span> <span class='word'>the</span> <span class='word'>algorithm</span> <span class='word'>can</span> <span class='word'>search</span> <span class='word'>multiple</span> <span class='word'>local</span> <span class='word'>maxima</span><span class='ignore'>/</span><span class='word'>minima</span> <span class='word'>at</span> <span class='word'>once</span> <span class='word'>with</span> <span class='word'>its</span> <span class='word'>population</span> <span class='word'>and</span> <span class='word'>start</span> <span class='word'>to</span> <span class='word'>identify</span> <span class='word'>the</span> <span class='word'>global</span> <span class='word'>maxima</span><span class='ignore'>/</span><span class='word'>minima</span> <span class='word'>as</span> <span class='word'>members</span> <span class='word'>become</span> <span class='word'>more</span> <span class='word'>optimized</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\section</span><span class='ignore'>{</span><span class='hword'>Brief</span> <span class='hword'>Task</span> <span class='hword'>Overview</span><span class='ignore'>}</span><span class='cmd'>\label</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>briefOverview</span><span class='ignore'>}</span><br>
<span class='word'>This</span> <span class='word'>work</span> <span class='word'>will</span> <span class='word'>look</span> <span class='word'>at</span> <span class='word'>discovering</span> <span class='word'>which</span> <span class='word'>sequences</span> <span class='word'>provide</span> <span class='word'>the</span> <span class='word'>highest</span> <span class='word'>score</span> <span class='word'>in</span> <span class='word'>a</span> <span class='word'>game</span> <span class='word'>for</span> <span class='word'>each</span> <span class='word'>given</span> <span class='word'>opponent</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>Iterated</span> <span class='word'>Prisoners</span> <span class='word'>Dilemma</span><span class='ignore'>.</span><br>
<span class='word'>For</span> <span class='word'>every</span> <span class='word'>game</span> <span class='word'>of</span> <span class='word'>length</span> <span class='mathgroup'>$</span><span class='math'>n</span><span class='mathgroup'>$</span> <span class='word'>there</span> <span class='word'>are</span> <span class='mathgroup'>$</span><span class='math'>2</span><span class='math'>^</span><span class='math'>n</span><span class='mathgroup'>$</span> <span class='word'>possible</span> <span class='word'>combinations</span> <span class='word'>of</span> <span class='word'>moves</span> <span class='word'>we</span> <span class='word'>can</span> <span class='word'>make</span> <span class='word'>against</span> <span class='word'>an</span> <span class='word'>opponent</span><span class='ignore'>,</span> <span class='word'>we</span> <span class='word'>are</span> <span class='word'>trying</span> <span class='word'>to</span> <span class='word'>find</span> <span class='word'>the</span> <span class='word'>sequence</span> <span class='word'>which</span> <span class='word'>provides</span> <span class='word'>us</span> <span class='word'>with</span> <span class='word'>the</span> <span class='word'>highest</span> <span class='word'>score</span> <span class='word'>overall</span><span class='ignore'>.</span><br>
<br>
<span class='word'>Analysis</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>focused</span> <span class='word'>on</span> <span class='word'>looking</span> <span class='word'>into</span> <span class='word'>just</span> <span class='word'>the</span> <span class='word'>single</span> <span class='word'>opponent</span> <span class='word'>use</span> <span class='word'>case</span><span class='ignore'>,</span> <span class='word'>but</span> <span class='word'>the</span> <span class='word'>idea</span> <span class='word'>of</span> <span class='word'>designing</span> <span class='word'>a</span> <span class='word'>sequence</span> <span class='word'>for</span> <span class='word'>a</span> <span class='word'>tournament</span> <span class='word'>for</span> <span class='word'>a</span> <span class='word'>given</span> <span class='word'>number</span> <span class='word'>of</span> <span class='word'>opponents</span> <span class='word'>is</span> <span class='word'>a</span> <span class='word'>potential</span> <span class='word'>follow</span> <span class='word'>on</span> <span class='word'>to</span> <span class='word'>this</span> <span class='word'>work</span><span class='ignore'>.</span><br>
<span class='word'>Our</span> <span class='word'>task</span> <span class='word'>is</span> <span class='word'>as</span> <span class='word'>follows</span><span class='ignore'>:</span><br>
<br>
<span class='word'>Problem</span><span class='ignore'>:</span><br>
<span class='envir'>\begin</span><span class='envir'>{center}</span><br>
<span class='cmd'>\itshape</span><span class='ignore'>~</span><span class='word'>When</span> <span class='word'>playing</span> <span class='word'>a</span> <span class='word'>given</span> <span class='word'>Iterated</span> <span class='word'>Prisoners</span> <span class='word'>Dilemma</span> <span class='word'>strategy</span> <span class='word'>as</span> <span class='word'>an</span> <span class='word'>opponent</span><span class='ignore'>,</span> <span class='word'>what</span> <span class='word'>is</span> <span class='word'>the</span> <span class='word'>best</span> <span class='word'>ordered</span> <span class='word'>sequence</span> <span class='word'>of</span> <span class='word'>moves</span> <span class='word'>to</span> <span class='word'>play</span> <span class='word'>in</span> <span class='word'>order</span> <span class='word'>for</span> <span class='word'>us</span> <span class='word'>to</span> <span class='word'>obtain</span> <span class='word'>the</span> <span class='word'>highest</span> <span class='word'>possible</span> <span class='word'>average</span> <span class='word'>score</span> <span class='word'>per</span> <span class='word'>move</span> <span class='word'>across</span> <span class='word'>the</span> <span class='word'>game</span><span class='ignore'>?</span> <span class='word'>Are</span> <span class='word'>there</span> <span class='word'>any</span> <span class='word'>patterns</span> <span class='word'>in</span> <span class='word'>these</span> <span class='word'>solution</span> <span class='word'>sequences</span><span class='ignore'>?</span><br>
<span class='envir'>\end</span><span class='envir'>{center}</span><br>
<br>
<span class='word'>For</span> <span class='word'>example</span> <span class='word'>an</span> <span class='word'>opponent</span> <span class='word'>known</span> <span class='word'>as</span> <span class='word'>Tit</span> <span class='word'>For</span> <span class='word'>Tat</span><span class='ignore'>,</span> <span class='word'>which</span> <span class='word'>cooperates</span> <span class='word'>on</span> <span class='word'>its</span> <span class='word'>first</span> <span class='word'>move</span> <span class='word'>and</span> <span class='word'>the</span> <span class='word'>copies</span> <span class='word'>your</span> <span class='word'>last</span> <span class='word'>move</span> <span class='word'>on</span> <span class='word'>subsequent</span> <span class='word'>moves</span><span class='ignore'>,</span> <span class='word'>will</span> <span class='word'>have</span> <span class='word'>a</span> <span class='word'>solution</span> <span class='word'>sequence</span> <span class='word'>of</span> <span class='word'>moves</span> <span class='word'>that</span> <span class='word'>are</span> <span class='word'>all</span> <span class='word'>cooperation</span> <span class='word'>apart</span> <span class='word'>from</span> <span class='word'>the</span> <span class='word'>last</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>is</span> <span class='word'>an</span> <span class='word'>easy</span> <span class='word'>example</span> <span class='word'>and</span> <span class='word'>a</span> <span class='word'>simple</span> <span class='word'>strategy</span> <span class='word'>to</span> <span class='word'>calculate</span> <span class='word'>a</span> <span class='word'>solution</span> <span class='word'>for</span> <span class='word'>due</span> <span class='word'>to</span> <span class='word'>its</span> <span class='word'>explicit</span> <span class='word'>rules</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>ultimate</span> <span class='word'>goal</span> <span class='word'>of</span> <span class='word'>this</span> <span class='word'>investigation</span> <span class='word'>is</span> <span class='word'>to</span> <span class='word'>look</span> <span class='word'>into</span> <span class='word'>every</span> <span class='word'>strategy</span> <span class='word'>as</span> <span class='word'>defined</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>open</span> <span class='word'>source</span> <span class='word'>Axelrod</span> <span class='word'>Library</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>axelrodproject</span><span class='ignore'>}</span><span class='ignore'>,</span> <span class='word'>where</span> <span class='word'>over</span> <span class='word'>200</span> <span class='word'>strategies</span> <span class='word'>are</span> <span class='word'>implemented</span><span class='ignore'>.</span><br>
<span class='word'>These</span> <span class='word'>opponents</span> <span class='word'>are</span> <span class='word'>listed</span> <span class='word'>in</span> <span class='word'>Appendix</span> <span class='word'>Section</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>apndx</span><span class='ignore'>:</span><span class='ignore'>opponents</span><span class='ignore'>}</span> <span class='word'>and</span> <span class='word'>definitions</span> <span class='word'>can</span> <span class='word'>be</span> <span class='word'>found</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>Axelrod</span> <span class='word'>Library</span> <span class='word'>codebase</span><span class='ignore'>.</span><br>
<br>
<span class='word'>This</span> <span class='word'>problem</span> <span class='word'>can</span> <span class='word'>be</span> <span class='word'>reduced</span> <span class='word'>to</span> <span class='word'>searching</span> <span class='word'>the</span> <span class='word'>solution</span> <span class='word'>space</span> <span class='word'>of</span> <span class='word'>every</span> <span class='word'>single</span> <span class='word'>possible</span> <span class='word'>combination</span> <span class='word'>of</span> <span class='mathgroup'>$</span><span class='math'>C</span><span class='mathgroup'>$</span> <span class='word'>and</span> <span class='mathgroup'>$</span><span class='math'>D</span><span class='mathgroup'>$</span> <span class='word'>moves</span><span class='ignore'>.</span><br>
<span class='word'>With</span> <span class='word'>a</span> <span class='word'>game</span> <span class='word'>length</span> <span class='word'>of</span> <span class='word'>200</span> <span class='word'>however</span><span class='ignore'>,</span> <span class='word'>this</span> <span class='word'>is</span> <span class='word'>an</span> <span class='word'>infeasibly</span> <span class='word'>large</span> <span class='word'>number</span> <span class='word'>of</span> <span class='word'>checks</span> <span class='word'>and</span> <span class='word'>so</span> <span class='word'>we</span> <span class='word'>will</span> <span class='word'>use</span> <span class='word'>the</span> <span class='word'>genetic</span> <span class='word'>algorithm</span> <span class='word'>to</span> <span class='word'>selectively</span> <span class='word'>improve</span> <span class='word'>potential</span> <span class='word'>sequences</span><span class='ignore'>.</span> <br>
<span class='word'>Some</span> <span class='word'>examples</span> <span class='word'>of</span> <span class='word'>5</span> <span class='word'>move</span> <span class='word'>games</span> <span class='word'>against</span> <span class='word'>Tit</span> <span class='word'>For</span> <span class='word'>Tat</span> <span class='word'>are</span> <span class='word'>shown</span> <span class='word'>below</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>optimal</span> <span class='word'>solution</span> <span class='word'>for</span> <span class='word'>Tit</span> <span class='word'>For</span> <span class='word'>Tat</span> <span class='word'>in</span> <span class='word'>this</span> <span class='word'>scenario</span> <span class='word'>is</span> <span class='word'>game</span> <span class='word'>5</span> <span class='word'>whereas</span> <span class='word'>all</span> <span class='word'>the</span> <span class='word'>others</span> <span class='word'>are</span> <span class='word'>sub</span> <span class='word'>optimal</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>is</span> <span class='word'>the</span> <span class='word'>process</span> <span class='word'>we</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>doing</span> <span class='word'>for</span> <span class='word'>every</span> <span class='word'>opponent</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>context</span> <span class='word'>of</span> <span class='word'>a</span> <span class='word'>genetic</span> <span class='word'>algorithm</span><span class='ignore'>.</span><br>
<br>
<span class='mathgroup'>$$</span><br>
<span class='exclenv'>\begin</span><span class='exclenv'>{matrix}</span><br>
<span class='mathcmd'>\textbf</span><span class='ignore'>{</span><span class='math'>Game</span> <span class='math'>1</span><span class='ignore'>}</span><span class='math'>&</span> <span class='math'>1</span> <span class='math'>&</span> <span class='math'>2</span> <span class='math'>&</span> <span class='math'>3</span> <span class='math'>&</span> <span class='math'>4</span> <span class='math'>&</span> <span class='math'>5</span> <span class='math'>&</span> <span class='math'>&</span><span class='mathcmd'>\text</span><span class='ignore'>{</span><span class='math'>final</span> <span class='math'>score</span><span class='ignore'>}</span> <span class='math'>&</span> <span class='mathcmd'>\text</span><span class='ignore'>{</span><span class='math'>per</span> <span class='math'>turn</span><span class='ignore'>}</span><span class='mathcmd'>\\</span><br>
<span class='mathcmd'>\text</span><span class='ignore'>{</span><span class='math'>Tit</span> <span class='math'>For</span> <span class='math'>Tat</span><span class='math'>:</span><span class='ignore'>}</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span><span class='math'>&</span> <span class='math'>4</span> <span class='math'>&</span> <span class='math'>0.8</span><span class='mathcmd'>\\</span><br>
<span class='mathcmd'>\text</span><span class='ignore'>{</span><span class='math'>Solution</span><span class='math'>:</span><span class='ignore'>}</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span><span class='math'>&</span> <span class='math'>9</span> <span class='math'>&</span> <span class='math'>1.8</span><br>
<span class='exclenv'>\end</span><span class='exclenv'>{matrix}</span><br>
<span class='mathgroup'>$$</span><br>
<span class='mathgroup'>$$</span><br>
<span class='exclenv'>\begin</span><span class='exclenv'>{matrix}</span><br>
<span class='mathcmd'>\textbf</span><span class='ignore'>{</span><span class='math'>Game</span> <span class='math'>2</span><span class='ignore'>}</span><span class='math'>&</span> <span class='math'>1</span> <span class='math'>&</span> <span class='math'>2</span> <span class='math'>&</span> <span class='math'>3</span> <span class='math'>&</span> <span class='math'>4</span> <span class='math'>&</span> <span class='math'>5</span> <span class='math'>&</span> <span class='math'>&</span><span class='mathcmd'>\text</span><span class='ignore'>{</span><span class='math'>final</span> <span class='math'>score</span><span class='ignore'>}</span> <span class='math'>&</span> <span class='mathcmd'>\text</span><span class='ignore'>{</span><span class='math'>per</span> <span class='math'>turn</span><span class='ignore'>}</span><span class='mathcmd'>\\</span><br>
<span class='mathcmd'>\text</span><span class='ignore'>{</span><span class='math'>Tit</span> <span class='math'>For</span> <span class='math'>Tat</span><span class='math'>:</span><span class='ignore'>}</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span><span class='math'>&</span> <span class='math'>7</span> <span class='math'>&</span> <span class='math'>1.4</span><span class='mathcmd'>\\</span><br>
<span class='mathcmd'>\text</span><span class='ignore'>{</span><span class='math'>Solution</span><span class='math'>:</span><span class='ignore'>}</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span><span class='math'>&</span> <span class='math'>12</span> <span class='math'>&</span> <span class='math'>2.4</span><br>
<span class='exclenv'>\end</span><span class='exclenv'>{matrix}</span><br>
<span class='mathgroup'>$$</span><br>
<span class='mathgroup'>$$</span><br>
<span class='exclenv'>\begin</span><span class='exclenv'>{matrix}</span><br>
<span class='mathcmd'>\textbf</span><span class='ignore'>{</span><span class='math'>Game</span> <span class='math'>3</span><span class='ignore'>}</span><span class='math'>&</span> <span class='math'>1</span> <span class='math'>&</span> <span class='math'>2</span> <span class='math'>&</span> <span class='math'>3</span> <span class='math'>&</span> <span class='math'>4</span> <span class='math'>&</span> <span class='math'>5</span> <span class='math'>&</span> <span class='math'>&</span><span class='mathcmd'>\text</span><span class='ignore'>{</span><span class='math'>final</span> <span class='math'>score</span><span class='ignore'>}</span> <span class='math'>&</span> <span class='mathcmd'>\text</span><span class='ignore'>{</span><span class='math'>per</span> <span class='math'>turn</span><span class='ignore'>}</span><span class='mathcmd'>\\</span><br>
<span class='mathcmd'>\text</span><span class='ignore'>{</span><span class='math'>Tit</span> <span class='math'>For</span> <span class='math'>Tat</span><span class='math'>:</span><span class='ignore'>}</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span><span class='math'>&</span> <span class='math'>10</span> <span class='math'>&</span> <span class='math'>2.0</span><span class='mathcmd'>\\</span><br>
<span class='mathcmd'>\text</span><span class='ignore'>{</span><span class='math'>Solution</span><span class='math'>:</span><span class='ignore'>}</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span><span class='math'>&</span> <span class='math'>15</span> <span class='math'>&</span> <span class='math'>3.0</span><br>
<span class='exclenv'>\end</span><span class='exclenv'>{matrix}</span><br>
<span class='mathgroup'>$$</span><br>
<span class='mathgroup'>$$</span><br>
<span class='exclenv'>\begin</span><span class='exclenv'>{matrix}</span><br>
<span class='mathcmd'>\textbf</span><span class='ignore'>{</span><span class='math'>Game</span> <span class='math'>4</span><span class='ignore'>}</span><span class='math'>&</span> <span class='math'>1</span> <span class='math'>&</span> <span class='math'>2</span> <span class='math'>&</span> <span class='math'>3</span> <span class='math'>&</span> <span class='math'>4</span> <span class='math'>&</span> <span class='math'>5</span> <span class='math'>&</span> <span class='math'>&</span><span class='mathcmd'>\text</span><span class='ignore'>{</span><span class='math'>final</span> <span class='math'>score</span><span class='ignore'>}</span> <span class='math'>&</span> <span class='mathcmd'>\text</span><span class='ignore'>{</span><span class='math'>per</span> <span class='math'>turn</span><span class='ignore'>}</span><span class='mathcmd'>\\</span><br>
<span class='mathcmd'>\text</span><span class='ignore'>{</span><span class='math'>Tit</span> <span class='math'>For</span> <span class='math'>Tat</span><span class='math'>:</span><span class='ignore'>}</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span><span class='math'>&</span> <span class='math'>11</span> <span class='math'>&</span> <span class='math'>2.2</span><span class='mathcmd'>\\</span><br>
<span class='mathcmd'>\text</span><span class='ignore'>{</span><span class='math'>Solution</span><span class='math'>:</span><span class='ignore'>}</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span><span class='math'>&</span> <span class='math'>16</span> <span class='math'>&</span> <span class='math'>3.2</span> <br>
<span class='exclenv'>\end</span><span class='exclenv'>{matrix}</span><br>
<span class='mathgroup'>$$</span><br>
<span class='mathgroup'>$$</span><br>
<span class='exclenv'>\begin</span><span class='exclenv'>{matrix}</span><br>
<span class='mathcmd'>\textbf</span><span class='ignore'>{</span><span class='math'>Game</span> <span class='math'>5</span><span class='ignore'>}</span><span class='math'>&</span> <span class='math'>1</span> <span class='math'>&</span> <span class='math'>2</span> <span class='math'>&</span> <span class='math'>3</span> <span class='math'>&</span> <span class='math'>4</span> <span class='math'>&</span> <span class='math'>5</span> <span class='math'>&</span> <span class='math'>&</span><span class='mathcmd'>\text</span><span class='ignore'>{</span><span class='math'>final</span> <span class='math'>score</span><span class='ignore'>}</span> <span class='math'>&</span> <span class='mathcmd'>\text</span><span class='ignore'>{</span><span class='math'>per</span> <span class='math'>turn</span><span class='ignore'>}</span><span class='mathcmd'>\\</span><br>
<span class='mathcmd'>\text</span><span class='ignore'>{</span><span class='math'>Tit</span> <span class='math'>For</span> <span class='math'>Tat</span><span class='math'>:</span><span class='ignore'>}</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span><span class='math'>&</span> <span class='math'>12</span> <span class='math'>&</span> <span class='math'>2.4</span><span class='mathcmd'>\\</span><br>
<span class='mathcmd'>\text</span><span class='ignore'>{</span><span class='math'>Solution</span><span class='math'>:</span><span class='ignore'>}</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>D</span> <span class='math'>&</span><span class='math'>&</span> <span class='math'>17</span> <span class='math'>&</span> <span class='math'>3.4</span> <br>
<span class='exclenv'>\end</span><span class='exclenv'>{matrix}</span><br>
<span class='mathgroup'>$$</span><br>
<span class='mathgroup'>$$</span><br>
<span class='exclenv'>\begin</span><span class='exclenv'>{matrix}</span><br>
<span class='mathcmd'>\textbf</span><span class='ignore'>{</span><span class='math'>Game</span> <span class='math'>6</span><span class='ignore'>}</span><span class='math'>&</span> <span class='math'>1</span> <span class='math'>&</span> <span class='math'>2</span> <span class='math'>&</span> <span class='math'>3</span> <span class='math'>&</span> <span class='math'>4</span> <span class='math'>&</span> <span class='math'>5</span> <span class='math'>&</span> <span class='math'>&</span><span class='mathcmd'>\text</span><span class='ignore'>{</span><span class='math'>final</span> <span class='math'>score</span><span class='ignore'>}</span> <span class='math'>&</span> <span class='mathcmd'>\text</span><span class='ignore'>{</span><span class='math'>per</span> <span class='math'>turn</span><span class='ignore'>}</span><span class='mathcmd'>\\</span><br>
<span class='mathcmd'>\text</span><span class='ignore'>{</span><span class='math'>Tit</span> <span class='math'>For</span> <span class='math'>Tat</span><span class='math'>:</span><span class='ignore'>}</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span><span class='math'>&</span> <span class='math'>15</span> <span class='math'>&</span> <span class='math'>3.0</span><span class='mathcmd'>\\</span><br>
<span class='mathcmd'>\text</span><span class='ignore'>{</span><span class='math'>Solution</span><span class='math'>:</span><span class='ignore'>}</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span> <span class='math'>C</span> <span class='math'>&</span><span class='math'>&</span> <span class='math'>15</span> <span class='math'>&</span> <span class='math'>3.0</span> <br>
<span class='exclenv'>\end</span><span class='exclenv'>{matrix}</span><br>
<span class='mathgroup'>$$</span><br>
<br>
<span class='cmd'>\section</span><span class='ignore'>{</span><span class='hword'>Conclusion</span> <span class='cmd'>\&</span> <span class='hword'>Structure</span> <span class='hword'>of</span> <span class='hword'>This</span> <span class='hword'>Report</span><span class='ignore'>}</span><br>
<span class='word'>This</span> <span class='word'>chapter</span> <span class='word'>contained</span> <span class='word'>introduction</span> <span class='word'>to</span> <span class='word'>content</span> <span class='word'>that</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>needed</span> <span class='word'>and</span> <span class='word'>expanded</span> <span class='word'>on</span> <span class='word'>moving</span> <span class='word'>forward</span><span class='ignore'>.</span> <br>
<span class='word'>All</span> <span class='word'>the</span> <span class='word'>topics</span> <span class='word'>mentioned</span> <span class='word'>in</span> <span class='word'>this</span> <span class='word'>chapter</span> <span class='word'>are</span> <span class='word'>core</span> <span class='word'>themes</span> <span class='word'>to</span> <span class='word'>what</span> <span class='word'>the</span> <span class='word'>rest</span> <span class='word'>of</span> <span class='word'>this</span> <span class='word'>report</span> <span class='word'>focuses</span> <span class='word'>on</span><span class='ignore'>.</span><br>
<br>
<span class='word'>This</span> <span class='word'>report</span> <span class='word'>will</span> <span class='word'>contain</span> <span class='word'>the</span> <span class='word'>following</span> <span class='word'>chapters</span><span class='ignore'>:</span><br>
<span class='envir'>\begin</span><span class='envir'>{itemize}</span><br>
<span class='cmd'>\item</span> <span class='word'>Chapter</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>ch</span><span class='ignore'>:</span><span class='ignore'>taskBackground</span><span class='ignore'>}</span> <span class='word'>describes</span> <span class='word'>methodology</span> <span class='word'>and</span> <span class='word'>concepts</span> <span class='word'>that</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>needed</span> <span class='word'>to</span> <span class='word'>understand</span> <span class='word'>results</span> <span class='word'>in</span> <span class='word'>later</span> <span class='word'>chapters</span><span class='ignore'>.</span><br>
<span class='cmd'>\item</span> <span class='word'>Chapter</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>ch</span><span class='ignore'>:</span><span class='ignore'>literature</span><span class='ignore'>}</span> <span class='word'>looks</span> <span class='word'>into</span> <span class='word'>previous</span> <span class='word'>work</span> <span class='word'>and</span> <span class='word'>literature</span> <span class='word'>behind</span> <span class='word'>the</span> <span class='word'>concepts</span> <span class='word'>we</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>working</span> <span class='word'>with</span><span class='ignore'>.</span><br>
<span class='cmd'>\item</span> <span class='word'>Chapter</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>ch</span><span class='ignore'>:</span><span class='ignore'>developingthecodebase</span><span class='ignore'>}</span> <span class='word'>looks</span> <span class='word'>into</span> <span class='word'>the</span> <span class='word'>programming</span> <span class='word'>and</span> <span class='word'>development</span> <span class='word'>that</span> <span class='word'>was</span> <span class='word'>undertaken</span> <span class='word'>to</span> <span class='word'>complete</span> <span class='word'>the</span> <span class='word'>work</span><span class='ignore'>.</span><br>
<span class='cmd'>\item</span> <span class='word'>Chapter</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>ch</span><span class='ignore'>:</span><span class='ignore'>implementation</span><span class='ignore'>}</span> <span class='word'>looks</span> <span class='word'>at</span> <span class='word'>how</span> <span class='word'>the</span> <span class='word'>application</span> <span class='word'>of</span> <span class='word'>our</span> <span class='word'>genetic</span> <span class='word'>algorithm</span> <span class='word'>was</span> <span class='word'>improved</span> <span class='word'>before</span> <span class='word'>running</span> <span class='word'>the</span> <span class='word'>final</span> <span class='word'>analysis</span><span class='ignore'>.</span><br>
<span class='cmd'>\item</span> <span class='word'>Chapter</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>ch</span><span class='ignore'>:</span><span class='ignore'>results</span><span class='ignore'>}</span> <span class='word'>is</span> <span class='word'>the</span> <span class='word'>chapter</span> <span class='word'>communicating</span> <span class='word'>the</span> <span class='word'>results</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>final</span> <span class='word'>analysis</span><span class='ignore'>.</span><br>
<span class='cmd'>\item</span> <span class='word'>Chapter</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>ch</span><span class='ignore'>:</span><span class='ignore'>conclusions</span><span class='ignore'>}</span> <span class='word'>describes</span> <span class='word'>the</span> <span class='word'>takeaways</span> <span class='word'>and</span> <span class='word'>possible</span> <span class='word'>applications</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>work</span><span class='ignore'>.</span><br>
<span class='envir'>\end</span><span class='envir'>{itemize}</span></p></div>
<br>
<div class='count'>
<ul class='count'>
<li class='header'>File: C:/dev/projects/uni/FinalYearProject/LaTeX/tex/chapters/01-introduction.tex</li>
<li>Encoding: ascii</li>
<li>Words in text: 1739</li>
<li>Words in headers: 22</li>
<li>Words outside text (captions, etc.): 25</li>
<li>Number of headers: 6</li>
<li>Number of floats/tables/figures: 1</li>
<li>Number of math inlines: 25</li>
<li>Number of math displayed: 10</li>
<li>Subcounts:</li>
<span class='subcount'>
<li class='fielddesc'> text+headers+captions (#headers/#floats/#inlines/#displayed)</li>
<li>175+1+0 (1/0/2/0) Chapter: Introduction}\label{ch:intro</li>
<li>345+7+0 (1/0/7/4) Section: The Prisoners Dilemma and Its Iterated Form}\label{sec:iteratedPrisonersDilemma</li>
<li>202+4+0 (1/0/0/0) Section: Machine Learning \& Computer Intelligence}\label{sec:machineLearningAndcomputerIntelligence</li>
<li>552+2+25 (1/1/12/0) Subsection: Genetic Algorithms}\label{subsec:geneticAlgorithms</li>
<li>342+3+0 (1/0/4/6) Section: Brief Task Overview}\label{sec:briefOverview</li>
<li>123+5+0 (1/0/0/0) Section: Conclusion \& Structure of This Report</li>
</span>
</ul>
</div>
</div>
<div class='filegroup'>
<h2>File: C:/dev/projects/uni/FinalYearProject/LaTeX/tex/chapters/02-background.tex</h2>
<div class='parse'><p>
<span class='comment'>% !TEX root = ../main.tex</span><br>
<br>
<span class='cmd'>\chapter</span><span class='ignore'>{</span><span class='hword'>Task</span> <span class='hword'>Methodology</span><span class='ignore'>}</span><span class='cmd'>\label</span><span class='ignore'>{</span><span class='ignore'>ch</span><span class='ignore'>:</span><span class='ignore'>taskBackground</span><span class='ignore'>}</span><br>
<span class='word'>This</span> <span class='word'>chapter</span> <span class='word'>will</span> <span class='word'>cover</span> <span class='word'>sections</span> <span class='word'>on</span> <span class='word'>how</span> <span class='word'>each</span> <span class='word'>part</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>project</span> <span class='word'>will</span> <span class='word'>been</span> <span class='word'>carried</span> <span class='word'>out</span><span class='ignore'>.</span><br>
<span class='word'>Useful</span> <span class='word'>notation</span> <span class='word'>is</span> <span class='word'>given</span> <span class='word'>in</span> <span class='word'>Section</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>notation</span><span class='ignore'>}</span> <span class='word'>which</span> <span class='word'>will</span> <span class='word'>allow</span> <span class='word'>us</span> <span class='word'>to</span> <span class='word'>describe</span> <span class='word'>a</span> <span class='word'>given</span> <span class='word'>sequence</span> <span class='word'>or</span> <span class='word'>set</span> <span class='word'>of</span> <span class='word'>sequences</span> <span class='word'>in</span> <span class='word'>a</span> <span class='word'>compact</span> <span class='word'>form</span><span class='ignore'>.</span><br>
<span class='word'>We</span> <span class='word'>will</span> <span class='word'>also</span> <span class='word'>look</span> <span class='word'>at</span> <span class='word'>our</span> <span class='word'>specific</span> <span class='word'>instance</span> <span class='word'>of</span> <span class='word'>genetic</span> <span class='word'>algorithm</span> <span class='word'>that</span> <span class='word'>exists</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>Axelrod</span> <span class='word'>Dojo</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>dojoV008</span><span class='ignore'>}</span> <span class='word'>codebase</span><span class='ignore'>.</span><br>
<span class='word'>Lastly</span> <span class='word'>Section</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>solutionForm</span><span class='ignore'>}</span> <span class='word'>looks</span> <span class='word'>at</span> <span class='word'>the</span> <span class='word'>what</span> <span class='word'>results</span> <span class='word'>we</span> <span class='word'>will</span> <span class='word'>expect</span> <span class='word'>from</span> <span class='word'>the</span> <span class='word'>analysis</span> <span class='word'>and</span> <span class='word'>the</span> <span class='word'>potential</span> <span class='word'>problems</span> <span class='word'>we</span> <span class='word'>want</span> <span class='word'>to</span> <span class='word'>mitigate</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\section</span><span class='ignore'>{</span><span class='hword'>Notation</span><span class='ignore'>}</span><span class='cmd'>\label</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>notation</span><span class='ignore'>}</span><br>
<span class='word'>As</span> <span class='word'>described</span> <span class='word'>in</span> <span class='word'>Chapter</span> <span class='word'>1</span> <span class='word'>this</span> <span class='word'>work</span> <span class='word'>focuses</span> <span class='word'>on</span> <span class='word'>identifying</span> <span class='word'>best</span> <span class='word'>responses</span> <span class='word'>against</span> <span class='word'>the</span> <span class='word'>set</span> <span class='word'>of</span> <span class='word'>opponents</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>Python</span> <span class='word'>Axelrod</span> <span class='word'>Library</span><span class='ignore'>.</span> <span class='word'>The</span> <span class='word'>best</span> <span class='word'>responses</span> <span class='word'>are</span> <span class='word'>in</span> <span class='word'>a</span> <span class='word'>sequence</span> <span class='word'>format</span> <span class='word'>with</span> <span class='word'>a</span> <span class='word'>length</span> <span class='word'>of</span> <span class='word'>200</span><span class='ignore'>.</span> <span class='word'>In</span> <span class='word'>this</span> <span class='word'>Section</span> <span class='word'>we</span> <span class='word'>introduce</span> <span class='word'>a</span> <span class='word'>notation</span> <span class='word'>which</span> <span class='word'>will</span> <span class='word'>allow</span> <span class='word'>us</span> <span class='word'>to</span> <span class='word'>represent</span> <span class='word'>such</span> <span class='word'>sequences</span> <span class='word'>in</span> <span class='word'>a</span> <span class='word'>concise</span> <span class='word'>format</span><span class='ignore'>.</span><br>
<br>
<span class='word'>Let</span> <span class='mathgroup'>$</span><span class='math'>S</span><span class='mathgroup'>$</span> <span class='word'>be</span> <span class='word'>a</span> <span class='word'>sequence</span> <span class='word'>where</span> <span class='mathgroup'>\(</span><span class='math'>S</span><span class='mathcmd'>\in</span><span class='math'>\{</span><span class='math'>C</span><span class='math'>,</span> <span class='math'>D</span><span class='math'>\}</span><span class='math'>^</span><span class='math'>L</span><span class='mathgroup'>\)</span> <span class='word'>and</span> <span class='mathgroup'>\(</span><span class='math'>C</span><span class='mathgroup'>\)</span><span class='ignore'>,</span> <span class='mathgroup'>\(</span><span class='math'>D</span><span class='mathgroup'>\)</span> <span class='word'>represents</span> <span class='word'>a</span> <span class='word'>cooperation</span><span class='ignore'>,</span> <span class='word'>defection</span> <span class='word'>respectively</span><span class='ignore'>.</span><br>
<span class='mathgroup'>\(</span><span class='math'>L</span><span class='math'>=</span><span class='math'>200</span><span class='mathgroup'>\)</span> <span class='word'>is</span> <span class='word'>used</span> <span class='word'>throughout</span> <span class='word'>this</span> <span class='word'>report</span><span class='ignore'>.</span><br>
<span class='word'>Using</span> <span class='word'>the</span> <span class='word'>binary</span> <span class='word'>nature</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>sequence</span> <span class='word'>elements</span><span class='ignore'>,</span> <span class='word'>we</span> <span class='word'>can</span> <span class='word'>split</span> <span class='word'>up</span> <span class='word'>sequences</span> <span class='word'>into</span> <span class='word'>blocks</span> <span class='word'>of</span> <span class='word'>consecutive</span> <span class='word'>move</span> <span class='word'>elements</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>same</span> <span class='word'>type</span><span class='ignore'>.</span><br>
<span class='word'>We</span> <span class='word'>will</span> <span class='word'>use</span> <span class='mathgroup'>\(</span><span class='math'>B</span><span class='math'>_</span><span class='math'>i</span><span class='mathgroup'>\)</span> <span class='word'>to</span> <span class='word'>denote</span> <span class='word'>the</span> <span class='word'>block</span> <span class='word'>after</span> <span class='mathgroup'>\(</span><span class='math'>i</span><span class='mathgroup'>\)</span> <span class='word'>changes</span> <span class='word'>of</span> <span class='word'>move</span> <span class='word'>type</span> <span class='word'>from</span> <span class='word'>the</span> <span class='word'>explicitly</span> <span class='word'>stated</span> <span class='word'>starting</span> <span class='word'>move</span> <span class='word'>type</span><span class='ignore'>.</span><br>
<span class='envir'>\begin</span><span class='envir'>{itemize}</span><br>
<span class='cmd'>\item</span> <span class='word'>Every</span> <span class='word'>move</span> <span class='word'>in</span> <span class='word'>a</span> <span class='word'>block</span> <span class='word'>is</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>same</span> <span class='word'>type</span><span class='ignore'>,</span> <span class='mathgroup'>$</span><span class='math'>C</span><span class='mathgroup'>$</span> <span class='word'>or</span> <span class='mathgroup'>$</span><span class='math'>D</span><span class='mathgroup'>$</span><span class='ignore'>;</span><br>
<span class='word'>the</span> <span class='word'>type</span> <span class='word'>is</span> <span class='word'>implicit</span> <span class='word'>based</span> <span class='word'>on</span> <span class='word'>whether</span> <span class='mathgroup'>\(</span><span class='math'>i</span><span class='mathgroup'>\)</span> <span class='word'>is</span> <span class='word'>even</span> <span class='word'>or</span> <span class='word'>not</span> <span class='word'>and</span> <span class='word'>what</span> <span class='word'>the</span> <span class='word'>starting</span> <span class='word'>move</span> <span class='word'>type</span> <span class='word'>was</span><span class='ignore'>.</span><br>
