single-winner-voting-systems.nb

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Cell["Comparison of Single-Winner Voting Systems", "Title",
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Cell["Adam Rumpf, 10/31/2017", "Text",
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Cell["Introduction", "Section",
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Cell[TextData[{
 "This demonstration simulates a single-winner election between 2-5 \
candidates using four common voting systems. The voting population is assumed \
to follow a normal distribution along a single ideological spectrum, while \
each candiate\[CloseCurlyQuote]s position lies at a specific point on the \
spectrum. The bell curve can be colored to display either regions where each \
candidate is the favorite, or intervals over which voters approve of each \
candidate. The text underneath the bell curve shows which candidate would win \
under each method for the given voter distribution, while the colored bars \
show which candidate would win if the mean voter (a.k.a. the ",
 StyleBox["center of opinion",
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 ", indicated by a gray line underneath the bell curve) were centered at each \
point on the spectrum."
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Cell["\<\
The purpose of this demonstration is to show how much of an impact the voting \
system can have on an election. In a single-winner election with only two \
candidates there is obviously only one fair way to decide on the winner: ask \
each voter which candidate they prefer and take the candidate that more \
voters prefer. When there are more than two candidates it is not nearly as \
straightforward, and different voting systems can produce different results.\
\>", "Text",
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Cell[TextData[{
 "Four common voting methods are included in this Notebook: ",
 StyleBox["first-past-the-post",
  FontSlant->"Italic"],
 " (FPTP, a.k.a. ",
 StyleBox["plurality voting",
  FontSlant->"Italic"],
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 StyleBox["instant-runoff voting",
  FontSlant->"Italic"],
 " (IRV, a.k.a. ",
 StyleBox["alternative vote,",
  FontSlant->"Italic"],
 " ",
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  FontSlant->"Italic"],
 ", or ",
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 StyleBox["Borda Count",
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 ", and ",
 StyleBox["Approval Voting",
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 " voting systems while approval voting is a ",
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 " system. Their rules are as follows:"
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Cell["\<\
FPTP: This is the system that most people are familiar with. Each voter gets \
to vote for one candidate, and the candidate with the most votes wins. We \
assume that each voter votes for the candidate ideologically closest to them\
\>", "Item",
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Cell["\<\
IRV: Each voter gets to rank the candidates from their first choice to their \
last chocie, with no ties allowed. The votes are then tallied by applying all \
of the first-choice votes to the appropriate candidates. If one candidate has \
a strict majority, then they win, and otherwise the candidate in last place \
is eliminated. All of the votes that originally ranked this candidate first \
are then moved to their second choice candidates and the process is repeated. \
Candidates are iteratively dropped and their votes are moved down the \
preference lists until one candidate achieves a majority. We assume that each \
voter ranks candidates from ideologically closest to ideologically farthest.\
\>", "Item",
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Cell[TextData[{
 "Borda Count: Each voter ranks candidates exactly as in IRV. The votes are \
then tallied by giving each candidate points based on their rankings on each \
ballot. If there are ",
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 " point. The candidate with the highest total score wins. We again assume \
that each voter ranks candidates from ideologically closest to ideologically \
farthest."
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Cell[TextData[{
 "Approval Voting: This is a very minor modification of FPTP. Each voter gets \
to vote for ",
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assume that each voter has a set ",
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vote for any and all candidates within that range and no candidate outside of \
this range. If no candidate falls within their approval range, then they \
abstain."
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Cell["\<\
The Code section of this Notebook includes three sections: initialization \
code, a collection of particular examples that illustrate particular \
phemonena, and finally a Manipulate environment that allows the user to play \
with an interactive model.\
\>", "Text",
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FPTP and IRV both tend to squeeze out candidates flanked closely by other \
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Cell["Non-Monotonicity of IRV", "Subsubsection",
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IRV can also display some unusual properties that the other voting systems do \
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the center of opinion moves farther from them: this property is called \
monotonicity. In IRV, that is not always the case.\
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$CellContext`intlist[$CellContext`\[Mu], $CellContext`\[Sigma]^2, \
$CellContext`breaks]; $CellContext`tot = \
$CellContext`totscore[$CellContext`pref, $CellContext`votes, 
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$CellContext`winner[$CellContext`tot], 
           
