Lab05.ipynb

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   "cell_type": "markdown",
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   "source": [
    "# Game theory\n",
    "\n",
    "### Theory Recap\n",
    "\n",
    "A **game** in strategic form is a triplet $\\left(\\mathcal V, \\mathcal A, \\{u_i\\}_{i \\in \\mathcal V}\\right)$ where:\n",
    "- $\\mathcal V$ is the finite set of **players**\n",
    "- $\\mathcal A$ is the set of **actions** available to each player. Actions $x_i \\in \\mathcal A$ chosen  by each player are collected in a vector $x = (x_1,x_2, \\ldots, x_n)$ called **configuration** of the game. We denote the set of all configuration by $\\mathcal X = \\mathcal A^{\\mathcal V}$.\n",
    "- for each player $i \\in \\mathcal V$, the utility function $u_i: \\mathcal X \\rightarrow \\mathbb{R}$ describes the **payoff** the player gets in each configuration: the utility of a player depends on its own action but also on the actions of other players.\t\n",
    "\n",
    "To each player $i \\in \\mathcal V$ we associate a **best response** function, which gives, for each configuration of the other players, the best action for $i$ (i.e., the actions that maximize its utility)\n",
    "$$\n",
    "\\mathcal B_i(x_{-i})= \\arg\\max_{x_i \\in \\mathcal A} u_i(x_i,x_{-i})\n",
    "$$\n",
    "\n",
    "A **Nash equilibrium** is a configuration $x^* \\in \\mathcal X$ such that\n",
    "$$\n",
    "x_i^* \\in \\mathcal B_i (x^*_{-i}), \\quad i \\in \\mathcal V\n",
    "$$\n",
    "\n",
    "A game is **potential** if there exists a potential function $\\Phi: \\mathcal X \\rightarrow \\mathbb{R}$ which describes the way the utility function of each player changes when that player unilaterally varies its action. That is, $\\forall i \\in \\mathcal V$, $\\forall x,y \\in \\mathcal X$\n",
    "$$\n",
    "x_{-i}=y_{-i} \\implies u_i(y)-u_i(x) = \\Phi(y)-\\Phi(x)\n",
    "$$"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "# Discrete time best response dynamics\n",
    "In this section we present and analyze a first game-theoretic learning process, the discrete time best response dynamics.\n",
    "\n",
    "Consider a game $\\left(\\mathcal V, \\mathcal A, \\{u_i\\}_{i \\in \\mathcal V}\\right)$. \n",
    "\n",
    "The **discrete-time asynchronous best response dynamics** is a Markov chain $X(t)$ with state space $\\mathcal X= \\mathcal A^{\\mathcal V}$ coinciding with the configuration space of the game.\n",
    "At time instant $t$, when the chain is in $X(t)$, one player $i$ is selected uniformly at random in $\\mathcal V$ and she changes her action to a best response chosen uniformly at random in the set $B_i(X(t)_{-i})$.\n",
    "\n",
    "In other words, the best response dynamics is random walk on the transition graph $\\mathcal G_{B}$, i.e., the graph with nodes corresponding to configurations $\\mathcal X=\\mathcal A^{\\mathcal V}$ and links $(x,y)$ from configurations $x$ and $y$ that differ in exactly one entry $i$ if and only if $y_i \\in B_i(x_{-i})$."
   ]
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   "cell_type": "markdown",
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   "source": [
    "## Example: Rock-Scissor-Paper game\n",
    "The Rock-Scissor-Paper game is a two-player symmetric game with action set $\\mathcal A =\\{R, S, P\\}$ and utility matrix\n",
    "\n",
    "![utility-matrix](RSP-utility.png)\n",
    "\n",
    "1. Write a function that implements the best response correspondence for both players.\n",
    "2. Does the game admit Nash equilibria?\n",
    "3. Is the game potential?\n",
    "4. Consider the best response dynamics for the RSP game. Plot the transition graph $\\mathcal G_{B}$.\n",
    "5. Simulate the (discrete time) best response dynamics and estimate the invariant probability distribution $\\pi$.\n",
    "6. Compute and plot the condensation graph of $\\mathcal G_{B}$.\n",
    "7. Compute analytically the invariant distribution $\\pi$ of the best response dynamics.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> 1. Write a function that implements the best response for both players."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "# Define action vector\n",
    "# we identify actions with numbers: R->0, S->1, P->2\n",
    "actions = np.array([0,1,2])\n",
    "\n",
    "# Define utility matrix\n",
    "# first index is the player whose utility is being computed 0=first/row player,\n",
    "# 1=second/column player,\n",
    "# second index is the first player action 0=R, 1=S, 2=P,\n",
    "# third index is the second player action 0=R, 1=S, 2=P\n",
    "utility = np.array([[[0, 1, -1],\n",
    "                     [-1, 0, 1],\n",
    "                     [1, -1, 0]],\n",
    "                    [[0, -1, 1],\n",
    "                     [1, 0, -1],\n",
    "                     [-1, 1, 0]]\n",
    "                   ])\n",
    "\n",
    "\n",
    "# Best response function for 2 player games \n",
    "def best_response(player, configuration):\n",
    "    other_player = 1-player\n",
    "    other_action = configuration[other_player]\n",