<span class='cmd'>\item</span> <span class='word'>We</span> <span class='word'>can</span> <span class='word'>use</span> <span class='word'>the</span> <span class='word'>notation</span> <span class='mathgroup'>\(</span><span class='math'>|</span><span class='math'>B</span><span class='math'>_</span><span class='math'>i</span><span class='math'>|</span><span class='mathgroup'>\)</span> <span class='word'>to</span> <span class='word'>denote</span> <span class='word'>the</span> <span class='word'>length</span> <span class='word'>of</span> <span class='word'>the</span> <span class='mathgroup'>\(</span><span class='math'>i</span><span class='mathgroup'>\)</span><span class='word'>th</span> <span class='word'>block</span> <span class='ignore'>(</span><span class='word'>number</span> <span class='word'>of</span> <span class='word'>moves</span> <span class='word'>within</span> <span class='word'>the</span> <span class='word'>block</span><span class='ignore'>)</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>sequence</span><span class='ignore'>.</span><span class='mathgroup'>\(</span><span class='math'>|</span><span class='math'>B</span><span class='math'>_</span><span class='math'>i</span><span class='math'>|</span> <span class='mathcmd'>\in</span> <span class='mathcmd'>\mathbb</span><span class='ignore'>{</span><span class='math'>Z</span><span class='ignore'>}</span><span class='mathgroup'>\)</span><br>
<span class='envir'>\end</span><span class='envir'>{itemize}</span><br>
<br>
<span class='word'>This</span> <span class='word'>means</span> <span class='word'>we</span> <span class='word'>can</span> <span class='word'>write</span> <span class='word'>a</span> <span class='word'>sequence</span> <span class='word'>as</span> <span class='word'>a</span> <span class='word'>series</span> <span class='word'>of</span> <span class='word'>blocks</span><span class='ignore'>:</span><br>
<span class='mathgroup'>\[</span><span class='math'>S</span><span class='math'>=</span> <span class='math'>B</span><span class='math'>_</span><span class='math'>1</span> <span class='math'>B</span><span class='math'>_</span><span class='math'>2</span><span class='math'>,</span><span class='mathcmd'>\ldots</span><span class='math'>,</span><span class='math'>B</span><span class='math'>_</span><span class='math'>n</span><span class='mathgroup'>\]</span><br>
<span class='word'>A</span> <span class='word'>sequence</span> <span class='word'>can</span> <span class='word'>also</span> <span class='word'>be</span> <span class='word'>represented</span> <span class='word'>shorthand</span> <span class='word'>by</span> <span class='word'>specifying</span> <span class='word'>the</span> <span class='word'>starting</span> <span class='word'>move</span> <span class='word'>and</span> <span class='word'>the</span> <span class='word'>length</span> <span class='word'>of</span> <span class='word'>subsequent</span> <span class='word'>blocks</span><span class='ignore'>:</span><br>
<span class='mathgroup'>\[</span><span class='math'>S</span> <span class='math'>=</span> <span class='math'>C</span><span class='mathcmd'>\ </span><span class='math'>|</span><span class='math'>B</span><span class='math'>_</span><span class='math'>1</span><span class='math'>|</span><span class='math'>,</span><span class='math'>|</span><span class='math'>B</span><span class='math'>_</span><span class='math'>2</span><span class='math'>|</span><span class='math'>,</span><span class='mathcmd'>\ldots</span><span class='math'>,</span><span class='math'>|</span><span class='math'>B</span><span class='math'>_</span><span class='math'>n</span><span class='math'>|</span><span class='mathcmd'>\ </span><span class='mathcmd'>\quad</span> <span class='mathcmd'>\Rightarrow</span> <span class='mathcmd'>\quad</span> <span class='math'>S</span><span class='math'>=</span><span class='mathcmd'>\overbrace</span><span class='ignore'>{</span><span class='math'>C</span><span class='mathcmd'>\ldots</span> <span class='math'>C</span><span class='ignore'>}</span><span class='math'>^</span><span class='ignore'>{</span><span class='math'>|</span><span class='math'>B</span><span class='math'>_</span><span class='math'>1</span><span class='math'>|</span><span class='ignore'>}</span><span class='mathcmd'>\overbrace</span><span class='ignore'>{</span><span class='math'>D</span><span class='mathcmd'>\ldots</span> <span class='math'>D</span><span class='ignore'>}</span><span class='math'>^</span><span class='ignore'>{</span><span class='math'>|</span><span class='math'>B</span><span class='math'>_</span><span class='math'>2</span><span class='math'>|</span><span class='ignore'>}</span><span class='mathcmd'>\ldots</span><span class='mathcmd'>\overbrace</span><span class='ignore'>{</span><span class='math'>(</span><span class='math'>C</span><span class='math'>|</span><span class='math'>D</span><span class='math'>)</span><span class='mathcmd'>\ldots</span> <span class='math'>(</span><span class='math'>C</span><span class='math'>|</span><span class='math'>D</span><span class='math'>)</span><span class='ignore'>}</span><span class='math'>^</span><span class='ignore'>{</span><span class='math'>|</span><span class='math'>B</span><span class='math'>_</span><span class='math'>n</span><span class='math'>|</span><span class='ignore'>}</span> <span class='mathgroup'>\]</span><br>
<span class='word'>We</span> <span class='word'>can</span> <span class='word'>also</span> <span class='word'>construct</span> <span class='word'>sequences</span> <span class='word'>from</span> <span class='word'>repetitions</span> <span class='word'>of</span> <span class='word'>a</span> <span class='word'>sequence</span> <span class='word'>of</span> <span class='word'>blocks</span> <span class='word'>when</span> <span class='word'>it</span> <span class='word'>makes</span> <span class='word'>sense</span><span class='ignore'>:</span><br>
<span class='mathgroup'>\[</span><span class='math'>C</span><span class='mathcmd'>\ </span><span class='mathcmd'>\big</span><span class='ignore'>{</span><span class='math'>(</span><span class='ignore'>}</span><span class='math'>|</span><span class='math'>B</span><span class='math'>_</span><span class='math'>1</span><span class='math'>|</span><span class='math'>,</span><span class='math'>|</span><span class='math'>B</span><span class='math'>_</span><span class='math'>2</span><span class='math'>|</span><span class='math'>,</span><span class='mathcmd'>\ </span><span class='mathcmd'>\ldots</span><span class='math'>,</span><span class='mathcmd'>\ </span><span class='math'>|</span><span class='math'>B</span><span class='math'>_</span><span class='math'>m</span><span class='math'>|</span><span class='mathcmd'>\big</span><span class='ignore'>{</span><span class='math'>)</span><span class='ignore'>}</span><span class='math'>^</span><span class='ignore'>{</span><span class='math'>k</span><span class='ignore'>}</span><span class='mathcmd'>\ </span><span class='mathcmd'>\quad</span> <span class='mathcmd'>\Rightarrow</span> <span class='mathcmd'>\quad</span> <span class='mathcmd'>\big</span><span class='ignore'>{</span><span class='math'>(</span><span class='ignore'>}</span><span class='mathcmd'>\small</span><span class='ignore'>{</span><span class='mathcmd'>\overbrace</span><span class='ignore'>{</span><span class='math'>C</span><span class='mathcmd'>\ldots</span> <span class='math'>C</span><span class='ignore'>}</span><span class='math'>^</span><span class='ignore'>{</span><span class='math'>|</span><span class='math'>B</span><span class='math'>_</span><span class='math'>1</span><span class='math'>|</span><span class='ignore'>}</span><span class='mathcmd'>\overbrace</span><span class='ignore'>{</span><span class='math'>D</span><span class='mathcmd'>\ldots</span> <span class='math'>D</span><span class='ignore'>}</span><span class='math'>^</span><span class='ignore'>{</span><span class='math'>|</span><span class='math'>B</span><span class='math'>_</span><span class='math'>2</span><span class='math'>|</span><span class='ignore'>}</span><span class='mathcmd'>\ldots</span><span class='mathcmd'>\overbrace</span><span class='ignore'>{</span><span class='math'>D</span><span class='mathcmd'>\ldots</span> <span class='math'>D</span><span class='ignore'>}</span><span class='math'>^</span><span class='ignore'>{</span><span class='math'>|</span><span class='math'>B</span><span class='math'>_</span><span class='math'>m</span><span class='math'>|</span><span class='ignore'>}</span><span class='ignore'>}</span><span class='mathcmd'>\big</span><span class='ignore'>{</span><span class='math'>)</span><span class='ignore'>}</span><span class='math'>^</span><span class='ignore'>{</span><span class='math'>k-times</span><span class='ignore'>}</span><span class='mathgroup'>\]</span><br>
<span class='word'>The</span> <span class='word'>two</span> <span class='word'>notations</span> <span class='word'>can</span> <span class='word'>be</span> <span class='word'>combined</span> <span class='word'>to</span> <span class='word'>add</span> <span class='word'>starting</span> <span class='word'>and</span> <span class='word'>ending</span> <span class='word'>blocks</span> <span class='word'>to</span> <span class='word'>a</span> <span class='word'>repeating</span> <span class='word'>sequence</span> <span class='ignore'>(</span><span class='word'>shown</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>examples</span><span class='ignore'>)</span><span class='ignore'>.</span><br>
<br>
<span class='word'>It</span> <span class='word'>is</span> <span class='word'>also</span> <span class='word'>possible</span> <span class='word'>to</span> <span class='word'>define</span> <span class='word'>sets</span> <span class='word'>of</span> <span class='word'>sequences</span> <span class='word'>by</span> <span class='word'>adding</span> <span class='word'>variables</span> <span class='word'>to</span> <span class='word'>parameters</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>sequence</span><span class='ignore'>.</span><br>
<span class='word'>Appropriate</span> <span class='word'>selection</span> <span class='word'>of</span> <span class='word'>parameters</span> <span class='word'>mean</span> <span class='word'>the</span> <span class='word'>length</span> <span class='word'>of</span> <span class='word'>sequences</span> <span class='word'>should</span> <span class='word'>not</span> <span class='word'>grow</span><span class='ignore'>.</span><br>
<span class='mathgroup'>\[</span> <span class='math'>\{</span><span class='math'>Ci</span><span class='math'>,</span><span class='math'>l-i</span><span class='math'>\}</span> <span class='mathcmd'>\quad</span> <span class='math'>i</span><span class='mathcmd'>\in</span> <span class='option'>[a,b]</span> <span class='mathcmd'>\Rightarrow</span> <span class='math'>\{</span><span class='mathcmd'>\underbrace</span><span class='ignore'>{</span><span class='math'>C</span><span class='mathcmd'>\ldots</span> <span class='math'>C</span><span class='ignore'>}</span><span class='math'>_</span><span class='ignore'>{</span><span class='math'>a</span><span class='ignore'>}</span><span class='mathcmd'>\overbrace</span><span class='ignore'>{</span><span class='math'>D</span><span class='mathcmd'>\ldots</span> <span class='math'>D</span><span class='ignore'>}</span><span class='math'>^</span><span class='ignore'>{</span><span class='math'>l-a</span><span class='ignore'>}</span><span class='math'>,</span><span class='mathcmd'>\ </span><span class='mathcmd'>\underbrace</span><span class='ignore'>{</span><span class='math'>C</span><span class='mathcmd'>\ldots</span> <span class='math'>C</span><span class='ignore'>}</span><span class='math'>_</span><span class='ignore'>{</span><span class='math'>a</span><span class='math'>+</span><span class='math'>1</span><span class='ignore'>}</span><span class='mathcmd'>\overbrace</span><span class='ignore'>{</span><span class='math'>D</span><span class='mathcmd'>\ldots</span> <span class='math'>D</span><span class='ignore'>}</span><span class='math'>^</span><span class='ignore'>{</span><span class='math'>l</span><span class='math'>-</span><span class='math'>(</span><span class='math'>a</span><span class='math'>+</span><span class='math'>1</span><span class='math'>)</span><span class='ignore'>}</span><span class='math'>,</span><span class='mathcmd'>\ldots</span> <span class='math'>,</span><span class='mathcmd'>\ </span><span class='mathcmd'>\underbrace</span><span class='ignore'>{</span><span class='math'>C</span><span class='mathcmd'>\ldots</span> <span class='math'>C</span><span class='ignore'>}</span><span class='math'>_</span><span class='ignore'>{</span><span class='math'>b</span><span class='ignore'>}</span><span class='mathcmd'>\overbrace</span><span class='ignore'>{</span><span class='math'>D</span><span class='mathcmd'>\ldots</span> <span class='math'>D</span><span class='ignore'>}</span><span class='math'>^</span><span class='ignore'>{</span><span class='math'>l-b</span><span class='ignore'>}</span><span class='math'>\}</span> <span class='mathgroup'>\]</span><br>
<span class='word'>For</span> <span class='word'>long</span> <span class='word'>sequences</span> <span class='word'>where</span> <span class='word'>there</span> <span class='word'>is</span> <span class='word'>no</span> <span class='word'>recognisable</span> <span class='word'>pattern</span> <span class='word'>it</span> <span class='word'>is</span> <span class='word'>typically</span> <span class='word'>easier</span> <span class='word'>to</span> <span class='word'>describe</span> <span class='word'>the</span> <span class='word'>solution</span><span class='ignore'>.</span><br>
<span class='word'>Otherwise</span> <span class='word'>we</span> <span class='word'>use</span> <span class='word'>the</span> <span class='word'>notation</span> <span class='word'>to</span> <span class='word'>describe</span> <span class='word'>a</span> <span class='word'>sequence</span><span class='ignore'>.</span><br>
<span class='word'>When</span> <span class='word'>we</span> <span class='word'>look</span> <span class='word'>at</span> <span class='word'>solutions</span> <span class='word'>there</span> <span class='word'>too</span> <span class='word'>long</span> <span class='word'>to</span> <span class='word'>write</span> <span class='word'>the</span> <span class='word'>sequence</span> <span class='word'>directly</span><span class='ignore'>,</span> <span class='word'>in</span> <span class='word'>which</span> <span class='word'>case</span> <span class='word'>we</span> <span class='word'>may</span> <span class='word'>abbreviate</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>following</span> <span class='word'>style</span><span class='ignore'>:</span> <span class='mathgroup'>$</span><span class='math'>C1</span><span class='math'>,</span><span class='math'>5</span><span class='math'>,</span><span class='math'>2</span><span class='math'>,</span><span class='math'>3</span><span class='math'>,</span><span class='math'>5</span><span class='math'>,</span><span class='math'>6</span><span class='math'>,</span><span class='math'>.</span><span class='math'>.</span><span class='math'>.</span><span class='math'>=</span><span class='math'>CDDDDDCCDDDCCCCCDDDDDD</span><span class='math'>.</span><span class='math'>.</span><span class='math'>.</span><span class='mathgroup'>$</span><span class='ignore'>.</span><br>
<br>
<span class='word'>Examples</span><span class='ignore'>:</span><br>
<span class='envir'>\begin</span><span class='envir'>{align}</span><br>
<span class='math'>C1</span><span class='math'>,</span><span class='math'>4</span><span class='math'>,</span><span class='math'>3</span><span class='math'>,</span><span class='math'>2</span> <span class='math'>&</span><span class='math'>=</span> <span class='math'>CDDDDCCCDD</span><span class='mathcmd'>\\</span><br>
<span class='math'>D</span><span class='math'>(</span><span class='math'>1</span><span class='math'>,</span><span class='math'>1</span><span class='math'>)</span><span class='math'>^</span><span class='ignore'>{</span><span class='math'>5</span><span class='ignore'>}</span> <span class='math'>&</span><span class='math'>=</span> <span class='math'>DCDCDCDCDC</span><span class='mathcmd'>\\</span><br>
<span class='math'>C1</span><span class='math'>,</span><span class='math'>(</span><span class='math'>2</span><span class='math'>,</span><span class='math'>1</span><span class='math'>)</span><span class='math'>^</span><span class='ignore'>{</span><span class='math'>2</span><span class='ignore'>}</span><span class='math'>,</span><span class='math'>2</span><span class='math'>,</span><span class='math'>1</span> <span class='math'>&</span><span class='math'>=</span> <span class='math'>CDDCDDCDDC</span><span class='mathcmd'>\\</span><br>
<span class='math'>\{</span><span class='math'>D</span><span class='math'>[</span><span class='math'>i</span><span class='math'>,</span><span class='math'>5-i</span><span class='math'>]</span><span class='math'>\}</span> <span class='mathcmd'>\quad</span> <span class='math'>i</span><span class='mathcmd'>\in</span> <span class='option'>[2,4]</span> <span class='math'>&</span><span class='math'>=</span> <span class='math'>\{</span><span class='math'>DDCCC</span><span class='math'>,</span><span class='mathcmd'>\ </span><span class='math'>DDDCC</span><span class='math'>,</span><span class='mathcmd'>\ </span><span class='math'>DDDCC</span><span class='math'>\}</span><span class='mathcmd'>\\</span><br>
<span class='envir'>\end</span><span class='envir'>{align}</span><br>
<br>
<span class='word'>To</span> <span class='word'>avoid</span> <span class='word'>confusion</span> <span class='word'>between</span> <span class='word'>a</span> <span class='word'>sequence</span> <span class='word'>for</span> <span class='word'>a</span> <span class='word'>given</span> <span class='word'>opponent</span><span class='ignore'>,</span> <span class='mathgroup'>$</span><span class='math'>S</span><span class='math'>_</span><span class='ignore'>{</span><span class='math'>O</span><span class='math'>_</span><span class='math'>i</span><span class='ignore'>}</span><span class='mathgroup'>$</span><span class='ignore'>,</span> <span class='word'>and</span> <span class='word'>the</span> <span class='word'>score</span> <span class='word'>for</span> <span class='word'>a</span> <span class='word'>given</span> <span class='word'>sequence</span> <span class='word'>we</span> <span class='word'>define</span> <span class='word'>the</span> <span class='word'>score</span> <span class='word'>function</span> <span class='mathgroup'>$</span><span class='math'>f</span><span class='math'>(</span><span class='math'>S</span><span class='math'>_</span><span class='ignore'>{</span><span class='math'>O</span><span class='math'>_</span><span class='math'>i</span><span class='ignore'>}</span><span class='math'>)</span><span class='mathgroup'>$</span> <span class='word'>as</span> <span class='word'>follows</span><span class='ignore'>.</span><br>
<span class='mathgroup'>$$</span><span class='math'>f</span><span class='math'>(</span><span class='math'>S</span><span class='math'>_</span><span class='ignore'>{</span><span class='math'>O</span><span class='math'>_</span><span class='math'>i</span><span class='ignore'>}</span><span class='math'>)</span><span class='math'>:</span><span class='math'>S</span><span class='math'>_</span><span class='ignore'>{</span><span class='math'>O</span><span class='math'>_</span><span class='math'>i</span><span class='ignore'>}</span> <span class='mathcmd'>\rightarrow</span> <span class='option'>[0,5]</span><span class='mathgroup'>$$</span><br>
<span class='word'>This</span> <span class='word'>score</span> <span class='word'>function</span> <span class='word'>represents</span> <span class='word'>playing</span> <span class='word'>the</span> <span class='word'>sequence</span> <span class='word'>against</span> <span class='word'>the</span> <span class='word'>given</span> <span class='word'>opponent</span> <span class='word'>and</span> <span class='word'>calculating</span> <span class='word'>the</span> <span class='word'>score</span> <span class='word'>per</span> <span class='word'>turn</span> <span class='word'>over</span> <span class='word'>200</span> <span class='word'>turns</span><span class='ignore'>.</span> <br>
<span class='word'>In</span> <span class='word'>this</span> <span class='word'>report</span> <span class='word'>the</span> <span class='word'>score</span> <span class='word'>will</span> <span class='word'>not</span> <span class='word'>be</span> <span class='word'>given</span> <span class='word'>its</span> <span class='word'>own</span> <span class='word'>notation</span> <span class='word'>other</span> <span class='word'>than</span> <span class='word'>the</span> <span class='word'>result</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>function</span> <span class='mathgroup'>$</span><span class='math'>f</span><span class='mathgroup'>$</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\section</span><span class='ignore'>{</span><span class='hword'>Customizing</span> <span class='hword'>an</span> <span class='hword'>Algorithm</span><span class='ignore'>}</span><span class='cmd'>\label</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>buildingTheAlgorithm</span><span class='ignore'>}</span><br>
<span class='word'>In</span> <span class='word'>order</span> <span class='word'>to</span> <span class='word'>generate</span> <span class='word'>a</span> <span class='word'>solution</span> <span class='word'>sequence</span> <span class='word'>we</span> <span class='word'>have</span> <span class='word'>to</span> <span class='word'>train</span> <span class='word'>against</span> <span class='word'>each</span> <span class='word'>Axelrod</span> <span class='word'>opponent</span><span class='ignore'>.</span><br>
<span class='word'>We</span> <span class='word'>will</span> <span class='word'>expand</span> <span class='word'>on</span> <span class='word'>the</span> <span class='word'>GA</span> <span class='word'>described</span> <span class='word'>in</span> <span class='word'>Section</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>subsec</span><span class='ignore'>:</span><span class='ignore'>geneticAlgorithms</span><span class='ignore'>}</span> <span class='word'>to</span> <span class='word'>produce</span> <span class='word'>our</span> <span class='word'>own</span><span class='ignore'>.</span><br>
<span class='word'>Our</span> <span class='word'>implementation</span> <span class='word'>of</span> <span class='word'>a</span> <span class='word'>GA</span> <span class='word'>has</span> <span class='word'>the</span> <span class='word'>following</span> <span class='word'>steps</span><span class='ignore'>:</span><br>
<span class='envir'>\begin</span><span class='envir'>{enumerate}</span><br>
<span class='cmd'>\item</span> <span class='word'>Start</span> <span class='word'>with</span> <span class='word'>a</span> <span class='word'>predefined</span> <span class='word'>population</span><span class='ignore'>,</span> <span class='word'>supplemented</span> <span class='word'>with</span> <span class='word'>randomly</span> <span class='word'>generated</span> <span class='word'>member</span> <span class='word'>until</span> <span class='word'>to</span> <span class='word'>size</span><span class='ignore'>.</span><br>
<span class='cmd'>\item</span> <span class='word'>Each</span> <span class='word'>member</span> <span class='word'>plays</span> <span class='word'>the</span> <span class='word'>given</span> <span class='word'>opponent</span> <span class='word'>with</span> <span class='word'>their</span> <span class='word'>sequence</span> <span class='word'>and</span> <span class='word'>returns</span> <span class='word'>with</span> <span class='word'>the</span> <span class='word'>average</span> <span class='word'>score</span> <span class='word'>per</span> <span class='word'>turn</span><span class='ignore'>.</span><br>
<span class='cmd'>\item</span> <span class='word'>Members</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>population</span> <span class='word'>are</span> <span class='word'>ranked</span> <span class='word'>by</span> <span class='word'>this</span> <span class='word'>average</span> <span class='word'>score</span> <span class='word'>per</span> <span class='word'>turn</span> <span class='word'>and</span> <span class='word'>the</span> <span class='word'>highest</span> <span class='word'>scoring</span> <span class='word'>25</span><span class='ignore'>\%</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>kept</span> <span class='word'>for</span> <span class='word'>the</span> <span class='word'>next</span> <span class='word'>round</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>remaining</span> <span class='word'>75</span><span class='ignore'>\%</span> <span class='word'>are</span> <span class='word'>killed</span> <span class='word'>off</span><span class='ignore'>.</span><br>
<span class='cmd'>\item</span> <span class='word'>The</span> <span class='word'>remaining</span> <span class='word'>population</span> <span class='word'>will</span> <span class='word'>then</span> <span class='word'>be</span> <span class='word'>copied</span> <span class='word'>and</span> <span class='word'>these</span> <span class='word'>copies</span> <span class='word'>are</span> <span class='word'>mutated</span> <span class='word'>to</span> <span class='word'>create</span> <span class='word'>unique</span> <span class='word'>sequences</span> <span class='word'>before</span> <span class='word'>being</span> <span class='word'>merged</span> <span class='word'>back</span> <span class='word'>in</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>main</span> <span class='word'>population</span><span class='ignore'>.</span><br>
<span class='cmd'>\item</span> <span class='word'>The</span> <span class='word'>remaining</span> <span class='word'>50</span><span class='ignore'>\%</span> <span class='word'>difference</span> <span class='word'>is</span> <span class='word'>then</span> <span class='word'>made</span> <span class='word'>up</span> <span class='word'>of</span> <span class='word'>mutated</span> <span class='word'>results</span> <span class='word'>of</span> <span class='word'>crossovers</span> <span class='word'>from</span> <span class='word'>members</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>current</span> <span class='word'>population</span> <span class='word'>or</span> <span class='word'>random</span> <span class='word'>new</span> <span class='word'>members</span><span class='ignore'>,</span> <span class='word'>depending</span> <span class='word'>on</span> <span class='word'>a</span> <span class='word'>random</span> <span class='word'>selection</span> <span class='word'>algorithm</span><span class='ignore'>.</span><span class='cmd'>\footnote</span><span class='ignore'>{</span><span class='oword'>This</span> <span class='oword'>algorithm</span> <span class='oword'>had</span> <span class='oword'>a</span> <span class='oword'>bug</span> <span class='oword'>which</span> <span class='oword'>would</span> <span class='oword'>change</span> <span class='oword'>the</span> <span class='oword'>size</span> <span class='oword'>of</span> <span class='oword'>the</span> <span class='oword'>population</span> <span class='oword'>in</span> <span class='oword'>the</span> <span class='oword'>first</span> <span class='oword'>generation</span><span class='ignore'>.</span><br>
<span class='oword'>This</span> <span class='oword'>was</span> <span class='oword'>fixed</span> <span class='oword'>after</span> <span class='oword'>Section</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>conclusionOfApproach</span><span class='ignore'>}</span> <span class='oword'>was</span> <span class='oword'>written</span><span class='ignore'>.</span><br>
<span class='oword'>See</span><span class='ignore'>:</span> <span class='oword'>https</span><span class='ignore'>:</span><span class='ignore'>/</span><span class='ignore'>/</span><span class='oword'>github.com</span><span class='ignore'>/</span><span class='oword'>Axelrod-Python</span><span class='ignore'>/</span><span class='oword'>axelrod-dojo</span><span class='ignore'>/</span><span class='oword'>issues</span><span class='ignore'>/</span><span class='oword'>43</span><span class='ignore'>}</span><br>
<span class='cmd'>\item</span> <span class='word'>A</span> <span class='word'>generation</span> <span class='word'>has</span> <span class='word'>now</span> <span class='word'>concluded</span><span class='ignore'>.</span><br>
<span class='word'>Repeat</span> <span class='word'>from</span> <span class='word'>step</span> <span class='word'>2</span> <span class='word'>until</span> <span class='word'>the</span> <span class='word'>desired</span> <span class='word'>number</span> <span class='word'>of</span> <span class='word'>generations</span> <span class='word'>are</span> <span class='word'>finished</span> <span class='word'>and</span> <span class='word'>a</span> <span class='word'>final</span> <span class='word'>best</span> <span class='word'>sequence</span> <span class='word'>is</span> <span class='word'>returned</span><span class='ignore'>.</span><br>
<span class='envir'>\end</span><span class='envir'>{enumerate}</span><br>
<br>
<span class='envir'>\begin</span><span class='envir'>{figure}</span><span class='option'>[ht]</span><br>
<span class='exclcmd'>\includegraphics</span><span class='option'>[width=1.0\textwidth, center]</span><span class='ignore'>{</span><span class='ignore'>.</span><span class='ignore'>/</span><span class='ignore'>img</span><span class='ignore'>/</span><span class='ignore'>flows</span><span class='ignore'>/</span><span class='ignore'>custom</span><span class='ignore'>_</span><span class='ignore'>ga</span><span class='ignore'>_</span><span class='ignore'>cycle</span><span class='ignore'>}</span><br>
<span class='cmd'>\caption</span><span class='ignore'>{</span><span class='oword'>High</span> <span class='oword'>Level</span> <span class='oword'>Genetic</span> <span class='oword'>Algorithm</span> <span class='oword'>Cycle</span><span class='ignore'>.</span> <span class='oword'>Extension</span> <span class='oword'>of</span> <span class='oword'>Figure</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>genericGACycle</span><span class='ignore'>}</span><span class='ignore'>}</span><span class='exclcmd'>\label</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>customGAcycle</span><span class='ignore'>}</span><br>
<span class='envir'>\end</span><span class='envir'>{figure}</span><br>
<br>
<span class='word'>Figure</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>customGAcycle</span><span class='ignore'>}</span> <span class='word'>shows</span> <span class='word'>a</span> <span class='word'>flow</span> <span class='word'>diagram</span> <span class='word'>of</span> <span class='word'>our</span> <span class='word'>cycle</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>is</span> <span class='word'>the</span> <span class='word'>algorithm</span> <span class='word'>we</span> <span class='word'>will</span> <span class='word'>use</span> <span class='word'>in</span> <span class='word'>Chapter</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>ch</span><span class='ignore'>:</span><span class='ignore'>implementation</span><span class='ignore'>}</span> <span class='word'>when</span> <span class='word'>analysing</span> <span class='word'>the</span> <span class='word'>algorithms</span> <span class='word'>parameters</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>model</span> <span class='word'>means</span> <span class='word'>we</span> <span class='word'>can</span> <span class='word'>create</span> <span class='word'>a</span> <span class='word'>population</span> <span class='word'>of</span> <span class='word'>Cycler</span> <span class='word'>players</span><span class='ignore'>,</span> <span class='word'>described</span> <span class='word'>in</span> <span class='word'>Section</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>strategiesOfInterest</span><span class='ignore'>}</span><span class='ignore'>,</span> <span class='word'>and</span> <span class='word'>input</span> <span class='word'>a</span> <span class='word'>sequence</span> <span class='word'>of</span> <span class='word'>length</span> <span class='word'>200</span> <span class='word'>as</span> <span class='word'>a</span> <span class='word'>parameter</span> <span class='word'>to</span> <span class='word'>set</span> <span class='word'>off</span> <span class='word'>our</span> <span class='word'>genetic</span> <span class='word'>algorithm</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>subsequent</span> <span class='word'>inputs</span> <span class='word'>for</span> <span class='word'>the</span> <span class='word'>populations</span> <span class='word'>Cycler</span> <span class='word'>players</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>created</span> <span class='word'>using</span> <span class='word'>the</span> <span class='word'>genetic</span> <span class='word'>mutation</span> <span class='word'>and</span> <span class='word'>crossover</span> <span class='word'>techniques</span><span class='ignore'>,</span> <span class='word'>see</span> <span class='word'>Section</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>subsec</span><span class='ignore'>:</span><span class='ignore'>geneticAlgorithms</span><span class='ignore'>}</span> <span class='word'>for</span> <span class='word'>details</span><span class='ignore'>.</span><br>
<br>
<span class='word'>The</span> <span class='word'>looping</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>the</span> <span class='word'>basis</span> <span class='word'>of</span> <span class='word'>creating</span> <span class='word'>the</span> <span class='word'>optimal</span> <span class='word'>strategy</span> <span class='word'>for</span> <span class='word'>each</span> <span class='word'>other</span> <span class='word'>opponent</span><span class='ignore'>.</span><br>
<span class='word'>Each</span> <span class='word'>step</span> <span class='word'>is</span> <span class='word'>defined</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>Axelrod</span> <span class='word'>Dojo</span> <span class='cmd'>\mintinline</span><span class='ignore'>{</span><span class='word'>python</span><span class='ignore'>}</span><span class='ignore'>{</span><span class='word'>Population</span><span class='ignore'>}</span> <span class='word'>and</span> <span class='cmd'>\mintinline</span><span class='ignore'>{</span><span class='word'>python</span><span class='ignore'>}</span><span class='ignore'>{</span><span class='word'>CyclerParams</span><span class='ignore'>}</span> <span class='word'>classes</span><span class='ignore'>.</span><br>
<span class='word'>Rather</span> <span class='word'>than</span> <span class='word'>store</span> <span class='word'>all</span> <span class='word'>the</span> <span class='word'>functionality</span> <span class='word'>in</span> <span class='word'>one</span> <span class='word'>place</span> <span class='word'>we</span> <span class='word'>are</span> <span class='word'>able</span> <span class='word'>split</span> <span class='word'>up</span> <span class='word'>aspects</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>flow</span> <span class='word'>to</span> <span class='word'>allow</span> <span class='word'>for</span> <span class='word'>flexibility</span> <span class='word'>in</span> <span class='word'>what</span> <span class='word'>type</span> <span class='word'>of</span> <span class='word'>population</span> <span class='word'>can</span> <span class='word'>be</span> <span class='word'>used</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\section</span><span class='ignore'>{</span><span class='hword'>Solution</span> <span class='hword'>Form</span><span class='ignore'>}</span><span class='cmd'>\label</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>solutionForm</span><span class='ignore'>}</span><br>
<span class='word'>In</span> <span class='word'>this</span> <span class='word'>research</span> <span class='word'>our</span> <span class='word'>goal</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>to</span> <span class='word'>use</span> <span class='word'>the</span> <span class='word'>algorithm</span><span class='ignore'>,</span> <span class='word'>described</span> <span class='word'>in</span> <span class='word'>Subsection</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>buildingTheAlgorithm</span><span class='ignore'>}</span><span class='ignore'>,</span> <span class='word'>to</span> <span class='word'>produce</span> <span class='word'>an</span> <span class='word'>arbitrary</span> <span class='word'>sequence</span> <span class='word'>for</span> <span class='word'>an</span> <span class='word'>opponent</span><span class='ignore'>,</span> <span class='mathgroup'>\(</span><span class='math'>S</span><span class='math'>_</span><span class='math'>o</span><span class='mathgroup'>\)</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>sequence</span> <span class='word'>will</span> <span class='word'>represent</span> <span class='word'>the</span> <span class='word'>moves</span> <span class='word'>we</span> <span class='word'>should</span> <span class='word'>play</span> <span class='word'>against</span> <span class='word'>the</span> <span class='word'>opponent</span> <span class='word'>to</span> <span class='word'>get</span> <span class='word'>our</span> <span class='word'>largest</span> <span class='word'>potential</span> <span class='word'>score</span> <span class='word'>per</span> <span class='word'>turn</span> <span class='word'>for</span> <span class='word'>a</span> <span class='word'>single</span> <span class='word'>game</span> <span class='word'>of</span> <span class='word'>200</span> <span class='word'>turns</span><span class='ignore'>.</span><br>
<br>
<span class='word'>This</span> <span class='word'>investigation</span> <span class='word'>focuses</span> <span class='word'>on</span> <span class='word'>sequences</span> <span class='word'>that</span> <span class='word'>will</span> <span class='word'>allow</span> <span class='word'>us</span> <span class='word'>to</span> <span class='word'>maximise</span> <span class='word'>our</span> <span class='word'>score</span> <span class='word'>overall</span><span class='ignore'>,</span> <span class='word'>rather</span> <span class='word'>than</span> <span class='word'>just</span> <span class='word'>beating</span> <span class='word'>any</span> <span class='word'>given</span> <span class='word'>opponent</span><span class='ignore'>.</span><br>
<span class='word'>An</span> <span class='word'>analogy</span> <span class='word'>of</span> <span class='word'>this</span> <span class='word'>concept</span> <span class='word'>is</span> <span class='word'>a</span> <span class='word'>team</span> <span class='word'>playing</span> <span class='word'>a</span> <span class='word'>football</span> <span class='word'>tournament</span><span class='ignore'>,</span> <span class='word'>but</span> <span class='word'>instead</span> <span class='word'>of</span> <span class='word'>a</span> <span class='word'>knockout</span> <span class='word'>competition</span> <span class='word'>our</span> <span class='word'>team</span> <span class='word'>is</span> <span class='word'>placed</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>standings</span> <span class='word'>based</span> <span class='word'>off</span> <span class='word'>the</span> <span class='word'>total</span> <span class='word'>goals</span> <span class='word'>they</span> <span class='word'>have</span> <span class='word'>scored</span> <span class='word'>across</span> <span class='word'>the</span> <span class='word'>tournament</span><span class='ignore'>.</span><br>
<span class='word'>More</span> <span class='word'>real</span> <span class='word'>world</span> <span class='word'>applications</span> <span class='word'>of</span> <span class='word'>these</span> <span class='word'>results</span> <span class='word'>are</span> <span class='word'>discussed</span> <span class='word'>in</span> <span class='word'>Chapter</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>ch</span><span class='ignore'>:</span><span class='ignore'>conclusions</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<br>
<span class='word'>Each</span> <span class='word'>solution</span> <span class='word'>sequence</span> <span class='word'>is</span> <span class='word'>uniquely</span> <span class='word'>generated</span> <span class='word'>for</span> <span class='word'>each</span> <span class='word'>opponent</span><span class='ignore'>.</span><br>