           3, {$CellContext`pref, $CellContext`breaks} = \
$CellContext`prefbreaks[$CellContext`p]; $CellContext`votes = \
$CellContext`intlist[$CellContext`\[Mu], $CellContext`\[Sigma]^2, \
$CellContext`breaks]; $CellContext`tot = \
$CellContext`totscore[$CellContext`pref, $CellContext`votes, 
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$CellContext`winner[$CellContext`tot], 
           4, $CellContext`breaks = $CellContext`appbreaks[$CellContext`p, \
$CellContext`\[Epsilon]]; $CellContext`tot = \
$CellContext`intlist[$CellContext`\[Mu], $CellContext`\[Sigma]^2, \
$CellContext`breaks]; $CellContext`out = \
$CellContext`winner[$CellContext`tot], 
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$CellContext`prefbreaks[$CellContext`p]; $CellContext`votes = \
$CellContext`intlist[$CellContext`\[Mu], $CellContext`\[Sigma]^2, \
$CellContext`breaks]; $CellContext`out = \
$CellContext`irvwinner[$CellContext`pref, $CellContext`votes]; 
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{}, $CellContext`int = {}, $CellContext`temp}, $CellContext`pair = 
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$CellContext`mid = Table[
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            Length[$CellContext`pref]}]; {$CellContext`used, \
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         Pattern[$CellContext`\[Sigma]s, 
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         Pattern[$CellContext`list, 
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          Part[$CellContext`list, $CellContext`i, 2]], {$CellContext`i, 1, 
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          Blank[]], 
         Pattern[$CellContext`\[Sigma]s, 
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         Pattern[$CellContext`a, 
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         Pattern[$CellContext`b, 
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       N[(1/2) (
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          Erf[($CellContext`a - $CellContext`\[Mu])/(
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          Blank[]], 
         Pattern[$CellContext`votes, 
          Blank[]], 
         Pattern[$CellContext`score, 
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            $CellContext`listscore[
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             Part[$CellContext`votes, $CellContext`i], $CellContext`score]], \
{$CellContext`i, 1, 
            
            Length[$CellContext`pref]}]; $CellContext`tot], \
$CellContext`listscore[
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          Blank[]], 
         Pattern[$CellContext`votes, 
          Blank[]], 
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            Part[$CellContext`score, $CellContext`i], {$CellContext`i, 1, 
            
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$CellContext`scorefptp[
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         Pattern[$CellContext`\[Epsilon], 
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           Part[$CellContext`p, $CellContext`i] - $CellContext`\[Epsilon]], 
          Min[
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{$CellContext`i, 1, 
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          Blank[]], 
         Pattern[$CellContext`votes, 
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                Part[$CellContext`pref, $CellContext`i] = Delete[
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         Pattern[$CellContext`\[Sigma], 
          Blank[]], 
         Pattern[$CellContext`pos, 
          Blank[]]] := 
       Module[{$CellContext`order, $CellContext`breaks}, $CellContext`order = 
          Ordering[$CellContext`pos]; $CellContext`breaks = Join[{-3}, 
            Table[
             N[(Part[$CellContext`pos, 
                 Part[$CellContext`order, $CellContext`i + 1]] + 
               Part[$CellContext`pos, 
                 Part[$CellContext`order, $CellContext`i]])/
              2], {$CellContext`i, 1, Length[$CellContext`order] - 1}], {3}]; 
         Show[
           Table[
            Plot[
             PDF[
              
              NormalDistribution[$CellContext`\[Mu], \
$CellContext`\[Sigma]^2], $CellContext`x], {$CellContext`x, 
              Part[$CellContext`breaks, $CellContext`i], 
              Part[$CellContext`breaks, $CellContext`i + 1]}, Axes -> False, 
             AxesOrigin -> {0, 0}, PlotStyle -> Opacity[0], Filling -> Axis, 
             FillingStyle -> Part[$CellContext`colors, 
               Part[$CellContext`order, $CellContext`i]]], {$CellContext`i, 1,
              Length[$CellContext`breaks] - 1}], PlotRange -> 
           All]], $CellContext`colors = {
         RGBColor[1, 0, 0], 
         RGBColor[0, 1, 0], 
         RGBColor[1, 0.8, 0], 
         RGBColor[0, 0, 1], 
         RGBColor[1, 0, 1]}, $CellContext`bellcurveshade[
         Pattern[$CellContext`\[Mu], 
          Blank[]], 
         Pattern[$CellContext`\[Sigma], 
          Blank[]], 
         Pattern[$CellContext`p, 
          Blank[]], 
         Pattern[$CellContext`\[Epsilon], 
          Blank[]]] := 
       Module[{$CellContext`int}, $CellContext`int = \
$CellContext`appbreaks[$CellContext`p, $CellContext`\[Epsilon]]; Show[
           Join[{
             Plot[
              PDF[
               
               NormalDistribution[$CellContext`\[Mu], \
$CellContext`\[Sigma]^2], $CellContext`x], {$CellContext`x, -3, 3}, Axes -> 
              False, AxesOrigin -> {0, 0}, PlotStyle -> LightGray, Filling -> 
              Axis, FillingStyle -> LightGray]}, 
            Table[
             Plot[
              PDF[
               