    "    if player == 0:\n",
    "        return [np.argmax(utility[player, :, other_action])]\n",
    "    if player == 1:\n",
    "        return [np.argmax(utility[player, other_action, :])]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> 2. Does the game admit Nash equilibria?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Configurations: [(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)]\n",
      "The game does not posses Nash equilibria.\n"
     ]
    }
   ],
   "source": [
    "from itertools import product\n",
    "\n",
    "# Compute all game's configurations\n",
    "# product function computes the cartesian product of actions with itself\n",
    "# returns an iterator of tuples of actions\n",
    "configurations = list(product(actions, repeat=2))\n",
    "print(\"Configurations:\", configurations)\n",
    "\n",
    "nash_list = []\n",
    "\n",
    "# we cycle over configurations:\n",
    "# each configuration is a Nash unless there exists (at least) one player who\n",
    "# is not playing a best response to the configuration of other players\n",
    "for configuration in configurations:\n",
    "    is_nash = True\n",
    "    for player in [0,1]:\n",
    "        if configuration[player] not in best_response(player,configuration):\n",
    "            is_nash = False\n",
    "            break\n",
    "    if is_nash:\n",
    "        nash_list.append(configuration)\n",
    "        \n",
    "if len(nash_list) != 0:\n",
    "    print(\"The game admits the following Nash equilibria\", nash_list)\n",
    "else:\n",
    "    print(\"The game does not posses Nash equilibria.\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The game does not posses any Nash equilibrium. In each configuration at least one player has an incentive in changing is current action to a best response, which gives him utility $+1$. There are no configurations where both players have utility $+1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> 3. Is the game potential?\n",
    "\n",
    "If a finite game (i.e., a  game with a finite configuration set) is potential, then the set of its Nash equilibria is non-empty as it contains the maxima of the potential function $\\Phi$.\n",
    "The RSP game does not posses Nash equilibria, so it is not potential."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> 4. Consider the best response dynamics for the RSP game. Plot the transition graph $\\mathcal G_{B}$ (i.e., the graph with nodes corresponding to configurations $\\mathcal X=\\mathcal A^{\\mathcal V}$ and links $(x,y)$ between configurations $x$ and $y$ that differ in exactly one entry $i$ if and only if $y_i \\in B_i(x_{-i})$)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import networkx as nx\n",
    "\n",
    "# transition_graph is a directed graph\n",
    "transition_graph = nx.DiGraph()\n",
    "# nodes of transition_graph are configurations\n",
    "transition_graph.add_nodes_from(configurations)\n",
    "\n",
    "# we visit each configuration x and add links to neighboring configurations.\n",
    "# Neighboring configurations y of x are those that differ in exactly one entry,\n",
    "# the action of a single player i, and such that y_i is a best response to\n",
    "# x_i for player i\n",
    "for configuration in configurations:\n",
    "    for player in [0,1]:\n",
    "        for other_action in actions:\n",
    "            if other_action != configuration[player] and other_action in best_response(player,configuration):\n",
    "                other_config = list(configuration)\n",
    "                other_config[player] = other_action\n",
    "                other_config = tuple(other_config)\n",
    "                transition_graph.add_edge(configuration,other_config)\n",
    "                \n",
    "pos = nx.spring_layout(transition_graph)\n",
    "nx.draw(transition_graph, pos=pos, with_labels=True)              "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The transition graph has nodes corresponding to the $9$ game configurations and directed links which represent admissible transitions for the best response dynamics. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> 5. Simulate the (discrete time) best response dynamics and estimate the invariant probability distribution $\\pi$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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gxAhtaxNVT4JPCFFpSim2bNmCXq/nl19+4bnnnmP16tV06tRJ69KKcXWF3FzzOXn16+u4elV6eTWF3OMTQlTYtWvXWLBgAeHh4bi4uBAWFsaIESOoU6eO1qUV89Zb8OWXshu6Jcg9PiFEjaKUIioqCr1ez+rVq3n88ceZO3cuPXr00Gyi+Z0U3w1dMWKEjkWLNCxKaEZ6fEKIMklLS2Px4sXo9XrS09MJDQ1l1KhRNGjQQOvSStWsGZw/bz5a09FRR36+llXZv4JNaO21xyfz+IQQt3Xw4EHCwsJo3rw5mzZt4vPPPyc+Pp633nrLZkNv82bTgtJFoWeak/fLLxJ6lmV/oQdyqVMIUYKsrCy+++479Ho958+fZ8yYMRw6dIgmTZpoXdodOTqC0Wg+eKVlSx2nTtnnD2lheXKpUwhR6MSJE+j1ehYuXEhgYCChoaEMGjQIJyfb/x35+edh+fKavRu6tRRd6gR7TBC51ClEDZeXl8eKFSt49NFH6dmzJ87OzuzZs4cNGzbw5JNP2nzoFeyGbgq9ont5f/+7TEQXJbPt72ghRJU5f/48ERERzJkzh9atWxMWFsbQoUNxdXXVurQyq18frl83DzxXV53dryUpqpb0+ISoQQwGAxs2bOCJJ54gICCAlJQUNm3axI4dOxg2bJjdhF5kpGnwSlHomQavnDoloSfuTHp8QtQAV65cYe7cuYSHh1O/fn3CwsJYunQp7u7uWpdWbkULShcNXunSRUd0tFzWtDaNd5CqMAk+IaoppRQ7duxAr9ezceNGhg4dynfffUdgYKDWpVXIQw/Brl2yoLQtsZHNNcpNgk+IaiY5OZmFCxei1+sBCA0NZebMmdStW1fjyiqm5N3QFXq9AyEh2tZW09ngynRlIsEnRDWxd+9e9Ho9K1euZMCAAej1enr16mUzy4hdSzhP/NQZOMUdwjktjTxPT/L9OtD2ndep1+LeEj/j7g6ZmeaXNevU0ZGSYhvnVNPdc4/WFVSMBJ8QdiwjI4OlS5ei1+u5du0aISEhHD9+nEaNGmldWqH4H7eQPmkyfrG/EaDT4ZafW/haVtRmdLO/Zl/AQ3hMep/7BvcF4JNPYOLE4utrXr/uIAtK25AHH9S6goqRCexC2KG4uDj0ej1LliyhZ8+ehIaG0r9/fxwcrDtQe8KECdx1112MHz++xNf3vD0F/+lTcMnLxZHSf9QY0JHr7MLB8RPp/vlEbt0NPTBwLa1aLWHZsmUWPwdRPsnJ4O1t+v+zf79pM197I9MZhLATOTk5LFmyhIcffph+/frh7e1NbGwsq1evZuDAgVYPvcTERBYuXEjITTfaNm/eTLt27ahduzZdmrWk7teTccvLuW3oATii+DQvh9Gff4AOR+AjQKHTmVZe2bv3CQ4fPszBgwer9qTEHS1eDAW9cHsMPZDgE8LmnT59mnfeeYdmzZoxb9483njjDc6ePcvHH39M06ZNNatr/vz5BAcHF+6sfvXqVYYOHcrkyZPZE7mShy+e4+WbLmveSWvgCxTBOmjilsCqVTqMxqLXhw0bRkREhIXPQpTXmjVaV1B5EnxC2KD8/HxWrVrFgAED6NatGwaDgV9//ZVNmzbx9NNP4+zsrHWJbNiwgd69exc+XrlyJX5+fjz77LPkfTqVj4wGDgDHytjey8BAwEsZGVR3B08+af56UFAQ69ats1D1oqJOnNC6gsqTwS1C2JCLFy8ye/ZsZs2aRfPmzQkNDWXVqlXUqlVL69KKOXToEG3bti18HBcXR8eOHbmWcB6/2N9wBVoBcUC7crSrAxr+eY7rCRfMRnv6+vpy5swZUlNTbXKH95ri2jWtK6g86fEJoTGj0cimTZsYOnQoHTp04PLly6xbt47ffvuNkSNH2mTogWm+oKenZ+Hj9PR0vLy8iJ86A+ONKRReQFoF2lbA8WkzzJ4rOFZycnIFKxaWkJWldQWVJz0+ITRy9epV5s+fT3h4OO7u7oSFhbFgwQKzMLFl3t7epKUVxZqHhwepqak4xZ0pnLKQClTkbJyVEae4w2bPFRzLXifiVxfVYSNf6fEJYUVKKX777TdGjBhB69atOXz4MIsWLWL//v2EhITYTegB+Pv7Ex8fX/jYz8+PAwcO4HwjoDKAU4BfBdt3Tksxe3z06FF8fHzkMqfGqsMEOAk+IawgNTWV//3vf3Ts2JHRo0fTpUsXTp8+zfz58+nevbvNrK5SHsHBwWzfvr3w8ZAhQzh8+DA/5WSQDXwM+FN0f28+4HOb9vKAbMAI5APp7h4YDIbC17dv387AgQMteQqihpIJ7EJUof3796PX61m+fDn9+vUjNDSUPn362GXQ3erq1asEBARw4sSJwikNv/zyC688+zxXkq/TDfOwm4xphOfiUtobBSy45bl58+YxatQoADp06EBkZCQdO3a05GmIcir61lV2u0C4BJ8QFpaZmcny5cvR6/VcunSJsWPHMnr0aO6x14UNb+O9996jUaNGZiu3XEs4j0ebVrga8sze+xjwb8C3DO1mOzqTeeJ04ajOtWvXsmjRIpYvX2654kWFSPAJIQodO3aM8PBwFi1aRLdu3QgLC2PgwIE42uumZZWwL7APHWO233HFlpIY0HEgMIjOe7dUQWWisqpD8Mk9PiEqITc3l+XLl9O3b1+CgoJwc3MjOjqadevWMXjw4BoZegDukyaS61yxzdpynV3wnPS+hSsSlmP/fSWZziBEBZw5c4ZZs2Yxd+5c2rVrR2hoKEOGDMHFXnfmtICTJ420aWMEHIG+/L1FGJMvzMQtL6fMbWQ5u3Jw/ES6DepTZXUKS7HP3h5Ij0+IMjMYDPz4448MHjyYLl26kJGRwdatW9m6dSvPP/98jQ29WrVAp1O0aaPD9Lu0acPYLxOmcnD8RLKcXTHc4YekAR1ZTjdCb9pEa5QtajDp8QlxB3/++Sdz5swhIiKCu+++m9DQUJYvX07t2rW1Ls0m5ORA8d/+FSdOONG69URO9H6ItEmTuX//TtSt+/E5uaBTiu33tmVSejKPuGTTOS/PJtYiFdWXDG4RogRKKbZt28bMmTPZtGkTzz77LKGhoXTu3Fnr0mxOyTMzig98uJ5wgePTZuAUdxjntBTyPL3I92tP27fH8cGXn/HNN9/g6upK06ZNWb58OZ06dbJK/aJ8dLqivRLtNT2kxyfETa5fv87ChQvR6/U4OTkRFhbGrFmz8PLy0ro0G2bk1rsmOp2BW3+81GtxLw/O/KzEFgp2jM/JyeHUqVP06tWLpKQk6fmJKiHBJ2o8pRS///47M2fOZNWqVQwePJjZs2fz0EMPVYuJ5lVJp7s59Ap6Aopr18r3o6Vu3bo4OTmhlKJ58+asW7dOQk9UGRncIuzGtYTz7A59h729gokN6MXeXsHsDn2H6wkXKtReeno6ERERdOnSheHDh+Pn58eJEyeIjIykZ8+eEnp3YB56Rq5fN329XFx0eHuXr62WLVvSrFkzvvnmG1JTU6lXr55FaxWW52DH6SH3+ITNi/9xC+mTJuMX+xvGUgZHHAl4CI9J73Pf4L53bO/QoUPo9XqWLl1K7969CQsL49FHH8XBnv8lW1lR6ClM9/Ms97X729/+RkZGBuHh4RZrU1hOwT0+V1fIzta6moqR4BOamjBhAnfddZfZklc32/P2FPynT8ElL/e2q4AY0JHr7FLqcPjvv/+eL774AicnJ86cOcOYMWN45ZVXuPfee0toTdxOVYYemPbba9euHRs3biQgIMCibYvKKwi+unUhKUnraipGgk9oJjExkYCAAE6ePFm4yPHmzZt57bXXOHfuHL4N7mbJpYu0vamHdztngJd1OvY4OuHTqiUzZszAx8eH8PBwFixYQGZmJlOmTGHcuHE4Ocnt7Yqo6tArEB4eztKlS9m6datccrYxBcHXti0cO6Z1NRUj13aEZubPn09wcHBh6F29epWhQ4cyefJk9kSu5OGL53i5jKEHMAzoohR/oGNgx64MHDiQ7t274+DgwO7du5kwYQInT56U0Ksga4UewKuvvkpSUhIrVqyosmOIyrn/fq0rqDgJPqGZDRs20Lt378LHK1euxM/Pj2effZa8T6fykdHAAUxb2dxJPLAP+Ajwys9j4O+xtGrVikmTJjF16lRatWpFUFAQ69atq5qTqeasGXoAjo6O/Pvf/+att94iKyurSo8lKua557SuoOIk+IRmDh06RNu2bQsfx8XF0bFjR64lnMcv9jfqAK2AuDK0FQe0BDwBRxQPnz9Gz67dOX78eOF7fH19OXPmDKmpqZY9kWrO2qFXICgoiMDAQL766iurHE/cmen3RtOUlQEDNC6mEiT4hGaSk5Px9PQsfJyeno6XlxfxU2dgvHFfxwtIK0Nb6TfeW0DpdOQdjictrejTBcdKTk6ufPE1hOl+jvVDr8Dnn3/OV199xcWLF616XFGyZcuK/l63rnZ1VJYEn9CMt7e3WTB5eHiQmpqKU9yhwikLqZh6cXficeO9Bdzyc8n68w+zYC04Vl17/hdrJUlJNy9NpU3oAbRo0YLQ0FAmTJhg9WOL4vbt07oCy5DgE5rx9/cnPj6+8LGfnx8HDhzA+UZAZQCnAL8ytOUHnMa8d3gyNQk/v6JPHz16FB8fH+rUqWOB6quvpCSoV0/70CswYcIEtmzZQlRUlGY1CJPLl7WuwDIk+IRmgoOD2b59e+HjIUOGcPjwYX7KySAb+BjwB9rdeH0+4FNKW/cBAZgGt2QDPwAnsjN5+umnC9+zfft2Bg4caOGzqF5sLfTAdCXg008/5Y033sBoNGpaS02XkaF1BZYhwSc089JLL7F+/frCUXsNGzZkxYoV/O/Ps3gDe4CbbilwHnjoNu0tA6IBb+Ad4JOBT9OwYcPC15cuXUpISIiFz6L6sMXQKzBixAgAFi9erHElNVtu2WcX2TSZwC409d5779GoUSOzlVuuJZzHo00rXA15Zu99DPg34FuGdrMdnck8cZp6LUwrs6xdu5ZFixaxfPlyyxVfjdhy6BWIiorimWee4dixY3h4eGhdTo2SnMwt66+avldmzIDXXtOoqEqQ4BM2aV9gHzrGbL/tMmWlMaDjQGAQnfduqYLKqh97CL0CI0eOpHnz5kyZMkXrUmoUR0co7SqzPSaIbX53ixrPY9L75Dq5VOizuc4ueE5638IVVU/2FHoAn332GXq9noSEBK1LqVFGjQJK/CXUPu+52u53uKjRrtbry3v3vEemk2u5Ppfl7MrB8RNpM6hPFVVWfRQPPWw69ACaNGnC+PHj+cc//qF1KTXKnDklPatYvdo+g08udQqbc+wYBAZmk5FRi697fkTInn9VencGYa7k0LOPxaCzsrLw9fVl/vz5BAUFaV1OjdGsWTbnz7ti+p4BMKCUo5YlVZht/3onapw//oBevTLJyKjFI49kMG7rh1z4YQMHAoPIdnQm65bLn1lOLmQ7OnMgMIgLP2yQ0CsDew49ADc3Nz7//HPeeOMNDAaD1uXUGOfO1brpkeL6dfuND+nxCZuRkgKBgemcPOlB+/bp7NnjQe3aRa9fT7jA8WkzcIo7jHNaCnmeXuT7taft2+MKR2+K27P30CuglCIoKIgXX3xRpqhYkaNjFkZjLSAXpcp3G8KWSPAJm5CbCw8/nMGePe40aZJBbKw7DRpoXVX1Ul1Cr0BsbCwDBgzg2LFjsgydFe3cuZOePXtqXUalSPAJzRmNMHRoJqtX18bLK5P9+2vTooXWVVUv1S30CowdOxYPDw/ZwaGKXUs4T/zUGTjFHcI5LY08T0/y/TrQ9p3X7fJqiwSf0Nxf/5rDf//riotLDlFRrnTqpHVF1Ut1DT2AK1eu4Ofnx86dO822uBKWEf/jFtInTcYv9