<span class='word'>If</span> <span class='word'>there</span> <span class='word'>are</span> <span class='word'>two</span> <span class='word'>similar</span> <span class='word'>opponents</span><span class='ignore'>,</span> <span class='word'>say</span> <span class='word'>Grudger</span> <span class='word'>and</span> <span class='word'>Collective</span> <span class='word'>Strategy</span><span class='ignore'>,</span> <span class='word'>these</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>independently</span> <span class='word'>analysed</span> <span class='word'>and</span> <span class='word'>individual</span> <span class='word'>solutions</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>generated</span><span class='ignore'>.</span><br>
<br>
<span class='word'>If</span> <span class='word'>needed</span><span class='ignore'>,</span> <span class='word'>we</span> <span class='word'>may</span> <span class='word'>re-run</span> <span class='word'>analysis</span> <span class='word'>on</span> <span class='word'>some</span> <span class='word'>opponents</span> <span class='word'>if</span> <span class='word'>there</span> <span class='word'>are</span> <span class='word'>bugs</span> <span class='word'>that</span> <span class='word'>arise</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>code</span><span class='ignore'>.</span><br>
<span class='word'>In</span> <span class='word'>this</span> <span class='word'>case</span> <span class='word'>older</span> <span class='word'>results</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>overwritten</span> <span class='word'>and</span> <span class='word'>previous</span> <span class='word'>analysis</span> <span class='word'>discarded</span><span class='ignore'>.</span><br>
<br>
<span class='word'>When</span> <span class='word'>working</span> <span class='word'>with</span> <span class='word'>stochastic</span> <span class='word'>players</span><span class='ignore'>,</span> <span class='word'>we</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>seeding</span> <span class='word'>the</span> <span class='word'>random</span> <span class='word'>numbers</span> <span class='word'>they</span> <span class='word'>use</span> <span class='word'>in</span> <span class='word'>order</span> <span class='word'>to</span> <span class='word'>determine</span> <span class='word'>the</span> <span class='word'>best</span> <span class='word'>sequence</span><span class='ignore'>.</span><br>
<span class='word'>Each</span> <span class='word'>stochastic</span> <span class='word'>player</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>considered</span> <span class='word'>under</span> <span class='word'>a</span> <span class='word'>variety</span> <span class='word'>of</span> <span class='word'>seeds</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>motive</span> <span class='word'>to</span> <span class='word'>allow</span> <span class='word'>this</span> <span class='word'>to</span> <span class='word'>happen</span> <span class='word'>is</span> <span class='word'>described</span> <span class='word'>in</span> <span class='word'>Section</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>stochasticOpponents</span><span class='ignore'>}</span><span class='ignore'>.</span> <span class='word'>Non-stochastic</span> <span class='word'>opponents</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>seeded</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>code</span><span class='ignore'>,</span> <span class='word'>however</span> <span class='word'>there</span> <span class='word'>is</span> <span class='word'>no</span> <span class='word'>change</span> <span class='word'>in</span> <span class='word'>their</span> <span class='word'>behaviour</span> <span class='word'>because</span> <span class='word'>of</span> <span class='word'>this</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\section</span><span class='ignore'>{</span><span class='hword'>Conclusion</span><span class='ignore'>}</span><br>
<span class='word'>Here</span> <span class='word'>we</span> <span class='word'>have</span> <span class='word'>looked</span> <span class='word'>at</span> <span class='word'>the</span> <span class='word'>background</span> <span class='word'>required</span> <span class='word'>to</span> <span class='word'>answer</span> <span class='word'>a</span> <span class='word'>specific</span> <span class='word'>question</span><span class='ignore'>,</span> <span class='word'>build</span> <span class='word'>a</span> <span class='word'>relevant</span> <span class='word'>model</span> <span class='word'>and</span> <span class='word'>understand</span> <span class='word'>what</span> <span class='word'>were</span> <span class='word'>looking</span> <span class='word'>for</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>concepts</span> <span class='word'>themselves</span> <span class='word'>are</span> <span class='word'>extensions</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>basic</span> <span class='word'>models</span> <span class='word'>brought</span> <span class='word'>up</span> <span class='word'>in</span> <span class='word'>Chapter</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>ch</span><span class='ignore'>:</span><span class='ignore'>intro</span><span class='ignore'>}</span> <span class='word'>and</span> <span class='word'>should</span> <span class='word'>allow</span> <span class='word'>us</span> <span class='word'>to</span> <span class='word'>provide</span> <span class='word'>relevant</span> <span class='word'>results</span> <span class='word'>in</span> <span class='word'>a</span> <span class='word'>uniform</span> <span class='word'>manner</span><span class='ignore'>.</span><br>
<br>
<span class='word'>We</span> <span class='word'>have</span> <span class='word'>also</span> <span class='word'>looked</span> <span class='word'>at</span> <span class='word'>what</span> <span class='word'>problems</span> <span class='word'>may</span> <span class='word'>arise</span> <span class='word'>from</span> <span class='word'>certain</span> <span class='word'>areas</span> <span class='word'>of</span> <span class='word'>solving</span> <span class='word'>the</span> <span class='word'>task</span><span class='ignore'>,</span> <span class='word'>such</span> <span class='word'>as</span> <span class='word'>understanding</span> <span class='word'>what</span> <span class='word'>a</span> <span class='word'>best</span> <span class='word'>response</span> <span class='word'>for</span> <span class='word'>a</span> <span class='word'>stochastic</span> <span class='word'>opponents</span> <span class='word'>could</span> <span class='word'>be</span> <span class='word'>or</span> <span class='word'>how</span> <span class='word'>to</span> <span class='word'>represent</span> <span class='word'>a</span> <span class='word'>complex</span> <span class='word'>solution</span> <span class='word'>in</span> <span class='word'>a</span> <span class='word'>result</span><span class='ignore'>.</span><br>
<span class='word'>Because</span> <span class='word'>of</span> <span class='word'>this</span> <span class='word'>premeditation</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>problem</span> <span class='word'>and</span> <span class='word'>its</span> <span class='word'>components</span> <span class='word'>Chapters</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>ch</span><span class='ignore'>:</span><span class='ignore'>implementation</span><span class='ignore'>}</span> <span class='word'>and</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>ch</span><span class='ignore'>:</span><span class='ignore'>results</span><span class='ignore'>}</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>more</span> <span class='word'>concise</span><span class='ignore'>.</span><br>
<br>
<span class='word'>After</span> <span class='word'>we</span> <span class='word'>conclude</span> <span class='word'>the</span> <span class='word'>runtime</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>GA</span> <span class='word'>and</span> <span class='word'>obtain</span> <span class='word'>a</span> <span class='word'>solution</span> <span class='word'>we</span> <span class='word'>will</span> <span class='word'>have</span> <span class='word'>an</span> <span class='word'>output</span> <span class='word'>sequence</span> <span class='word'>for</span> <span class='word'>each</span> <span class='word'>opponent</span><span class='ignore'>.</span><br>
<span class='word'>These</span> <span class='word'>solution</span> <span class='word'>sequences</span> <span class='word'>can</span> <span class='word'>then</span> <span class='word'>be</span> <span class='word'>compared</span> <span class='word'>on</span> <span class='word'>how</span> <span class='word'>similar</span> <span class='word'>they</span> <span class='word'>are</span> <span class='word'>to</span> <span class='word'>each</span> <span class='word'>other</span><span class='ignore'>;</span><br>
<span class='word'>what</span> <span class='word'>type</span> <span class='word'>of</span> <span class='word'>opponents</span> <span class='word'>and</span> <span class='word'>their</span> <span class='word'>corresponding</span> <span class='ignore'>`</span><span class='word'>best</span> <span class='word'>scores</span><span class='ignore'>'</span> <span class='word'>are</span> <span class='word'>and</span> <span class='word'>whether</span> <span class='word'>there</span> <span class='word'>are</span> <span class='word'>patterns</span> <span class='word'>to</span> <span class='word'>how</span> <span class='word'>to</span> <span class='word'>group</span> <span class='word'>strategies</span><span class='ignore'>,</span> <span class='word'>Chapter</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>ch</span><span class='ignore'>:</span><span class='ignore'>results</span><span class='ignore'>}</span> <span class='word'>contains</span> <span class='word'>further</span> <span class='word'>details</span><span class='ignore'>.</span></p></div>
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<li>78+2+0 (1/0/0/0) Chapter: Task Methodology}\label{ch:taskBackground</li>
<li>371+1+0 (1/0/17/6) Section: Notation}\label{sec:notation</li>
<li>308+3+39 (1/1/0/0) Section: Customizing an Algorithm}\label{sec:buildingTheAlgorithm</li>
<li>240+2+0 (1/0/1/0) Section: Solution Form}\label{sec:solutionForm</li>
<li>163+1+0 (1/0/0/0) Section: Conclusion</li>
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<span class='cmd'>\chapter</span><span class='ignore'>{</span><span class='hword'>Literature</span> <span class='hword'>Review</span><span class='ignore'>}</span><span class='cmd'>\label</span><span class='ignore'>{</span><span class='ignore'>ch</span><span class='ignore'>:</span><span class='ignore'>literature</span><span class='ignore'>}</span><br>
<span class='word'>In</span> <span class='word'>this</span> <span class='word'>chapter</span> <span class='word'>we</span> <span class='word'>will</span> <span class='word'>look</span> <span class='word'>at</span> <span class='word'>previous</span> <span class='word'>works</span> <span class='word'>in</span> <span class='word'>areas</span> <span class='word'>relevant</span> <span class='word'>to</span> <span class='word'>this</span> <span class='word'>report</span><span class='ignore'>.</span><br>
<span class='word'>We</span> <span class='word'>will</span> <span class='word'>look</span> <span class='word'>at</span> <span class='word'>how</span> <span class='word'>these</span> <span class='word'>pieces</span> <span class='word'>of</span> <span class='word'>work</span> <span class='word'>developed</span> <span class='word'>standards</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>areas</span> <span class='word'>we</span> <span class='word'>are</span> <span class='word'>interested</span> <span class='word'>in</span> <span class='word'>and</span> <span class='word'>the</span> <span class='word'>history</span> <span class='word'>of</span> <span class='word'>studying</span> <span class='word'>them</span> <span class='word'>with</span> <span class='word'>computer</span> <span class='word'>models</span><span class='ignore'>.</span> <br>
<br>
<span class='word'>Section</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>strategiesOfInterest</span><span class='ignore'>}</span> <span class='word'>describes</span> <span class='word'>interesting</span> <span class='word'>opponents</span> <span class='word'>we</span> <span class='word'>will</span> <span class='word'>look</span> <span class='word'>at</span> <span class='word'>in</span> <span class='word'>more</span> <span class='word'>detail</span> <span class='word'>during</span> <span class='word'>Chapter</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>ch</span><span class='ignore'>:</span><span class='ignore'>implementation</span><span class='ignore'>}</span> <span class='word'>to</span> <span class='word'>provide</span> <span class='word'>our</span> <span class='word'>algorithm</span> <span class='word'>a</span> <span class='word'>variety</span> <span class='word'>of</span> <span class='word'>opponents</span> <span class='word'>to</span> <span class='word'>be</span> <span class='word'>developed</span> <span class='word'>against</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\section</span><span class='ignore'>{</span><span class='hword'>Background</span><span class='ignore'>}</span><span class='cmd'>\label</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>background</span><span class='ignore'>}</span><br>
<span class='word'>The</span> <span class='word'>PD</span> <span class='word'>and</span> <span class='word'>its</span> <span class='word'>a</span> <span class='word'>large</span> <span class='word'>area</span> <span class='word'>of</span> <span class='word'>repeated</span> <span class='word'>games</span> <span class='word'>in</span> <span class='word'>GT</span> <span class='word'>and</span> <span class='word'>has</span> <span class='word'>applications</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>real</span> <span class='word'>world</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>is</span> <span class='word'>due</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>game</span> <span class='word'>being</span> <span class='word'>a</span> <span class='word'>good</span> <span class='word'>example</span> <span class='word'>of</span> <span class='word'>strategies</span> <span class='word'>that</span> <span class='word'>give</span> <span class='word'>a</span> <span class='word'>cooperative</span> <span class='word'>benefit</span> <span class='word'>to</span> <span class='word'>repeated</span> <span class='word'>games</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>PD</span> <span class='word'>was</span> <span class='word'>first</span> <span class='word'>formally</span> <span class='word'>presented</span> <span class='word'>by</span> <span class='word'>Albert</span> <span class='word'>W</span><span class='ignore'>.</span> <span class='word'>Tucker</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>cambell2016thesis</span><span class='ignore'>,</span> <span class='ignore'>gass2005annotated</span><span class='ignore'>}</span> <span class='word'>and</span> <span class='word'>the</span> <span class='word'>iterated</span> <span class='word'>version</span> <span class='word'>was</span> <span class='word'>made</span> <span class='word'>famous</span> <span class='word'>by</span> <span class='word'>Robert</span> <span class='word'>Axelrods</span> <span class='word'>work</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>1980s</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>axelrod1980effective</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<span class='word'>It</span> <span class='word'>has</span> <span class='word'>been</span> <span class='word'>used</span> <span class='word'>to</span> <span class='word'>describe</span> <span class='word'>actions</span> <span class='word'>of</span> <span class='word'>people</span> <span class='word'>and</span> <span class='word'>governments</span> <span class='word'>in</span> <span class='word'>situations</span> <span class='word'>stretching</span> <span class='word'>from</span> <span class='word'>warfare</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>tooby1988war</span><span class='ignore'>,</span><span class='ignore'>aumann1992handbook</span><span class='ignore'>}</span> <span class='word'>and</span> <span class='word'>finance</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>cable1997finance</span><span class='ignore'>}</span> <span class='word'>to</span> <span class='word'>politics</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>snidal1985Politics</span><span class='ignore'>}</span> <span class='word'>and</span> <span class='word'>relationships</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>low2015sex</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<br>
<span class='word'>The</span> <span class='word'>Prisoners</span> <span class='word'>dilemma</span> <span class='word'>became</span> <span class='word'>a</span> <span class='word'>large</span> <span class='word'>research</span> <span class='word'>area</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>combination</span> <span class='word'>of</span> <span class='word'>mathematics</span> <span class='word'>and</span> <span class='word'>computer</span> <span class='word'>science</span> <span class='word'>after</span> <span class='word'>Robert</span> <span class='word'>Axelrod</span> <span class='word'>published</span> <span class='word'>his</span> <span class='word'>work</span> <span class='word'>named</span> <span class='ignore'>`</span><span class='word'>Effective</span> <span class='word'>Choice</span> <span class='word'>In</span> <span class='word'>The</span> <span class='word'>Prisoners</span> <span class='word'>Dilemma</span><span class='ignore'>'</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>axelrod1980effective</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<span class='word'>In</span> <span class='word'>it</span> <span class='word'>he</span> <span class='word'>makes</span> <span class='word'>an</span> <span class='word'>introduction</span> <span class='word'>to</span> <span class='word'>how</span> <span class='word'>tournaments</span> <span class='word'>are</span> <span class='word'>run</span> <span class='word'>and</span> <span class='word'>the</span> <span class='word'>properties</span> <span class='word'>of</span> <span class='word'>successful</span> <span class='word'>results</span><span class='ignore'>.</span><br>
<span class='word'>His</span> <span class='word'>method</span> <span class='word'>of</span> <span class='word'>experimenting</span> <span class='word'>became</span> <span class='word'>the</span> <span class='word'>standard</span> <span class='word'>for</span> <span class='word'>handling</span> <span class='word'>the</span> <span class='word'>IDP</span> <span class='word'>problem</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>was</span> <span class='word'>also</span> <span class='word'>the</span> <span class='word'>first</span> <span class='word'>example</span> <span class='word'>of</span> <span class='word'>computer</span> <span class='word'>tournaments</span><span class='ignore'>,</span> <span class='word'>for</span> <span class='word'>which</span> <span class='word'>he</span> <span class='word'>asked</span> <span class='word'>for</span> <span class='word'>strategies</span> <span class='word'>as</span> <span class='word'>computer</span> <span class='word'>programs</span> <span class='word'>which</span> <span class='word'>had</span> <span class='word'>inputs</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>history</span> <span class='word'>for</span> <span class='word'>both</span> <span class='word'>players</span> <span class='word'>and</span> <span class='word'>resulted</span> <span class='word'>in</span> <span class='word'>an</span> <span class='word'>output</span> <span class='word'>move</span> <span class='word'>for</span> <span class='word'>that</span> <span class='word'>next</span> <span class='word'>turn</span><span class='ignore'>.</span><br>
<span class='word'>After</span> <span class='word'>the</span> <span class='word'>tournament</span> <span class='word'>is</span> <span class='word'>complete</span> <span class='word'>he</span> <span class='word'>describes</span> <span class='word'>what</span> <span class='word'>successful</span> <span class='word'>strategies</span> <span class='word'>had</span> <span class='word'>in</span> <span class='word'>common</span><span class='ignore'>;</span> <br>
<span class='word'>It</span> <span class='word'>turns</span> <span class='word'>out</span> <span class='word'>the</span> <span class='word'>majority</span> <span class='word'>of</span> <span class='word'>successful</span> <span class='word'>strategies</span><span class='ignore'>,</span> <span class='word'>including</span> <span class='word'>the</span> <span class='word'>winner</span><span class='ignore'>,</span> <span class='word'>Tit</span> <span class='word'>For</span> <span class='word'>Tat</span><span class='ignore'>,</span> <span class='word'>have</span> <span class='word'>properties</span> <span class='word'>of</span> <span class='word'>niceness</span> <span class='word'>and</span> <span class='word'>forgiveness</span><span class='ignore'>.</span><br>
<span class='word'>Niceness</span> <span class='word'>is</span> <span class='word'>the</span> <span class='word'>property</span> <span class='word'>of</span> <span class='word'>not</span> <span class='word'>defecting</span> <span class='word'>to</span> <span class='word'>start</span> <span class='word'>and</span> <span class='word'>forgiveness</span> <span class='word'>is</span> <span class='word'>the</span> <span class='word'>property</span> <span class='word'>of</span> <span class='word'>forgetting</span> <span class='word'>previous</span> <span class='word'>defections</span> <span class='word'>in</span> <span class='word'>a</span> <span class='word'>timely</span> <span class='word'>manner</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>allowed</span> <span class='word'>them</span> <span class='word'>to</span> <span class='word'>thrive</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>tournament</span> <span class='word'>and</span> <span class='word'>have</span> <span class='word'>overall</span> <span class='word'>scores</span> <span class='word'>that</span> <span class='word'>rose</span> <span class='word'>above</span> <span class='word'>strategies</span> <span class='word'>without</span> <span class='word'>niceness</span> <span class='word'>or</span> <span class='word'>forgiveness</span><span class='ignore'>.</span><br>
<br>
<span class='word'>Since</span> <span class='word'>Axelrods</span><span class='ignore'>'</span> <span class='word'>original</span> <span class='word'>tournament</span> <span class='word'>there</span> <span class='word'>have</span> <span class='word'>been</span> <span class='word'>many</span> <span class='word'>research</span> <span class='word'>papers</span> <span class='word'>on</span> <span class='word'>what</span> <span class='word'>makes</span> <span class='word'>a</span> <span class='word'>successful</span> <span class='word'>strategy</span> <span class='word'>for</span> <span class='word'>a</span> <span class='word'>specific</span> <span class='word'>objective</span><span class='ignore'>.</span><br>
<span class='word'>For</span> <span class='word'>example</span> <span class='word'>William</span> <span class='word'>H</span> <span class='word'>Press</span> <span class='word'>and</span> <span class='word'>Freeman</span> <span class='word'>J</span> <span class='word'>Dyson</span><span class='ignore'>,</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>press2012iterated</span><span class='ignore'>}</span><span class='ignore'>,</span> <span class='word'>looked</span> <span class='word'>at</span> <span class='word'>how</span> <span class='word'>to</span> <span class='word'>it</span> <span class='word'>is</span> <span class='word'>possible</span> <span class='word'>to</span> <span class='word'>deterministically</span> <span class='word'>set</span> <span class='word'>an</span> <span class='word'>opponents</span> <span class='word'>score</span> <span class='word'>and</span> <span class='word'>Shashi</span> <span class='word'>Mittal</span> <span class='word'>and</span> <span class='word'>Kalyanmoy</span> <span class='word'>Deb</span> <span class='word'>have</span> <span class='word'>looked</span> <span class='word'>at</span> <span class='word'>satisfying</span> <span class='word'>a</span> <span class='word'>range</span> <span class='word'>of</span> <span class='word'>different</span> <span class='word'>objectives</span> <span class='word'>at</span> <span class='word'>once</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>mittal2009optimal</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<span class='word'>These</span> <span class='word'>specific</span> <span class='word'>objectives</span> <span class='word'>are</span> <span class='word'>the</span> <span class='word'>core</span> <span class='word'>part</span> <span class='word'>of</span> <span class='word'>applying</span> <span class='word'>GT</span> <span class='word'>and</span> <span class='word'>the</span> <span class='word'>PD</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>real</span> <span class='word'>word</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>rehmeyer2012climateNegotiations</span><span class='ignore'>,</span><span class='ignore'>osang2013environmental</span><span class='ignore'>,</span><span class='ignore'>Schneier2012doping</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<span class='word'>Objectives</span> <span class='word'>are</span> <span class='word'>the</span> <span class='word'>wrapper</span> <span class='word'>for</span> <span class='word'>which</span> <span class='word'>we</span> <span class='word'>can</span> <span class='word'>work</span> <span class='word'>with</span> <span class='word'>real</span> <span class='word'>world</span> <span class='word'>scenarios</span><span class='ignore'>,</span> <span class='word'>for</span> <span class='word'>example</span> <span class='word'>in</span> <span class='word'>a</span> <span class='word'>the</span> <span class='word'>cold</span> <span class='word'>war</span> <span class='word'>it</span> <span class='word'>was</span> <span class='word'>not</span> <span class='word'>the</span> <span class='word'>goal</span> <span class='word'>to</span> <span class='word'>get</span> <span class='word'>as</span> <span class='word'>many</span> <span class='word'>missiles</span> <span class='word'>as</span> <span class='word'>possible</span> <span class='word'>to</span> <span class='word'>pass</span> <span class='word'>an</span> <span class='word'>oppositions</span> <span class='word'>defence</span> <span class='word'>but</span> <span class='word'>to</span> <span class='word'>not</span> <span class='word'>allow</span> <span class='word'>any</span> <span class='word'>missiles</span> <span class='word'>through</span> <span class='word'>your</span> <span class='word'>own</span> <span class='word'>defence</span><span class='ignore'>;</span> <span class='word'>i.e.minimising</span> <span class='word'>your</span> <span class='word'>opponents</span> <span class='word'>score</span><span class='ignore'>.</span><br>
<span class='word'>In</span> <span class='word'>a</span> <span class='word'>football</span> <span class='word'>tournament</span> <span class='word'>where</span> <span class='word'>winning</span> <span class='word'>is</span> <span class='word'>the</span> <span class='word'>number</span> <span class='word'>or</span> <span class='word'>goals</span> <span class='word'>your</span> <span class='word'>team</span> <span class='word'>scores</span> <span class='word'>it</span> <span class='word'>does</span> <span class='word'>not</span> <span class='word'>matter</span> <span class='word'>how</span> <span class='word'>many</span> <span class='word'>goals</span> <span class='word'>you</span> <span class='word'>get</span> <span class='word'>in</span> <span class='word'>so</span> <span class='word'>long</span> <span class='word'>as</span> <span class='word'>you</span> <span class='word'>score</span> <span class='word'>the</span> <span class='word'>most</span> <span class='word'>overall</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>leads</span> <span class='word'>to</span> <span class='word'>observing</span> <span class='word'>that</span> <span class='word'>the</span> <span class='word'>IPD</span> <span class='word'>describes</span> <span class='word'>that</span> <span class='word'>a</span> <span class='word'>players</span> <span class='word'>world</span> <span class='word'>is</span> <span class='word'>better</span> <span class='word'>if</span> <span class='word'>their</span> <span class='word'>opponent</span> <span class='word'>is</span> <span class='word'>cooperative</span><span class='ignore'>.</span> <br>
<span class='word'>There</span> <span class='word'>are</span> <span class='word'>also</span> <span class='word'>some</span> <span class='word'>very</span> <span class='word'>useful</span> <span class='word'>applications</span> <span class='word'>levering</span> <span class='word'>these</span> <span class='word'>ideas</span><span class='ignore'>,</span> <span class='word'>such</span> <span class='word'>as</span> <span class='word'>modelling</span> <span class='word'>rent</span> <span class='word'>splitting</span> <span class='word'>or</span> <span class='word'>work</span> <span class='word'>assignments</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>goldman2015spliddit</span><span class='ignore'>}</span><span class='ignore'>;</span> <span class='word'>all</span> <span class='word'>of</span> <span class='word'>which</span> <span class='word'>were</span> <span class='word'>programmed</span> <span class='word'>and</span> <span class='word'>deployed</span> <span class='word'>to</span> <span class='word'>a</span> <span class='word'>website</span> <span class='word'>for</span> <span class='word'>use</span> <span class='word'>online</span> <span class='word'>at</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>spliddit</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\section</span><span class='ignore'>{</span><span class='hword'>Strategy</span> <span class='hword'>Structures</span><span class='ignore'>}</span><span class='cmd'>\label</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>stratergyStructures</span><span class='ignore'>}</span><br>
<span class='word'>A</span> <span class='word'>strategy</span> <span class='word'>is</span> <span class='word'>a</span> <span class='word'>way</span> <span class='word'>of</span> <span class='word'>defining</span> <span class='word'>how</span> <span class='word'>to</span> <span class='word'>play</span> <span class='word'>the</span> <span class='word'>IPD</span><span class='ignore'>.</span> <br>
<span class='word'>It</span> <span class='word'>is</span> <span class='word'>a</span> <span class='word'>way</span> <span class='word'>of</span> <span class='word'>identifying</span> <span class='word'>which</span> <span class='word'>move</span> <span class='word'>to</span> <span class='word'>play</span> <span class='word'>next</span> <span class='word'>based</span> <span class='word'>on</span> <span class='word'>some</span> <span class='word'>parameters</span><span class='ignore'>;</span> <span class='word'>for</span> <span class='word'>example</span> <span class='word'>the</span> <span class='word'>strategy</span> <span class='word'>of</span> <span class='word'>all</span> <span class='mathgroup'>$</span><span class='math'>C</span><span class='mathgroup'>$</span> <span class='word'>moves</span> <span class='word'>is</span> <span class='word'>valid</span><span class='ignore'>,</span> <span class='word'>as</span> <span class='word'>is</span> <span class='word'>a</span> <span class='word'>wildly</span> <span class='word'>complicated</span> <span class='word'>description</span> <span class='word'>of</span> <span class='word'>what</span> <span class='word'>to</span> <span class='word'>do</span> <span class='word'>based</span> <span class='word'>on</span> <span class='word'>the</span> <span class='word'>last</span> <span class='word'>20</span> <span class='word'>moves</span> <span class='word'>of</span> <span class='word'>an</span> <span class='word'>opponent</span><span class='ignore'>.</span><br>
<span class='word'>Some</span> <span class='word'>examples</span><span class='ignore'>,</span> <span class='word'>further</span> <span class='word'>looked</span> <span class='word'>into</span> <span class='word'>in</span> <span class='word'>Section</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>strategiesOfInterest</span><span class='ignore'>}</span><span class='ignore'>,</span> <span class='word'>include</span> <span class='word'>creating</span> <span class='word'>strategies</span> <span class='word'>that</span> <span class='word'>extort</span> <span class='word'>others</span> <span class='word'>to</span> <span class='word'>create</span> <span class='word'>overall</span> <span class='word'>scores</span> <span class='word'>it</span> <span class='word'>desires</span> <span class='word'>or</span> <span class='word'>strategies</span> <span class='word'>with</span> <span class='word'>handshakes</span> <span class='word'>for</span> <span class='word'>self</span> <span class='word'>identification</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>knight2017evolution</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<br>
<span class='word'>Individual</span> <span class='word'>strategies</span> <span class='word'>can</span> <span class='word'>be</span> <span class='word'>defined</span> <span class='word'>in</span> <span class='word'>multiple</span> <span class='word'>ways</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>harper2017reinforcement</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<span class='word'>Each</span> <span class='word'>method</span> <span class='word'>of</span> <span class='word'>representing</span> <span class='word'>a</span> <span class='word'>strategies</span> <span class='word'>has</span> <span class='word'>its</span> <span class='word'>benefits</span> <span class='word'>and</span> <span class='word'>drawbacks</span><span class='ignore'>,</span> <span class='word'>typically</span> <span class='word'>due</span> <span class='word'>to</span> <span class='word'>fundamental</span> <span class='word'>properties</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>strategy</span> <span class='ignore'>(</span><span class='word'>like</span> <span class='word'>being</span> <span class='word'>stochastic</span><span class='ignore'>)</span> <span class='word'>or</span> <span class='word'>that</span> <span class='word'>it</span> <span class='word'>is</span> <span class='word'>more</span> <span class='word'>concise</span> <span class='word'>to</span> <span class='word'>write</span> <span class='word'>in</span> <span class='word'>a</span> <span class='word'>certain</span> <span class='word'>way</span><span class='ignore'>.</span><br>
<span class='word'>Ways</span> <span class='word'>of</span> <span class='word'>structuring</span> <span class='word'>a</span> <span class='word'>strategy</span> <span class='word'>are</span> <span class='word'>shown</span> <span class='word'>below</span><span class='ignore'>:</span><br>
<br>
<span class='envir'>\begin</span><span class='envir'>{itemize}</span><br>
<span class='cmd'>\item</span> <span class='word'>LookerUpper</span><span class='ignore'>,</span> <span class='word'>for</span> <span class='word'>example</span> <span class='word'>figure</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>tit</span><span class='ignore'>_</span><span class='ignore'>for</span><span class='ignore'>_</span><span class='ignore'>tat</span><span class='ignore'>_</span><span class='ignore'>LUD</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<span class='cmd'>\item</span> <span class='word'>Gambler</span><span class='ignore'>,</span> <span class='word'>for</span> <span class='word'>example</span> <span class='word'>the</span> <span class='word'>Stochastic</span> <span class='word'>strategies</span><span class='ignore'>;</span> <span class='word'>examples</span> <span class='word'>can</span> <span class='word'>be</span> <span class='word'>found</span> <span class='word'>in</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>press2012iterated</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<span class='cmd'>\item</span> <span class='word'>Neural</span> <span class='word'>Networks</span><span class='ignore'>.</span><br>
<span class='cmd'>\item</span> <span class='word'>Finite</span> <span class='word'>State</span> <span class='word'>Machines</span><span class='ignore'>,</span> <span class='word'>for</span> <span class='word'>example</span> <span class='word'>figure</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>tit</span><span class='ignore'>_</span><span class='ignore'>for</span><span class='ignore'>_</span><span class='ignore'>tat</span><span class='ignore'>_</span><span class='ignore'>FSD</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<span class='cmd'>\item</span> <span class='word'>Hidden</span> <span class='word'>Markov</span> <span class='word'>Models</span><span class='ignore'>.</span><br>
<span class='cmd'>\item</span> <span class='word'>Explicit</span> <span class='word'>Move</span> <span class='word'>List</span><span class='ignore'>,</span> <span class='word'>for</span> <span class='word'>example</span> <span class='word'>the</span> <span class='word'>solutions</span> <span class='word'>given</span> <span class='word'>in</span> <span class='word'>Appendix</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>apndx</span><span class='ignore'>:</span><span class='ignore'>solutionGroups</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<span class='cmd'>\item</span> <span class='word'>Mixtures</span><span class='ignore'>.</span><br>
<span class='envir'>\end</span><span class='envir'>{itemize}</span><br>
<br>
<span class='cmd'>\subsection</span><span class='ignore'>{</span><span class='hword'>Equivalent</span> <span class='hword'>Strategies</span><span class='ignore'>}</span><span class='cmd'>\label</span><span class='ignore'>{</span><span class='ignore'>subsec</span><span class='ignore'>:</span><span class='ignore'>equivalentStrategies</span><span class='ignore'>}</span><br>
<span class='word'>Looking</span> <span class='word'>at</span> <span class='word'>certain</span> <span class='word'>opponents</span> <span class='word'>there</span> <span class='word'>are</span> <span class='word'>occasions</span> <span class='word'>some</span> <span class='word'>strategies</span> <span class='word'>that</span> <span class='word'>look</span> <span class='word'>indistinguishable</span> <span class='word'>from</span> <span class='word'>others</span><span class='ignore'>;</span><br>
<span class='word'>for</span> <span class='word'>example</span> <span class='word'>two</span> <span class='word'>stochastic</span> <span class='word'>opponents</span> <span class='word'>can</span> <span class='word'>can</span> <span class='word'>be</span> <span class='word'>incredibly</span> <span class='word'>similar</span><span class='ignore'>,</span> <span class='word'>but</span> <span class='word'>identifying</span> <span class='word'>which</span> <span class='word'>one</span> <span class='word'>is</span> <span class='word'>which</span> <span class='word'>from</span> <span class='word'>their</span> <span class='word'>play</span> <span class='word'>history</span> <span class='word'>could</span> <span class='word'>be</span> <span class='word'>impossible</span><span class='ignore'>.</span><br>
<span class='word'>One</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>ways</span> <span class='word'>we</span> <span class='word'>are</span> <span class='word'>able</span> <span class='word'>to</span> <span class='word'>identify</span> <span class='word'>strategies</span> <span class='word'>is</span> <span class='word'>the</span> <span class='word'>process</span> <span class='word'>of</span> <span class='word'>fingerprinting</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>Ashlock2004</span><span class='ignore'>,</span><span class='ignore'>Ashlock2008</span><span class='ignore'>,</span><span class='ignore'>cambell2016thesis</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<span class='word'>However</span> <span class='word'>this</span> <span class='word'>can</span> <span class='word'>be</span> <span class='word'>inconclusive</span> <span class='word'>and</span> <span class='word'>less</span> <span class='word'>accurate</span> <span class='word'>than</span> <span class='word'>desired</span><span class='ignore'>.</span><br>