               NormalDistribution[$CellContext`\[Mu], \
$CellContext`\[Sigma]^2], $CellContext`x], {$CellContext`x, 
               Part[$CellContext`int, $CellContext`i, 1], 
               Part[$CellContext`int, $CellContext`i, 2]}, Axes -> False, 
              AxesOrigin -> {0, 0}, PlotStyle -> {
                Part[$CellContext`colors, $CellContext`i], 
                Opacity[0.25]}, Filling -> Axis, FillingStyle -> {
                Part[$CellContext`colors, $CellContext`i], 
                Opacity[0.25]}], {$CellContext`i, 1, 
              Length[$CellContext`p]}]]]], $CellContext`partylines[
         Pattern[$CellContext`pos, 
          Blank[]]] := Graphics[
         Flatten[
          Table[{Black, 
            Thickness[0.008], 
            Line[{{
               Part[$CellContext`pos, $CellContext`i], -0.1}, {
               Part[$CellContext`pos, $CellContext`i], 0.75}}], 
            Part[$CellContext`colors, $CellContext`i], 
            Thickness[0.004], 
            Line[{{
               Part[$CellContext`pos, $CellContext`i], -0.1}, {
               Part[$CellContext`pos, $CellContext`i], 
               0.75}}]}, {$CellContext`i, 1, 
            Length[$CellContext`pos]}]]], $CellContext`\[Mu]sweepcolor[
         Pattern[$CellContext`p, 
          Blank[]], 
         Pattern[$CellContext`\[Sigma], 
          Blank[]], 
         Pattern[$CellContext`\[Epsilon], 
          Blank[]], 
         Pattern[$CellContext`w, 
          Blank[]], 
         Optional[
          Pattern[$CellContext`n, 
           Blank[]], 101], 
         Optional[
          Pattern[$CellContext`xr, 
           Blank[]], {-1, 1}], 
         Optional[
          Pattern[$CellContext`yr, 
           Blank[]], {-1, 1}]] := ArrayPlot[{
          $CellContext`\[Mu]sweep[$CellContext`p, $CellContext`\[Sigma], \
$CellContext`\[Epsilon], $CellContext`w, $CellContext`n]}, ColorRules -> 
         Table[$CellContext`i -> 
           Part[$CellContext`colors, $CellContext`i], {$CellContext`i, 1, 
            Length[$CellContext`colors]}], 
         DataRange -> {$CellContext`xr, $CellContext`yr}], \
$CellContext`\[Mu]sweep[
         Pattern[$CellContext`p$, 
          Blank[]], 
         Pattern[$CellContext`\[Sigma]$, 
          Blank[]], 
         Pattern[$CellContext`\[Epsilon]$, 
          Blank[]], 
         Pattern[$CellContext`w$, 
          Blank[]], 
         Optional[
          Pattern[$CellContext`n$, 
           Blank[]], 101]] := Table[
         $CellContext`driver1[$CellContext`p$, 
          FE`\[Mu]$$33, $CellContext`\[Sigma]$, $CellContext`\[Epsilon]$, \
$CellContext`w$], {FE`\[Mu]$$33, -3, 3, 6/($CellContext`n$ - 1)}], 
       Attributes[$CellContext`p$] = {Temporary}, 
       Attributes[$CellContext`\[Sigma]$] = {Temporary}, 
       Attributes[$CellContext`\[Epsilon]$] = {Temporary}, 
       Attributes[$CellContext`w$] = {Temporary}, 
       Attributes[$CellContext`n$] = {Temporary}, FE`\[Mu]$$33 = 
       0, $CellContext`let[1] := "A", $CellContext`let[2] := 
       "B", $CellContext`let[3] := "C", $CellContext`let[4] := 
       "D", $CellContext`let[5] := "E"}; Typeset`initDone$$ = True),
    SynchronousInitialization->True,
    UndoTrackedVariables:>{Typeset`show$$, Typeset`bookmarkMode$$},
    UnsavedVariables:>{Typeset`initDone$$},
    UntrackedVariables:>{Typeset`size$$}], "Manipulate",
   Deployed->True,
   StripOnInput->False],
  Manipulate`InterpretManipulate[1]]], "Output",
 CellChangeTimes->{3.7770515367171426`*^9, 3.777053418649888*^9, 
  3.7770550650820446`*^9, 3.777057041075801*^9}]
}, {2}]]
}, Open  ]]
}, Open  ]]
}, Open  ]]
},
WindowSize->{759, 833},
WindowMargins->{{99, Automatic}, {Automatic, 70}},
FrontEndVersion->"10.4 for Microsoft Windows (64-bit) (April 11, 2016)",
StyleDefinitions->"Default.nb"
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