jeMOl3h4vFgur+uU4ojAQ/hMel97hvcV8NKy8d+706KauGLL/L5739dcXAwsHati4SehZ0+XX1DD6BRo0ZMmDCBv/3tb1qXYncmTJjA9OnTS319z9tTaDo0mI4x23E15JmFHph2QKllyKNjzHaaDg1mz9slLyqwZs0aXnjhBYvWXmlKCI0sWWJQpnUflFqwwKB1OdXOqVNKgfHG19iowKh1SVUiJydH3XfffWr9+vVal2I3rly5oho3bqwyMzOVUmxsufsAACAASURBVKav4dNPP62aN2+uAPXN86NVprOrKvwHeps/l0G9AOpuUO4urqpHjx4qKirK7Hh+fn7qwIEDWpxqiaTHJzSxdSuMHGma/DplSh4vvSTfipZ0+jS0alV9e3o3c3Fx4auvvuLNN98kLy/vzh8QzJ8/n+DgYNzc3Aqf69mzJ5GRkTT0rker7xfhlpdTprbSgQeAfcCfRsWAzt0ZNGgQ6enphe8ZNmwYERERlj2JSpCfNsLqDh6E4OAcDAYnxo7N5r33nLUuqVqpSaFXIDg4GB8fH7755hutS7ELGzZsoHfv3oWPXVxcGD9+PD179kSXno6Toey/QLQE/gbcA7jl5zEoaj+5ubkcP3688D1BQUGsW7fOcidQSRJ8wqrOnYOgoEyys10JDs7kf/+rha56/0y2qpoYegA6nY6vv/6aTz75hMTERK3LsXmHDh0qcTDQtYTzOOflUtH1WBxR5O37ldycXFq3bl34vK+vL2fOnCE1NbWCLVuWBJ+wmuvXoVevDJKSatOlSzorVtTG0T5XPLJJNTX0Cvj6+jJ8+HA++OADrUuxecnJyXh6ehZ7Pn7qjArsh1IkFXjFaOD/Arrh5eVV+HzBsZKTkyvRuuVI8AmryMqCRx7J4Nw5d3x8Mti0yYNate78OVE2MTE1O/QKfPjhh6xcuZKDBw9qXYpN8/b2Ji0trdjzTnGHqOh3TRbwOPAgitGu5qFacCxbWWhAgk9UOYMBhgzJJDbWnfr1M/n1V/dbNrUUlRETA4GBN4eerkaGHph+oE+aNInx48ejZIpyqVq2bMnjjz/OqFGj+Pbbb/nzzz8BcC4hDMsiB3gKaAKEA85pKWavHz16FB8fH+rUqVOpui1Fgk9UKaVg7NhsfvqpNm5u2WzbVpt77W+hB5tVcuhpXJTGxowZQ2JiIj/88IPWpdisfv36ceXKFRYsWMCoUaNo0qQJnp6epLubFsfNBbIp2oFvPuBTSlt5wDOAG7AQU6jkeXqZvWf79u0MHDjQ0qdRYbJyi6hSH3+cx4cfOuPomM+WLU48/LDWFVUfEnql27x5M2PGjOHIkSPUkmvqgOly4++//87u3bvZtm0bmzdvLnzN2dkZg+EujMYLZp9JwBR4k4FjwOIS2t0OBGEKvoKelNHJmZ+2bKZXr14AdOjQgcjISDp27GjZk6og6fGJKjNvnpEPP3QGjCxZ4iChZ0ESerf3yCOPEBAQwNdff611KZpQSnHy5EkWLVpEWFgYAQEB3H333XzwwQekpKTwl7/8haZNm6LT6YBE8vJyMBrPcZfbObIcnVGYvqt8brT3K1Dahl+9b7w3E9OcvquOzlyIP10YemvXrsXX19dmQg+kxyeqyIYNisGDjRiNjnz1VT5vvumkdUnVxubN8OijRaGn0+kw2udG2FXq9OnTdO3alYMHD9K4cWOty6lSmZmZREdHs2vXLnbv3s3u3btxcXGhR48ePPjgg/To0YOAgABcXYu2EnJwyL9pI9miQVE/Nu7LgD+233bj59IY0HEgMIjOe7dY4KyqjgSfsLiYGOjRI5fcXBfeeCOH6dPtd98uWyOhVz4TJkzg0qVLzJ8/X+tSLEYpxdmzZwsDbteuXRw9epT27dsXBt2DDz5I06ZNi302OZkbA8sKvocK/g7e3jquXzctTN10aHCZV265WZazKxd+2ECbQX0qfH7WIMEnLOrUKejcOYvUVDeefjqT5ctr4yAX1C1CQq/80tLSaNeuHT/88ANdu3bVupwKyc7OZt++fYUht3v3boxGo1nIdenSxWz5sVvFxkKnTgU/6s0Dz99fx4ED5u/f8/YU/KdPKVf4ZTm7cnD8RLpNK+2iqO2Q4BMWk5gIAQEZ/PGHOz16ZLB1qzsuLlpXVT1I6FXc/PnzCQ8PZ9euXTfuadm2ixcvmoXcwYMHadeuXWHI9ejRAx8fnzKdy9y58MorRkzfNzcHnuLZZx1Yvrz0zxaEn0te7m0vexrQkevsYjehBxJ8wkIyMqBr13SOHPHgvvvSiY72oISFIUQFrF4NTz0loVdRRqORbt26MX78eIYPH651OWZyc3OJjY01u2yZmZlZGHAPPvggDzzwAO7u7uVq95VXYO7ckgNv6lQH3n67bO2cWLeVtEmTuX//TlRp+/F16onnpPdt/vLmzST4RKXl58Ojj2awfbs7jRplcOCAO3ffrXVV1cOtoefgoMNg0Loq+7Nr1y6ef/55jh07Vu4QsaTLly+b9eb2799Pq1atzHpzrVu3rnDP9MEHISqq5EuaW7fqCAqqWN3XEy5wfNoMnOIO45yWQp6nF/l+7Wn79jjZgV3UPErB8OFZLF3qhrt7Fvv2uXHffVpXVT1I6FnW8OHDadWqFR9//LFVjpefn8+hQ4fMRlpev36d7t27F4Zc165dLbKaSePGcOlSSYGnIykJbGSlMJshwScq5d13c5k61QUnp1x+/dWF7t21rqh6iIyEkSOLQs/RUUd+vtZV2bcLFy4QEBBATEwMzZs3t3j7V69eJSoqqrBHFx0dTdOmTc0uW7Zr1w4HC472cnOD7OzigSe/JN2eBJ+osP/9z8Brrzmi0xlZtUrHE0/Y/sABeyChV3U+/vhj4uLi+PbbbyvVjsFg4MiRI2aXLS9dukS3bt0KL1t2794d7ypYlDY5GerV48ZapOaXM93dddy0/6sohQSfqJBVqxRDhyqUcmDmzHxCQ2WCuiXcGnpOTjpkU3HLyczMxNfXl0WLFvFwOZYSSk5OLuzN7d69mz179tCoUSN69OhR2Jvz8/PDsQr32TpzBlq0gJLm4N17r47z56vs0NWOBJ8ot9274eGH88jPd2bChFw+/VTmLFhCeDiEhkroVbVvv/2Wzz77jOjo6BKDymg0cvz4cbORlufOnaNLly6FIde9e3caNmxolXrXrYPBg0sesBIUpGPrVquUUa1I8IlyOXYMAgOzycioxYgRWSxc6CY7qFuAhJ71KKV4+OGHefnll3n11VcLF28uuGQZFRVF3bp1zUZa+vv74+Rk3asaH38MH35Y8pSEN9904KuvrFpOtSLBJ8rs0iXo2DGTxMTa9OmTwc8/u2PlnwXV0q2h5+ysIzf3Tp+yfdcSzhM/dQZOcYdwTksjz9OTfL8OtH3ndc2GwCulOHXqFEuWLOGzzz6jZcuWJCQk0KlTJ7OVUO7WcD7OoEGwfn1JgQdLl+p44QWtKqs+JPhEmaSmQpcu6Zw86UH79ulERXmg4XSoamP6dHjzzaLQc3HRkVP+JRJtSvyPW0ifNBm/2N8wljbpOeAhPCa9z32D+1ZpLZmZmezdu9dsEEqtWrXo0aMHZ8+epXXr1sydOxcXG1hiqE0bOHmy5CkJCQng46NNXdWRBJ+4o9xc6N07g6god5o0yWD/fnesdHujWquOoaflMlcFizffPG/u6NGjdOjQwWxKwb03dkK+fPky7du357fffuM+DSefenlBaqrMwbMmCT5xW0YjPPNMJj/8UJs6dbLYv9+Nli21rsr+3Rp6rq46srO1rurOJkyYwF133cX48eOLvWbJhY3XrFnDkiVLWLZsWamfzc7OJiYmxqw3B5iFXJcuXW67Ee3nn3/Ojh07WLt2bZlrthQnJzAYbg4809+rwy9ANk8JcRt//Wu2AqVcXLJVTIzW1VQPU6YoBUZlWvfGqFxdta6obK5cuaIaN26sMjMzlVJK7d69Wz366KPK29tbedfxUkN0DuoP00nd8c9lUC+AugdUHVCdff1UVFSU2fH8/PzUgQMHCh+fP39eLV++XI0fP15169ZN1a5dW3Xp0kWNGzdOLV68WCUkJCij0Viuc8rOzlatW7dWGzdurPwXqAySklTh//eiL4dRgVHVr2+VEoRSSnp8olRffpnPW2854eBgYP16B/r3l+GblfXJJzBxYlFPz81NR2am1lWVzeeff058fDyzZs0CYMOGDaSnp9O/f38OBQ1m9v5fuQRsLENbp4FVwDCgPvBx87bo069y5swZPDw8yM3N5a9//StxcXE0btyYXbt2kZ2dbdabCwwMtMi6m2vWrOHdd9/lwIEDODs7V7q9kpR3WyBRxbROXmGbli41KDAoUGr+fIPW5VQL//yneU+vdm2tKyqfPn36qEWLFhV7/urpcyrb0VnFgPIoY4/v1j9Zjs7KrVYtNXLkSPXQQw8pd3d31apVK+Xp6akWLFig4uPjy92bKyuj0aj69eun/vOf/1i87Rkz1I1/R7f28Azq2WctfjhRRhJ8opitW5VydMxToNTkyblal1Mt2HvoKaVUgwYN1O+//17s+V0hb6tMJxf1NahuFQy+3Y5OyknnoN577z31yy+/qNTUVHXt2jUFqJSUlCo/t8OHD6uGDRuqxMREi7Q3enTpgTd1qkUOISpBLnUKM4cOQdeu2WRn12LMmGzCw2vJBPVKeust+PLLosub9rqeorOzM4cOHaJdu3Zmz+/tFYzrzg0EAauBXuVsNxV4CHjYpy3fJBwrfD4vLw8XFxfOnj1Ls2bNKld8Gbz++usYjUa++eabCrfxwAMQHW35bYGEZVlumXBh986fh6CgTLKzaxEcnMnMmRJ6lVVdQg/A29ubtLS0Ys9funqFgcC/KX/oZQGPA92BMXXN58gUHKuulcbzf/TRR3z33XccOnSo3J+9+27Q6dSN0CuYeG4KvKQkHUpJ6NkSCT4BwPXr0KtXBtev16Zz53RWrKhNFa63WyPcGnoeHvYbegD+/v7Ex8ebPXf27FlCE47wPjCynO3lAE8BTYBwIM/Ty+z1o0eP4uPjY5H96sqiXr16fPDBB7z55puU9UKYm5sp8C5fNg88BwdQSodSMg/PFknwCbKzTTuonz3rjo9POr/84sFtpj6JMhg3zjz0PD11lNBZsivBwcFs37698PHFixfp27cvQwO68rJT8ZVP5gM+pbSVBzwDuAELgRwnF/L92pu9Z/v27QwcONAitZdVaGgof/75J2vWrCn1PcnJ4OBgCjzT3MuiS5oeHqbAk73wbJzG9xiFxvLzlRowIEOBUvXqZahz57SuyP699pr5QJY6dbSuyDISExNVkyZNCufxvfvuuwpQDg4Oyh0K/xSM5vgY1IulDGbZdmO1ZbebPudeu7basWNH4fHat2+vYmNjrX6eP//8s2rZsqXKzs42ez4hofQ5eM2bW71MUQnS46vBlIKQkGw2bqxNrVo5bN9em6ZNta7Kvr36KnzzTVFPr25dHSkpWldlGQ0aNGDkyJG89957vPLKK+j1ep577jk2bdrEjs5BpKDj5iu5vwKlLUbWG1PyZQIp6NgR2If0jAx69TLdJVy7di2+vr507NixSs+pJP369aN9+/ZMnz4dMG0LpNMpWrS4dZUVI489Zrp/d+aM1csUlaF18grtfPxxrgLT1IVt27Suxv698op5T69ePa0rspzU1FQ1c+ZMFRAQoFq0aKH+9a9/qT///LPw9eNrN6tMZ9cKTWXIdHZV8T9u0fDsijtx4oRydJxS6pSECRO0rlBUhgRfDTVvnuHGP2SDWrZMJqhX1ogR1TP0oqOj1ZgxY1TdunXV0KFD1U8//aQMhpK/X6L+Mbnc4Zfp7Kqi/jHZymd1e489VtocPKNaulTr6oQlyG5qNdDGjYpXXjFdtvnySyPPPy/fBpUxciRERhZd3qxfX8fVq1pXVXHp6eksW7aM8PBwrly5wpgxYwqXDrudbtMmsgc0252hslq2hISEgpoL7gKZLmkmJDjKtkDVidbJK6wrOlopF5ccBaYFqEXlPPeceU+vYUOtK6q42NhYFRYWpry9vdUTTzyh1q1bp/Lz88vdTvyPW1RMYB+V5eisMp1czHt4Ti4qy9FZxQT2sZnLm56eRT26m3t4Op1SX3wxW/Xo0aPKlksT2pCVW2qQ06ehU6csUlPdGDo0k+++q42DDG+qsKeegtWri3p6jRrpuHxZ66rKJzMzk+XLlxMeHs758+d59dVXeeWVV2hqgVFO1xMucHzaDJziDuOclkKepxf5fu1p+/Y4zXZgv5mjIxiNBf//oGDC+c1bRBkMBrp27cpbb73FsGHDNKlTWJ4EXw2RmAidOmVw8aI7Dz6YwbZt7tjAptN2y95DLy4ujvDwcBYvXkz37t0JCQkhODgYJ6fqfdk7ORm8vQGKB17DhjquXCn+mZ07d/Liiy9y7NgxateubaVKRVWS3/drgIwM6NMnnYsX3WnTJp2NGyX0KmPwYPPQu/tu+wi97OxsIiMj6dmzJ/369cPLy4t9+/axbt06nnjiiWodetu2maYkeHvfOiVBERBgmpJQUugB9OzZkx49ejBt2jTrFCuqnPT4qrn8fHjssQy2bnWnUaMMYmPduecerauyX4MHw7p1RaHXpImOCxe0rur2jh07RkREBIsWLaJz586EhITw+OOPV9nec7bkm29g3DgjRcuJQUHgjRjhwKJFZWvn3LlzdOrUif3791tlwWxRtaTHV40pBS+/nMXWre64u2ezY4eEXmX062ceek2b2m7o5eTksGzZMoKCgggKCsLV1ZU9e/bw008/MXTo0GofeiNHgk5nZNw4henHXNGk8xkzdChV9tADaNasGePGjeOdd96pmoKFVUmPrxp7771c/vUvF5ycctmxw4UHH9S6IvvVrx/88ktR6Pn46EhI0Lqq4k6ePElERAQLFiygffv2hIaG8uSTT+JSQ65td+oEsbElbwu0f7+OgICKt52RkYGvry9LliyhZ8+elapTaKv6XtSv4WbOzOdf/3JBpzPy/ffOEnqV0KcPbNtmu6GXm5vL6tWrCQ8P5+DBg7z88svs3LmTNm3aaF2a1TRoAKZ9a8E88HQkJeksskOCu7s7n332GW+88QZ79+7FQYZE2y3p8VVDa9YonnpKoZQD//tfPmFh8vtNRT30EOzaVRR6LVvqOHVK66pMEhISmDVrFnPnzqVdu3aEhIQwdOhQXF1dtS7NalxdITe3eOA5OurIz7f88ZRS9OzZk1deeYXRo0db/gDCKiT4qpmoKOjVK5f8fBfefdd0qVNUjC2GXn5+PmvXriU8PJzo6GhGjhzJ2LFj8fX11bYwK7rdlARPTx2pqVV7/OjoaB5//HGOHz9utb0ChWVJ8FUjx49Dly5ZZGS48eKLWURGuskO6hXUvTvs2VMUem3a6LhlD1arOnfuHLNnz2bOnDm0aNGCkJAQnnnmGdzc3LQrysrOnIEWLaCkwGvRQsfp09ar5f/+7/9o1KgRU6dOtd5BhcVI8FUTly5BQEAGV66406dPBj//7E41npZVpW4NPV9fHUeOWL8Og8HA+vXrCQ8PZ/fu3QwfPpyxY8fSvn37O3+4Glm2DIYNK+n+neKxxxz46Sfr13Tp0iU6dOhAVFQUrVu3tn4BolIk+KqB1FQIDMzgxAl3/PzS2bPHA3d3rauyT4GBEBNTFHp+fjoOH7ZuDRcvXmT27NnMnj2bJk2aEBISwvPPP1/jVg3529/g669LnoM3YYIDn36qXW0An332GVFRUaxatUrbQkS5SfDZudxcCArKYPdud5o0yWD/fncaNtS6Kvt0a+h17KgjNtY6xzYYDPz888+Eh4ezY8cOXnjhBUJCQjTZiFVrRaNo4dZLmj/+qGPQIE3KKiY7Oxs/Pz/0ej39+vXTuhxRDhJ8dsxohGefzWTlytrUqZPF/v1utGypdVX2KSAADhywfuhdunSJuXPnMmvWLBo2bEhISAgvvPACHh4eVX9wG+PjA2fPljwlISEBm9wWaNWqVUycOJHY2FhSzl8ifuoMnOIO4ZyWRp6nJ/l+HWj7zus2sSi3KCLBZ8fGj8/h3/92xcUlh127XOnSReuK7FP79hAXVxR6XbroiI6uuuMZjUY2b95MeHg4mzdv5tlnnyUkJIQuNfR/oKcnpKcXDzydTsf161hkDl5VUUrxau8B/F9iEg+ciMWo0+GWn1v4epaTCzqlOBLwEB6T3ue+wX01rFYUkOCzU199lc/f/+6Eg4OBdescGDBAhm9WxP33w9Gj1gm9K1euMG/ePCIiIqhTpw4hISG8+OKLNXZIfGnbAtWqpSMrS7OyymXP21Pw/3oKLvn2t/FuTSZLD9ihb7818ve/m/7XzZmjk9CroFtDr1s3y4eeUoqtW7fy/PPP07ZtW44fP87SpUvZt28foaGhNS70kpNBpzPtlGA0ws27JNx1l2mXBFsKvQkTJjB9+vQSX9vz9hT8p0/BLT/ntqEH4IjCLS8H/+lT2PP2lGKvr1mzhhdeeMEiNYsysMp2t8Jitm1TytExT4FSH3+cq3U5dqtNG/Od03v0sGz7iYmJ6osvvlD33Xefat++vfrvf/+rkpKSLHsQO7J1a8m7nINRBQZqXV3Jrly5oho3bqwyMzOVUkrl5OSop59+WjVv3lwBaqOjs9nu8nf6MxFUe1COoMYNe6nY8fz8/NSBAwesfZo1kvT47MjhwzBgQA4GgxOvvJLNxInVe4X9qtKqFZw4UdTT69FDx2+/Vb5dpRQ7duxg+PDhtGnThoMHDzJv3jwOHjzIuHHjqGvLN6uqyLRppl0S+vQp+HoX7ZIwerSph7d3r7Y1lmb+/PkEBwebLRLQs2dPIiMjqe/kgpMhr1zttQamAcFA7rYdxV4fNmwYERERlStalI3WySvK5tw5perVy1Cg1IAB6So/X+uK7FPLluY9vaCgyrd5/fp1NX36dOXr66vatWunvv76a3Xt2rXKN2zHnn1WKTCU0MMzqBkztK6ubPr06aMWLVpU7Pmrp8+pJqC2lqO3d/Of4aD+qXNQ106fN2t3586dysfHx1qnV6NJj88OJCXBww9ncP16bTp1ymDlSnccHbWuyv60aAGnTxf19IKCdGzdWrG2lFLs2rWLl19+mZYtW/L777+j1+s5cuQI48ePp169epYs3W507Gi6f/fdd7fug6fYv9+0D95rr2lbY1kdOnSItm3bFns+fuqMO9zRuzMFHJ82w+w5X19fzpw5Q2pVLzYqZFsiW5edDf36pXPmjAfNm6ezebMHNWh5Rotp1gzOny8KvUcf1bFpU/nbSUlJYdGiRURERJCdnU1ISAhffvklDRo0sHTJdsUa2wJZW3JyMp6ensWed4o7RGWHkzkrI05x5ksCFRwrOTm5xg16sjYJPhtmMMDTT2cSE+NBvXqZ7NjhcWNVelEe994LFy9WPPSUUuzdu5fw8HBWrlzJY489xvTp0+nTpw+6Gr4KuIsL5OXdHHim3p1pWyD7/tp4e3uTlpZW7HnnEp6rCOe0FLPHBceqifeCrU2Cz0YpBWFh2axfX5tatbLZtq02zZppXZX9uece+PPPotAbNEjHjz+W7bNpaWksXryY8PBwUlNTGTt2LMePH6dRo0ZVWbLNK74tUFHg1amjIyXldp+2H/7+/sTHx/PAAw8ApmXljh49SlKl+3smeZ5eZo+PHj2Kj4+P9PasQILPRn36aR6zZtXC0TGfDRtq0aGD1hXZn1tD78kndZRlPeGYmBjCw8P57rvveOSRR5g2bRqPPPJIjd9xu/i2QAWBB61b6zhxQrPSLM5oNPLAAw8wZ84coqOjiY6OJjY2lrvuuou/1vNBAblANuCK6SsxH5gEnCmlzTzAABiBLJ0DGW3bYTAYcLxxw3779u0MHDiwSs9LmMjKLTZowQIjo0Y5AEaWLoUXXqjZP3BvdS3h/B3XRLzrLrhypeyhl56ezrJlywgPDycxMZExY8YwevRo7rnnHqucky0zbQtU8i4JwcEOrFunXW2WoJTizJkzhQG3d+9e9u3bR506dbh69SrvvfcePXr0oHPnzgQEBHD27FmzzycAPsBk4BiwuJTjjAIW3PLcvHnzGDVqFAAdOnQgMjKyRi5MbnWajikVxWzcaFQODvkKlPriizyty7Epx9duVjFdglS2o7PKdHIxGyKe6eSishydVUyXINW/xWazKQvPPWf6/J49e1RycrJZm7GxsSosLEx5e3urJ598Uq1fv17ly1wRpZRSb75Z+pSEjz7SurqKMRqN6sKFC2rVqlXqn//8p+rfv7+qX7++aty4sXriiSfU5MmT1YYNG9SVK1eUUkpNmDBBff3118XaiekSpPLRmX0P9gN1pAzTGfLRqZjAPmbtrVmzRj377LNW+RoIpST4bEhMjFIuLtkKlHr99Wyty7EpUf+YrDKdXYv9sCnph0qGk6v6672TzUJv69atytHRUb3zzjsqIyNDzZs3T3Xr1k3de++96qOPPlIXLlzQ9gRtSM+epa+y8uOPWldXPpcvX1br1q1TH330kXr88cfV3XffrRo0aKAGDhyo3n//fbV69Wp18eLFcrd7fO1mlensWqF5fJnOrir+xy1VcLairORSp404fRo6d84iJcWNIUMy+f772tSkW0oTJkzgrrvuYvz48cVeK1wTMS+nzO1lOrkS0X0i43+dSFxcHN27dyc9PR0nJyd0Oh2PPfYYISEhDBw4ECfZqh6Apk3hwoWSpyTY6rZAN0tKSiImJqbwcmV0dDQpKSkEBgYW/nnggQdo1qyZRUbjVuT7MsvZVRaqtgESfDbg6lUICMjg4kV3unfPYPt2d1xctK7KehITEwkICODkyZO4ubkRFRXF+++/T0xMDBiMBKWl8Y0yUta7be8Dq4CjwEtPPM33W38uHCru4uJCgwYN2LBhA/7+/lV0Rvaldm3IyioeeA4OOgwGraq6vbS0NPbv318YcNHR0fz555906tSpMOACAwNp1apVlQ5KKgg/lzzZncGeSPBpLCMDundP5/BhD1q3TicmxoOaNpr5888/Jz4+nlmzZgGwYcMG0tPT6d+/P4eCBjN7/69cAjaWsb0FQCNgJuDl1YjIlCs4OztTq1YtcnJyaNasGf3792fGjBl3aKl6s5dtgbKysjhw4EBhyO3du5ezZ8/i7+9v1ptr165d4QhJazqxbitpkyZz//6dqNL24+vUE89J79NmUB+rdi0sAAAAFe1JREFU1yeKk+DTUH4+9O+fwZYt7jRsmMmBA7WpiYMI+/bty+jRoxkxYoTZ89cSzuPRphVxhjx6A+WdNjwC8NE58MbJBHR1anP16lUSExM5c+YMH3zwAQkJCZY6BbtRfA5ewd/hnnt0/PGHRoXdkJuby6FDh8xGWMbHx+Pr62vWk/Pz88PZ2bYWab+ecIHj02bgFHcY57QU8jy9yPdrT9u3x8kO7DZGgk8jSsFLL2URGemGu3s20dG1aNdO66q00bBhQ9avX184UbjA7tB3CJgznfD8XJYBUeVsdwTQXOfA4JB/8ODMzwqfv379OvXr1yclJaXGTBbeto0bOyTArYHXvbuO3butX1N+fj5Hjx41uyd3+PBhWrVqZRZy/v7+1KpVy/oFimpL7uprZOLEXCIj3XByyuPnn2tu6MHt10Q8kZ/Lx8DqCrZd09dEnDYN3nmn5Dl4o0c7MGeOdeowGo3Ex8eb9eQOHDhAkyZNCgPuxRdfpFOnTri7u1unKFFjSfBpQK838OmnLuh0Rr77zokePbSuSFulrYl46eoVwoB/A70q0X5NXBPxuefgu+8KAq9gcIcp8ObMcWD06KpbR1MpRUJCQrEJ4fXr1y+8HzdlyhQ6d+6Ml5fXnRsUwsIk+Kxs7VrFX/5i+qEzY4biqadq0JyFUty6JiLA2bNnCU04wgfAyEq2X5PWRPTzgyNHCi5p3hx4sH+/joAAywaeUoqLFy+aXa6Mjo7Gzc2t8HLlu+++S5cuXWr8DhbCdkjwWVFUFAwdmodSLrzzTi5/+UsNmrNwG8HBwWzfvp3hw4cDcPHiRfr27cvQgK68HLMbbholB7ImYkm8vSE5ueq3Bbp8+XJhuBWEXcG6loGBgYwbN47AwEBZ6k3YNBncYiXx8dC5czYZGbUYNiyLxYvdqOE72gDc+BokAp2AeMAN+Aj4CLdatXDIzi58b/qN/9r7mohlWWu0NAXfM0lJULduadsCgZOTjry8ytV5/fp1YmJizHpyaWlpZpPBAwMDadq0aY3fnknYFwk+K/jzT+jYMYMrV9zp3TuDTZvcsbGR2Jop+nn5HqbZdwUrtyiOHj1O5ogwOsZsN5sc/Bim+36+d2jbgI4DgUF03rul8Lm1a9eyaNEili9fbqlTKLP4H7eQPmkyfrG/YSxtvlfAQ3hMep/7Bvct9nmdzkjR5UsoaUpC3bo6kpLKX1taWhr79u0zC7nLly/TuXPnwoArmBAuISfsnQRfFUtLg8DAdOLjPbj//nR+/90DGbRm8sknMHHizT+8CyjAgFJOxP+4haZDg8u1LFSBLGdXLvywwSYmDVd2hQ9X1zxyc50o+WsFbdvqOHasbLVkZmYSGxtrdrny3LlzdOzY0aw3d99992kyIVyIqibBV4Vyc6FPnwx27XKnceMMYmPdadhQ66q016oVnD596/2omxlRqqhnYw9rIlp6rdGb6+/SJYV9++pg+lqtAZYAywAjQ4Y4sHJl6e3k5ORw6NAhs57ciRMnuP/++80uV95///02NyFciKoiwVdFjEZ47rlMVqyoTZ06Wezb50arVlpXpZ2kJKhf3zQK0MT8Et3Nj5UqHoa2vCbirWuN5ubm8uKLLxIdHc3Zs2fZ6OhMf0PZb7idAf4P2APc3aQpCRfnAP1uekd7TOHnz83/ev+/vXsPjrK+9zj+fjYXsrmAgeIVLDdBkoqLgnAK0oAgk+CAijrQgh6CtThgDxfHc9SgYu206kzxDNJRzFAq4TKgFeVyZpQoF5kx1IgSiRKEhAKVlgBRkmyySfY5fzxsNpvdQEI2u8nu5/UP5snmeX7LMH7y/e3v9/3V19dTXFzs09qruLiYQYMG+UxXDhs2jG7dugXhnYt0TVrV2UGWLKnl3XcTiY93kZ8fvaGXlwezZzcNN88CDBPDMDh71mDPHrj3XusV584F/vxo1Cs5HPnFmFb3RBwVwunNNWvWkJWVhd1ub7w2duxYFi5cyL3j7yK22arUy5kJ/AewFVgZl8z/8CDWwh/PdMEM4E1gBWvXrmsMuoMHD9K3b9/GgJs9ezYOh4PExMRgvE2RiKGKrwMsX17P4sWx2GwNbNtmIzMz+hYDOBzw1VeBOoZA374G//jHld+7s/VEvFSv0VsH3EgekNHKe5UAtwDlQApQExNHxvBbMWJj+eKLL3C5vCE6YMAAn+nK2267LSL3JooEmyq+INu0yc3ixdbnU7m5RtSFXlwc1NcH2kBt8tJLNp59tv3P6Nm/j0/vzXArKipiyJAhftdLXn79EpOygR0CBmCFHoBpGCR/f478U8f8jtc5cOCAgk7kCqhtSBDt2QO//GUDYGPZsjrmzImOv978fGtbgmGY1NdD8wrv3DkD0wxO6HVGl+o12tZfeyqBpn1m7PUu+toMHnjgAXJzc5k6dSrJycmNzxWRtouO/zOHwNdfw+TJtTQ0xJGdXcPSpZG/Qm7SJGtv2cSJzTdQm/TqZWCaBqbpOQYncrXUazQuwLXLSQZ+bHbN6azm2muvZc6cObz//vuNxylFcq9RkY6k4AuCEycgI6OamppuTJ5czapVCRHdlSUhwarudu40sf4JeQLPzdy5VuCVl4d3jKHk6TXq4TnzrqTcvwq8nHTgGL5nDx521ZKenu79+vDhiO01KhIKCr52qqiAceOqOHs2EYejivfeSyQS9/weO+adzqytBd8VmnD0qDWdmZsbxkGGyfjxWcyZsxvDMDEMk9RUk4qKWg4YQzEBF1CDd+PGGqBfC/caDDiwmrbVABttsXxXU8X06dMbXxOJvUZFQkmLW9qhpgYmTqykrCyZG2+sIj8/iSYr2iPC7NmQl9d0daY37JKSDCorI7i0bUFZGaSlgdPpibLZWL1Ga7B6jQIM4Y8njwMw+eKVUqzAOwGMucT9N2L1G00F+rrr+Wvu2/Ru0vlgw4YN5OXlBeOtiEQlVXxXyO2G6dOrKSxMJjW1mr17k+jZM9yjCp4ePazqLi/Pfzpz2jRrOrOy8tL3iBRlZd7pXcMw6d/fxOkE7y8CvbHC7w3AJCUFzp8vwzRNCm/PoB4DE2+Vtxe41Nb6fsAuoBKDjSPGc//smY3f27p1K0OHDg1bg22RSKB9fFfANGHevBpWrUogIaGWgoJuDBsW7lG13/nzXAzv5v0zra8//xxuvz0sQwupL7+EO+6AujpPpQu00CMTrMbQpaUEPPonUnqNikQSVXxX4A9/qGPVqgRiYurZsaPrh96TT1qrM3v29F+dGR9P4+rMSA29XbsgNtb6OzAMk+HDzYtH+ngqXe/fB0Dv3nD+vGfVqtF4RFAgg++ZwMGFOTjj2tYizNOrU6EnEnyq+Nro7bfdPPKI9fvC+vVuZs7sur87XHcdnD4dqHemyc9/bmPfvjANrINt3w5Tp4Lb3ZqKzuD66+HUqfY9szP3GhWJNgq+NvjwQ5PMTDdudwyvvlrPk092vbVBvtOZ0Hyz+ZYtBtOmhWFgHWj1avj1r1sfdP37W6tYg+3I9k9a3WtUlZ5Ix1HwtdKBAzB6dC0uVzcWLKhlxYqu1d3+tddg0aLAYRcTY1zsuBIZVq6EJ54A02xd0KWlwaFDoRtfZ+s1KhJtFHytUFoKw4c7+eEHO9OmVfO3vyVi6yIznIMHw5EjgQNv6FCD4uKwDCuoXnwRnn8e8JlCbDnoHA7rFxkRiU4KvssoL4fhw6s4eTKJUaOq2L07ia5wlJnN1vLZd2+8YfCb34RlWEGxeDEsXw6tDbqxY2Hv3pAMTUS6AAXfJVRXw+jRlRQVJTNoUCWFhcl05i5R/mffgScArLPvumbfzLlzrc/pWg463+tZWdYCFhGRQLre6owQqa+HqVOrKCpKpnfvKnbv7ryhN2IEFBYGPvvuhhsMTp4M18iuzEMPwebN4B90vitPPdcffNBg06YQDlBEurQu8klVaJkmZGc7yc9PIjGxhj17krj++nCPyl98vNVNpLDQv7vKs89ae8y6QuhNmeLtA2oYJps3Nz+xHZqGXXa2dw+daaLQE5E2UcUXwNKlLtautRMbW8dHHyVw883hHpFXfj5MnAje7ipNN1cbnDtnkJrauftn3nknfPoptLaiW7TI4E9/CuEARSSi6TO+Zt58s4F582IwDDfvvmtw332dI0QmTYKdOwNPZ151ldU9pLMaORI+/xz89w56+AbgsmXw3HMhGZqIRCFVfE1s3Wry+OPWf69YYXLffeGfCU5KgupqTzB4xmNVRHPnds5jgNLTubhNonnQ+QY2mBiGjRUrDObPD+UIRSSaqeK7qKAAxo6to74+jqeecvHyy/FhG8uxYzBwILTULProURgwICxDC2jAAGuv4+UrOhObzcZbb0F2dsiGJyLiQ8EHlJTA7bc7qay0M2OGk/Xr7WE5Qd3/7DvwhEZiokFVVejHFMgNN8A//wmtDbp162DGjJANT0TkkqI++E6fBoejin/9K4lx46rYuTOJuLjQjiE1FSoqAjeLnjLFxrZtoR1PUxUVVveXM2c8Y4JLBV1MjI2dOyEjI1QjFBFpm6gOvgsXYOTISg4fTmbo0Er2708mOTk0z750s2iDnTvhrrtCM5amKiqgf/+mQdx0bB7eoIuLs7F/PzgcIRqgiEg7hX/1Rpi4XJCZWcXhw8lcd10Vu3aFJvRycgKdfQdWiHjPvgtV6FVUQEqKdw9daqpJRYVnXM23S5gkJEBpqWcPnQ2XS6EnIl1LVK7qNE341a+c7NuXREqKk717k7j66o59pu/Zd76rM0eNsvHZZx37fI+yMkhLA6ezpT10nnFZ7HaD4mKDfv1CMz4RkY4WlRXfkiW1vPOOnfh4Fx9/bL+4gjL4zp/3diSxQs+3glq71qqaOjL0ysogIcFb0fXvb+J0QksVXUqK7+ni1dUo9EQkokRd8L32Wj3Ll3fDMBrYsiWOESM64hlW0ASazrROTbBCZdas4D/7yy89rczcjUFXW+sZQ/Ogg6uu8g26H3+0romIRKqomurcvNnNokVW1r/1FmRmBnfPQloafPNN4NMRbrrJoKQkqI8DYNcuq4VZQ0PzQ1eb/k7jHdM118C33xoKNxGJWlFT8e3dCzNnNgA2XnjBxdy5MUG7d0yMVeFZoedbVS1fblVSwQq9jRs9z7MquvHjTRoawNuk2rei69MHn4bOp0+rohOR6BYV2xkOHYKRI2twOhPIzq4hNzeh3RvU338f7r03cHVnNYsOztl3q1fDo4+CaTav6JryPnfQIDhypP3PFRGJVBEffCdPgsNRzdmzidx9dxU7diQR045ib/RoKCgI3F3l2msNvv++feNduRIWLABoXdClpVnBLiIirRPRU50VFTBuXCVnzybicFSyZcuVh163btZ0ZkGB/9l3S5ZYU4lXEnovvuh7Ft2CBU23PPhPXY4Y4Tt1qdATEWmbiFvcsmWLFQa//S1MmlRFaWkyN95YRX5+MnZ72+5VWMjFVZ9NP7vzfn0lZ98tXgzLl3vu4dHyWXQZGfDJJ53jaCQRkUgQUVOdLpdJr15uKitj+MlPGigvjyE1tZoDBxL56U9bf5977oHt2wNPZ3bvbvDDD62/19y51ud0/kHn4Xs9Kwu2b2/9/UVEpG0iaqpz9eqvqay05jLLy63DZN95p/Whl5RkTTlu3+4/nTlrljW9eLnQe+gh36nL1aubLoDxr+o89/VMXSr0REQ6VkRNda5ff8bna9O0MX8+HDxIiycu+J995zudefSowYABLU81Tp4MH37oeb1Hy1OXjz9u8Oc/t+FNiYhIUEVUxVdQ4H86a0MD2AK8y0cftfbCDRzYtCKzQspu9zaLbn7g6513+lZ0Vuh5ft4/7J5+2reiU+iJiIRXl6n4zpaeoOTl14k9VETchQvUpaRQn34LQ/77CXr270N5+XlcLu+cZvfuf2f+/Hief/5Wn5WcvXrBuXOBm0VPnGjjo498nzt8uNUGzP/4oOZ796xry5YZPPdc+9+viIh0jE6/uKVk28dUvvA70r/ch9swsNe7Gr/njI3HME2KHWP49p5MZr2YRp8+/fjLX1K46y5vCLbl7Lubb4bDhwO9liY/Y/1pGDZWrID584P3fkVEpGN16uAreOolhr32EvF1LmJoeZgNGLji4vnqv3IY/WpO4/WcHPj97wOvzoyNNairs04eOH7ce/1SQWez2Vi3DmbMaO87ExGRcAlr8D399NNcc801LFy40O97ntCz19W2+n7OuG4cXJjDgxtzOHGiaZB9AKwHNmAY1mdtrQ26Dz6AKVPa8KZERKRTC1vwnTlzBofDwXfffYfdbuezzz5j6dKlFBYWQoObjAsXWGm6ua6V9ysD5gAFgGnrS417NTAR72rNnwHrgFub/aQ36GJjrc/4MjLa++5ERKSzCtuqzjVr1pCVlYX9YjuV8+fP89hjj1FWVsbWgcPoYbqZ04b7zQSGA/8GFvboDjwAnMFb0c0E3sK74tJNXBwcOOBZcWmjrk6hJyIS6cJW8U2YMIHs7GxmNTuN9WzpCZJvGsihhjp+AVxoxb1KgFuAciAFqImJo7vbQZ2ZDcy7+Kp9wCyglM77qaaIiHS0sFV8RUVFDBkyxO96ycuv4zYM9gDprbzXIWAAVugBmIbBuOtdwNd4K7ybsSZEf2znyEVEpCsLW/BVVFSQkpLidz32UBFH6l28CLzayntVAj2afG2vd9HXVskjj1Q2bh7fv787ADt2VLR36CIi0oWFLfhSU1O5cMF/IvP78n+TCfwvcGcr75WMfx3ndFb7BOvAgdazxozR8eMiItEsbME3bNgwSkpKfK4dP36ceaXFLAVmt+Fe6cAxfD8PPOyqJT3dO1n6zTff0K9fP7p3796OUYuISFcXtuDLyspi9+7djV+fOnWKCRMmcL/jDh6Jjfd7/RqgXwv3Ggw4gGVADbDRFst3NVVMnz698TW7d+8mMzMzaOMXEZGuKWy9Oh9++GEcDgdOpxO73U5ubi7Hjh1jzenTrGnSlqzy4p8ngDGXuN9G4D+BVKCvu56/5r5N7969G7+/YcMG8vLygv4+RESkawlr55ZnnnmGq6++2q9zyxcjxnNr4W6fNmV3Y33uN/Qy92zA4KsRGdz2948br23dupW1a9eyadOm4A1eRES6pE7Zq7Nk28f0vT+rTe3KPJxx3Tj53v9x05TxHTAyERHp6jrleXyD75nAwYU5OOO6tennPL06FXoiItKSTlnxebT1dIaDC3MY9UpOi68TERHp1MEHcGT7J1x44XekHfgUs6Xz+IaPJeWFpar0RETksjp98HmcKz3J4VdeJ/bQ18Rd+IG6lB7Up/+MIU8toGf/PuEenoiIdBFdJvhERESCoVMubhEREekoCj4REYkqCj4REYkqCj4REYkqCj4REYkqCj4REYkqCj4REYkqCj4REYkqCj4REYkqCj4REYkqCj4REYkqCj4REYkqCj4REYkqCj4REYkqCj4REYkqCj4REYkqCj4REYkqCj4REYkqCj4REYkqCj4REYkqCj4REYkq/w/iGTc+24oJEgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from numpy.random import choice, rand \n",
    "\n",
    "# Simulates the best response dynamics as a random walk on the transition graph G\n",
    "# starting from node xi.\n",
    "# if till_first_return = True the random walk stops the first time\n",
    "# it returns to the starting node xi.\n",
    "# Otherwise, it goes on for num_steps steps.\n",
    "\n",
    "def RandomWalk(G, xi, num_steps, till_first_return = False):\n",
    "    # nodeSeq stores the sequence of visited nodes\n",
    "    nodeSeq = []\n",
    "    nodeSeq.append(xi)\n",
    "    \n",
    "    # if the walk ends at the first return to xi\n",
    "    if till_first_return:\n",
    "        # stores the initial position to check if the \n",
    "        # walk returns to it\n",
    "        x_init = xi\n",
    "        \n",
    "        while True:\n",
    "            # compute the next visited node xi by chosing uniformly\n",
    "            # at random a neighbor of the current one\n",
    "            neighbors = list(G.neighbors(xi))\n",
    "            index = choice(len(neighbors),1)[0] \n",
    "            xi = neighbors[index] \n",
    "            nodeSeq.append(xi)\n",
    "            \n",
    "            # check if the walk has returned to the starting node\n",
    "            # if so, end the walk\n",
    "            if xi == x_init:\n",
    "                return nodeSeq\n",
    "    \n",
    "    # if the walk ends after num_steps steps\n",
    "    else:\n",
    "        for i in range(num_steps):\n",
    "            neighbors = list(G.neighbors(xi))\n",
    "            index = choice(len(neighbors),1)[0] \n",
    "            xi = neighbors[index]\n",
    "            nodeSeq.append(xi)\n",
    "        return nodeSeq\n",
    "\n",
    "# Simulate the BR dynamics \n",