<span class='word'>An</span> <span class='word'>effective</span> <span class='word'>method</span> <span class='word'>of</span> <span class='word'>identifying</span> <span class='word'>equivalent</span> <span class='word'>strategies</span> <span class='word'>is</span> <span class='word'>to</span> <span class='word'>write</span> <span class='word'>them</span> <span class='word'>down</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>form</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>other</span><span class='ignore'>.</span><br>
<span class='word'>For</span> <span class='word'>example</span> <span class='word'>if</span> <span class='word'>we</span> <span class='word'>can</span> <span class='word'>write</span> <span class='word'>any</span> <span class='word'>strategy</span> <span class='word'>in</span> <span class='word'>one</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>forms</span> <span class='word'>given</span> <span class='word'>in</span> <span class='word'>figures</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>tit</span><span class='ignore'>_</span><span class='ignore'>for</span><span class='ignore'>_</span><span class='ignore'>tat</span><span class='ignore'>_</span><span class='ignore'>FSD</span><span class='ignore'>}</span><span class='ignore'>,</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>tit</span><span class='ignore'>_</span><span class='ignore'>for</span><span class='ignore'>_</span><span class='ignore'>tat</span><span class='ignore'>_</span><span class='ignore'>LUD</span><span class='ignore'>}</span> <span class='word'>we</span> <span class='word'>know</span> <span class='word'>its</span> <span class='word'>an</span> <span class='word'>equivalent</span> <span class='word'>to</span> <span class='word'>Tit</span> <span class='word'>For</span> <span class='word'>Tat</span><span class='ignore'>.</span><br>
<br>
<span class='word'>The</span> <span class='word'>work</span> <span class='word'>being</span> <span class='word'>completed</span> <span class='word'>in</span> <span class='word'>this</span> <span class='word'>paper</span> <span class='word'>is</span> <span class='word'>another</span> <span class='word'>method</span> <span class='word'>of</span> <span class='word'>identifying</span> <span class='word'>strategies</span><span class='ignore'>;</span> <span class='word'>identifying</span> <span class='word'>the</span> <span class='word'>best</span> <span class='word'>response</span> <span class='word'>to</span> <span class='word'>them</span> <span class='word'>may</span> <span class='word'>lead</span> <span class='word'>to</span> <span class='word'>observations</span> <span class='word'>about</span> <span class='word'>how</span> <span class='word'>and</span> <span class='word'>why</span> <span class='word'>different</span> <span class='word'>strategies</span> <span class='word'>act</span> <span class='word'>in</span> <span class='word'>similar</span> <span class='word'>manners</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>wont</span> <span class='word'>lead</span> <span class='word'>to</span> <span class='word'>show</span> <span class='word'>strategy</span> <span class='word'>equivalence</span><span class='ignore'>,</span> <span class='word'>but</span> <span class='word'>it</span> <span class='word'>will</span> <span class='word'>show</span> <span class='word'>how</span> <span class='word'>solution</span> <span class='word'>equivalence</span><span class='ignore'>.</span><br>
<br>
<span class='envir'>\begin</span><span class='envir'>{figure}</span><span class='option'>[ht]</span><br>
<span class='exclcmd'>\centering</span><br>
<span class='exclenv'>\begin</span><span class='exclenv'>{minipage}</span><span class='ignore'>{</span><span class='ignore'>0.48</span><span class='exclcmd'>\textwidth</span><span class='ignore'>}</span><br>
<span class='exclcmd'>\centering</span><br>
<span class='exclcmd'>\tikzset</span><span class='ignore'>{</span><span class='ignore'>EdgeStyle</span><span class='ignore'>/</span><span class='ignore'>.</span><span class='ignore'>append</span> <span class='ignore'>style</span> <span class='ignore'>=</span> <span class='ignore'>{</span><span class='ignore'>-</span><span class='ignore'>></span><span class='ignore'>}</span> <span class='ignore'>,</span><br>
<span class='ignore'>LabelStyle</span><span class='ignore'>/</span><span class='ignore'>.</span><span class='ignore'>style</span> <span class='ignore'>=</span> <span class='ignore'>{</span><span class='ignore'>rectangle</span><span class='ignore'>,</span> <span class='ignore'>rounded</span> <span class='ignore'>corners</span><span class='ignore'>,</span> <span class='ignore'>draw</span><span class='ignore'>,</span> <span class='ignore'>fill</span> <span class='ignore'>=</span> <span class='ignore'>black</span><span class='ignore'>!</span><span class='ignore'>5</span><span class='ignore'>}</span><span class='ignore'>}</span><br>
<span class='exclenv'>\begin</span><span class='exclenv'>{tikzpicture}</span><span class='ignore'>[</span><br>
<span class='ignore'> </span><span class='ignore'>startnode</span><span class='ignore'>/</span><span class='ignore'>.</span><span class='ignore'>style</span><span class='ignore'>=</span><span class='ignore'>{</span><span class='ignore'>rectangle</span><span class='ignore'>,</span><span class='ignore'> </span><span class='ignore'>rounded</span><span class='ignore'> </span><span class='ignore'>corners</span><span class='ignore'>,</span><span class='ignore'> </span><span class='ignore'>draw</span><span class='ignore'>,</span><span class='ignore'> </span><span class='ignore'>fill</span><span class='ignore'> </span><span class='ignore'>=</span><span class='ignore'> </span><span class='ignore'>blue</span><span class='ignore'>!</span><span class='ignore'>5</span><span class='ignore'>}</span><span class='ignore'>,</span><br>
<span class='ignore'> </span><span class='ignore'>roundnode</span><span class='ignore'>/</span><span class='ignore'>.</span><span class='ignore'>style</span><span class='ignore'>=</span><span class='ignore'>{</span><span class='ignore'>circle</span><span class='ignore'>,</span><span class='ignore'> </span><span class='ignore'>draw</span><span class='ignore'>=</span><span class='ignore'>black</span><span class='ignore'>!</span><span class='ignore'>60</span><span class='ignore'>,</span><span class='ignore'> </span><span class='ignore'>fill</span><span class='ignore'>=</span><span class='ignore'>green</span><span class='ignore'>!</span><span class='ignore'>10</span><span class='ignore'>,</span><span class='ignore'> </span><span class='ignore'>very</span><span class='ignore'> </span><span class='ignore'>thick</span><span class='ignore'>,</span><span class='ignore'> </span><span class='ignore'>minimum</span><span class='ignore'> </span><span class='ignore'>size</span><span class='ignore'>=</span><span class='ignore'>7mm</span><span class='ignore'>}</span><br>
<span class='ignore'> </span><span class='ignore'>]</span><br>
<span class='ignore'> </span><span class='comment'>%Nodes</span><br>
<span class='ignore'> </span><span class='ignore'>\node</span><span class='ignore'>[</span><span class='ignore'>roundnode</span><span class='ignore'>]</span><span class='ignore'>(</span><span class='ignore'>initial</span><span class='ignore'>_</span><span class='ignore'>state</span><span class='ignore'>)</span><span class='ignore'>{</span><span class='ignore'>1</span><span class='ignore'>}</span><span class='ignore'>;</span><span class='ignore'> </span><br>
<span class='ignore'> </span><span class='ignore'>\node</span><span class='ignore'>[</span><span class='ignore'>startnode</span><span class='ignore'>]</span><span class='ignore'>(</span><span class='ignore'>start</span><span class='ignore'>_</span><span class='ignore'>node</span><span class='ignore'>)</span><span class='ignore'>[</span><span class='ignore'>left</span><span class='ignore'>=</span><span class='ignore'>of</span><span class='ignore'> </span><span class='ignore'>initial</span><span class='ignore'>_</span><span class='ignore'>state</span><span class='ignore'>]</span><span class='ignore'>{</span><span class='ignore'>Start</span><span class='ignore'>}</span><span class='ignore'>;</span><br>
<span class='ignore'> </span><span class='comment'>%Lines</span><br>
<span class='ignore'> </span><span class='ignore'>\Edge</span><span class='ignore'>[</span><span class='ignore'>label</span><span class='ignore'> </span><span class='ignore'>=</span><span class='ignore'> </span><span class='ignore'>$</span><span class='ignore'>C</span><span class='ignore'>$</span><span class='ignore'>]</span><span class='ignore'>(</span><span class='ignore'>start</span><span class='ignore'>_</span><span class='ignore'>node</span><span class='ignore'>)</span><span class='ignore'>(</span><span class='ignore'>initial</span><span class='ignore'>_</span><span class='ignore'>state</span><span class='ignore'>)</span><br>
<span class='ignore'> </span><span class='ignore'>\Loop</span><span class='ignore'>[</span><span class='ignore'>dist</span><span class='ignore'> </span><span class='ignore'>=</span><span class='ignore'> </span><span class='ignore'>3cm</span><span class='ignore'>,</span><span class='ignore'> </span><span class='ignore'>dir</span><span class='ignore'> </span><span class='ignore'>=</span><span class='ignore'> </span><span class='ignore'>NO</span><span class='ignore'>,</span><span class='ignore'> </span><span class='ignore'>label</span><span class='ignore'> </span><span class='ignore'>=</span><span class='ignore'> </span><span class='ignore'>$</span><span class='ignore'>C</span><span class='ignore'>/</span><span class='ignore'>C</span><span class='ignore'>$</span><span class='ignore'>]</span><span class='ignore'>(</span><span class='ignore'>initial</span><span class='ignore'>_</span><span class='ignore'>state.north</span><span class='ignore'>)</span><br>
<span class='ignore'> </span><span class='ignore'>\Loop</span><span class='ignore'>[</span><span class='ignore'>dist</span><span class='ignore'> </span><span class='ignore'>=</span><span class='ignore'> </span><span class='ignore'>3cm</span><span class='ignore'>,</span><span class='ignore'> </span><span class='ignore'>dir</span><span class='ignore'> </span><span class='ignore'>=</span><span class='ignore'> </span><span class='ignore'>SO</span><span class='ignore'>,</span><span class='ignore'> </span><span class='ignore'>label</span><span class='ignore'> </span><span class='ignore'>=</span><span class='ignore'> </span><span class='ignore'>$</span><span class='ignore'>D</span><span class='ignore'>/</span><span class='ignore'>D</span><span class='ignore'>$</span><span class='ignore'>]</span><span class='ignore'>(</span><span class='ignore'>initial</span><span class='ignore'>_</span><span class='ignore'>state.south</span><span class='ignore'>)</span><br>
<span class='ignore'> </span><span class='exclenv'>\end</span><span class='exclenv'></span><span class='exclenv'>{tikzpicture}</span><br>
<span class='cmd'>\caption</span><span class='ignore'>{</span><span class='oword'>Finite</span> <span class='oword'>State</span> <span class='oword'>diagram</span> <span class='oword'>of</span> <span class='oword'>strategy</span> <span class='oword'>Tit</span> <span class='oword'>For</span> <span class='oword'>Tat</span><span class='ignore'>}</span><span class='exclcmd'>\label</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>tit</span><span class='ignore'>_</span><span class='ignore'>for</span><span class='ignore'>_</span><span class='ignore'>tat</span><span class='ignore'>_</span><span class='ignore'>FSD</span><span class='ignore'>}</span><br>
<span class='exclenv'>\end</span><span class='exclenv'>{minipage}</span><span class='exclcmd'>\hfill</span><br>
<span class='exclenv'>\begin</span><span class='exclenv'>{minipage}</span><span class='ignore'>{</span><span class='ignore'>0.48</span><span class='exclcmd'>\textwidth</span><span class='ignore'>}</span><br>
<span class='exclcmd'>\centering</span><br>
<span class='exclcmd'>\includegraphics</span><span class='option'>[width=0.6\textwidth, center]</span><span class='ignore'>{</span><span class='ignore'>.</span><span class='ignore'>/</span><span class='ignore'>img</span><span class='ignore'>/</span><span class='ignore'>examples</span><span class='ignore'>/</span><span class='ignore'>tit</span><span class='ignore'>_</span><span class='ignore'>for</span><span class='ignore'>_</span><span class='ignore'>tat</span><span class='ignore'>_</span><span class='ignore'>LUD.pdf</span><span class='ignore'>}</span><br>
<span class='cmd'>\caption</span><span class='ignore'>{</span><span class='oword'>Look</span> <span class='oword'>Up</span> <span class='oword'>diagram</span> <span class='oword'>of</span> <span class='oword'>strategy</span> <span class='oword'>Tit</span> <span class='oword'>For</span> <span class='oword'>Tat</span><span class='ignore'>}</span><span class='exclcmd'>\label</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>tit</span><span class='ignore'>_</span><span class='ignore'>for</span><span class='ignore'>_</span><span class='ignore'>tat</span><span class='ignore'>_</span><span class='ignore'>LUD</span><span class='ignore'>}</span><br>
<span class='exclenv'>\end</span><span class='exclenv'>{minipage}</span><br>
<span class='envir'>\end</span><span class='envir'>{figure}</span><br>
<br>
<span class='cmd'>\section</span><span class='ignore'>{</span><span class='hword'>Strategies</span> <span class='hword'>Of</span> <span class='hword'>Interest</span><span class='ignore'>}</span><span class='cmd'>\label</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>strategiesOfInterest</span><span class='ignore'>}</span><br>
<span class='cmd'>\textbf</span><span class='ignore'>{</span><span class='word'>Tit</span> <span class='word'>For</span> <span class='word'>Tat</span><span class='ignore'>}</span> <span class='word'>was</span> <span class='word'>the</span> <span class='word'>winner</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>original</span> <span class='word'>axelrod</span> <span class='word'>tournament</span> <span class='word'>in</span> <span class='word'>1980</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>axelrod1980effective</span><span class='ignore'>}</span><span class='ignore'>.</span> <span class='word'>It</span> <span class='word'>s</span> <span class='word'>a</span> <span class='word'>very</span> <span class='word'>basic</span> <span class='word'>opponent</span> <span class='word'>who</span> <span class='word'>has</span> <span class='word'>strong</span> <span class='word'>forgiveness</span> <span class='ignore'>(</span><span class='word'>it</span> <span class='word'>will</span> <span class='word'>forgive</span> <span class='word'>a</span> <span class='word'>defection</span> <span class='word'>after</span> <span class='word'>1</span> <span class='word'>move</span><span class='ignore'>)</span> <span class='word'>and</span> <span class='word'>generosity</span> <span class='ignore'>(</span><span class='word'>it</span> <span class='word'>will</span> <span class='word'>start</span> <span class='word'>with</span> <span class='word'>a</span> <span class='word'>cooperation</span><span class='ignore'>)</span> <span class='word'>which</span> <span class='word'>is</span> <span class='word'>thought</span> <span class='word'>to</span> <span class='word'>give</span> <span class='word'>its</span> <span class='word'>well</span> <span class='word'>overall</span> <span class='word'>score</span> <span class='word'>in</span> <span class='word'>tournaments</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\textbf</span><span class='ignore'>{</span><span class='word'>Alternator</span><span class='ignore'>}</span> <span class='word'>is</span> <span class='word'>a</span> <span class='ignore'>`</span><span class='word'>dumb</span><span class='ignore'>'</span> <span class='word'>opponent</span><span class='ignore'>,</span> <span class='word'>i.e</span><span class='ignore'>.</span> <span class='word'>no</span> <span class='word'>strategic</span> <span class='word'>method</span> <span class='word'>at</span> <span class='word'>all</span><span class='ignore'>.</span> <span class='word'>All</span> <span class='word'>it</span> <span class='word'>will</span> <span class='word'>do</span> <span class='word'>is</span> <span class='word'>alternate</span> <span class='word'>between</span> <span class='word'>cooperation</span> <span class='word'>and</span> <span class='word'>defection</span> <span class='word'>until</span> <span class='word'>the</span> <span class='word'>game</span> <span class='word'>ends</span><span class='ignore'>.</span> <br>
<span class='word'>Playing</span> <span class='word'>against</span> <span class='word'>an</span> <span class='word'>Alternator</span> <span class='word'>effectively</span> <span class='word'>is</span> <span class='word'>to</span> <span class='word'>defect</span> <span class='word'>the</span> <span class='word'>whole</span> <span class='word'>game</span><span class='ignore'>;</span> <span class='word'>identifying</span> <span class='word'>an</span> <span class='word'>alternator</span> <span class='word'>to</span> <span class='word'>play</span> <span class='word'>this</span> <span class='word'>sequence</span> <span class='word'>of</span> <span class='word'>defections</span> <span class='word'>without</span> <span class='word'>backlash</span> <span class='word'>can</span> <span class='word'>be</span> <span class='word'>difficult</span><span class='ignore'>.</span><br>
<span class='word'>If</span> <span class='word'>we</span> <span class='word'>are</span> <span class='word'>playing</span> <span class='word'>a</span> <span class='word'>similar</span> <span class='word'>starting</span> <span class='word'>strategy</span> <span class='word'>such</span> <span class='word'>as</span> <span class='word'>Collective</span> <span class='word'>Strategy</span> <span class='word'>defecting</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>first</span> <span class='word'>two</span> <span class='word'>moves</span> <span class='word'>will</span> <span class='word'>cause</span> <span class='word'>us</span> <span class='word'>to</span> <span class='word'>sore</span> <span class='word'>badly</span> <span class='word'>overall</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\textbf</span><span class='ignore'>{</span><span class='word'>Grudger</span><span class='ignore'>}</span> <span class='word'>can</span> <span class='word'>be</span> <span class='word'>considered</span> <span class='word'>as</span> <span class='word'>the</span> <span class='word'>most</span> <span class='word'>unforgiving</span> <span class='word'>strategy</span> <span class='word'>that</span> <span class='word'>exists</span><span class='ignore'>.</span><br>
<span class='word'>Starting</span> <span class='word'>by</span> <span class='word'>cooperating</span><span class='ignore'>,</span> <span class='word'>if</span> <span class='word'>you</span> <span class='word'>defect</span> <span class='word'>even</span> <span class='word'>once</span> <span class='word'>then</span> <span class='word'>the</span> <span class='word'>Grudger</span> <span class='word'>will</span> <span class='word'>defect</span> <span class='word'>until</span> <span class='word'>the</span> <span class='word'>end</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>game</span><span class='ignore'>.</span><br>
<span class='word'>In</span> <span class='word'>line</span> <span class='word'>with</span> <span class='word'>Tit</span> <span class='word'>For</span> <span class='word'>Tat</span><span class='ignore'>,</span> <span class='word'>our</span> <span class='word'>best</span> <span class='word'>score</span> <span class='word'>will</span> <span class='word'>come</span> <span class='word'>from</span> <span class='word'>not</span> <span class='ignore'>`</span><span class='word'>upsetting</span><span class='ignore'>'</span> <span class='word'>the</span> <span class='word'>opponent</span> <span class='word'>until</span> <span class='word'>the</span> <span class='word'>last</span> <span class='word'>move</span> <span class='word'>where</span> <span class='word'>it</span> <span class='word'>cant</span> <span class='word'>retaliate</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\textbf</span><span class='ignore'>{</span><span class='word'>Random</span><span class='ignore'>}</span> <span class='word'>is</span> <span class='word'>the</span> <span class='word'>most</span> <span class='word'>basic</span> <span class='word'>stochastic</span> <span class='word'>opponent</span><span class='ignore'>,</span> <span class='word'>and</span> <span class='word'>like</span> <span class='word'>Alternator</span> <span class='word'>and</span> <span class='word'>Cycler</span> <span class='word'>it</span> <span class='word'>is</span> <span class='ignore'>`</span><span class='word'>dumb</span><span class='ignore'>'</span><span class='ignore'>.</span> <br>
<span class='word'>With</span> <span class='word'>a</span> <span class='word'>probability</span> <span class='mathgroup'>$</span><span class='math'>p</span><span class='mathgroup'>$</span> <span class='word'>of</span> <span class='word'>cooperating</span> <span class='word'>and</span> <span class='mathgroup'>$</span><span class='math'>1-p</span><span class='mathgroup'>$</span> <span class='word'>of</span> <span class='word'>defecting</span><span class='ignore'>,</span> <span class='word'>we</span> <span class='word'>can</span> <span class='word'>just</span> <span class='word'>defect</span> <span class='word'>the</span> <span class='word'>whole</span> <span class='word'>time</span> <span class='word'>to</span> <span class='word'>beat</span> <span class='word'>this</span> <span class='word'>opponent</span><span class='ignore'>;</span> <span class='word'>picking</span> <span class='word'>up</span> <span class='word'>bonus</span> <span class='word'>points</span> <span class='word'>on</span> <span class='word'>its</span> <span class='word'>cooperation</span> <span class='word'>moves</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\textbf</span><span class='ignore'>{</span><span class='word'>EvolvedFSM16</span><span class='ignore'>}</span> <span class='word'>is</span> <span class='word'>a</span> <span class='word'>16</span> <span class='word'>state</span> <span class='word'>Finite</span> <span class='word'>State</span> <span class='word'>Machine</span> <span class='ignore'>(</span><span class='word'>FSM</span><span class='ignore'>)</span> <span class='word'>that</span> <span class='word'>has</span> <span class='word'>been</span> <span class='word'>trained</span> <span class='word'>with</span> <span class='word'>an</span> <span class='word'>evolutionary</span> <span class='word'>algorithm</span><span class='ignore'>.</span> <span class='word'>It</span> <span class='word'>has</span> <span class='word'>been</span> <span class='word'>optimised</span> <span class='word'>for</span> <span class='word'>tournament</span> <span class='word'>matches</span><span class='ignore'>,</span> <span class='word'>the</span> <span class='word'>definition</span> <span class='word'>can</span> <span class='word'>be</span> <span class='word'>found</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>axelrod</span> <span class='word'>documentation</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>axelrodproject</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\textbf</span><span class='ignore'>{</span><span class='word'>CollectiveStrategy</span><span class='ignore'>}</span> <span class='word'>was</span> <span class='word'>defined</span> <span class='word'>in</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>li2009strategy</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<span class='ignore'>‘</span><span class='word'>It</span> <span class='word'>always</span> <span class='word'>cooperates</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>first</span> <span class='word'>move</span> <span class='word'>and</span> <span class='word'>defects</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>second</span> <span class='word'>move</span><span class='ignore'>.</span><br>
<span class='word'>If</span> <span class='word'>the</span> <span class='word'>opponent</span> <span class='word'>also</span> <span class='word'>cooperates</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>first</span> <span class='word'>move</span> <span class='word'>and</span> <span class='word'>defects</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>second</span> <span class='word'>move</span><span class='ignore'>,</span> <span class='word'>CS</span> <span class='word'>will</span> <span class='word'>cooperate</span> <span class='word'>until</span> <span class='word'>the</span> <span class='word'>opponent</span> <span class='word'>defects</span><span class='ignore'>.</span> <span class='word'>Otherwise</span><span class='ignore'>,</span> <span class='word'>CS</span> <span class='word'>will</span> <span class='word'>always</span> <span class='word'>defect</span><span class='ignore'>.</span><span class='ignore'>’</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\textbf</span><span class='ignore'>{</span><span class='word'>ZDExtortion</span><span class='ignore'>}</span> <span class='word'>is</span> <span class='word'>first</span> <span class='word'>mentioned</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>paper</span> <span class='cmd'>\textit</span><span class='ignore'>{</span><span class='word'>Iterated</span> <span class='word'>Prisoner</span><span class='ignore'>’</span><span class='word'>s</span> <span class='word'>Dilemma</span> <span class='word'>contains</span> <span class='word'>strategies</span> <span class='word'>that</span> <span class='word'>dominate</span> <span class='word'>any</span> <span class='word'>evolutionary</span> <span class='word'>opponent</span><span class='ignore'>}</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>press2012iterated</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<span class='word'>In</span> <span class='word'>it</span> <span class='word'>the</span> <span class='word'>authors</span> <span class='word'>discuss</span> <span class='word'>methods</span> <span class='word'>of</span> <span class='word'>extorting</span> <span class='word'>score</span> <span class='word'>from</span> <span class='word'>another</span> <span class='word'>player</span><span class='ignore'>.</span> <span class='word'>We</span> <span class='word'>use</span> <span class='word'>this</span> <span class='word'>player</span> <span class='word'>an</span> <span class='word'>example</span> <span class='word'>of</span> <span class='word'>a</span> <span class='word'>difficult</span> <span class='word'>to</span> <span class='word'>beat</span> <span class='word'>player</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\textbf</span><span class='ignore'>{</span><span class='word'>Cycler</span><span class='ignore'>}</span> <span class='word'>is</span> <span class='word'>a</span> <span class='word'>strategy</span> <span class='word'>which</span> <span class='word'>cycles</span> <span class='word'>and</span> <span class='word'>repeats</span> <span class='word'>a</span> <span class='word'>sequence</span> <span class='word'>until</span> <span class='word'>the</span> <span class='word'>end</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>match</span><span class='ignore'>.</span> <br>
<span class='word'>For</span> <span class='word'>example</span> <span class='cmd'>\mintinline</span><span class='ignore'>{</span><span class='word'>python</span><span class='ignore'>}</span><span class='ignore'>{</span><span class='word'>cycler</span><span class='ignore'>(</span><span class='ignore'>"</span><span class='word'>CCD</span><span class='ignore'>"</span><span class='ignore'>)</span><span class='ignore'>}</span> <span class='word'>will</span> <span class='word'>play</span> <span class='word'>C</span><span class='mathgroup'>$</span><span class='math'>2</span><span class='math'>,</span><span class='math'>1</span><span class='math'>,</span><span class='math'>2</span><span class='math'>,</span><span class='math'>1</span><span class='math'>,</span><span class='math'>2</span><span class='math'>,</span><span class='mathcmd'>\ldots</span><span class='mathgroup'>$</span> <span class='word'>for</span> <span class='word'>as</span> <span class='word'>long</span> <span class='word'>as</span> <span class='word'>necessary</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>Cycler</span> <span class='word'>strategy</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>directly</span> <span class='word'>used</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>GA</span> <span class='word'>by</span> <span class='word'>using</span> <span class='word'>the</span> <span class='word'>current</span> <span class='word'>solution</span><span class='ignore'>,</span> <span class='word'>of</span> <span class='word'>length</span> <span class='word'>200</span><span class='ignore'>,</span> <span class='word'>as</span> <span class='word'>the</span> <span class='word'>parameter</span> <span class='word'>to</span> <span class='word'>play</span> <span class='word'>a</span> <span class='word'>specific</span> <span class='word'>absolute</span> <span class='word'>sequence</span> <span class='word'>during</span> <span class='word'>a</span> <span class='word'>generations</span> <span class='word'>scoring</span> <span class='word'>round</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>cycler</span> <span class='word'>class</span> <span class='word'>will</span> <span class='word'>contain</span> <span class='word'>our</span> <span class='word'>solution</span> <span class='word'>sequence</span> <span class='word'>and</span> <span class='word'>the</span> <span class='word'>sequence</span> <span class='word'>can</span> <span class='word'>be</span> <span class='word'>extracted</span> <span class='word'>for</span> <span class='word'>mutation</span> <span class='cmd'>\&</span> <span class='word'>crossover</span> <span class='word'>as</span> <span class='word'>needed</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\section</span><span class='ignore'>{</span><span class='hword'>Genetic</span> <span class='hword'>Algorithms</span><span class='ignore'>}</span><br>
<span class='word'>This</span> <span class='word'>work</span> <span class='word'>will</span> <span class='word'>focus</span> <span class='word'>on</span> <span class='word'>GAs</span><span class='ignore'>,</span> <span class='word'>which</span> <span class='word'>are</span> <span class='word'>a</span> <span class='word'>specific</span> <span class='word'>form</span> <span class='word'>of</span> <span class='word'>ML</span><span class='ignore'>.</span><br>
<span class='word'>Advanced</span> <span class='word'>techniques</span> <span class='word'>of</span> <span class='word'>ML</span> <span class='word'>can</span> <span class='word'>be</span> <span class='word'>combined</span> <span class='word'>and</span> <span class='word'>used</span> <span class='word'>together</span><span class='cmd'>\footnote</span><span class='ignore'>{</span><span class='oword'>An</span> <span class='oword'>example</span> <span class='oword'>includes</span> <span class='oword'>techniques</span> <span class='oword'>for</span> <span class='oword'>teaching</span> <span class='oword'>and</span> <span class='oword'>developing</span> <span class='oword'>static</span> <span class='oword'>algorithms</span><span class='ignore'>.</span> <span class='oword'>Building</span> <span class='oword'>a</span> <span class='oword'>video</span> <span class='oword'>game</span> <span class='oword'>AI</span> <span class='oword'>where</span> <span class='oword'>the</span> <span class='oword'>core</span> <span class='oword'>goal</span> <span class='oword'>of</span> <span class='oword'>the</span> <span class='oword'>AI</span> <span class='oword'>is</span> <span class='oword'>fine</span> <span class='oword'>tuned</span> <span class='oword'>using</span> <span class='oword'>a</span> <span class='oword'>GA</span> <span class='oword'>in</span> <span class='oword'>an</span> <span class='oword'>development</span> <span class='oword'>environment</span><span class='ignore'>,</span> <span class='oword'>then</span> <span class='oword'>only</span> <span class='oword'>the</span> <span class='oword'>trained</span> <span class='oword'>algorithm</span> <span class='oword'>and</span> <span class='oword'>not</span> <span class='oword'>the</span> <span class='oword'>training</span> <span class='oword'>is</span> <span class='oword'>implemented</span> <span class='oword'>into</span> <span class='oword'>the</span> <span class='oword'>game</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>bakkes2009rapid</span><span class='ignore'>}</span><span class='ignore'>.</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>field</span> <span class='word'>of</span> <span class='word'>mathematics</span> <span class='word'>research</span> <span class='word'>is</span> <span class='word'>one</span> <span class='word'>which</span> <span class='word'>has</span> <span class='word'>plenty</span> <span class='word'>of</span> <span class='word'>examples</span> <span class='word'>of</span> <span class='word'>GAs</span> <span class='word'>in</span> <span class='word'>action</span><span class='ignore'>;</span><br>
<span class='word'>for</span> <span class='word'>example</span> <span class='word'>the</span> <span class='word'>same</span> <span class='word'>techniques</span> <span class='word'>as</span> <span class='word'>we</span> <span class='word'>will</span> <span class='word'>using</span> <span class='word'>is</span> <span class='word'>used</span> <span class='word'>in</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>chu1997genetic</span><span class='ignore'>}</span> <span class='word'>to</span> <span class='word'>solve</span> <span class='word'>the</span> <span class='word'>Generalised</span> <span class='word'>Assignment</span> <span class='word'>Problem</span><span class='ignore'>.</span><br>
<span class='word'>In</span> <span class='word'>it</span> <span class='word'>the</span> <span class='word'>authors</span> <span class='word'>discuss</span> <span class='word'>how</span> <span class='word'>to</span> <span class='word'>solve</span> <span class='word'>an</span> <span class='word'>NP-complete</span> <span class='word'>problem</span> <span class='word'>by</span> <span class='word'>exploring</span> <span class='word'>solutions</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>problem</span> <span class='word'>space</span> <span class='word'>using</span> <span class='word'>a</span> <span class='word'>GA</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>is</span> <span class='word'>similar</span> <span class='word'>to</span> <span class='word'>what</span> <span class='word'>the</span> <span class='word'>result</span> <span class='word'>of</span> <span class='word'>this</span> <span class='word'>report</span> <span class='word'>will</span> <span class='word'>contain</span><span class='ignore'>;</span><br>
<span class='word'>we</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>searching</span> <span class='word'>the</span> <span class='word'>problem</span> <span class='word'>space</span> <span class='ignore'>(</span><span class='word'>Every</span> <span class='word'>possible</span> <span class='word'>sequence</span> <span class='word'>permutation</span> <span class='word'>of</span> <span class='mathgroup'>$</span><span class='math'>C</span><span class='mathgroup'>$</span> <span class='word'>and</span> <span class='mathgroup'>$</span><span class='math'>D</span><span class='mathgroup'>$</span> <span class='word'>with</span> <span class='word'>length</span> <span class='word'>200</span><span class='ignore'>)</span> <span class='word'>to</span> <span class='word'>create</span> <span class='word'>solutions</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>problem</span> <span class='word'>described</span> <span class='word'>in</span> <span class='word'>Chapter</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>ch</span><span class='ignore'>:</span><span class='ignore'>taskBackground</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<br>
<span class='word'>GAs</span> <span class='word'>are</span> <span class='word'>not</span> <span class='word'>limited</span> <span class='word'>to</span> <span class='word'>mathematics</span><span class='ignore'>,</span> <span class='word'>in</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>hauptpractical</span><span class='ignore'>}</span> <span class='word'>there</span> <span class='word'>are</span> <span class='word'>discussions</span> <span class='word'>around</span> <span class='word'>the</span> <span class='word'>applications</span> <span class='word'>of</span> <span class='word'>GAs</span> <span class='word'>in</span> <span class='word'>art</span> <span class='word'>and</span> <span class='word'>music</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>goals</span> <span class='word'>of</span> <span class='word'>a</span> <span class='word'>GA</span> <span class='word'>have</span> <span class='word'>to</span> <span class='word'>be</span> <span class='word'>defined</span> <span class='word'>beforehand</span> <span class='word'>and</span> <span class='word'>can</span> <span class='word'>be</span> <span class='word'>as</span> <span class='word'>abstract</span> <span class='word'>as</span> <span class='word'>we</span> <span class='word'>like</span><span class='ignore'>,</span> <span class='word'>both</span> <span class='word'>art</span> <span class='word'>and</span> <span class='word'>music</span> <span class='word'>are</span> <span class='word'>abstract</span> <span class='word'>and</span> <span class='word'>would</span> <span class='word'>require</span> <span class='word'>careful</span> <span class='word'>consideration</span> <span class='word'>for</span> <span class='word'>what</span> <span class='word'>the</span> <span class='word'>algorithms</span> <span class='word'>fitness</span> <span class='word'>function</span> <span class='word'>would</span> <span class='word'>do</span><span class='ignore'>.</span> <br>
<span class='word'>Because</span> <span class='word'>of</span> <span class='word'>their</span> <span class='word'>flexibility</span> <span class='word'>GAs</span> <span class='word'>are</span> <span class='word'>used</span> <span class='word'>in</span> <span class='word'>a</span> <span class='word'>wide</span> <span class='word'>range</span> <span class='word'>of</span> <span class='word'>application</span> <span class='word'>in</span> <span class='word'>many</span> <span class='word'>areas</span> <span class='word'>including</span> <span class='word'>antenna</span> <span class='word'>design</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>lohn2004evolutionary</span><span class='ignore'>}</span> <span class='word'>and</span> <span class='word'>creating</span> <span class='word'>cost</span> <span class='word'>effective</span> <span class='word'>networks</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>savic1997genetic</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\section</span><span class='ignore'>{</span><span class='hword'>Conclusion</span><span class='ignore'>}</span><br>
<span class='word'>In</span> <span class='word'>this</span> <span class='word'>chapter</span> <span class='word'>we</span> <span class='word'>have</span> <span class='word'>looked</span> <span class='word'>at</span> <span class='word'>previous</span> <span class='word'>work</span> <span class='word'>in</span> <span class='word'>areas</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>IPD</span> <span class='word'>and</span> <span class='word'>GAs</span><span class='ignore'>.</span><br>
<span class='word'>Importantly</span> <span class='word'>we</span> <span class='word'>have</span> <span class='word'>looked</span> <span class='word'>at</span> <span class='word'>structures</span> <span class='word'>of</span> <span class='word'>strategies</span> <span class='word'>and</span> <span class='word'>some</span> <span class='word'>various</span> <span class='word'>examples</span> <span class='word'>that</span> <span class='word'>would</span> <span class='word'>be</span> <span class='word'>good</span> <span class='word'>to</span> <span class='word'>peruse</span> <span class='word'>when</span> <span class='word'>looking</span> <span class='word'>into</span> <span class='word'>constructing</span> <span class='word'>an</span> <span class='word'>effective</span> <span class='word'>GA</span><span class='ignore'>.</span><br>
<span class='word'>We</span> <span class='word'>have</span> <span class='word'>also</span> <span class='word'>outlined</span> <span class='word'>how</span> <span class='word'>strategies</span> <span class='word'>can</span> <span class='word'>be</span> <span class='word'>linked</span> <span class='word'>together</span> <span class='word'>through</span> <span class='word'>certain</span> <span class='word'>properties</span><span class='ignore'>,</span> <span class='word'>such</span> <span class='word'>as</span> <span class='word'>their</span> <span class='word'>structure</span><span class='ignore'>.</span><br>
<span class='word'>As</span> <span class='word'>we</span> <span class='word'>continue</span> <span class='word'>looking</span> <span class='word'>at</span> <span class='word'>solutions</span> <span class='word'>for</span> <span class='word'>certain</span> <span class='word'>strategies</span> <span class='word'>we</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>constructing</span> <span class='word'>another</span> <span class='word'>grouping</span> <span class='word'>of</span> <span class='word'>strategies</span> <span class='word'>in</span> <span class='word'>their</span> <span class='word'>solution</span> <span class='word'>sequences</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>may</span> <span class='word'>lead</span> <span class='word'>to</span> <span class='word'>finding</span> <span class='word'>similarities</span> <span class='word'>to</span> <span class='word'>opponents</span> <span class='word'>who</span> <span class='word'>seem</span> <span class='word'>to</span> <span class='word'>have</span> <span class='word'>vastly</span> <span class='word'>different</span> <span class='word'>in</span> <span class='word'>structure</span><span class='ignore'>.</span><br>
</p></div>
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<li class='header'>File: C:/dev/projects/uni/FinalYearProject/LaTeX/tex/chapters/03-literature.tex</li>
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<li>Words in text: 1562</li>
<li>Words in headers: 13</li>
<li>Words outside text (captions, etc.): 62</li>
<li>Number of headers: 7</li>
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<li>67+2+0 (1/0/0/0) Chapter: Literature Review}\label{ch:literature</li>
<li>465+1+0 (1/0/0/0) Section: Background}\label{sec:background</li>
<li>168+2+0 (1/0/1/0) Section: Strategy Structures}\label{sec:stratergyStructures</li>
<li>152+2+16 (1/1/0/0) Subsection: Equivalent Strategies}\label{subsec:equivalentStrategies</li>
<li>420+3+0 (1/0/3/0) Section: Strategies Of Interest}\label{sec:strategiesOfInterest</li>
<li>195+2+46 (1/0/2/0) Section: Genetic Algorithms</li>
<li>95+1+0 (1/0/0/0) Section: Conclusion</li>
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<h2>File: C:/dev/projects/uni/FinalYearProject/LaTeX/tex/chapters/04-code.tex</h2>
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<span class='cmd'>\chapter</span><span class='ignore'>{</span><span class='hword'>Developing</span> <span class='hword'>The</span> <span class='hword'>Codebase</span><span class='ignore'>}</span><span class='cmd'>\label</span><span class='ignore'>{</span><span class='ignore'>ch</span><span class='ignore'>:</span><span class='ignore'>developingthecodebase</span><span class='ignore'>}</span><br>
<span class='word'>This</span> <span class='word'>chapter</span> <span class='word'>will</span> <span class='word'>discuss</span> <span class='word'>the</span> <span class='word'>development</span> <span class='word'>practices</span> <span class='word'>encountered</span> <span class='word'>during</span> <span class='word'>the</span> <span class='word'>project</span> <span class='word'>and</span> <span class='word'>a</span> <span class='word'>walkthrough</span> <span class='word'>of</span> <span class='word'>setting</span> <span class='word'>up</span> <span class='word'>a</span> <span class='word'>research</span> <span class='word'>environment</span><span class='ignore'>.</span><br>
<span class='word'>Details</span> <span class='word'>on</span> <span class='word'>the</span> <span class='word'>set</span> <span class='word'>up</span> <span class='word'>and</span> <span class='word'>reasoning</span> <span class='word'>behind</span> <span class='word'>using</span> <span class='word'>specific</span> <span class='word'>development</span> <span class='word'>environments</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>given</span> <span class='word'>along</span> <span class='word'>with</span> <span class='word'>a</span> <span class='word'>look</span> <span class='word'>at</span> <span class='word'>the</span> <span class='word'>open</span> <span class='word'>source</span> <span class='word'>contributions</span> <span class='word'>made</span><span class='ignore'>.</span><br>
<span class='word'>A</span> <span class='word'>brief</span> <span class='word'>look</span> <span class='word'>at</span> <span class='word'>version</span> <span class='word'>control</span> <span class='word'>will</span> <span class='word'>also</span> <span class='word'>show</span> <span class='word'>how</span> <span class='word'>modern</span> <span class='word'>code</span> <span class='word'>is</span> <span class='word'>developed</span> <span class='word'>to</span> <span class='word'>be</span> <span class='word'>sustainable</span> <span class='word'>and</span> <span class='word'>reproducible</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\section</span><span class='ignore'>{</span><span class='hword'>Codebase</span> <span class='hword'>Contributions</span><span class='ignore'>}</span><br>
<span class='word'>Throughout</span> <span class='word'>the</span> <span class='word'>project</span> <span class='word'>I</span> <span class='word'>had</span> <span class='word'>split</span> <span class='word'>time</span> <span class='word'>between</span> <span class='word'>writing</span> <span class='word'>my</span> <span class='word'>own</span> <span class='word'>code</span> <span class='word'>and</span> <span class='word'>expanding</span> <span class='word'>the</span> <span class='word'>Axelrod</span> <span class='word'>Dojo</span> <span class='word'>codebase</span><span class='ignore'>.</span> <br>
<span class='word'>In</span> <span class='word'>modern</span> <span class='word'>software</span> <span class='word'>development</span> <span class='word'>there</span> <span class='word'>is</span> <span class='word'>a</span> <span class='word'>commitment</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>developer</span> <span class='word'>community</span> <span class='word'>to</span> <span class='word'>track</span> <span class='word'>and</span> <span class='word'>reuse</span> <span class='word'>as</span> <span class='word'>much</span> <span class='word'>code</span> <span class='word'>as</span> <span class='word'>possible</span><span class='ignore'>,</span> <span class='word'>typically</span> <span class='word'>using</span> <span class='word'>a</span> <span class='word'>version</span> <span class='word'>control</span> <span class='word'>system</span> <span class='word'>such</span> <span class='word'>as</span> <span class='word'>Git</span> <span class='word'>or</span> <span class='word'>Subversion</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>majority</span> <span class='word'>of</span> <span class='word'>open</span> <span class='word'>source</span> <span class='word'>community</span> <span class='word'>development</span> <span class='word'>is</span> <span class='word'>conducted</span> <span class='word'>on</span> <span class='word'>GitHub</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>GitHub</span><span class='ignore'>}</span> <span class='word'>using</span> <span class='word'>Git</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>Axelrod</span> <span class='word'>libraries</span><span class='ignore'>,</span> <span class='word'>along</span> <span class='word'>with</span> <span class='word'>most</span> <span class='word'>other</span> <span class='word'>libraries</span> <span class='word'>I</span> <span class='word'>used</span><span class='ignore'>,</span> <span class='word'>are</span> <span class='word'>primarily</span> <span class='word'>hosted</span> <span class='word'>here</span><span class='ignore'>.</span><br>
<br>
<span class='word'>When</span> <span class='word'>writing</span> <span class='word'>code</span> <span class='word'>in</span> <span class='word'>a</span> <span class='word'>professional</span> <span class='word'>environment</span> <span class='word'>there</span> <span class='word'>is</span> <span class='word'>a</span> <span class='word'>predetermined</span> <span class='word'>scope</span> <span class='word'>that</span> <span class='word'>all</span> <span class='word'>parties</span> <span class='word'>agree</span> <span class='word'>upon</span> <span class='word'>before</span> <span class='word'>the</span> <span class='word'>work</span> <span class='word'>commences</span><span class='ignore'>.</span> <br>
<span class='word'>However</span> <span class='word'>research</span> <span class='word'>development</span> <span class='word'>is</span> <span class='word'>more</span> <span class='word'>organic</span><span class='ignore'>;</span><br>
<span class='word'>final</span> <span class='word'>products</span> <span class='word'>that</span> <span class='word'>are</span> <span class='word'>created</span> <span class='word'>when</span> <span class='word'>conducting</span> <span class='word'>research</span> <span class='word'>for</span> <span class='word'>papers</span> <span class='word'>are</span> <span class='word'>highly</span> <span class='word'>personalised</span> <span class='word'>and</span> <span class='word'>typically</span> <span class='word'>cannot</span> <span class='word'>be</span> <span class='word'>used</span> <span class='word'>in</span> <span class='word'>other</span> <span class='word'>areas</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>leads</span> <span class='word'>to</span> <span class='word'>more</span> <span class='word'>flexible</span> <span class='word'>products</span> <span class='word'>that</span> <span class='word'>operate</span> <span class='word'>more</span> <span class='word'>as</span> <span class='word'>platforms</span> <span class='word'>or</span> <span class='word'>tools</span> <span class='word'>for</span> <span class='word'>further</span> <span class='word'>research</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>Axelrod</span><span class='ignore'>,</span> <span class='word'>Axelrod</span> <span class='word'>Dojo</span> <span class='word'>and</span> <span class='word'>other</span> <span class='word'>libraries</span> <span class='word'>mentioned</span> <span class='word'>in</span> <span class='word'>table</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>table</span><span class='ignore'>:</span><span class='ignore'>functionalLibrares</span><span class='ignore'>}</span> <span class='word'>exist</span> <span class='word'>as</span> <span class='word'>platforms</span> <span class='word'>because</span> <span class='word'>of</span> <span class='word'>this</span> <span class='word'>and</span> <span class='word'>the</span> <span class='word'>open</span> <span class='word'>source</span> <span class='word'>drive</span> <span class='word'>for</span> <span class='word'>reusable</span> <span class='word'>code</span><span class='ignore'>.</span><br>
<span class='word'>My</span> <span class='word'>final</span> <span class='word'>code</span> <span class='word'>will</span> <span class='word'>only</span> <span class='word'>be</span> <span class='word'>used</span> <span class='word'>for</span> <span class='word'>researching</span> <span class='word'>the</span> <span class='word'>IPD</span> <span class='word'>in</span> <span class='word'>my</span> <span class='word'>projects</span> <span class='word'>specific</span> <span class='word'>direction</span><span class='ignore'>,</span> <span class='word'>as</span> <span class='word'>I</span> <span class='word'>worked</span> <span class='word'>on</span> <span class='word'>the</span> <span class='word'>project</span> <span class='word'>I</span> <span class='word'>had</span> <span class='word'>to</span> <span class='word'>extend</span> <span class='word'>the</span> <span class='word'>platforms</span> <span class='word'>in</span> <span class='word'>order</span> <span class='word'>to</span> <span class='word'>handle</span> <span class='word'>what</span> <span class='word'>my</span> <span class='word'>project</span> <span class='word'>needed</span> <span class='word'>them</span> <span class='word'>to</span> <span class='word'>do</span><span class='ignore'>,</span> <span class='word'>subsection</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>ssec</span><span class='ignore'>:</span><span class='ignore'>versioncontrol</span><span class='ignore'>}</span> <span class='word'>looks</span> <span class='word'>into</span> <span class='word'>how</span> <span class='word'>this</span> <span class='word'>was</span> <span class='word'>completed</span> <span class='word'>in</span> <span class='word'>my</span> <span class='word'>project</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\subsection</span><span class='ignore'>{</span><span class='hword'>Version</span> <span class='hword'>Control</span><span class='ignore'>}</span><span class='cmd'>\label</span><span class='ignore'>{</span><span class='ignore'>ssec</span><span class='ignore'>:</span><span class='ignore'>versioncontrol</span><span class='ignore'>}</span><br>
<span class='word'>During</span> <span class='word'>the</span> <span class='word'>project</span> <span class='word'>I</span> <span class='word'>created</span> <span class='word'>a</span> <span class='ignore'>`</span><span class='word'>fork</span><span class='ignore'>'</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>Axelrod</span> <span class='word'>Dojo</span> <span class='word'>in</span> <span class='word'>order</span> <span class='word'>to</span> <span class='word'>add</span> <span class='word'>content</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>open</span> <span class='word'>source</span> <span class='word'>community</span><span class='ignore'>.</span> <br>
<span class='word'>This</span> <span class='word'>resulted</span> <span class='word'>in</span> <span class='word'>using</span> <span class='word'>Git</span> <span class='word'>to</span> <span class='word'>create</span> <span class='word'>a</span> <span class='word'>new</span> <span class='ignore'>`</span><span class='word'>branch</span><span class='ignore'>'</span> <span class='word'>on</span> <span class='word'>which</span> <span class='word'>to</span> <span class='word'>write</span> <span class='word'>my</span> <span class='word'>code</span> <span class='word'>before</span> <span class='word'>asking</span> <span class='word'>the</span> <span class='word'>owners</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>repository</span> <span class='word'>to</span> <span class='word'>pull</span> <span class='word'>my</span> <span class='word'>work</span> <span class='word'>back</span> <span class='word'>into</span> <span class='word'>the</span> <span class='word'>core</span> <span class='word'>product</span> <span class='word'>using</span> <span class='word'>a</span> <span class='ignore'>`</span><span class='word'>pull</span> <span class='word'>request</span><span class='ignore'>'</span><span class='ignore'>(</span><span class='word'>PR</span><span class='ignore'>)</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>PR</span> <span class='word'>opened</span> <span class='word'>for</span> <span class='word'>my</span> <span class='word'>code</span><span class='cmd'>\footnote</span><span class='ignore'>{</span><span class='oword'>https</span><span class='ignore'>:</span><span class='ignore'>/</span><span class='ignore'>/</span><span class='oword'>github.com</span><span class='ignore'>/</span><span class='oword'>Axelrod-Python</span><span class='ignore'>/</span><span class='oword'>axelrod-dojo</span><span class='ignore'>/</span><span class='oword'>pull</span><span class='ignore'>/</span><span class='oword'>45</span><span class='ignore'>}</span> <span class='word'>resulted</span> <span class='word'>in</span> <span class='word'>changes</span> <span class='word'>to</span> <span class='word'>457</span> <span class='word'>lines</span> <span class='word'>of</span> <span class='word'>code</span> <span class='word'>and</span> <span class='word'>14</span> <span class='word'>files</span><span class='ignore'>,</span> <span class='word'>adding</span> <span class='word'>classes</span> <span class='word'>and</span> <span class='word'>fixing</span> <span class='word'>bug</span> <span class='word'>that</span> <span class='word'>had</span> <span class='word'>been</span> <span class='word'>previously</span> <span class='word'>flagged</span><span class='ignore'>.</span><br>
<span class='word'>Figure</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>PR-open</span><span class='ignore'>}</span> <span class='word'>shows</span> <span class='word'>the</span> <span class='word'>initial</span> <span class='word'>scope</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>PR</span> <span class='word'>and</span> <span class='word'>Figure</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>commit-log</span><span class='ignore'>}</span> <span class='word'>shows</span> <span class='word'>a</span> <span class='word'>section</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>commits</span> <span class='word'>made</span> <span class='word'>before</span> <span class='word'>merging</span> <span class='word'>the</span> <span class='word'>branch</span><span class='ignore'>.</span><br>
<br>
<span class='envir'>\begin</span><span class='envir'>{figure}</span><span class='option'>[ht]</span><br>
<span class='exclcmd'>\centering</span><br>
<span class='exclenv'>\begin</span><span class='exclenv'>{minipage}</span><span class='ignore'>{</span><span class='ignore'>0.48</span><span class='exclcmd'>\textwidth</span><span class='ignore'>}</span><br>
<span class='exclcmd'>\centering</span><br>
<span class='exclcmd'>\includegraphics</span><span class='option'>[width=1.0\textwidth, center]</span><span class='ignore'>{</span><span class='ignore'>.</span><span class='ignore'>/</span><span class='ignore'>img</span><span class='ignore'>/</span><span class='ignore'>vcs</span><span class='ignore'>/</span><span class='ignore'>PR-Open.png</span><span class='ignore'>}</span><br>
<span class='cmd'>\caption</span><span class='ignore'>{</span><span class='oword'>Description</span> <span class='oword'>and</span> <span class='oword'>commits</span> <span class='oword'>for</span> <span class='oword'>PR</span> <span class='oword'>on</span> <span class='oword'>Github</span><span class='ignore'>}</span><span class='exclcmd'>\label</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>PR-open</span><span class='ignore'>}</span><br>
<span class='exclenv'>\end</span><span class='exclenv'>{minipage}</span><span class='exclcmd'>\hfill</span><br>
<span class='exclenv'>\begin</span><span class='exclenv'>{minipage}</span><span class='ignore'>{</span><span class='ignore'>0.48</span><span class='exclcmd'>\textwidth</span><span class='ignore'>}</span><br>
<span class='exclcmd'>\centering</span><br>
<span class='exclcmd'>\includegraphics</span><span class='option'>[width=1.0\textwidth, center,keepaspectratio]</span><span class='ignore'>{</span><span class='ignore'>.</span><span class='ignore'>/</span><span class='ignore'>img</span><span class='ignore'>/</span><span class='ignore'>vcs</span><span class='ignore'>/</span><span class='ignore'>commit-log.png</span><span class='ignore'>}</span> <br>
<span class='cmd'>\caption</span><span class='ignore'>{</span><span class='oword'>Tail</span> <span class='oword'>of</span> <span class='oword'>code</span> <span class='oword'>commit</span> <span class='oword'>log</span> <span class='oword'>as</span> <span class='oword'>shown</span> <span class='oword'>on</span> <span class='oword'>Github</span><span class='ignore'>}</span><span class='exclcmd'>\label</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>commit-log</span><span class='ignore'>}</span><br>
<span class='exclenv'>\end</span><span class='exclenv'>{minipage}</span><br>
<span class='envir'>\end</span><span class='envir'>{figure}</span><br>
<br>
<span class='word'>After</span> <span class='word'>a</span> <span class='word'>request</span> <span class='word'>is</span> <span class='word'>opened</span> <span class='word'>there</span> <span class='word'>is</span> <span class='word'>a</span> <span class='word'>period</span> <span class='word'>of</span> <span class='word'>reviews</span> <span class='word'>by</span> <span class='word'>the</span> <span class='word'>owners</span><span class='ignore'>,</span> <span class='word'>this</span> <span class='word'>is</span> <span class='word'>to</span> <span class='word'>ensure</span> <span class='word'>code</span> <span class='word'>quality</span> <span class='word'>and</span> <span class='word'>scope</span> <span class='word'>coverage</span><span class='ignore'>.</span> <br>
<span class='word'>As</span> <span class='word'>my</span> <span class='word'>work</span> <span class='word'>progressed</span> <span class='word'>I</span> <span class='word'>continued</span> <span class='word'>to</span> <span class='word'>add</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>PR</span> <span class='word'>which</span> <span class='word'>lead</span> <span class='word'>to</span> <span class='word'>more</span> <span class='word'>and</span> <span class='word'>more</span> <span class='word'>requests</span> <span class='word'>being</span> <span class='word'>added</span> <span class='word'>for</span> <span class='word'>features</span> <span class='word'>that</span> <span class='word'>fell</span> <span class='word'>outside</span> <span class='word'>the</span> <span class='word'>scope</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>PR</span><span class='ignore'>.</span><br>
<span class='word'>Eventually</span> <span class='word'>we</span> <span class='word'>decided</span> <span class='word'>to</span> <span class='word'>create</span> <span class='word'>another</span> <span class='word'>fork</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>PR</span> <span class='word'>that</span> <span class='word'>contained</span> <span class='word'>specialist</span> <span class='word'>code</span> <span class='word'>for</span> <span class='word'>my</span> <span class='word'>project</span> <span class='word'>that</span> <span class='word'>would</span> <span class='word'>not</span> <span class='word'>benefit</span> <span class='word'>the</span> <span class='word'>codebase</span> <span class='word'>as</span> <span class='word'>a</span> <span class='word'>whole</span><span class='ignore'>.</span><br>
<span class='word'>Figures</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>PR-discussion</span><span class='ignore'>}</span> <span class='cmd'>\&</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>PR-scope-discussion</span><span class='ignore'>}</span> <span class='word'>show</span> <span class='word'>examples</span> <span class='word'>of</span> <span class='word'>requests</span> <span class='word'>and</span> <span class='word'>discussions</span> <span class='word'>around</span> <span class='word'>features</span> <span class='word'>and</span> <span class='word'>code</span> <span class='word'>quality</span><span class='ignore'>.</span><br>
<span class='word'>Because</span> <span class='word'>of</span> <span class='word'>this</span> <span class='word'>review</span> <span class='word'>process</span> <span class='word'>the</span> <span class='word'>overall</span> <span class='word'>quality</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>codebase</span> <span class='word'>can</span> <span class='word'>be</span> <span class='word'>kept</span> <span class='word'>high</span> <span class='word'>and</span> <span class='word'>the</span> <span class='word'>owners</span> <span class='word'>can</span> <span class='word'>decide</span> <span class='word'>on</span> <span class='word'>how</span> <span class='word'>their</span> <span class='word'>platforms</span> <span class='word'>are</span> <span class='word'>developed</span><span class='ignore'>.</span><br>
<br>
<span class='envir'>\begin</span><span class='envir'>{figure}</span><span class='option'>[ht]</span><br>
<span class='exclcmd'>\centering</span><br>
<span class='exclenv'>\begin</span><span class='exclenv'>{minipage}</span><span class='ignore'>{</span><span class='ignore'>0.48</span><span class='exclcmd'>\textwidth</span><span class='ignore'>}</span><br>
<span class='exclcmd'>\centering</span><br>
<span class='exclcmd'>\includegraphics</span><span class='option'>[width=1.0\textwidth, center,keepaspectratio]</span><span class='ignore'>{</span><span class='ignore'>.</span><span class='ignore'>/</span><span class='ignore'>img</span><span class='ignore'>/</span><span class='ignore'>vcs</span><span class='ignore'>/</span><span class='ignore'>feature-discussion.png</span><span class='ignore'>}</span><br>
<span class='cmd'>\caption</span><span class='ignore'>{</span><span class='oword'>Feature</span> <span class='oword'>discussions</span> <span class='oword'>in</span> <span class='oword'>PR</span> <span class='oword'>on</span> <span class='oword'>GitHub</span><span class='ignore'>}</span><span class='exclcmd'>\label</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>PR-discussion</span><span class='ignore'>}</span><br>
<span class='exclenv'>\end</span><span class='exclenv'>{minipage}</span><span class='exclcmd'>\hfill</span><br>
<span class='exclenv'>\begin</span><span class='exclenv'>{minipage}</span><span class='ignore'>{</span><span class='ignore'>0.48</span><span class='exclcmd'>\textwidth</span><span class='ignore'>}</span><br>
<span class='exclcmd'>\centering</span><br>
<span class='exclcmd'>\includegraphics</span><span class='option'>[width=1.0\textwidth, center,keepaspectratio]</span><span class='ignore'>{</span><span class='ignore'>.</span><span class='ignore'>/</span><span class='ignore'>img</span><span class='ignore'>/</span><span class='ignore'>vcs</span><span class='ignore'>/</span><span class='ignore'>scope-discussion.png</span><span class='ignore'>}</span> <br>
<span class='cmd'>\caption</span><span class='ignore'>{</span><span class='oword'>Scope</span> <span class='oword'>Discussion</span> <span class='oword'>in</span> <span class='oword'>PR</span> <span class='oword'>on</span> <span class='oword'>GitHub</span><span class='ignore'>}</span><span class='exclcmd'>\label</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>PR-scope-discussion</span><span class='ignore'>}</span><br>
<span class='exclenv'>\end</span><span class='exclenv'>{minipage}</span><br>
<span class='envir'>\end</span><span class='envir'>{figure}</span><br>
<br>
<span class='cmd'>\subsection</span><span class='ignore'>{</span><span class='hword'>Testing</span><span class='ignore'>}</span><span class='cmd'>\label</span><span class='ignore'>{</span><span class='ignore'>ssec</span><span class='ignore'>:</span><span class='ignore'>testing</span><span class='ignore'>}</span><br>
<span class='word'>Code</span> <span class='word'>testing</span> <span class='word'>and</span> <span class='word'>version</span> <span class='word'>control</span> <span class='word'>go</span> <span class='word'>hand</span> <span class='word'>in</span> <span class='word'>hand</span><span class='ignore'>.</span> <br>
<span class='word'>When</span> <span class='word'>new</span> <span class='word'>releases</span> <span class='word'>of</span> <span class='word'>code</span> <span class='word'>are</span> <span class='word'>made</span> <span class='word'>it</span> <span class='word'>is</span> <span class='word'>important</span> <span class='word'>to</span> <span class='word'>ensure</span> <span class='word'>that</span> <span class='word'>the</span> <span class='word'>new</span> <span class='word'>changes</span> <span class='word'>dont</span> <span class='word'>break</span> <span class='word'>previous</span> <span class='word'>functionality</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>is</span> <span class='word'>kept</span> <span class='word'>in</span> <span class='word'>check</span> <span class='word'>by</span> <span class='word'>the</span> <span class='word'>presence</span> <span class='word'>of</span> <span class='word'>continues</span> <span class='word'>integrations</span> <span class='ignore'>(</span><span class='word'>CI</span><span class='ignore'>)</span> <span class='word'>and</span> <span class='word'>test</span> <span class='word'>environments</span><span class='ignore'>.</span><br>
<span class='word'>As</span> <span class='word'>of</span> <span class='word'>this</span> <span class='word'>project</span> <span class='word'>the</span> <span class='word'>CI</span> <span class='word'>for</span> <span class='word'>the</span> <span class='word'>Axelrod</span> <span class='word'>Dojo</span> <span class='word'>keeps</span> <span class='word'>the</span> <span class='word'>libraries</span> <span class='word'>tests</span> <span class='word'>running</span> <span class='word'>on</span> <span class='word'>a</span> <span class='word'>linux</span> <span class='word'>environment</span> <span class='word'>and</span> <span class='word'>the</span> <span class='word'>output</span> <span class='word'>fed</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>Github</span> <span class='word'>page</span><span class='ignore'>.</span><br>
<span class='word'>During</span> <span class='word'>my</span> <span class='word'>development</span> <span class='word'>I</span> <span class='word'>had</span> <span class='word'>to</span> <span class='word'>implement</span> <span class='word'>tests</span> <span class='word'>for</span> <span class='word'>any</span> <span class='word'>functionality</span> <span class='word'>I</span> <span class='word'>wrote</span> <span class='word'>to</span> <span class='word'>ensure</span> <span class='word'>the</span> <span class='word'>library</span> <span class='word'>still</span> <span class='word'>worked</span> <span class='word'>as</span> <span class='word'>intended</span><span class='ignore'>.</span><br>
<span class='word'>Following</span> <span class='word'>the</span> <span class='word'>mixed</span> <span class='word'>practice</span> <span class='word'>of</span> <span class='word'>test</span> <span class='word'>driven</span> <span class='word'>design</span> <span class='ignore'>(</span><span class='word'>TDD</span><span class='ignore'>)</span> <span class='word'>and</span> <span class='word'>behaviour</span> <span class='word'>driven</span> <span class='word'>design</span> <span class='ignore'>(</span><span class='word'>BDD</span><span class='ignore'>)</span> <span class='word'>the</span> <span class='word'>code</span> <span class='word'>written</span> <span class='word'>to</span> <span class='word'>extend</span> <span class='word'>the</span> <span class='word'>Axelrod</span> <span class='word'>Dojo</span> <span class='word'>had</span> <span class='word'>tests</span> <span class='word'>to</span> <span class='word'>cover</span> <span class='word'>examples</span> <span class='word'>of</span> <span class='word'>what</span> <span class='word'>happens</span> <span class='word'>in</span> <span class='word'>a</span> <span class='word'>production</span> <span class='word'>environment</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>hong2015top</span><span class='ignore'>,</span> <span class='ignore'>prlic2012ten</span><span class='ignore'>,</span> <span class='ignore'>sandve2013ten</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<span class='word'>Snippet</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>code</span><span class='ignore'>:</span><span class='ignore'>test-example</span><span class='ignore'>}</span> <span class='word'>shows</span> <span class='word'>an</span> <span class='word'>example</span> <span class='word'>of</span> <span class='word'>a</span> <span class='word'>test</span><span class='ignore'>.</span><br>
<br>
<span class='envir'>\begin</span><span class='envir'>{figure}</span><span class='option'>[ht]</span><br>
<span class='exclcmd'>\inputminted</span><span class='ignore'>{</span><span class='ignore'>python</span><span class='ignore'>}</span><span class='ignore'>{</span><span class='ignore'>code</span><span class='ignore'>_</span><span class='ignore'>snippets</span><span class='ignore'>/</span><span class='ignore'>dev-examples</span><span class='ignore'>/</span><span class='ignore'>example-test.py</span><span class='ignore'>}</span><br>
<span class='cmd'>\caption</span><span class='ignore'>{</span><span class='oword'>An</span> <span class='oword'>Example</span> <span class='oword'>of</span> <span class='oword'>a</span> <span class='oword'>test</span> <span class='oword'>in</span> <span class='oword'>the</span> <span class='oword'>Axelrod</span> <span class='oword'>Dojo</span><span class='ignore'>}</span><span class='exclcmd'>\label</span><span class='ignore'>{</span><span class='ignore'>code</span><span class='ignore'>:</span><span class='ignore'>test-example</span><span class='ignore'>}</span><br>
<span class='envir'>\end</span><span class='envir'>{figure}</span><br>
<br>
<span class='cmd'>\section</span><span class='ignore'>{</span><span class='hword'>Libraries</span> <span class='hword'>And</span> <span class='hword'>External</span> <span class='hword'>Modules</span><span class='ignore'>}</span><br>
<span class='cmd'>\paragraph</span><span class='ignore'>{</span><span class='hword'>Main</span> <span class='hword'>Research</span> <span class='hword'>Libraries</span><span class='ignore'>}</span><br>
<span class='envir'>\begin</span><span class='envir'>{itemize}</span><br>
<span class='cmd'>\item</span> <span class='cmd'>\textbf</span><span class='ignore'>{</span><span class='word'>Axelrod</span><span class='ignore'>}</span> <span class='ignore'>-</span><span class='ignore'>-</span><span class='ignore'>-</span> <span class='word'>Used</span> <span class='word'>for</span> <span class='word'>the</span> <span class='word'>core</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>prisoners</span> <span class='word'>dilemma</span> <span class='word'>and</span> <span class='word'>iterated</span> <span class='word'>prisoners</span> <span class='word'>dilemma</span> <span class='word'>functionality</span> <span class='word'>code</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>axelrodproject</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<span class='cmd'>\item</span> <span class='cmd'>\textbf</span><span class='ignore'>{</span><span class='word'>Axelrod</span> <span class='word'>Dojo</span><span class='ignore'>}</span> <span class='ignore'>-</span><span class='ignore'>-</span><span class='ignore'>-</span> <span class='word'>Applied</span> <span class='word'>machine</span> <span class='word'>learning</span> <span class='word'>techniques</span> <span class='word'>that</span> <span class='word'>revolve</span> <span class='word'>around</span> <span class='word'>generating</span> <span class='word'>solutions</span> <span class='word'>to</span> <span class='word'>questions</span> <span class='word'>relating</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>Axelrod</span> <span class='word'>library</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>dojoV008</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<span class='envir'>\end</span><span class='envir'>{itemize}</span><br>
<br>
<span class='cmd'>\paragraph</span><span class='ignore'>{</span><span class='hword'>Functional</span> <span class='hword'>libraries</span><span class='ignore'>}</span><br>
<span class='word'>Table</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>table</span><span class='ignore'>:</span><span class='ignore'>functionalLibrares</span><span class='ignore'>}</span> <span class='word'>shows</span> <span class='word'>the</span> <span class='word'>external</span> <span class='word'>functional</span> <span class='word'>libraries</span> <span class='word'>used</span><span class='ignore'>,</span> <span class='word'>while</span> <span class='word'>Table</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>table</span><span class='ignore'>:</span><span class='ignore'>builtinmodules</span><span class='ignore'>}</span> <span class='word'>shows</span> <span class='word'>the</span> <span class='word'>internal</span> <span class='word'>python</span> <span class='word'>built</span> <span class='word'>in</span> <span class='word'>modules</span> <span class='word'>that</span> <span class='word'>where</span> <span class='word'>leveraged</span> <span class='word'>during</span> <span class='word'>development</span><span class='ignore'>.</span><br>
<span class='word'>These</span> <span class='word'>are</span> <span class='word'>libraries</span> <span class='word'>which</span> <span class='word'>are</span> <span class='word'>not</span> <span class='word'>involved</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>core</span> <span class='word'>functionality</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>IPD</span><span class='cmd'>\@</span><span class='ignore'>.</span><br>
<span class='envir'>\begin</span><span class='envir'>{table*}</span><span class='option'>[ht]</span><br>
<span class='exclcmd'>\centering</span><br>
<span class='exclenv'>\begin</span><span class='exclenv'>{tabular}</span><span class='ignore'>{</span><span class='ignore'>ccc</span><span class='ignore'>}</span><br>
<span class='exclcmd'>\toprule</span><br>
<span class='ignore'>Library</span> <span class='ignore'>&</span> <span class='ignore'>Reason</span> <span class='ignore'>&</span> <span class='ignore'>Reference</span><span class='exclcmd'>\\</span><br>
<span class='exclcmd'>\midrule</span><br>
<span class='exclcmd'>\textbf</span><span class='ignore'>{</span><span class='ignore'>matplotlib</span> <span class='ignore'>pyplot</span><span class='ignore'>}</span> <span class='ignore'>&</span> <span class='ignore'>For</span> <span class='ignore'>plotting</span> <span class='ignore'>graphs</span> <span class='ignore'>and</span> <span class='ignore'>images</span> <span class='ignore'>with</span> <span class='ignore'>data</span> <span class='ignore'>&</span> <span class='exclcmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>hunter2007matplotlib</span><span class='ignore'>}</span><span class='exclcmd'>\\</span><br>
<span class='exclcmd'>\textbf</span><span class='ignore'>{</span><span class='ignore'>pandas</span><span class='ignore'>}</span> <span class='ignore'>&</span> <span class='ignore'>For</span> <span class='ignore'>data</span> <span class='ignore'>manipulation</span><span class='ignore'>.</span> <span class='ignore'>&</span> <span class='exclcmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>PandasGithub</span><span class='ignore'>,</span><span class='ignore'>Mckinney2010pandas</span><span class='ignore'>}</span><span class='exclcmd'>\\</span><br>
<span class='exclcmd'>\textbf</span><span class='ignore'>{</span><span class='ignore'>numpy</span><span class='ignore'>}</span> <span class='ignore'>&</span> <span class='ignore'>For</span> <span class='ignore'>reducing</span> <span class='ignore'>complexity</span> <span class='ignore'>of</span> <span class='ignore'>numerical</span> <span class='ignore'>calculations</span><span class='ignore'>.</span><span class='ignore'>&</span> <span class='exclcmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>oliphant2006numpy</span><span class='ignore'>}</span><span class='exclcmd'>\\</span><br>
<span class='exclcmd'>\textbf</span><span class='ignore'>{</span><span class='ignore'>SciKit</span> <span class='ignore'>Learn</span><span class='ignore'>}</span> <span class='ignore'>&</span> <span class='ignore'>For</span> <span class='ignore'>existing</span> <span class='ignore'>ML</span> <span class='ignore'>tools</span><span class='ignore'>.</span> <span class='ignore'>&</span> <span class='exclcmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>pedregosa2011scikit</span><span class='ignore'>}</span><span class='exclcmd'>\\</span><br>
<span class='exclcmd'>\bottomrule</span><br>
<span class='exclenv'>\end</span><span class='exclenv'>{tabular}</span><br>
<span class='cmd'>\caption</span><span class='ignore'>{</span><span class='oword'>Functional</span> <span class='oword'>Python</span> <span class='oword'>libraries</span> <span class='oword'>for</span> <span class='oword'>analysis</span><span class='ignore'>}</span><span class='exclcmd'>\label</span><span class='ignore'>{</span><span class='ignore'>table</span><span class='ignore'>:</span><span class='ignore'>functionalLibrares</span><span class='ignore'>}</span><br>
<span class='envir'>\end</span><span class='envir'>{table*}</span><br>
<span class='envir'>\begin</span><span class='envir'>{table*}</span><span class='option'>[ht]</span><br>
<span class='exclcmd'>\centering</span><br>
<span class='exclenv'>\begin</span><span class='exclenv'>{tabular}</span><span class='ignore'>{</span><span class='ignore'>cc</span><span class='ignore'>}</span><br>
<span class='exclcmd'>\toprule</span><br>
<span class='ignore'>Module</span> <span class='ignore'>&</span> <span class='ignore'>Reason</span><span class='exclcmd'>\\</span><br>
<span class='exclcmd'>\midrule</span><br>
<span class='exclcmd'>\textbf</span><span class='ignore'>{</span><span class='ignore'>os</span><span class='ignore'>}</span> <span class='ignore'>&</span> <span class='ignore'>For</span> <span class='ignore'>operating</span> <span class='ignore'>system</span> <span class='ignore'>functionality</span><span class='ignore'>.</span><span class='exclcmd'>\\</span><br>
<span class='exclcmd'>\textbf</span><span class='ignore'>{</span><span class='ignore'>time</span><span class='ignore'>}</span> <span class='ignore'>&</span> <span class='ignore'>For</span> <span class='ignore'>time</span> <span class='ignore'>calculations</span><span class='ignore'>.</span><span class='exclcmd'>\\</span><br>
<span class='exclcmd'>\textbf</span><span class='ignore'>{</span><span class='ignore'>itertools</span><span class='ignore'>}</span> <span class='ignore'>&</span> <span class='ignore'>For</span> <span class='ignore'>easier</span> <span class='ignore'>iterations</span> <span class='ignore'>over</span> <span class='ignore'>data</span> <span class='ignore'>structures</span><span class='ignore'>.</span><span class='exclcmd'>\\</span><br>
<span class='exclcmd'>\bottomrule</span><br>
<span class='exclenv'>\end</span><span class='exclenv'>{tabular}</span><br>
<span class='cmd'>\caption</span><span class='ignore'>{</span><span class='oword'>Internal</span> <span class='oword'>built</span> <span class='oword'>in</span> <span class='oword'>python</span> <span class='oword'>modules</span> <span class='oword'>used</span><span class='ignore'>}</span><span class='exclcmd'>\label</span><span class='ignore'>{</span><span class='ignore'>table</span><span class='ignore'>:</span><span class='ignore'>builtinmodules</span><span class='ignore'>}</span><br>
<span class='envir'>\end</span><span class='envir'>{table*}</span><br>
<br>
<span class='cmd'>\section</span><span class='ignore'>{</span><span class='hword'>Reproducing</span> <span class='hword'>Analysis</span><span class='ignore'>}</span><span class='cmd'>\label</span><span class='ignore'>{</span><span class='ignore'>subsec</span><span class='ignore'>:</span><span class='ignore'>settingUpAResearchEnvironment</span><span class='ignore'>}</span><br>
<span class='word'>A</span> <span class='word'>mix</span> <span class='word'>of</span> <span class='word'>Jupyter</span> <span class='word'>Notebooks</span> <span class='word'>and</span> <span class='word'>integrated</span> <span class='word'>development</span> <span class='word'>environments</span><span class='cmd'>\footnote</span><span class='ignore'>{</span><span class='oword'>Pycharm</span> <span class='oword'>Professional</span> <span class='oword'>Edition</span> <span class='oword'>and</span> <span class='oword'>Microsoft</span> <span class='oword'>VS</span> <span class='oword'>Code</span><span class='ignore'>.</span><span class='ignore'>}</span> <span class='word'>were</span> <span class='word'>used</span> <span class='word'>to</span> <span class='word'>write</span> <span class='word'>and</span> <span class='word'>execute</span> <span class='word'>code</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>main</span> <span class='word'>analysis</span> <span class='word'>was</span> <span class='word'>run</span> <span class='word'>using</span> <span class='word'>a</span> <span class='word'>factory</span> <span class='word'>class</span> <span class='word'>pattern</span><span class='ignore'>,</span> <span class='word'>called</span> <span class='cmd'>\mintinline</span><span class='ignore'>{</span><span class='word'>python</span><span class='ignore'>}</span><span class='ignore'>{</span><span class='word'>AnalysisRun</span><span class='ignore'>}</span> <span class='word'>in</span> <span class='word'>the</span> <span class='cmd'>\mintinline</span><span class='ignore'>{</span><span class='word'>python</span><span class='ignore'>}</span><span class='ignore'>{</span><span class='word'>full</span><span class='ignore'>_</span><span class='word'>analysis</span><span class='ignore'>}</span> <span class='word'>module</span><span class='ignore'>,</span> <span class='word'>show</span> <span class='word'>in</span> <span class='word'>Appendix</span> <span class='word'>Figure</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>apcode</span><span class='ignore'>:</span><span class='ignore'>AnalysisRun.py</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>class</span> <span class='word'>was</span> <span class='word'>used</span> <span class='word'>to</span> <span class='word'>wrap</span> <span class='word'>a</span> <span class='word'>query</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>Axelrod</span> <span class='word'>Dojo</span> <span class='word'>functionality</span> <span class='word'>and</span> <span class='word'>subsequently</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>Axelrod</span> <span class='word'>Library</span> <span class='word'>in</span> <span class='word'>such</span> <span class='word'>a</span> <span class='word'>way</span> <span class='word'>that</span> <span class='word'>was</span> <span class='word'>easy</span> <span class='word'>to</span> <span class='word'>control</span> <span class='word'>batch</span> <span class='word'>executions</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>analysis</span> <span class='word'>itself</span> <span class='word'>was</span> <span class='word'>done</span> <span class='word'>using</span> <span class='word'>native</span> <span class='word'>multi-threading</span> <span class='word'>on</span> <span class='word'>a</span> <span class='word'>Linux</span> <span class='word'>OS</span> <span class='word'>to</span> <span class='word'>improve</span> <span class='word'>individual</span> <span class='word'>opponent</span> <span class='word'>analysis</span> <span class='word'>run</span> <span class='word'>times</span> <span class='word'>and</span> <span class='word'>the</span> <span class='word'>overall</span> <span class='word'>scalability</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>project</span><span class='ignore'>.</span><br>
<br>
<span class='word'>The</span> <span class='word'>remainder</span> <span class='word'>of</span> <span class='word'>this</span> <span class='word'>tutorial</span> <span class='word'>will</span> <span class='word'>also</span> <span class='word'>assume</span> <span class='word'>you're</span> <span class='word'>working</span> <span class='word'>with</span> <span class='word'>a</span> <span class='word'>local</span> <span class='word'>development</span> <span class='word'>station</span><span class='ignore'>;</span><br>
<span class='word'>analysis</span> <span class='word'>on</span> <span class='word'>remote</span> <span class='word'>cloud</span> <span class='word'>instances</span> <span class='word'>of</span> <span class='word'>Jupyter</span> <span class='word'>Notebooks</span> <span class='word'>is</span> <span class='word'>possible</span> <span class='word'>but</span> <span class='word'>the</span> <span class='word'>set</span> <span class='word'>up</span> <span class='word'>is</span> <span class='word'>different</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>steps</span> <span class='word'>for</span> <span class='word'>working</span> <span class='word'>on</span> <span class='word'>a</span> <span class='word'>cloud</span> <span class='word'>instance</span> <span class='word'>are</span> <span class='word'>described</span> <span class='word'>after</span> <span class='word'>the</span> <span class='word'>local</span> <span class='word'>set</span> <span class='word'>up</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\paragraph</span><span class='ignore'>{</span><span class='hword'>Installing</span> <span class='hword'>Basic</span> <span class='hword'>Libraries</span><span class='ignore'>}</span><br>
<span class='word'>Your</span> <span class='word'>first</span> <span class='word'>step</span> <span class='word'>should</span> <span class='word'>be</span> <span class='word'>to</span> <span class='word'>download</span> <span class='word'>and</span> <span class='word'>install</span> <span class='word'>the</span> <span class='word'>Anaconda</span> <span class='word'>distribution</span> <span class='word'>for</span> <span class='word'>your</span> <span class='word'>OS</span> <span class='word'>here</span><span class='ignore'>:</span> <span class='word'>https</span><span class='ignore'>:</span><span class='ignore'>/</span><span class='ignore'>/</span><span class='word'>www.anaconda.com</span><span class='ignore'>/</span><span class='word'>download</span><span class='ignore'>/</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>will</span> <span class='word'>allow</span> <span class='word'>you</span> <span class='word'>to</span> <span class='word'>use</span> <span class='word'>the</span> <span class='word'>integrated</span> <span class='word'>c</span><span class='ignore'>+</span><span class='ignore'>+</span> <span class='word'>libraries</span> <span class='word'>python</span> <span class='word'>has</span> <span class='word'>to</span> <span class='word'>offer</span> <span class='word'>without</span> <span class='word'>needing</span> <span class='word'>to</span> <span class='word'>mess</span> <span class='word'>about</span> <span class='word'>too</span> <span class='word'>much</span><span class='ignore'>.</span><br>
<span class='word'>Anaconda</span> <span class='word'>also</span> <span class='word'>has</span> <span class='word'>them</span> <span class='word'>majority</span> <span class='word'>of</span> <span class='word'>Functional</span> <span class='word'>Libraries</span> <span class='word'>above</span> <span class='word'>and</span> <span class='word'>Jupyter</span> <span class='word'>Notebooks</span> <span class='word'>pre</span> <span class='word'>installed</span> <span class='word'>to</span> <span class='word'>make</span> <span class='word'>setting</span> <span class='word'>up</span> <span class='word'>much</span> <span class='word'>easier</span><span class='ignore'>.</span><br>
<span class='word'>From</span> <span class='word'>here</span><span class='ignore'>,</span> <span class='word'>follow</span> <span class='word'>the</span> <span class='word'>instructions</span> <span class='word'>the</span> <span class='word'>install</span> <span class='word'>wizard</span> <span class='word'>has</span> <span class='word'>to</span> <span class='word'>add</span> <span class='word'>any</span> <span class='word'>environment</span> <span class='word'>variables</span> <span class='word'>to</span> <span class='word'>allow</span> <span class='word'>CMD</span><span class='ignore'>/</span><span class='word'>Bash</span> <span class='word'>access</span> <span class='word'>to</span> <span class='word'>binaries</span><span class='ignore'>.</span><br>
<br>
<span class='word'>Installing</span> <span class='word'>the</span> <span class='word'>Axelrod</span> <span class='word'>and</span> <span class='word'>Axelrod</span> <span class='word'>Dojo</span> <span class='word'>libraries</span> <span class='word'>uses</span> <span class='word'>the</span> <span class='word'>pip</span> <span class='word'>tool</span> <span class='word'>that</span> <span class='word'>already</span> <span class='word'>comes</span> <span class='word'>with</span> <span class='word'>Anaconda</span> <span class='word'>and</span> <span class='word'>should</span> <span class='word'>be</span> <span class='word'>ready</span> <span class='word'>to</span> <span class='word'>execute</span> <span class='word'>after</span> <span class='word'>the</span> <span class='word'>last</span> <span class='word'>step</span><span class='ignore'>.</span><br>
<span class='word'>Running</span> <span class='ignore'>`</span><span class='word'>pip</span> <span class='word'>install</span> <span class='word'>axelrod</span><span class='ignore'>'</span> <span class='word'>then</span> <span class='ignore'>`</span><span class='word'>pip</span> <span class='word'>install</span> <span class='word'>axelrod-dojo</span><span class='ignore'>'</span> <span class='word'>will</span> <span class='word'>install</span> <span class='word'>these</span><span class='ignore'>.</span><br>
<br>
<span class='word'>Once</span> <span class='word'>this</span> <span class='word'>is</span> <span class='word'>installed</span> <span class='word'>the</span> <span class='cmd'>\mintinline</span><span class='ignore'>{</span><span class='word'>python</span><span class='ignore'>}</span><span class='ignore'>{</span><span class='word'>full</span><span class='cmd'>\_</span><span class='word'>analysis.py</span><span class='ignore'>}</span> <span class='word'>file</span> <span class='word'>has</span> <span class='word'>to</span> <span class='word'>be</span> <span class='word'>downloaded</span> <span class='word'>from</span> <span class='word'>GitHub</span><span class='cmd'>\footnote</span><span class='ignore'>{</span><span class='oword'>https</span><span class='ignore'>:</span><span class='ignore'>/</span><span class='ignore'>/</span><span class='oword'>github.com</span><span class='ignore'>/</span><span class='oword'>GitToby</span><span class='ignore'>/</span><span class='oword'>FinalYearProject</span><span class='ignore'>}</span><span class='ignore'>,</span> <span class='word'>it</span> <span class='word'>can</span> <span class='word'>be</span> <span class='word'>found</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>code</span> <span class='word'>directory</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>can</span> <span class='word'>just</span> <span class='word'>be</span> <span class='word'>copied</span> <span class='word'>and</span> <span class='word'>pasted</span> <span class='word'>if</span> <span class='word'>needed</span> <span class='word'>all</span> <span class='word'>were</span> <span class='word'>interested</span> <span class='word'>in</span> <span class='word'>is</span> <span class='word'>the</span> <span class='word'>class</span> <span class='word'>to</span> <span class='word'>generate</span> <span class='word'>a</span> <span class='word'>sequence</span> <span class='word'>for</span> <span class='word'>an</span> <span class='word'>opponent</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\paragraph</span><span class='ignore'>{</span><span class='hword'>Running</span> <span class='hword'>a</span> <span class='hword'>Test</span><span class='ignore'>}</span><br>
<span class='word'>Figure</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>code</span><span class='ignore'>:</span><span class='ignore'>analysisExample</span><span class='ignore'>}</span> <span class='word'>is</span> <span class='word'>some</span> <span class='word'>sample</span> <span class='word'>code</span> <span class='word'>that</span> <span class='word'>will</span> <span class='word'>run</span> <span class='word'>an</span> <span class='word'>analysis</span> <span class='word'>with</span> <span class='word'>the</span> <span class='word'>following</span> <span class='word'>settings</span><span class='ignore'>:</span><br>
<span class='envir'>\begin</span><span class='envir'>{itemize}</span><br>
<span class='cmd'>\item</span> <span class='word'>Override</span> <span class='word'>the</span> <span class='word'>default</span> <span class='word'>mutation</span> <span class='word'>frequency</span> <span class='word'>of</span> <span class='mathgroup'>\(</span><span class='math'>0.1</span><span class='mathgroup'>\)</span> <span class='word'>to</span> <span class='mathgroup'>\(</span><span class='math'>0.33</span><span class='mathgroup'>\)</span><span class='ignore'>.</span><br>
<span class='cmd'>\item</span> <span class='word'>Set</span> <span class='word'>the</span> <span class='word'>prefix</span> <span class='word'>for</span> <span class='word'>all</span> <span class='word'>the</span> <span class='word'>files</span> <span class='word'>to</span> <span class='word'>be</span> <span class='ignore'>`</span><span class='word'>example</span><span class='ignore'>-</span><span class='ignore'>'</span><span class='ignore'>.</span><br>
<span class='cmd'>\item</span> <span class='word'>Analysing</span> <span class='word'>3</span> <span class='word'>opponents</span><span class='ignore'>.</span> <span class='ignore'>(</span><span class='word'>Random</span> <span class='word'>will</span> <span class='word'>automatically</span> <span class='word'>have</span> <span class='word'>multiple</span> <span class='word'>instances</span> <span class='word'>for</span> <span class='word'>different</span> <span class='word'>seeds</span><span class='ignore'>.</span><span class='ignore'>)</span> <br>
<span class='envir'>\end</span><span class='envir'>{itemize}</span><br>
<span class='envir'>\begin</span><span class='envir'>{figure}</span><span class='option'>[ht]</span><br>
<span class='exclcmd'>\centering</span><br>
<span class='exclcmd'>\inputminted</span><span class='ignore'>{</span><span class='ignore'>python</span><span class='ignore'>}</span><span class='ignore'>{</span><span class='ignore'>code</span><span class='ignore'>_</span><span class='ignore'>snippets</span><span class='ignore'>/</span><span class='ignore'>analysisExample.py</span><span class='ignore'>}</span><br>
<span class='cmd'>\caption</span><span class='ignore'>{</span><span class='oword'>Code</span> <span class='oword'>to</span> <span class='oword'>create</span> <span class='oword'>a</span> <span class='oword'>sequence</span> <span class='oword'>result</span> <span class='oword'>to</span> <span class='oword'>optimise</span> <span class='oword'>best</span> <span class='oword'>score</span> <span class='oword'>for</span> <span class='oword'>3</span> <span class='oword'>opponents</span><span class='ignore'>}</span><span class='exclcmd'>\label</span><span class='ignore'>{</span><span class='ignore'>code</span><span class='ignore'>:</span><span class='ignore'>analysisExample</span><span class='ignore'>}</span><br>
<span class='envir'>\end</span><span class='envir'>{figure}</span><br>
<br>
<span class='word'>After</span> <span class='word'>it</span> <span class='word'>has</span> <span class='word'>run</span> <span class='word'>the</span> <span class='word'>generated</span> <span class='word'>data</span> <span class='word'>output</span> <span class='word'>should</span> <span class='word'>be</span> <span class='word'>stored</span> <span class='word'>in</span> <span class='word'>the</span> <span class='ignore'>`</span><span class='ignore'>.</span><span class='ignore'>/</span><span class='word'>output</span><span class='ignore'>'</span> <span class='word'>directory</span><span class='ignore'>.</span><br>
<span class='word'>If</span> <span class='word'>the</span> <span class='word'>code</span> <span class='word'>fails</span> <span class='word'>to</span> <span class='word'>run</span> <span class='word'>there</span> <span class='word'>may</span> <span class='word'>be</span> <span class='word'>issues</span> <span class='word'>with</span> <span class='word'>this</span> <span class='word'>directory</span> <span class='word'>being</span> <span class='word'>created</span><span class='ignore'>.</span><br>
<span class='word'>There</span> <span class='word'>should</span> <span class='word'>be</span> <span class='word'>multiple</span> <span class='word'>output</span> <span class='word'>files</span><span class='ignore'>,</span> <span class='word'>each</span> <span class='word'>with</span> <span class='word'>one</span> <span class='word'>opponents</span> <span class='word'>evolution</span> <span class='word'>stages</span> <span class='word'>through</span> <span class='word'>the</span> <span class='word'>generations</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\paragraph</span><span class='ignore'>{</span><span class='hword'>Cloud</span> <span class='hword'>Notebook</span> <span class='hword'>Setup</span><span class='ignore'>}</span> <br>
<span class='word'>If</span> <span class='word'>you</span> <span class='word'>want</span> <span class='word'>to</span> <span class='word'>use</span> <span class='word'>a</span> <span class='word'>Cloud</span> <span class='word'>service</span><span class='ignore'>,</span> <span class='word'>such</span> <span class='word'>as</span> <span class='word'>Azure</span> <span class='word'>Notebooks</span> <span class='word'>or</span> <span class='word'>AWS</span> <span class='word'>Sagemaker</span><span class='ignore'>,</span> <span class='word'>the</span> <span class='word'>set</span> <span class='word'>up</span> <span class='word'>is</span> <span class='word'>similar</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>above</span> <span class='word'>just</span> <span class='word'>executed</span> <span class='word'>differently</span><span class='ignore'>.</span><br>
<span class='word'>Installing</span> <span class='word'>Anaconda</span> <span class='word'>is</span> <span class='word'>not</span> <span class='word'>needed</span><span class='ignore'>,</span> <span class='word'>the</span> <span class='word'>environment</span> <span class='word'>has</span> <span class='word'>the</span> <span class='word'>required</span> <span class='word'>installs</span><span class='ignore'>.</span><br>
<span class='word'>Using</span> <span class='word'>pip</span> <span class='word'>to</span> <span class='word'>install</span> <span class='word'>the</span> <span class='word'>Axelrod</span> <span class='word'>libraries</span> <span class='word'>and</span> <span class='word'>download</span> <span class='word'>the</span> <span class='word'>full</span> <span class='word'>analysis</span> <span class='word'>script</span> <span class='word'>can</span> <span class='word'>be</span> <span class='word'>done</span> <span class='word'>in</span> <span class='word'>an</span> <span class='word'>integrated</span> <span class='word'>web</span> <span class='word'>terminal</span> <span class='word'>or</span> <span class='word'>directly</span> <span class='word'>in</span> <span class='word'>a</span> <span class='word'>notebook</span><span class='ignore'>.</span><br>
<span class='word'>Figure</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>code</span><span class='ignore'>:</span><span class='ignore'>jupyterExample</span><span class='ignore'>}</span> <span class='word'>shows</span> <span class='word'>and</span> <span class='word'>example</span> <span class='word'>in</span> <span class='word'>azure</span><span class='ignore'>,</span> <span class='word'>copy</span> <span class='word'>these</span> <span class='word'>in</span> <span class='word'>your</span> <span class='word'>top</span> <span class='word'>few</span> <span class='word'>cells</span> <span class='word'>of</span> <span class='word'>your</span> <span class='word'>jupyter</span> <span class='word'>notebook</span> <span class='word'>and</span> <span class='word'>it</span> <span class='word'>will</span> <span class='word'>work</span> <span class='word'>as</span> <span class='word'>required</span><span class='ignore'>.</span><br>
<br>
<span class='envir'>\begin</span><span class='envir'>{figure}</span><span class='option'>[ht]</span><br>
<span class='exclcmd'>\inputminted</span><span class='ignore'>{</span><span class='ignore'>python</span><span class='ignore'>}</span><span class='ignore'>{</span><span class='ignore'>code</span><span class='ignore'>_</span><span class='ignore'>snippets</span><span class='ignore'>/</span><span class='ignore'>dev-examples</span><span class='ignore'>/</span><span class='ignore'>jupyterCells.py</span><span class='ignore'>}</span><br>
<span class='cmd'>\caption</span><span class='ignore'>{</span><span class='oword'>Cells</span> <span class='oword'>for</span> <span class='oword'>creating</span> <span class='oword'>the</span> <span class='oword'>jupyter</span> <span class='oword'>instance</span> <span class='oword'>of</span> <span class='oword'>a</span> <span class='oword'>research</span> <span class='oword'>environment</span><span class='ignore'>}</span><span class='exclcmd'>\label</span><span class='ignore'>{</span><span class='ignore'>code</span><span class='ignore'>:</span><span class='ignore'>jupyterExample</span><span class='ignore'>}</span><br>
<span class='envir'>\end</span><span class='envir'>{figure}</span><br>
<br>
<span class='cmd'>\section</span><span class='ignore'>{</span><span class='hword'>Conclusion</span><span class='ignore'>}</span><br>
<span class='word'>In</span> <span class='word'>this</span> <span class='word'>chapter</span> <span class='word'>we</span> <span class='word'>covered</span> <span class='word'>the</span> <span class='word'>contributions</span> <span class='word'>needed</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>Axelrod</span> <span class='word'>Dojo</span> <span class='word'>to</span> <span class='word'>carry</span> <span class='word'>on</span> <span class='word'>the</span> <span class='word'>work</span> <span class='word'>described</span> <span class='word'>in</span> <span class='word'>this</span> <span class='word'>report</span><span class='ignore'>.</span> <span class='word'>Moreover</span><span class='ignore'>,</span> <span class='word'>we</span> <span class='word'>covered</span> <span class='word'>the</span> <span class='word'>procedure</span> <span class='word'>of</span> <span class='word'>implementing</span> <span class='word'>and</span> <span class='word'>contributing</span> <span class='word'>to</span> <span class='word'>an</span> <span class='word'>open</span> <span class='word'>source</span> <span class='word'>package</span><span class='ignore'>.</span><br>
<span class='word'>Of</span> <span class='word'>the</span> <span class='word'>learning</span> <span class='word'>that</span> <span class='word'>was</span> <span class='word'>complexity</span> <span class='word'>during</span> <span class='word'>the</span> <span class='word'>project</span><span class='ignore'>,</span> <span class='word'>learning</span> <span class='word'>a</span> <span class='word'>new</span> <span class='word'>version</span> <span class='word'>control</span> <span class='word'>system</span> <span class='word'>was</span> <span class='word'>the</span> <span class='word'>hardest</span> <span class='word'>along</span> <span class='word'>with</span> <span class='word'>handling</span> <span class='word'>the</span> <span class='word'>organic</span> <span class='word'>growth</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>project</span><span class='ignore'>.</span><br>
<span class='word'>While</span> <span class='word'>writing</span> <span class='word'>code</span> <span class='word'>there</span> <span class='word'>where</span> <span class='word'>sections</span> <span class='word'>that</span> <span class='word'>needed</span> <span class='word'>changing</span> <span class='word'>or</span> <span class='word'>removing</span><span class='ignore'>,</span> <span class='word'>however</span> <span class='word'>there</span> <span class='word'>was</span> <span class='word'>no</span> <span class='word'>scope</span> <span class='word'>or</span> <span class='word'>predetermined</span> <span class='word'>set</span> <span class='word'>of</span> <span class='word'>outcomes</span> <span class='word'>causing</span> <span class='word'>my</span> <span class='word'>own</span> <span class='word'>analysis</span> <span class='word'>notebooks</span> <span class='word'>to</span> <span class='word'>swell</span> <span class='word'>and</span> <span class='word'>become</span> <span class='word'>unwieldy</span> <span class='word'>at</span> <span class='word'>times</span><span class='ignore'>.</span><br>
<span class='word'>These</span> <span class='word'>problems</span> <span class='word'>were</span> <span class='word'>solved</span> <span class='word'>by</span> <span class='word'>selectively</span> <span class='word'>isolating</span> <span class='word'>work</span> <span class='word'>and</span> <span class='word'>leveraging</span> <span class='word'>3rd</span> <span class='word'>party</span> <span class='word'>libraries</span> <span class='word'>more</span><span class='ignore'>.</span><br>
<br>
<span class='word'>The</span> <span class='word'>final</span> <span class='word'>codebase</span> <span class='word'>contributions</span> <span class='word'>were</span> <span class='word'>a</span> <span class='word'>success</span> <span class='word'>and</span> <span class='word'>version</span> <span class='word'>0.0.8</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>Axelrod</span> <span class='word'>Dojo</span> <span class='word'>has</span> <span class='word'>been</span> <span class='word'>released</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>python</span> <span class='word'>installer</span> <span class='word'>package</span><span class='ignore'>,</span> <span class='ignore'>`</span><span class='word'>pip</span><span class='ignore'>'</span><span class='ignore'>~</span><span class='cmd'>\cite</span><span class='ignore'>{</span><span class='ignore'>dojoV008</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<span class='word'>Following</span> <span class='word'>the</span> <span class='word'>steps</span> <span class='word'>in</span> <span class='word'>Section</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>subsec</span><span class='ignore'>:</span><span class='ignore'>settingUpAResearchEnvironment</span><span class='ignore'>}</span> <span class='word'>will</span> <span class='word'>allow</span> <span class='word'>a</span> <span class='word'>working</span> <span class='word'>environment</span> <span class='word'>to</span> <span class='word'>be</span> <span class='word'>set</span> <span class='word'>up</span> <span class='word'>on</span> <span class='word'>any</span> <span class='word'>platform</span><span class='ignore'>,</span> <span class='word'>letting</span> <span class='word'>this</span> <span class='word'>experiment</span> <span class='word'>be</span> <span class='word'>repeated</span><span class='ignore'>.</span><br>
<span class='word'>Providing</span> <span class='word'>a</span> <span class='word'>reproducible</span> <span class='word'>experiment</span> <span class='word'>allows</span> <span class='word'>the</span> <span class='word'>open</span> <span class='word'>source</span> <span class='word'>and</span> <span class='word'>research</span> <span class='word'>communities</span> <span class='word'>independently</span> <span class='word'>verify</span> <span class='word'>any</span> <span class='word'>findings</span> <span class='word'>in</span> <span class='word'>this</span> <span class='word'>work</span> <span class='word'>to</span> <span class='word'>ensure</span> <span class='word'>a</span> <span class='word'>robust</span> <span class='word'>foundation</span> <span class='word'>for</span> <span class='word'>future</span> <span class='word'>work</span><span class='ignore'>.</span><br>
<span class='word'>All</span> <span class='word'>finalised</span> <span class='word'>notebooks</span> <span class='word'>and</span> <span class='word'>final</span> <span class='word'>code</span> <span class='word'>are</span> <span class='word'>published</span> <span class='word'>online</span> <span class='word'>on</span> <span class='word'>GitHub</span><span class='ignore'>,</span> <span class='word'>online</span> <span class='word'>resources</span> <span class='word'>can</span> <span class='word'>be</span> <span class='word'>found</span> <span class='word'>in</span> <span class='word'>Appendix</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>apndx</span><span class='ignore'>:</span><span class='ignore'>resources</span><span class='ignore'>}</span><span class='ignore'>.</span></p></div>
<br>
<div class='count'>
<ul class='count'>
<li class='header'>File: C:/dev/projects/uni/FinalYearProject/LaTeX/tex/chapters/04-code.tex</li>
<li>Encoding: ascii</li>
<li>Words in text: 1379</li>
<li>Words in headers: 29</li>
<li>Words outside text (captions, etc.): 88</li>
<li>Number of headers: 12</li>
<li>Number of floats/tables/figures: 7</li>
<li>Number of math inlines: 2</li>
<li>Number of math displayed: 0</li>
<li>Subcounts:</li>
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<li class='fielddesc'> text+headers+captions (#headers/#floats/#inlines/#displayed)</li>
<li>64+3+0 (1/0/0/0) Chapter: Developing The Codebase}\label{ch:developingthecodebase</li>
<li>214+2+0 (1/0/0/0) Section: Codebase Contributions</li>
<li>223+2+34 (1/2/0/0) Subsection: Version Control}\label{ssec:versioncontrol</li>
<li>133+1+9 (1/1/0/0) Subsection: Testing}\label{ssec:testing</li>
<li>68+9+11 (3/2/0/0) Section: Libraries And External Modules</li>
<li>479+11+34 (4/2/2/0) Section: Reproducing Analysis}\label{subsec:settingUpAResearchEnvironment</li>
<li>198+1+0 (1/0/0/0) Section: Conclusion</li>
</span>
</ul>
</div>
</div>
<div class='filegroup'>
<h2>File: C:/dev/projects/uni/FinalYearProject/LaTeX/tex/chapters/05-implementation.tex</h2>
<div class='parse'><p>
<span class='comment'>% !TEX root = ../main.tex</span><br>
<span class='cmd'>\chapter</span><span class='ignore'>{</span><span class='hword'>Implementation</span> <span class='hword'>Of</span> <span class='hword'>Sequence</span> <span class='hword'>Discovery</span><span class='ignore'>}</span><span class='cmd'>\label</span><span class='ignore'>{</span><span class='ignore'>ch</span><span class='ignore'>:</span><span class='ignore'>implementation</span><span class='ignore'>}</span><br>
<span class='word'>Genetic</span> <span class='word'>algorithms</span> <span class='word'>can</span> <span class='word'>have</span> <span class='word'>many</span> <span class='word'>variations</span> <span class='word'>depending</span> <span class='word'>on</span> <span class='word'>parameters</span> <span class='word'>and</span> <span class='word'>implementations</span> <span class='word'>of</span> <span class='word'>any</span> <span class='word'>attached</span> <span class='word'>methods</span><span class='ignore'>.</span> <span class='word'>In</span> <span class='word'>this</span> <span class='word'>chapter</span><span class='ignore'>,</span> <span class='word'>consideration</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>given</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>process</span> <span class='word'>of</span> <span class='word'>finding</span> <span class='word'>the</span> <span class='word'>optimal</span> <span class='word'>sequence</span> <span class='word'>of</span> <span class='word'>moves</span> <span class='word'>against</span> <span class='word'>another</span> <span class='word'>player</span><span class='ignore'>.</span><br>
<span class='word'>There</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>various</span> <span class='word'>approaches</span> <span class='word'>used</span> <span class='word'>and</span> <span class='word'>a</span> <span class='word'>detailed</span> <span class='word'>analysis</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>optimisation</span> <span class='word'>procedures</span> <span class='word'>and</span> <span class='word'>parameters</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>described</span> <span class='word'>for</span> <span class='word'>each</span> <span class='word'>parameter</span><span class='ignore'>.</span><br>
<br>
<span class='word'>In</span> <span class='word'>this</span> <span class='word'>chapter</span> <span class='word'>the</span> <span class='word'>terms</span> <span class='word'>strategy</span> <span class='word'>and</span> <span class='word'>opponent</span> <span class='word'>are</span> <span class='word'>used</span> <span class='word'>interchangeably</span><span class='ignore'>.</span><br>
<span class='word'>An</span> <span class='word'>opponent</span> <span class='word'>has</span> <span class='word'>a</span> <span class='word'>set</span> <span class='word'>of</span> <span class='word'>rules</span> <span class='word'>for</span> <span class='word'>calculating</span> <span class='word'>its</span> <span class='word'>next</span> <span class='word'>move</span><span class='ignore'>,</span> <span class='word'>this</span> <span class='word'>is</span> <span class='word'>known</span> <span class='word'>as</span> <span class='word'>its</span> <span class='word'>strategy</span> <span class='word'>and</span> <span class='word'>is</span> <span class='word'>unique</span><span class='ignore'>.</span><br>
<span class='word'>Also</span> <span class='word'>in</span> <span class='word'>this</span> <span class='word'>chapter</span> <span class='word'>the</span> <span class='word'>terms</span> <span class='word'>solution</span> <span class='word'>and</span> <span class='word'>sequence</span> <span class='word'>are</span> <span class='word'>interchange</span><span class='ignore'>,</span> <span class='word'>often</span> <span class='word'>called</span> <span class='word'>the</span> <span class='word'>solution</span> <span class='word'>sequence</span><span class='ignore'>.</span><br>
<span class='word'>A</span> <span class='word'>solution</span> <span class='word'>is</span> <span class='word'>a</span> <span class='word'>sequence</span> <span class='word'>of</span> <span class='mathgroup'>$</span><span class='math'>C</span><span class='mathgroup'>$</span> <span class='word'>and</span> <span class='mathgroup'>$</span><span class='math'>D</span><span class='mathgroup'>$</span> <span class='word'>moves</span><span class='ignore'>,</span> <span class='word'>as</span> <span class='word'>described</span> <span class='word'>in</span> <span class='word'>Section</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>solutionForm</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\section</span><span class='ignore'>{</span><span class='hword'>Background</span><span class='ignore'>}</span><br>
<span class='word'>Before</span> <span class='word'>conducting</span> <span class='word'>the</span> <span class='word'>bulk</span> <span class='word'>calculations</span> <span class='word'>for</span> <span class='word'>the</span> <span class='word'>set</span> <span class='word'>of</span> <span class='word'>opponents</span> <span class='word'>listed</span> <span class='word'>in</span> <span class='word'>Appendix</span> <span class='word'>Section</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>apndx</span><span class='ignore'>:</span><span class='ignore'>opponents</span><span class='ignore'>}</span> <span class='word'>we</span> <span class='word'>will</span> <span class='word'>test</span> <span class='word'>values</span> <span class='word'>for</span> <span class='word'>the</span> <span class='word'>algorithms</span><span class='ignore'>'</span> <span class='word'>parameters</span> <span class='word'>to</span> <span class='word'>see</span> <span class='word'>what</span> <span class='word'>values</span> <span class='word'>of</span> <span class='word'>parameters</span> <span class='word'>are</span> <span class='word'>best</span> <span class='word'>for</span> <span class='word'>finding</span> <span class='word'>solution</span> <span class='word'>sequences</span><span class='ignore'>.</span><br>
<span class='word'>We</span> <span class='word'>will</span> <span class='word'>run</span> <span class='word'>a</span> <span class='word'>series</span> <span class='word'>of</span> <span class='word'>tests</span> <span class='word'>against</span> <span class='word'>pre</span> <span class='word'>selected</span> <span class='word'>strategies</span> <span class='word'>and</span> <span class='word'>review</span> <span class='word'>the</span> <span class='word'>results</span> <span class='word'>of</span> <span class='word'>their</span> <span class='word'>best</span> <span class='word'>score</span> <span class='word'>per</span> <span class='word'>turn</span> <span class='word'>before</span> <span class='word'>running</span> <span class='word'>the</span> <span class='word'>full</span> <span class='word'>analysis</span><span class='ignore'>.</span><br>
<span class='word'>These</span> <span class='word'>test</span> <span class='word'>opponents</span> <span class='word'>have</span> <span class='word'>been</span> <span class='word'>selected</span> <span class='word'>because</span> <span class='word'>they</span> <span class='word'>are</span> <span class='word'>interpretable</span> <span class='word'>and</span> <span class='word'>have</span> <span class='word'>either</span> <span class='word'>calculable</span> <span class='word'>solution</span> <span class='word'>sequences</span> <span class='word'>or</span> <span class='word'>are</span> <span class='word'>interesting</span> <span class='word'>stochastic</span> <span class='word'>opponents</span><span class='ignore'>.</span><br>
<span class='word'>Table</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>table</span><span class='ignore'>:</span><span class='ignore'>table</span><span class='ignore'>_</span><span class='ignore'>test</span><span class='ignore'>_</span><span class='ignore'>opsponents</span><span class='ignore'>}</span> <span class='word'>shows</span> <span class='word'>the</span> <span class='word'>opponents</span> <span class='word'>and</span> <span class='word'>their</span> <span class='word'>solution</span><span class='ignore'>.</span><br>
<span class='word'>Each</span> <span class='word'>one</span> <span class='word'>has</span> <span class='word'>a</span> <span class='word'>fundamentally</span> <span class='word'>different</span> <span class='word'>structure</span> <span class='word'>to</span> <span class='word'>how</span> <span class='word'>they</span> <span class='word'>work</span><span class='ignore'>,</span> <span class='word'>because</span> <span class='word'>of</span> <span class='word'>this</span> <span class='word'>we</span> <span class='word'>can</span> <span class='word'>confirm</span> <span class='word'>that</span> <span class='word'>the</span> <span class='word'>genetic</span> <span class='word'>algorithm</span> <span class='word'>can</span> <span class='word'>select</span> <span class='word'>the</span> <span class='word'>optimal</span> <span class='word'>sequence</span> <span class='word'>solution</span> <span class='word'>for</span> <span class='word'>each</span> <span class='word'>structure</span><span class='ignore'>.</span><br>
<br>
<span class='envir'>\begin</span><span class='envir'>{table*}</span><br>
<span class='exclcmd'>\centering</span><br>
<span class='exclenv'>\begin</span><span class='exclenv'>{tabular}</span><span class='ignore'>{</span><span class='ignore'>ccc</span><span class='ignore'>}</span><br>
<span class='exclcmd'>\toprule</span><br>
<span class='ignore'>Player</span> <span class='ignore'>&</span> <span class='ignore'>Optimal</span> <span class='ignore'>Sequence</span> <span class='ignore'>&</span> <span class='ignore'>Representation</span> <span class='ignore'>of</span> <span class='ignore'>game</span> <span class='ignore'>length</span> <span class='exclmath'>$</span><span class='math'>n</span><span class='exclmath'>$</span><span class='exclcmd'>\\</span><br>
<span class='exclcmd'>\midrule</span><br>
<span class='ignore'>axl.TitForTat</span><span class='ignore'>(</span><span class='ignore'>)</span><span class='ignore'>&</span><span class='exclcmd'>\(</span><span class='ignore'>CCC</span><span class='exclcmd'>\ldots</span> <span class='ignore'>CD</span><span class='exclcmd'>\)</span><span class='ignore'>&</span> <span class='exclmath'>$</span><span class='math'>Cn-1</span><span class='math'>,</span><span class='math'>1</span><span class='exclmath'>$</span><span class='exclcmd'>\\</span><br>
<span class='ignore'>axl.Alternator</span><span class='ignore'>(</span><span class='ignore'>)</span><span class='ignore'>&</span><span class='exclcmd'>\(</span><span class='ignore'>DDD</span><span class='exclcmd'>\ldots</span> <span class='ignore'>DD</span><span class='exclcmd'>\)</span><span class='ignore'>&</span><span class='exclmath'>$</span><span class='math'>Dn</span><span class='exclmath'>$</span><span class='exclcmd'>\\</span><br>
<span class='ignore'>axl.Grudger</span><span class='ignore'>(</span><span class='ignore'>)</span><span class='ignore'>&</span><span class='exclcmd'>\(</span><span class='ignore'>CCC</span><span class='exclcmd'>\ldots</span> <span class='ignore'>CD</span><span class='exclcmd'>\)</span><span class='ignore'>&</span><span class='exclmath'>$</span><span class='math'>Cn-1</span><span class='math'>,</span><span class='math'>1</span><span class='exclmath'>$</span><span class='exclcmd'>\\</span><br>
<span class='ignore'>axl.Random</span><span class='ignore'>(</span><span class='ignore'>)</span><span class='ignore'>&</span><span class='exclcmd'>\(</span><span class='ignore'>DDD</span><span class='exclcmd'>\ldots</span> <span class='ignore'>DD</span><span class='exclcmd'>\)</span><span class='ignore'>&</span><span class='exclmath'>$</span><span class='math'>Dn</span><span class='exclmath'>$</span><span class='exclcmd'>\\</span><br>
<span class='ignore'>axl.EvolvedFSM16</span><span class='ignore'>(</span><span class='ignore'>)</span><span class='ignore'>&</span> <span class='exclmath'>$</span><span class='math'>CDC</span><span class='mathcmd'>\ldots</span> <span class='math'>DD</span><span class='exclmath'>$</span> <span class='ignore'>&</span> <span class='exclmath'>$</span><span class='math'>C1</span><span class='math'>,</span><span class='math'>1</span><span class='math'>,</span><span class='math'>n-2</span><span class='exclmath'>$</span><span class='exclcmd'>\\</span><br>
<span class='ignore'>axl.CollectiveStrategy</span><span class='ignore'>(</span><span class='ignore'>)</span><span class='ignore'>&</span><span class='exclmath'>$</span><span class='math'>CDC</span><span class='mathcmd'>\ldots</span> <span class='math'>CD</span><span class='exclmath'>$</span><span class='ignore'>&</span><span class='exclmath'>$</span><span class='math'>C1</span><span class='math'>,</span><span class='math'>1</span><span class='math'>,</span><span class='math'>n-3</span><span class='math'>,</span><span class='math'>1</span><span class='exclmath'>$</span><span class='exclcmd'>\\</span><br>
<span class='ignore'>axl.Champion</span><span class='ignore'>(</span><span class='ignore'>)</span><span class='ignore'>&</span> <span class='ignore'>Various</span><span class='exclcmd'>\footnote</span><span class='ignore'>{</span><span class='ignore'>Champion</span> <span class='ignore'>is</span> <span class='ignore'>a</span> <span class='ignore'>stochastic</span> <span class='ignore'>opponent</span><span class='ignore'>.</span><span class='ignore'>}</span> <span class='ignore'>&</span> <span class='exclmath'>$</span><span class='math'>NA</span><span class='exclmath'>$</span><span class='exclcmd'>\\</span><br>
<span class='ignore'>axl.ZDExtort</span><span class='ignore'>(</span><span class='ignore'>)</span><span class='ignore'>&</span> <span class='ignore'>Various</span><span class='exclcmd'>\footnote</span><span class='ignore'>{</span><span class='ignore'>ZD</span> <span class='ignore'>Extort</span> <span class='ignore'>is</span> <span class='ignore'>a</span> <span class='ignore'>stochastic</span> <span class='ignore'>opponent</span><span class='ignore'>.</span><span class='ignore'>}</span> <span class='ignore'>&</span> <span class='exclmath'>$</span><span class='math'>NA</span><span class='exclmath'>$</span><span class='exclcmd'>\\</span><br>
<span class='exclcmd'>\bottomrule</span><br>
<span class='exclenv'>\end</span><span class='exclenv'>{tabular}</span><br>
<span class='cmd'>\caption</span><span class='ignore'>{</span><span class='oword'>Table</span> <span class='oword'>of</span> <span class='oword'>test</span> <span class='oword'>opponents</span><span class='ignore'>}</span><span class='exclcmd'>\label</span><span class='ignore'>{</span><span class='ignore'>table</span><span class='ignore'>:</span><span class='ignore'>table</span><span class='ignore'>_</span><span class='ignore'>test</span><span class='ignore'>_</span><span class='ignore'>opsponents</span><span class='ignore'>}</span><br>
<span class='envir'>\end</span><span class='envir'>{table*}</span><br>
<br>
<span class='word'>In</span> <span class='word'>this</span> <span class='word'>chapter</span> <span class='word'>sequences</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>loosely</span> <span class='word'>described</span> <span class='word'>as</span> <span class='ignore'>`</span><span class='word'>converged</span><span class='ignore'>'</span> <span class='word'>if</span> <span class='word'>the</span> <span class='word'>best</span> <span class='word'>score</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>population</span> <span class='word'>has</span> <span class='word'>reached</span> <span class='word'>a</span> <span class='word'>stable</span> <span class='word'>point</span> <span class='word'>considering</span> <span class='word'>the</span> <span class='word'>generations</span> <span class='word'>it</span> <span class='word'>has</span> <span class='word'>run</span><span class='ignore'>.</span><br>
<span class='word'>In</span> <span class='word'>order</span> <span class='word'>to</span> <span class='word'>find</span> <span class='word'>these</span> <span class='word'>settings</span> <span class='word'>we</span> <span class='word'>look</span> <span class='word'>at</span> <span class='word'>is</span> <span class='word'>how</span> <span class='word'>the</span> <span class='word'>best</span> <span class='word'>score</span> <span class='word'>in</span> <span class='word'>a</span> <span class='word'>population</span> <span class='word'>rises</span> <span class='word'>over</span> <span class='word'>the</span> <span class='word'>number</span> <span class='word'>generations</span> <span class='word'>after</span> <span class='word'>changing</span> <span class='word'>the</span> <span class='word'>parameter</span> <span class='word'>were</span> <span class='word'>observing</span><span class='ignore'>.</span><br>
<span class='word'>Once</span> <span class='word'>the</span> <span class='word'>best</span> <span class='word'>score</span> <span class='word'>per</span> <span class='word'>turn</span> <span class='word'>hits</span> <span class='word'>a</span> <span class='word'>maximum</span> <span class='word'>such</span> <span class='word'>that</span> <span class='word'>it</span> <span class='word'>doesn't</span> <span class='word'>appear</span> <span class='word'>to</span> <span class='word'>change</span> <span class='word'>no</span> <span class='word'>mater</span> <span class='word'>how</span> <span class='word'>many</span> <span class='word'>more</span> <span class='word'>generations</span> <span class='word'>are</span> <span class='word'>run</span> <span class='word'>this</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>described</span> <span class='word'>as</span> <span class='word'>an</span> <span class='word'>optimal</span> <span class='word'>solution</span> <span class='word'>sequence</span><span class='ignore'>.</span><br>
<span class='word'>Note</span> <span class='word'>that</span> <span class='word'>for</span> <span class='word'>a</span> <span class='word'>solution</span> <span class='word'>to</span> <span class='word'>be</span> <span class='word'>unique</span> <span class='word'>we</span> <span class='word'>must</span> <span class='word'>be</span> <span class='word'>playing</span> <span class='word'>a</span> <span class='word'>non</span> <span class='word'>stochastic</span> <span class='word'>opponent</span><span class='ignore'>,</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>event</span> <span class='word'>of</span> <span class='word'>playing</span> <span class='word'>a</span> <span class='word'>stochastic</span> <span class='word'>opponent</span> <span class='word'>we</span> <span class='word'>will</span> <span class='word'>seed</span> <span class='word'>the</span> <span class='word'>random</span> <span class='word'>element</span> <span class='word'>of</span> <span class='word'>their</span> <span class='word'>behaviour</span> <span class='word'>for</span> <span class='word'>reproducibility</span><span class='ignore'>.</span><br>
<br>
<span class='word'>During</span> <span class='word'>the</span> <span class='word'>investigation</span> <span class='word'>we</span> <span class='word'>may</span> <span class='word'>find</span> <span class='word'>solutions</span> <span class='word'>that</span> <span class='word'>are</span> <span class='word'>not</span> <span class='word'>optimal</span><span class='ignore'>,</span> <span class='word'>meaning</span> <span class='word'>that</span> <span class='word'>the</span> <span class='word'>algorithm</span> <span class='word'>will</span> <span class='word'>have</span> <span class='word'>found</span> <span class='word'>a</span> <span class='word'>solution</span> <span class='word'>that</span> <span class='word'>will</span> <span class='word'>do</span> <span class='word'>well</span> <span class='word'>against</span> <span class='word'>an</span> <span class='word'>opponent</span> <span class='word'>but</span> <span class='word'>hasn't</span> <span class='word'>found</span> <span class='word'>the</span> <span class='word'>best</span> <span class='word'>from</span> <span class='word'>the</span> <span class='word'>solution</span> <span class='word'>space</span><span class='ignore'>.</span><br>
<span class='word'>These</span> <span class='word'>sub</span> <span class='word'>optimal</span> <span class='word'>solutions</span> <span class='word'>are</span> <span class='word'>due</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>occurrence</span> <span class='word'>of</span> <span class='word'>local</span> <span class='word'>maxima</span> <span class='ignore'>(</span><span class='word'>due</span> <span class='word'>to</span> <span class='word'>an</span> <span class='word'>increasing</span> <span class='word'>fitness</span> <span class='word'>function</span><span class='ignore'>)</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>space</span> <span class='word'>of</span> <span class='word'>neighbouring</span> <span class='word'>solution</span> <span class='word'>sequences</span> <span class='word'>to</span> <span class='word'>members</span><span class='ignore'>.</span><br>
<span class='word'>Section</span> <span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>subsec</span><span class='ignore'>:</span><span class='ignore'>geneticAlgorithms</span><span class='ignore'>}</span> <span class='word'>discusses</span> <span class='word'>how</span> <span class='word'>GAs</span> <span class='word'>are</span> <span class='word'>designed</span> <span class='word'>to</span> <span class='word'>mitigate</span> <span class='word'>local</span> <span class='word'>maxima</span><span class='ignore'>.</span><br>
<br>
<span class='word'>Some</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>questions</span> <span class='word'>we</span> <span class='word'>will</span> <span class='word'>hope</span> <span class='word'>to</span> <span class='word'>answer</span> <span class='word'>in</span> <span class='word'>this</span> <span class='word'>chapter</span> <span class='word'>include</span><span class='ignore'>:</span><br>
<span class='envir'>\begin</span><span class='envir'>{itemize}</span><br>
<span class='cmd'>\item</span> <span class='word'>If</span> <span class='word'>we</span> <span class='word'>have</span> <span class='word'>a</span> <span class='word'>larger</span> <span class='word'>initial</span> <span class='word'>population</span> <span class='word'>sample</span> <span class='word'>to</span> <span class='word'>start</span> <span class='word'>with</span><span class='ignore'>,</span> <span class='word'>will</span> <span class='word'>we</span> <span class='word'>reach</span> <span class='word'>our</span> <span class='word'>maximum</span> <span class='word'>best</span> <span class='word'>score</span> <span class='word'>earlier</span><span class='ignore'>?</span><br>
<span class='cmd'>\item</span> <span class='word'>Will</span> <span class='word'>increasing</span> <span class='word'>the</span> <span class='word'>number</span> <span class='word'>of</span> <span class='word'>generations</span> <span class='word'>impact</span> <span class='word'>our</span> <span class='word'>ability</span> <span class='word'>to</span> <span class='word'>find</span> <span class='word'>an</span> <span class='word'>optimal</span> <span class='word'>solution</span><span class='ignore'>?</span> <span class='word'>Is</span> <span class='word'>there</span> <span class='word'>an</span> <span class='word'>optimal</span> <span class='word'>number</span> <span class='word'>of</span> <span class='word'>generations</span> <span class='word'>to</span> <span class='word'>run</span> <span class='word'>the</span> <span class='word'>algorithm</span> <span class='word'>for</span> <span class='word'>such</span> <span class='word'>that</span> <span class='word'>we</span> <span class='word'>always</span> <span class='word'>find</span> <span class='word'>a</span> <span class='word'>solution</span> <span class='word'>sequence</span><span class='ignore'>?</span><br>
<span class='cmd'>\item</span> <span class='word'>If</span> <span class='word'>we</span> <span class='word'>make</span> <span class='word'>each</span> <span class='word'>sequence</span> <span class='word'>more</span> <span class='word'>likely</span> <span class='word'>to</span> <span class='word'>mutate</span> <span class='word'>generation</span> <span class='word'>to</span> <span class='word'>generation</span> <span class='word'>what</span> <span class='word'>will</span> <span class='word'>happen</span><span class='ignore'>?</span><br>
<span class='word'>What</span> <span class='word'>about</span> <span class='word'>increasing</span> <span class='word'>how</span> <span class='word'>potent</span> <span class='word'>our</span> <span class='word'>mutations</span> <span class='word'>are</span><span class='ignore'>?</span><br>
<span class='cmd'>\item</span> <span class='word'>How</span> <span class='word'>can</span> <span class='word'>we</span> <span class='word'>overcome</span> <span class='word'>the</span> <span class='word'>possibility</span> <span class='word'>of</span> <span class='word'>our</span> <span class='word'>algorithm</span> <span class='word'>finding</span> <span class='word'>a</span> <span class='word'>local</span> <span class='word'>maximum</span> <span class='word'>rather</span> <span class='word'>than</span> <span class='word'>the</span> <span class='word'>global</span> <span class='word'>maximum</span><span class='ignore'>?</span><br>
<span class='envir'>\end</span><span class='envir'>{itemize}</span><br>
<br>
<span class='cmd'>\section</span><span class='ignore'>{</span><span class='hword'>Changing</span> <span class='hword'>Initial</span> <span class='hword'>Population</span> <span class='hword'>Size</span><span class='ignore'>}</span><span class='cmd'>\label</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>ChangingInitialPopulationSize</span><span class='ignore'>}</span><br>
<span class='word'>The</span> <span class='word'>initial</span> <span class='word'>population</span> <span class='word'>size</span> <span class='word'>is</span> <span class='word'>the</span> <span class='word'>number</span> <span class='word'>of</span> <span class='word'>starting</span> <span class='word'>sequences</span> <span class='word'>we</span> <span class='word'>use</span> <span class='word'>in</span> <span class='word'>our</span> <span class='word'>algorithms</span> <span class='word'>first</span> <span class='word'>generation</span><span class='ignore'>.</span><br>
<span class='word'>Once</span> <span class='word'>this</span> <span class='word'>generation</span> <span class='word'>concludes</span> <span class='word'>the</span> <span class='word'>population</span> <span class='word'>will</span> <span class='word'>go</span> <span class='word'>through</span> <span class='word'>the</span> <span class='word'>series</span> <span class='word'>of</span> <span class='word'>phases</span> <span class='word'>outlined</span> <span class='word'>in</span> <span class='word'>Figure</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>customGAcycle</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>will</span> <span class='word'>alter</span> <span class='word'>the</span> <span class='word'>population</span> <span class='word'>to</span> <span class='word'>keep</span> <span class='word'>the</span> <span class='word'>fittest</span> <span class='word'>members</span> <span class='word'>to</span> <span class='word'>continue</span> <span class='word'>on</span> <span class='word'>to</span> <span class='word'>subsequent</span> <span class='word'>generations</span><span class='ignore'>.</span><br>
<span class='word'>We</span> <span class='word'>will</span> <span class='word'>look</span> <span class='word'>into</span> <span class='word'>population</span> <span class='word'>size</span> <span class='word'>because</span><span class='ignore'>,</span> <span class='word'>during</span> <span class='word'>any</span> <span class='word'>generation</span> <span class='word'>the</span> <span class='word'>size</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>population</span> <span class='word'>influences</span> <span class='word'>the</span> <span class='word'>range</span> <span class='word'>of</span> <span class='word'>scores</span> <span class='word'>that</span> <span class='word'>we</span> <span class='word'>can</span> <span class='word'>achieve</span> <span class='word'>against</span> <span class='word'>our</span> <span class='word'>opponent</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>more</span> <span class='word'>unique</span> <span class='word'>members</span> <span class='word'>we</span> <span class='word'>have</span> <span class='word'>the</span> <span class='word'>more</span> <span class='word'>unique</span> <span class='word'>evaluations</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>fitness</span> <span class='word'>function</span> <span class='word'>exist</span><span class='ignore'>.</span><br>
<span class='word'>Because</span> <span class='word'>of</span> <span class='word'>this</span> <span class='word'>we</span> <span class='word'>can</span> <span class='word'>reasonably</span> <span class='word'>assume</span> <span class='word'>the</span> <span class='word'>larger</span> <span class='word'>our</span> <span class='word'>population</span> <span class='word'>gets</span> <span class='word'>the</span> <span class='word'>chance</span> <span class='word'>of</span> <span class='word'>finding</span> <span class='word'>the</span> <span class='word'>solution</span> <span class='word'>sequence</span> <span class='word'>with</span> <span class='word'>the</span> <span class='word'>optimal</span> <span class='word'>score</span> <span class='word'>will</span> <span class='word'>increase</span><span class='ignore'>.</span><br>
<span class='word'>Hence</span> <span class='word'>with</span> <span class='word'>a</span> <span class='word'>larger</span> <span class='word'>population</span> <span class='word'>we</span> <span class='word'>should</span> <span class='word'>converge</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>solution</span> <span class='word'>sequence</span> <span class='word'>in</span> <span class='word'>less</span> <span class='word'>generations</span><span class='ignore'>.</span><br>
<span class='word'>For</span> <span class='word'>this</span> <span class='word'>section</span> <span class='word'>we</span> <span class='word'>analysed</span> <span class='word'>a</span> <span class='word'>range</span> <span class='word'>of</span> <span class='word'>populations</span> <span class='word'>to</span> <span class='word'>understand</span> <span class='word'>how</span> <span class='word'>solutions</span> <span class='word'>are</span> <span class='word'>affected</span> <span class='word'>when</span> <span class='word'>we</span> <span class='word'>run</span> <span class='word'>our</span> <span class='word'>algorithm</span> <span class='word'>through</span> <span class='word'>the</span> <span class='word'>set</span> <span class='word'>of</span> <span class='word'>population</span> <span class='word'>sizes</span> <span class='mathgroup'>\(</span><span class='math'>|</span><span class='math'>P</span><span class='math'>|</span> <span class='mathcmd'>\in</span> <span class='option'>[25,50,100,150,200,250,500]</span><span class='mathgroup'>\)</span><span class='ignore'>.</span><br>
<br>
<span class='word'>The</span> <span class='word'>code</span> <span class='word'>used</span> <span class='word'>for</span> <span class='word'>this</span> <span class='word'>these</span> <span class='word'>tests</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>shown</span> <span class='word'>in</span> <span class='word'>Appendix</span> <span class='word'>Snippet</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>apcode</span><span class='ignore'>:</span><span class='ignore'>populationChecker.py</span><span class='ignore'>}</span> <span class='word'>as</span> <span class='word'>an</span> <span class='word'>implementation</span> <span class='word'>how</span> <span class='word'>this</span> <span class='word'>analysis</span> <span class='word'>was</span> <span class='word'>conducted</span><span class='ignore'>.</span><br>
<span class='word'>It</span> <span class='word'>leverages</span> <span class='word'>the</span> <span class='word'>use</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>function</span> <span class='cmd'>\mintinline</span><span class='ignore'>{</span><span class='word'>python</span><span class='ignore'>}</span><span class='ignore'>{</span><span class='word'>runGeneticAlgo</span><span class='ignore'>}</span> <span class='word'>show</span> <span class='word'>in</span> <span class='word'>Appendix</span> <span class='word'>Snippet</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>apcode</span><span class='ignore'>:</span><span class='ignore'>runGeneticAlgo.py</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>code</span> <span class='word'>will</span> <span class='word'>output</span> <span class='word'>data</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>form</span> <span class='word'>of</span> <span class='word'>Table</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>table</span><span class='ignore'>:</span><span class='ignore'>popCheckerDataTable</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<br>
<span class='word'>As</span> <span class='word'>a</span> <span class='word'>note</span> <span class='word'>on</span> <span class='word'>efficiency</span><span class='ignore'>;</span> <span class='word'>increasing</span> <span class='word'>the</span> <span class='word'>size</span> <span class='word'>of</span> <span class='word'>our</span> <span class='word'>population</span> <span class='word'>will</span> <span class='word'>have</span> <span class='word'>an</span> <span class='word'>impact</span> <span class='word'>on</span> <span class='word'>computation</span> <span class='word'>time</span><span class='ignore'>.</span><br>
<span class='word'>Each</span> <span class='word'>generation</span> <span class='word'>must</span> <span class='word'>process</span> <span class='word'>the</span> <span class='word'>full</span> <span class='word'>population</span> <span class='word'>in</span> <span class='word'>a</span> <span class='word'>linear</span> <span class='word'>fashion</span> <span class='word'>causing</span> <span class='word'>a</span> <span class='word'>computation</span> <span class='word'>overhead</span> <span class='word'>of</span> <span class='mathgroup'>\(</span><span class='math'>O</span><span class='math'>(</span><span class='math'>n</span><span class='math'>)</span><span class='mathgroup'>\)</span><span class='ignore'>.</span><br>
<span class='word'>For</span> <span class='word'>an</span> <span class='word'>increase</span> <span class='word'>to</span> <span class='word'>be</span> <span class='word'>useful</span> <span class='word'>a</span> <span class='word'>in</span> <span class='word'>any</span> <span class='word'>time</span> <span class='word'>restricted</span> <span class='word'>scenario</span> <span class='word'>our</span> <span class='word'>algorithm</span> <span class='word'>would</span> <span class='word'>need</span> <span class='word'>to</span> <span class='word'>show</span> <span class='word'>a</span> <span class='word'>higher</span> <span class='word'>order</span> <span class='word'>benefit</span> <span class='word'>in</span> <span class='word'>our</span> <span class='word'>generations</span> <span class='word'>to</span> <span class='word'>convergence</span><span class='ignore'>,</span> <span class='word'>or</span> <span class='word'>in</span> <span class='word'>an</span> <span class='word'>increase</span> <span class='word'>of</span> <span class='word'>average</span> <span class='word'>score</span> <span class='word'>per</span> <span class='word'>turn</span><span class='ignore'>.</span><br>
<span class='word'>However</span> <span class='word'>We</span> <span class='word'>are</span> <span class='word'>not</span> <span class='word'>working</span> <span class='word'>in</span> <span class='word'>a</span> <span class='word'>time</span> <span class='word'>restricted</span> <span class='word'>scenario</span><span class='ignore'>,</span> <span class='word'>and</span> <span class='word'>so</span> <span class='word'>we</span> <span class='word'>should</span> <span class='word'>just</span> <span class='word'>select</span> <span class='word'>the</span> <span class='word'>best</span> <span class='word'>overall</span> <span class='word'>initial</span> <span class='word'>population</span> <span class='word'>size</span> <span class='word'>independent</span> <span class='word'>of</span> <span class='word'>computation</span> <span class='word'>overhead</span><span class='ignore'>.</span><br>
<br>
<span class='envir'>\begin</span><span class='envir'>{table*}</span><br>
<span class='exclcmd'>\centering</span><br>
<span class='exclenv'>\begin</span><span class='exclenv'>{tabular}</span><span class='ignore'>{</span><span class='ignore'>ccccccc</span><span class='ignore'>}</span><br>
<span class='exclcmd'>\toprule</span><br>
<span class='ignore'>Best</span> <span class='ignore'>Score</span> <span class='ignore'>&</span> <span class='ignore'>Gen</span> <span class='ignore'>&</span> <span class='ignore'>Mean</span> <span class='ignore'>Score</span> <span class='ignore'>&</span> <span class='ignore'>Population</span> <span class='ignore'>&</span> <span class='ignore'>Sequence</span> <span class='ignore'>&</span> <span class='ignore'>Std</span> <span class='ignore'>Dev</span> <span class='ignore'>&</span> <span class='ignore'>Time</span> <span class='ignore'>Taken</span> <span class='exclcmd'>\\</span><br>
<span class='exclcmd'>\midrule</span><br>
<span class='ignore'>2.425</span> <span class='ignore'>&</span> <span class='ignore'>1</span> <span class='ignore'>&</span> <span class='ignore'>2.264</span> <span class='ignore'>&</span> <span class='exclcmd'>\textbf</span><span class='ignore'>{</span><span class='ignore'>25.0</span><span class='ignore'>}</span> <span class='ignore'>&</span> <span class='ignore'>DD</span><span class='exclcmd'>\ldots</span> <span class='ignore'>&</span> <span class='ignore'>0.067</span> <span class='ignore'>&</span> <span class='ignore'>6.646</span><span class='exclcmd'>\\</span><br>
<span class='ignore'>2.425</span> <span class='ignore'>&</span> <span class='ignore'>2</span> <span class='ignore'>&</span> <span class='ignore'>2.343</span> <span class='ignore'>&</span> <span class='exclcmd'>\textbf</span><span class='ignore'>{</span><span class='ignore'>25.0</span><span class='ignore'>}</span> <span class='ignore'>&</span> <span class='ignore'>DD</span><span class='exclcmd'>\ldots</span> <span class='ignore'>&</span> <span class='ignore'>0.046</span> <span class='ignore'>&</span> <span class='ignore'>6.646</span><span class='exclcmd'>\\</span><br>
<span class='ignore'>2.425</span> <span class='ignore'>&</span> <span class='ignore'>3</span> <span class='ignore'>&</span> <span class='ignore'>2.393</span> <span class='ignore'>&</span> <span class='exclcmd'>\textbf</span><span class='ignore'>{</span><span class='ignore'>25.0</span><span class='ignore'>}</span> <span class='ignore'>&</span> <span class='ignore'>DD</span><span class='exclcmd'>\ldots</span> <span class='ignore'>&</span> <span class='ignore'>0.038</span> <span class='ignore'>&</span> <span class='ignore'>6.646</span><span class='exclcmd'>\\</span><br>
<span class='exclcmd'>\ldots</span> <span class='ignore'>&</span> <span class='exclcmd'>\ldots</span> <span class='ignore'>&</span> <span class='exclcmd'>\ldots</span> <span class='ignore'>&</span> <span class='exclcmd'>\ldots</span> <span class='ignore'>&</span> <span class='exclcmd'>\ldots</span> <span class='ignore'>&</span> <span class='exclcmd'>\ldots</span> <span class='ignore'>&</span> <span class='exclcmd'>\ldots</span> <span class='exclcmd'>\\</span><br>
<span class='ignore'>2.830</span> <span class='ignore'>&</span> <span class='ignore'>102</span> <span class='ignore'>&</span> <span class='ignore'>2.782</span> <span class='ignore'>&</span> <span class='exclcmd'>\textbf</span><span class='ignore'>{</span><span class='ignore'>100.0</span><span class='ignore'>}</span> <span class='ignore'>&</span> <span class='ignore'>CC</span><span class='exclcmd'>\ldots</span> <span class='ignore'>&</span> <span class='ignore'>0.112</span> <span class='ignore'>&</span> <span class='ignore'>28.425</span><span class='exclcmd'>\\</span><br>
<span class='exclcmd'>\ldots</span> <span class='ignore'>&</span> <span class='exclcmd'>\ldots</span> <span class='ignore'>&</span> <span class='exclcmd'>\ldots</span> <span class='ignore'>&</span> <span class='exclcmd'>\ldots</span> <span class='ignore'>&</span> <span class='exclcmd'>\ldots</span> <span class='ignore'>&</span> <span class='exclcmd'>\ldots</span> <span class='ignore'>&</span> <span class='exclcmd'>\ldots</span> <span class='exclcmd'>\\</span><br>
<span class='ignore'>2.980</span> <span class='ignore'>&</span> <span class='ignore'>150</span> <span class='ignore'>&</span> <span class='ignore'>2.911</span> <span class='ignore'>&</span> <span class='exclcmd'>\textbf</span><span class='ignore'>{</span><span class='ignore'>500.0</span><span class='ignore'>}</span> <span class='ignore'>&</span> <span class='ignore'>CC</span><span class='exclcmd'>\ldots</span> <span class='ignore'>&</span> <span class='ignore'>0.158</span> <span class='ignore'>&</span> <span class='ignore'>152.684</span><span class='exclcmd'>\\</span><br>
<span class='exclcmd'>\ldots</span> <span class='ignore'>&</span> <span class='exclcmd'>\ldots</span> <span class='ignore'>&</span> <span class='exclcmd'>\ldots</span> <span class='ignore'>&</span> <span class='exclcmd'>\ldots</span> <span class='ignore'>&</span> <span class='exclcmd'>\ldots</span> <span class='ignore'>&</span> <span class='exclcmd'>\ldots</span> <span class='ignore'>&</span> <span class='exclcmd'>\ldots</span> <span class='exclcmd'>\\</span><br>
<span class='exclcmd'>\bottomrule</span><br>
<span class='exclenv'>\end</span><span class='exclenv'>{tabular}</span><br>
<span class='cmd'>\caption</span><span class='ignore'>{</span><span class='oword'>Output</span> <span class='oword'>data</span> <span class='oword'>table</span><span class='ignore'>}</span><span class='exclcmd'>\label</span><span class='ignore'>{</span><span class='ignore'>table</span><span class='ignore'>:</span><span class='ignore'>popCheckerDataTable</span><span class='ignore'>}</span><br>
<span class='envir'>\end</span><span class='envir'>{table*}</span><br>
<br>
<span class='word'>After</span> <span class='word'>running</span> <span class='word'>the</span> <span class='word'>tests</span> <span class='word'>for</span> <span class='word'>the</span> <span class='word'>populations</span> <span class='word'>stated</span> <span class='word'>above</span> <span class='word'>we</span> <span class='word'>can</span> <span class='word'>group</span> <span class='word'>the</span> <span class='word'>data</span> <span class='word'>by</span> <span class='word'>population</span> <span class='word'>size</span> <span class='word'>and</span> <span class='word'>observations</span> <span class='word'>on</span> <span class='word'>how</span> <span class='word'>changing</span> <span class='word'>this</span> <span class='word'>parameter</span> <span class='word'>affects</span> <span class='word'>different</span> <span class='word'>opponents</span> <span class='word'>can</span> <span class='word'>be</span> <span class='word'>made</span><span class='ignore'>.</span><br>
<span class='word'>Figure</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>INIT-POP-bs-v-gens-all</span><span class='ignore'>}</span> <span class='word'>shows</span> <span class='word'>the</span> <span class='word'>best</span> <span class='word'>score</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>population</span> <span class='word'>for</span> <span class='word'>each</span> <span class='word'>opponent</span> <span class='word'>as</span> <span class='word'>the</span> <span class='word'>generations</span> <span class='word'>increase</span> <span class='word'>for</span> <span class='word'>every</span> <span class='word'>population</span> <span class='word'>size</span><span class='ignore'>.</span><br>
<span class='word'>We</span> <span class='word'>see</span> <span class='word'>that</span> <span class='word'>the</span> <span class='word'>initial</span> <span class='word'>population</span> <span class='word'>size</span> <span class='word'>has</span> <span class='word'>a</span> <span class='word'>significant</span> <span class='word'>effect</span> <span class='word'>on</span> <span class='word'>finding</span> <span class='word'>better</span> <span class='word'>sequences</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>figure</span> <span class='word'>shows</span> <span class='word'>if</span> <span class='word'>there</span> <span class='word'>is</span> <span class='word'>a</span> <span class='word'>larger</span> <span class='word'>initial</span> <span class='word'>population</span> <span class='word'>there</span> <span class='word'>is</span> <span class='word'>typically</span> <span class='word'>a</span> <span class='word'>higher</span> <span class='word'>best</span> <span class='word'>score</span> <span class='word'>shown</span> <span class='word'>once</span> <span class='word'>concluding</span> <span class='word'>all</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>generations</span><span class='ignore'>.</span><br>
<span class='word'>It</span> <span class='word'>doesn't</span><span class='ignore'>,</span> <span class='word'>however</span><span class='ignore'>,</span> <span class='word'>ensure</span> <span class='word'>that</span> <span class='word'>we</span> <span class='word'>find</span> <span class='word'>the</span> <span class='word'>solution</span> <span class='word'>sequence</span><span class='ignore'>,</span> <span class='word'>as</span> <span class='word'>is</span> <span class='word'>shown</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>lack</span> <span class='word'>of</span> <span class='word'>long</span> <span class='word'>plateaus</span> <span class='word'>of</span> <span class='word'>best</span> <span class='word'>score</span> <span class='word'>across</span> <span class='word'>opponents</span><span class='ignore'>.</span><br>
<br>
<span class='envir'>\begin</span><span class='envir'>{figure}</span><span class='option'>[h]</span><br>
<span class='exclcmd'>\includegraphics</span><span class='option'>[width=0.8\textwidth, keepaspectratio, center]</span><span class='ignore'>{</span><span class='ignore'>.</span><span class='ignore'>/</span><span class='ignore'>img</span><span class='ignore'>/</span><span class='ignore'>plots</span><span class='ignore'>/</span><span class='ignore'>INIT</span><span class='ignore'>_</span><span class='ignore'>POP</span><span class='ignore'>_</span><span class='ignore'>bs</span><span class='ignore'>_</span><span class='ignore'>v</span><span class='ignore'>_</span><span class='ignore'>gens</span><span class='ignore'>_</span><span class='ignore'>all.pdf</span><span class='ignore'>}</span><br>
<span class='cmd'>\caption</span><span class='ignore'>{</span><span class='cmd'>\textbf</span><span class='ignore'>{</span><span class='oword'>Initial</span> <span class='oword'>Population</span> <span class='oword'>Size</span> <span class='oword'>Analysis</span><span class='ignore'>:</span><span class='ignore'>}</span> <span class='oword'>Best</span> <span class='oword'>score</span> <span class='oword'>per</span> <span class='oword'>turn</span> <span class='oword'>vs</span> <span class='oword'>generation</span> <span class='oword'>for</span> <span class='oword'>different</span> <span class='oword'>initial</span> <span class='oword'>population</span> <span class='oword'>sizes</span><span class='ignore'>}</span><span class='exclcmd'>\label</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>INIT-POP-bs-v-gens-all</span><span class='ignore'>}</span><br>
<span class='envir'>\end</span><span class='envir'>{figure}</span><br>
<br>
<span class='word'>The</span> <span class='word'>improvements</span> <span class='word'>from</span> <span class='word'>the</span> <span class='word'>effect</span> <span class='word'>of</span> <span class='word'>population</span> <span class='word'>increase</span> <span class='word'>are</span> <span class='word'>non-linear</span> <span class='word'>from</span> <span class='word'>observation</span> <span class='word'>of</span> <span class='word'>Figure</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>INIT-POP-mean-bs-v-init-pop-all</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>figure</span> <span class='word'>shows</span> <span class='word'>the</span> <span class='word'>change</span> <span class='word'>in</span> <span class='word'>mean</span> <span class='word'>best</span> <span class='word'>score</span> <span class='word'>across</span> <span class='word'>population</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>final</span> <span class='word'>generation</span> <span class='word'>against</span> <span class='word'>the</span> <span class='word'>population</span> <span class='word'>size</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>change</span> <span class='word'>in</span> <span class='word'>final</span> <span class='word'>best</span> <span class='word'>score</span> <span class='word'>for</span> <span class='word'>a</span> <span class='word'>population</span> <span class='word'>of</span> <span class='word'>50</span> <span class='word'>compared</span> <span class='word'>with</span> <span class='word'>a</span> <span class='word'>population</span> <span class='word'>of</span> <span class='word'>250</span> <span class='word'>is</span> <span class='word'>large</span> <span class='word'>in</span> <span class='word'>comparison</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>same</span> <span class='word'>relative</span> <span class='word'>increase</span> <span class='word'>from</span> <span class='word'>250</span> <span class='word'>to</span> <span class='word'>500</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>may</span> <span class='word'>suggest</span> <span class='word'>there</span> <span class='word'>are</span> <span class='word'>more</span> <span class='word'>efficient</span> <span class='word'>approaches</span> <span class='word'>to</span> <span class='word'>improving</span> <span class='word'>our</span> <span class='word'>final</span> <span class='word'>best</span> <span class='word'>score</span> <span class='word'>after</span> <span class='word'>a</span> <span class='word'>certain</span> <span class='word'>initial</span> <span class='word'>population</span> <span class='word'>is</span> <span class='word'>reached</span><span class='ignore'>.</span><br>
<br>
<span class='envir'>\begin</span><span class='envir'>{figure}</span><span class='option'>[h]</span><br>
<span class='exclcmd'>\includegraphics</span><span class='option'>[width=0.8\textwidth, keepaspectratio, center]</span><span class='ignore'>{</span><span class='ignore'>.</span><span class='ignore'>/</span><span class='ignore'>img</span><span class='ignore'>/</span><span class='ignore'>plots</span><span class='ignore'>/</span><span class='ignore'>INIT</span><span class='ignore'>_</span><span class='ignore'>POP</span><span class='ignore'>_</span><span class='ignore'>mean</span><span class='ignore'>_</span><span class='ignore'>bs</span><span class='ignore'>_</span><span class='ignore'>diff</span><span class='ignore'>_</span><span class='ignore'>v</span><span class='ignore'>_</span><span class='ignore'>init</span><span class='ignore'>_</span><span class='ignore'>pop</span><span class='ignore'>_</span><span class='ignore'>all.pdf</span><span class='ignore'>}</span><br>
<span class='cmd'>\caption</span><span class='ignore'>{</span><span class='cmd'>\textbf</span><span class='ignore'>{</span><span class='oword'>Initial</span> <span class='oword'>Population</span> <span class='oword'>Size</span> <span class='oword'>Analysis</span><span class='ignore'>:</span><span class='ignore'>}</span> <span class='oword'>Scatter</span> <span class='oword'>of</span> <span class='oword'>mean</span> <span class='oword'>best</span> <span class='oword'>score</span> <span class='oword'>vs</span> <span class='oword'>different</span> <span class='oword'>initial</span> <span class='oword'>populations</span><span class='ignore'>}</span><span class='exclcmd'>\label</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>INIT-POP-mean-bs-v-init-pop-all</span><span class='ignore'>}</span><br>
<span class='envir'>\end</span><span class='envir'>{figure}</span><br>
<br>
<span class='word'>None</span> <span class='word'>of</span> <span class='word'>these</span> <span class='word'>results</span> <span class='word'>have</span> <span class='word'>found</span> <span class='word'>a</span> <span class='word'>solution</span> <span class='word'>sequence</span> <span class='ignore'>(</span><span class='word'>or</span> <span class='word'>at</span> <span class='word'>least</span> <span class='word'>we</span> <span class='word'>cant</span> <span class='word'>tell</span> <span class='word'>from</span> <span class='word'>the</span> <span class='word'>graph</span><span class='ignore'>)</span><span class='ignore'>.</span><br>
<span class='word'>It</span> <span class='word'>is</span> <span class='word'>clear</span> <span class='word'>that</span> <span class='word'>larger</span> <span class='word'>initial</span> <span class='word'>populations</span> <span class='word'>do</span><span class='ignore'>,</span> <span class='word'>on</span> <span class='word'>a</span> <span class='word'>relative</span> <span class='word'>scale</span><span class='ignore'>,</span> <span class='word'>much</span> <span class='word'>better</span> <span class='word'>than</span> <span class='word'>small</span> <span class='word'>ones</span><span class='ignore'>.</span><br>
<span class='word'>There</span> <span class='word'>are</span> <span class='word'>no</span> <span class='word'>large</span> <span class='word'>plateaus</span> <span class='word'>for</span> <span class='word'>the</span> <span class='word'>graph</span><span class='ignore'>,</span> <span class='word'>so</span> <span class='word'>as</span> <span class='word'>we</span> <span class='word'>continue</span> <span class='word'>our</span> <span class='word'>research</span> <span class='word'>the</span> <span class='word'>initial</span> <span class='word'>population</span> <span class='word'>size</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>increased</span> <span class='word'>to</span> <span class='word'>150</span> <span class='word'>to</span> <span class='word'>keep</span> <span class='word'>test</span> <span class='word'>computation</span> <span class='word'>times</span> <span class='word'>manageable</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>actual</span> <span class='word'>parameter</span> <span class='word'>we</span> <span class='word'>will</span> <span class='word'>use</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>full</span> <span class='word'>analysis</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>given</span> <span class='word'>consideration</span> <span class='word'>in</span> <span class='word'>Section</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>conclusionOfApproach</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\section</span><span class='ignore'>{</span><span class='hword'>Generation</span> <span class='hword'>Length</span> <span class='hword'>Analysis</span><span class='ignore'>}</span><span class='cmd'>\label</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>generationlengthanalysis</span><span class='ignore'>}</span><br>
<span class='word'>Another</span> <span class='word'>major</span> <span class='word'>component</span> <span class='word'>parameter</span> <span class='word'>of</span> <span class='word'>a</span> <span class='word'>genetic</span> <span class='word'>algorithm</span> <span class='word'>is</span> <span class='word'>the</span> <span class='word'>number</span> <span class='word'>of</span> <span class='word'>generations</span> <span class='word'>it</span> <span class='word'>will</span> <span class='word'>run</span> <span class='word'>for</span> <span class='word'>before</span> <span class='word'>outputting</span> <span class='word'>a</span> <span class='word'>final</span> <span class='word'>set</span> <span class='word'>of</span> <span class='word'>members</span> <span class='word'>with</span> <span class='word'>solution</span> <span class='word'>sequences</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>number</span> <span class='word'>of</span> <span class='word'>generations</span> <span class='word'>has</span> <span class='word'>an</span> <span class='word'>influence</span> <span class='word'>on</span> <span class='word'>a</span> <span class='word'>number</span> <span class='word'>of</span> <span class='word'>different</span> <span class='word'>things</span> <span class='word'>within</span> <span class='word'>the</span> <span class='word'>algorithm</span><span class='ignore'>:</span><br>
<span class='envir'>\begin</span><span class='envir'>{itemize}</span><br>
<span class='cmd'>\item</span> <span class='ignore'>{</span><span class='word'>The</span> <span class='word'>total</span> <span class='word'>combinations</span> <span class='word'>of</span> <span class='word'>features</span> <span class='word'>that</span> <span class='word'>the</span> <span class='word'>algorithm</span> <span class='word'>can</span> <span class='word'>evaluate</span><span class='ignore'>.</span> <br>
<span class='word'>The</span> <span class='word'>more</span> <span class='word'>generations</span> <span class='word'>it</span> <span class='word'>runs</span> <span class='word'>for</span> <span class='word'>the</span> <span class='word'>more</span> <span class='word'>combinations</span> <span class='word'>of</span> <span class='word'>members</span> <span class='word'>we</span> <span class='word'>can</span> <span class='word'>evaluate</span> <span class='word'>a</span> <span class='word'>fitness</span> <span class='word'>for</span><span class='ignore'>.</span><span class='ignore'>}</span><br>
<span class='cmd'>\item</span> <span class='ignore'>{</span><span class='word'>The</span> <span class='word'>total</span> <span class='word'>number</span> <span class='word'>of</span> <span class='word'>low</span> <span class='word'>performers</span> <span class='word'>it</span> <span class='word'>removes</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>population</span> <span class='word'>over</span> <span class='word'>the</span> <span class='word'>course</span> <span class='word'>of</span> <span class='word'>its</span> <span class='word'>run</span><span class='ignore'>.</span> <br>
<span class='word'>Each</span> <span class='word'>generation</span> <span class='word'>removes</span> <span class='word'>a</span> <span class='word'>proportion</span> <span class='word'>of</span> <span class='word'>its</span> <span class='word'>lowest</span> <span class='word'>fitness</span> <span class='word'>members</span><span class='ignore'>.</span><br>
<span class='word'>By</span> <span class='word'>definition</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>algorithm</span> <span class='word'>we</span> <span class='word'>have</span> <span class='word'>a</span> <span class='word'>non-decreasing</span> <span class='word'>sequence</span> <span class='word'>of</span> <span class='word'>fitness</span> <span class='word'>scores</span><span class='ignore'>;</span> <span class='word'>removing</span> <span class='word'>more</span> <span class='word'>lower</span> <span class='word'>fitness</span> <span class='word'>members</span> <span class='word'>will</span> <span class='word'>lead</span> <span class='word'>to</span> <span class='word'>a</span> <span class='word'>population</span> <span class='word'>of</span> <span class='word'>better</span> <span class='word'>or</span> <span class='word'>equal</span> <span class='word'>scoring</span> <span class='word'>members</span><span class='ignore'>.</span> <span class='ignore'>}</span><br>
<span class='envir'>\end</span><span class='envir'>{itemize}</span><br>
<span class='word'>Generation</span> <span class='word'>size</span> <span class='word'>differs</span> <span class='word'>from</span> <span class='word'>other</span> <span class='word'>parameters</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>fact</span> <span class='word'>this</span> <span class='word'>is</span> <span class='word'>purely</span> <span class='word'>performance</span> <span class='word'>based</span><span class='ignore'>,</span> <span class='word'>there</span> <span class='word'>is</span> <span class='word'>no</span> <span class='word'>changes</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>inner</span> <span class='word'>mechanics</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>algorithm</span> <span class='word'>only</span> <span class='word'>the</span> <span class='word'>amount</span> <span class='word'>of</span> <span class='word'>time</span> <span class='word'>it</span> <span class='word'>will</span> <span class='word'>run</span> <span class='word'>for</span><span class='ignore'>.</span><br>
<span class='word'>A</span> <span class='word'>genetic</span> <span class='word'>algorithm</span> <span class='word'>with</span> <span class='word'>1</span> <span class='word'>generation</span> <span class='word'>is</span> <span class='word'>just</span> <span class='word'>a</span> <span class='word'>series</span> <span class='word'>of</span> <span class='word'>tests</span> <span class='word'>split</span> <span class='word'>into</span> <span class='word'>2</span> <span class='word'>results</span> <span class='word'>sets</span> <span class='ignore'>-</span><span class='ignore'>-</span><span class='ignore'>-</span> <span class='word'>good</span> <span class='word'>performers</span> <span class='word'>and</span> <span class='word'>bad</span> <span class='word'>performers</span><span class='ignore'>.</span><br>
<span class='word'>As</span> <span class='word'>we</span> <span class='word'>extend</span> <span class='word'>the</span> <span class='word'>number</span> <span class='word'>of</span> <span class='word'>generations</span> <span class='word'>we</span> <span class='word'>want</span> <span class='word'>be</span> <span class='word'>more</span> <span class='word'>focused</span> <span class='word'>on</span> <span class='word'>what</span> <span class='word'>happens</span> <span class='word'>to</span> <span class='word'>measured</span> <span class='word'>quantities</span> <span class='word'>normalised</span> <span class='word'>by</span> <span class='word'>generations</span><span class='ignore'>,</span> <span class='word'>rather</span> <span class='word'>than</span> <span class='word'>any</span> <span class='word'>absolute</span> <span class='word'>improvement</span><span class='ignore'>.</span><br>
<br>
<span class='word'>As</span> <span class='word'>a</span> <span class='word'>note</span> <span class='word'>on</span> <span class='word'>efficiency</span><span class='ignore'>,</span> <span class='word'>the</span> <span class='word'>goal</span> <span class='word'>of</span> <span class='word'>finding</span> <span class='word'>the</span> <span class='word'>optimal</span> <span class='word'>solution</span> <span class='word'>sequence</span> <span class='word'>for</span> <span class='word'>each</span> <span class='word'>opponent</span> <span class='word'>would</span> <span class='word'>be</span> <span class='word'>solved</span> <span class='word'>by</span> <span class='word'>extending</span> <span class='word'>the</span> <span class='word'>generations</span> <span class='word'>to</span> <span class='word'>infinity</span><span class='ignore'>,</span> <span class='word'>i.e</span> <span class='word'>exhaustively</span> <span class='word'>search</span> <span class='word'>the</span> <span class='word'>whole</span> <span class='word'>solution</span> <span class='word'>space</span><span class='ignore'>.</span><br>
<span class='word'>This</span><span class='ignore'>,</span> <span class='word'>however</span><span class='ignore'>,</span> <span class='word'>is</span> <span class='word'>not</span> <span class='word'>a</span> <span class='word'>feasible</span> <span class='word'>solution</span><span class='ignore'>,</span> <span class='word'>and</span> <span class='word'>so</span> <span class='word'>this</span> <span class='word'>section</span> <span class='word'>looks</span> <span class='word'>in</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>effect</span> <span class='word'>of</span> <span class='word'>increasing</span> <span class='word'>generations</span> <span class='word'>has</span> <span class='word'>on</span> <span class='word'>improving</span> <span class='word'>a</span> <span class='word'>solution</span> <span class='word'>sequence</span> <span class='word'>with</span> <span class='word'>respect</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>generations</span> <span class='word'>its</span> <span class='word'>run</span><span class='ignore'>.</span><br>
<span class='word'>Here</span> <span class='word'>we</span> <span class='word'>will</span> <span class='word'>take</span> <span class='word'>generation</span> <span class='word'>lengths</span> <span class='mathgroup'>\(</span><span class='math'>G</span> <span class='mathcmd'>\in</span> <span class='option'>[50,150,250,350,450,500]</span><span class='mathgroup'>\)</span> <span class='word'>during</span> <span class='word'>these</span> <span class='word'>tests</span> <span class='word'>and</span> <span class='word'>a</span> <span class='word'>population</span> <span class='word'>size</span> <span class='word'>of</span> <span class='word'>150</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>code</span> <span class='word'>in</span> <span class='word'>Appendix</span> <span class='word'>Snippet</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>apcode</span><span class='ignore'>:</span><span class='ignore'>generationChecker.py</span><span class='ignore'>}</span> <span class='word'>shows</span> <span class='word'>how</span> <span class='word'>the</span> <span class='word'>approach</span> <span class='word'>the</span> <span class='word'>generation</span> <span class='word'>analysis</span> <span class='word'>was</span> <span class='word'>undertaken</span><span class='ignore'>.</span><br>
<br>
<span class='word'>Figure</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>GENS-mean-bs-diff-v-gens-all</span><span class='ignore'>}</span> <span class='word'>shows</span> <span class='word'>how</span> <span class='word'>the</span> <span class='word'>mean</span> <span class='word'>best</span> <span class='word'>score</span> <span class='word'>across</span> <span class='word'>the</span> <span class='word'>population</span> <span class='word'>changes</span> <span class='word'>generation</span> <span class='word'>to</span> <span class='word'>generation</span> <span class='word'>normalized</span> <span class='word'>by</span> <span class='word'>generation</span><span class='ignore'>.</span><br>
<span class='word'>It</span> <span class='word'>shows</span> <span class='word'>that</span> <span class='word'>over</span> <span class='word'>150</span> <span class='word'>generations</span> <span class='word'>the</span> <span class='word'>best</span> <span class='word'>scores</span> <span class='word'>against</span> <span class='word'>Alternator</span> <span class='word'>across</span> <span class='word'>its</span> <span class='word'>population</span> <span class='word'>increased</span><span class='ignore'>,</span> <span class='word'>on</span> <span class='word'>average</span><span class='ignore'>,</span> <span class='word'>by</span> <span class='word'>0.0035</span> <span class='word'>per</span> <span class='word'>generation</span><span class='ignore'>.</span><br>
<span class='word'>From</span> <span class='word'>this</span> <span class='word'>we</span> <span class='word'>can</span> <span class='word'>observe</span> <span class='word'>how</span> <span class='word'>the</span> <span class='word'>number</span> <span class='word'>of</span> <span class='word'>generations</span> <span class='word'>has</span> <span class='word'>a</span> <span class='word'>declining</span> <span class='word'>effect</span> <span class='word'>the</span> <span class='word'>overall</span> <span class='word'>change</span> <span class='word'>in</span> <span class='word'>our</span> <span class='word'>mean</span> <span class='word'>best</span> <span class='word'>score</span> <span class='word'>per</span> <span class='word'>generation</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>trend</span> <span class='word'>is</span> <span class='word'>to</span> <span class='word'>be</span> <span class='word'>expected</span><span class='ignore'>,</span> <span class='word'>when</span> <span class='word'>we</span> <span class='word'>are</span> <span class='word'>close</span> <span class='word'>to</span> <span class='word'>a</span> <span class='word'>maximum</span> <span class='word'>it</span> <span class='word'>is</span> <span class='word'>more</span> <span class='word'>difficult</span> <span class='word'>to</span> <span class='word'>randomly</span> <span class='word'>select</span> <span class='word'>which</span> <span class='word'>element</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>sequence</span> <span class='word'>needs</span> <span class='word'>changing</span> <span class='word'>to</span> <span class='word'>improve</span> <span class='word'>a</span> <span class='word'>members</span> <span class='word'>score</span><span class='ignore'>.</span><br>
<span class='word'>On</span> <span class='word'>this</span> <span class='word'>result</span> <span class='word'>we</span> <span class='word'>can</span> <span class='word'>conclude</span> <span class='word'>as</span> <span class='word'>we</span> <span class='word'>increase</span> <span class='word'>generations</span> <span class='word'>there</span> <span class='word'>is</span> <span class='word'>less</span> <span class='word'>and</span> <span class='word'>less</span> <span class='word'>benefit</span> <span class='word'>per</span> <span class='word'>generation</span><span class='ignore'>.</span><br>
<br>
<span class='envir'>\begin</span><span class='envir'>{figure}</span><span class='option'>[h]</span><br>
<span class='exclcmd'>\includegraphics</span><span class='option'>[width=0.8\textwidth, keepaspectratio, center]</span><span class='ignore'>{</span><span class='ignore'>.</span><span class='ignore'>/</span><span class='ignore'>img</span><span class='ignore'>/</span><span class='ignore'>plots</span><span class='ignore'>/</span><span class='ignore'>GENS</span><span class='ignore'>_</span><span class='ignore'>mean</span><span class='ignore'>_</span><span class='ignore'>bs</span><span class='ignore'>_</span><span class='ignore'>diff</span><span class='ignore'>_</span><span class='ignore'>v</span><span class='ignore'>_</span><span class='ignore'>gens</span><span class='ignore'>_</span><span class='ignore'>all.pdf</span><span class='ignore'>}</span><br>
<span class='cmd'>\caption</span><span class='ignore'>{</span><span class='cmd'>\textbf</span><span class='ignore'>{</span><span class='oword'>Generation</span> <span class='oword'>Analysis</span><span class='ignore'>:</span><span class='ignore'>}</span> <span class='oword'>Mean</span> <span class='oword'>Best</span> <span class='oword'>Score</span> <span class='oword'>diff</span> <span class='oword'>vs</span> <span class='oword'>total</span> <span class='oword'>generation</span> <span class='oword'>lengths</span><span class='ignore'>}</span><span class='exclcmd'>\label</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>GENS-mean-bs-diff-v-gens-all</span><span class='ignore'>}</span><br>
<span class='envir'>\end</span><span class='envir'>{figure}</span><br>
<br>
<span class='word'>Figure</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>GENS-max-bs-v-gens-all</span><span class='ignore'>}</span> <span class='word'>shows</span> <span class='word'>the</span> <span class='word'>maximum</span> <span class='word'>best</span> <span class='word'>score</span> <span class='word'>in</span> <span class='word'>a</span> <span class='word'>population</span> <span class='word'>once</span> <span class='word'>the</span> <span class='word'>analysis</span> <span class='word'>has</span> <span class='word'>concluded</span><span class='ignore'>.</span><br>
<span class='word'>For</span> <span class='word'>most</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>opponents</span> <span class='word'>250</span> <span class='word'>generations</span> <span class='word'>seems</span> <span class='word'>reasonable</span> <span class='word'>to</span> <span class='word'>reach</span> <span class='word'>a</span> <span class='word'>solution</span> <span class='word'>sequence</span> <span class='word'>as</span> <span class='word'>shown</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>Alternator</span> <span class='word'>and</span> <span class='word'>Tit</span> <span class='word'>For</span> <span class='word'>Tat</span><span class='ignore'>.</span><br>
<span class='word'>However</span><span class='ignore'>,</span> <span class='word'>there</span> <span class='word'>are</span> <span class='word'>clear</span> <span class='word'>signs</span> <span class='word'>of</span> <span class='word'>local</span> <span class='word'>maximums</span> <span class='word'>occurring</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>EvolvedFSM16</span> <span class='word'>example</span><span class='ignore'>.</span><br>
<span class='word'>In</span> <span class='word'>Figure</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>GENS-max-bs-v-gens-all</span><span class='ignore'>}</span> <span class='word'>EvolvedFSM16</span> <span class='word'>has</span> <span class='word'>reached</span> <span class='word'>a</span> <span class='word'>better</span> <span class='word'>sequence</span> <span class='word'>in</span> <span class='word'>150</span> <span class='word'>generations</span> <span class='word'>than</span> <span class='word'>500</span><span class='cmd'>\footnote</span><span class='ignore'>{</span><span class='oword'>These</span> <span class='oword'>are</span> <span class='oword'>independent</span> <span class='oword'>trials</span> <span class='oword'>and</span> <span class='oword'>have</span> <span class='oword'>different</span> <span class='oword'>sequences</span><span class='ignore'>.</span><span class='ignore'>}</span><span class='ignore'>,</span> <span class='word'>meaning</span> <span class='word'>that</span> <span class='word'>increasing</span> <span class='word'>the</span> <span class='word'>generation</span> <span class='word'>length</span> <span class='word'>doesn't</span> <span class='word'>necessarily</span> <span class='word'>mean</span> <span class='word'>finding</span> <span class='word'>a</span> <span class='word'>global</span> <span class='word'>maximum</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>complexities</span> <span class='word'>with</span> <span class='word'>local</span> <span class='word'>maximums</span> <span class='word'>during</span> <span class='word'>the</span> <span class='word'>generations</span> <span class='word'>lie</span> <span class='word'>with</span> <span class='word'>mutation</span> <span class='word'>rates</span> <span class='word'>and</span> <span class='word'>crossovers</span><span class='ignore'>.</span><br>
<span class='word'>We</span> <span class='word'>will</span> <span class='word'>cover</span> <span class='word'>this</span> <span class='word'>in</span> <span class='word'>Section</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>mitigatingLocalMaximums</span><span class='ignore'>}</span><br>
<br>
<span class='envir'>\begin</span><span class='envir'>{figure}</span><span class='option'>[h]</span> <br>
<span class='exclcmd'>\includegraphics</span><span class='option'>[width=0.8\textwidth, keepaspectratio, center]</span><span class='ignore'>{</span><span class='ignore'>.</span><span class='ignore'>/</span><span class='ignore'>img</span><span class='ignore'>/</span><span class='ignore'>plots</span><span class='ignore'>/</span><span class='ignore'>GENS</span><span class='ignore'>_</span><span class='ignore'>max</span><span class='ignore'>_</span><span class='ignore'>bs</span><span class='ignore'>_</span><span class='ignore'>v</span><span class='ignore'>_</span><span class='ignore'>gens</span><span class='ignore'>_</span><span class='ignore'>all.pdf</span><span class='ignore'>}</span><br>
<span class='cmd'>\caption</span><span class='ignore'>{</span><span class='cmd'>\textbf</span><span class='ignore'>{</span><span class='oword'>Generation</span> <span class='oword'>Analysis</span><span class='ignore'>:</span><span class='ignore'>}</span> <span class='oword'>Max</span> <span class='oword'>best</span> <span class='oword'>score</span> <span class='oword'>vs</span> <span class='oword'>total</span> <span class='oword'>number</span> <span class='oword'>of</span> <span class='oword'>generations</span><span class='ignore'>}</span><span class='exclcmd'>\label</span><span class='ignore'>{</span><span class='ignore'>fig</span><span class='ignore'>:</span><span class='ignore'>GENS-max-bs-v-gens-all</span><span class='ignore'>}</span><br>
<span class='envir'>\end</span><span class='envir'>{figure}</span><br>
<br>
<span class='word'>It</span> <span class='word'>is</span> <span class='word'>clear</span> <span class='word'>that</span> <span class='word'>a</span> <span class='word'>higher</span> <span class='word'>number</span> <span class='word'>of</span> <span class='word'>generations</span> <span class='word'>is</span> <span class='word'>preferred</span> <span class='word'>to</span> <span class='word'>find</span> <span class='word'>a</span> <span class='word'>better</span> <span class='word'>solution</span> <span class='word'>sequence</span><span class='ignore'>.</span><br>
<span class='word'>As</span> <span class='word'>the</span> <span class='word'>number</span> <span class='word'>of</span> <span class='word'>generations</span> <span class='word'>increase</span> <span class='word'>each</span> <span class='word'>generation</span> <span class='word'>provides</span> <span class='word'>less</span> <span class='word'>of</span> <span class='word'>an</span> <span class='word'>improvement</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>best</span> <span class='word'>scores</span><span class='ignore'>.</span><br>
<span class='word'>This</span> <span class='word'>is</span> <span class='word'>due</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>probability</span> <span class='word'>of</span> <span class='word'>finding</span> <span class='word'>a</span> <span class='word'>better</span> <span class='word'>solution</span> <span class='word'>sequence</span> <span class='word'>decreasing</span> <span class='word'>as</span> <span class='word'>we</span> <span class='word'>continue</span> <span class='word'>to</span> <span class='word'>improve</span> <span class='word'>a</span> <span class='word'>population</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>benefits</span> <span class='word'>of</span> <span class='word'>extending</span> <span class='word'>the</span> <span class='word'>generations</span> <span class='word'>are</span> <span class='word'>incredibly</span> <span class='word'>useful</span> <span class='word'>and</span> <span class='word'>due</span> <span class='word'>to</span> <span class='word'>the</span> <span class='word'>amount</span> <span class='word'>of</span> <span class='word'>computation</span> <span class='word'>time</span> <span class='word'>we</span> <span class='word'>can</span> <span class='word'>probably</span> <span class='word'>incorporate</span> <span class='word'>a</span> <span class='word'>high</span> <span class='word'>number</span> <span class='word'>of</span> <span class='word'>generations</span><span class='ignore'>.</span> <br>
<span class='word'>However</span> <span class='word'>we</span> <span class='word'>may</span> <span class='word'>have</span> <span class='word'>better</span> <span class='word'>performance</span> <span class='word'>by</span> <span class='word'>altering</span> <span class='word'>another</span> <span class='word'>parameter</span> <span class='word'>of</span> <span class='word'>the</span> <span class='word'>algorithm</span><span class='ignore'>.</span><br>
<span class='word'>The</span> <span class='word'>actual</span> <span class='word'>parameter</span> <span class='word'>we</span> <span class='word'>will</span> <span class='word'>use</span> <span class='word'>in</span> <span class='word'>the</span> <span class='word'>full</span> <span class='word'>analysis</span> <span class='word'>will</span> <span class='word'>be</span> <span class='word'>given</span> <span class='word'>consideration</span> <span class='word'>in</span> <span class='word'>Section</span><span class='ignore'>~</span><span class='cmd'>\ref</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>conclusionOfApproach</span><span class='ignore'>}</span><span class='ignore'>.</span><br>
<br>
<span class='cmd'>\section</span><span class='ignore'>{</span><span class='hword'>Changing</span> <span class='hword'>Mutation</span> <span class='hword'>Rate</span><span class='ignore'>}</span><span class='cmd'>\label</span><span class='ignore'>{</span><span class='ignore'>sec</span><span class='ignore'>:</span><span class='ignore'>changeingmutationrate</span><span class='ignore'>}</span><br>
<span class='word'>This</span> <span class='word'>section</span> <span class='word'>looks</span> <span class='word'>at</span> <span class='word'>changing</span> <span class='word'>the</span> <span class='word'>amount</span> <span class='word'>of</span> <span class='word'>mutations</span> <span class='word'>that</span> <span class='word'>occur</span> <span class='word'>in</span> <span class='word'>our</span> <span class='word'>population</span> <span class='word'>and</span> <span class='word'>the</span> <span class='word'>number</span> <span class='word'>elements</span> <span class='word'>within</span> <span class='word'>a</span> <span class='word'>sequence</span> <span class='word'>each</span> <span class='word'>mutation</span> <span class='word'>effects</span><span class='ignore'>.</span> <br>
<span class='word'>The</span> <span class='word'>default</span> <span class='word'>settings</span> <span class='word'>are</span> <span class='word'>a</span> <span class='word'>mutation</span> <span class='word'>frequency</span><span class='ignore'>,</span> <span class='mathgroup'>$</span><span class='math'>M</span><span class='math'>_</span><span class='math'>f</span><span class='mathgroup'>$</span><span class='ignore'>,</span> <span class='word'>of</span> <span class='word'>0.1</span><span class='ignore'>,</span> <span class='word'>meaning</span> <span class='word'>for</span> <span class='word'>every</span> <span class='word'>10</span> <span class='word'>members</span> <span class='word'>of</span> <span class='word'>our</span> <span class='word'>population</span> <span class='word'>that</span> <span class='word'>continue</span> <span class='word'>into</span> <span class='word'>the</span> <span class='word'>next</span> <span class='word'>generation</span> <span class='word'>one</span> <span class='word'>of</span> <span class='word'>these</span> <span class='word'>has</span> <span class='word'>some</span> <span class='word'>elements</span> <span class='word'>in</span> <span class='word'>its</span> <span class='word'>sequence</span> <span class='word'>changed</span><span class='ignore'>.</span><br>
<span class='word'>And</span> <span class='word'>a</span> <span class='word'>mutation</span> <span class='word'>potency</span><span class='ignore'>,</span> <span class='mathgroup'>$</span><span class='math'>M</span><span class='math'>_</span><span class='math'>p</span><span class='mathgroup'>$</span><span class='ignore'>,</span> <span class='word'>of</span> <span class='word'>1</span><span class='ignore'>,</span> <span class='word'>meaning</span> <span class='word'>that</span> <span class='word'>every</span> <span class='word'>sequence</span> <span class='word'>that</span> <span class='word'>is</span> <span class='word'>mutated</span> <span class='word'>only</span> <span class='word'>has</span> <span class='word'>1</span> <span class='word'>element</span> <span class='word'>altered</span><span class='ignore'>.</span><br>
<span class='envir'>\begin</span><span class='envir'>{itemize}</span><br>
<span class='cmd'>\item</span> <span class='word'>Is</span> <span class='word'>it</span> <span class='word'>beneficial</span> <span class='word'>for</span> <span class='word'>more</span><span class='ignore'>/</span><span class='word'>less</span> <span class='word'>than</span> <span class='word'>1</span> <span class='word'>in</span> <span class='word'>10</span> <span class='word'>members</span> <span class='word'>to</span> <span class='word'>be</span> <span class='word'>mutated</span> <span class='word'>generation</span> <span class='word'>to</span> <span class='word'>generation</span><span class='ignore'>?</span> <span class='ignore'>(</span><span class='word'>higher</span> <span class='word'>frequent</span> <span class='word'>mutation</span><span class='ignore'>)</span><br>
<span class='cmd'>\item</span> <span class='word'>Is</span> <span class='word'>changing</span> <span class='word'>one</span> <span class='word'>or</span> <span class='word'>more</span> <span class='word'>actions</span> <span class='word'>of</span> <span class='word'>a</span> <span class='word'>members</span><span class='ignore'>'</span> <span class='word'>sequence</span> <span class='word'>the</span> <span class='word'>best</span> <span class='word'>way</span> <span class='word'>of</span> <span class='word'>mutating</span> <span class='word'>a</span> <span class='word'>member</span><span class='ignore'>?</span> <span class='ignore'>(</span><span class='word'>higher</span> <span class='word'>potent</span> <span class='word'>mutation</span><span class='ignore'>)</span><br>
<span class='envir'>\end</span><span class='envir'>{itemize}</span><br>
<br>
<span class='word'>These</span> <span class='word'>are</span> <span class='word'>two</span> <span class='word'>separate</span> <span class='word'>questions</span><span class='ignore'>;</span> <span class='word'>first</span> <span class='word'>we</span> <span class='word'>will</span> <span class='word'>look</span> <span class='word'>at</span> <span class='word'>increasing</span> <span class='word'>the</span> <span class='word'>potency</span> <span class='word'>of</span> <span class='word'>our</span> <span class='word'>mutation</span><span class='ignore'>.</span><br>
<span class='word'>Once</span> <span class='word'>we</span> <span class='word'>have</span> <span class='word'>found</span> <span class='word'>some</span> <span class='word'>information</span> <span class='word'>on</span> <span class='word'>how</span> <span class='word'>this</span> <span class='word'>effects</span> <span class='word'>our</span> <span class='word'>solution</span><span class='ignore'>,</span> <span class='word'>we</span> <span class='word'>can</span> <span class='word'>look</span> <span class='word'>into</span> <span class='word'>the</span> <span class='word'>frequency</span> <span class='word'>of</span> <span class='word'>our</span> <span class='word'>mutations</span> <span class='word'>with</span> <span class='word'>the</span> <span class='word'>new</span> <span class='word'>potency</span> <span class='word'>as</span> <span class='word'>a</span> <span class='word'>permanent</span> <span class='word'>setting</span><span class='ignore'>.</span><br>
<br>