    "nodeSeq = RandomWalk(transition_graph, xi=(0,0), num_steps=100, till_first_return=False)\n",
    "edgeSeq = [(nodeSeq[i-1], nodeSeq[i]) for i in range(1,len(nodeSeq))]\n",
    "\n",
    "# Draw G and represent the random walk by colouring the edge sequence\n",
    "# first draw all nodes and links\n",
    "nx.draw(transition_graph, pos)\n",
    "# then, on the previous picture, add node labels and highlight the edge sequence\n",
    "nx.draw(transition_graph, pos, with_labels=True, edgelist = edgeSeq, edge_color='blue', node_color='red', width=2)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see that the dynamics starts at configuration (R,R) (represented by (0,0)) and after one step it exit from it. Then it continue cycling over best response cycle $(0,2) \\to (1,2) \\to (1,0) \\to (2,0) \\to (2,1) \\to (0,1) \\to (0,2)$. In particular, it never goes back to $(0,0)$ and $(1,1),(2,2)$ are never visited."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Frequencies: [0.000999   0.16583417 0.16683317 0.16683317 0.         0.16683317\n",
      " 0.16683317 0.16583417 0.        ]\n",
      "The approximate invariant distribution is supported on nodes:\n",
      "(0, 0)\n",
      "(0, 1)\n",
      "(0, 2)\n",
      "(1, 0)\n",
      "(1, 2)\n",
      "(2, 0)\n",
      "(2, 1)\n"
     ]
    }
   ],
   "source": [
    "# Esitimate the invariant measure pi of the BR dynamics by \n",
    "# simulating a \"long\" random walk\n",
    "\n",
    "nodeSeq = RandomWalk(transition_graph, (0,0), 1000, False)\n",
    "\n",
    "\n",
    "# Compute empirical frequencies\n",
    "\n",
    "# Define a dictionary to store frequencies of visit to each node\n",
    "frequencies = {node:0 for node in transition_graph.nodes}\n",
    "# count the visits to each node\n",
    "for node in nodeSeq:\n",
    "    frequencies[node] += 1\n",
    "# transform dict values to a np.array\n",
    "frequencies = list(frequencies[node] for node in transition_graph.nodes) \n",
    "frequencies = np.array(frequencies, dtype=\"double\")\n",
    "# normalize the counts to obtain frequencies\n",
    "frequencies /= len(nodeSeq)\n",
    "print(\"Frequencies:\", frequencies)\n",
    "\n",
    "# Print the indices of nodes where pi is positive\n",
    "print(\"The approximate invariant distribution is supported on nodes:\")\n",
    "for index,node in enumerate(transition_graph.nodes):\n",
    "    if frequencies[index]>0:\n",
    "        print(node)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The approximate invariant distribution appears to be supported on nodes $\\{(0,0),(0,1),(0,2),(1,0),(1,2),(2,0),(2,1)\\}$ of the transition graph. However, the measure of node $(0,0)$ is negligible with respect to that of $(0,1),(0,2),(1,0),(1,2),(2,0),(2,1)$, which are almost equal to each other. Actually, the little weight associated to node $(0,0)$ appears to depend on the fact that it has been visited once since it is the starting point of the chain. We guess that when the length on the walk tends to infinity the weight associated to node $(0,0)$ vanishes and the approximation of $\\pi$ is supported and uniformly distributed on $\\{(0,1),(0,2),(1,0),(1,2),(2,0),(2,1)\\}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> 6. Compute and plot the condensation graph of $\\mathcal G_{B}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{0: {'members': {(0, 1), (1, 2), (2, 1), (2, 0), (0, 2), (1, 0)}}, 1: {'members': {(0, 0)}}, 2: {'members': {(1, 1)}}, 3: {'members': {(2, 2)}}}\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "CG = nx.algorithms.components.condensation(transition_graph)\n",
    "nx.draw(CG, with_labels=True)\n",
    "print(dict(CG.nodes))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The condensation graph is shown above, where the hypernode $0$ contains all configurations except for $(R,R), (S,S), (P,P)$ (the ones that correspond to a draw). More precisely, the hypernode $0$ contains the $6$-cycle $(0,2) \\to (1,2) \\to (1,0) \\to (2,0) \\to (2,1) \\to (0,1) \\to (0,2)$. The hypernodes $1,2,3$ instead contain the singletons $(R,R), (S,S), (P,P)$. The condensation graph contains only one sink, that is $s_{\\text{transition_graph}} = 1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "7. Compute analytically the invariant distribution $\\pi$ of the best response dynamics."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We know that the invariant probability distribution is supported on the nodes contained in the unique sink of the condensation graph. The subgraph corresponding to such sink is balanced (all nodes have in-degree $1$ and out-degree $1$, so the sink is $1$-regular), then the extremal invariant probability distribution supported on that sink is obtained by normalizing the out-degree vector. In this way we find that $\\pi_x = 1/6$, for $x \\in \\{RS, RP, SR, SP, PR, PS\\}$ and $\\pi_x = 0$ for $x \\in \\{RR, SS, PP\\}$.\n",
    "\n",
    "We can check that this is correct with the following code."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "pi= [-0.          0.16666667  0.16666667  0.16666667 -0.          0.16666667\n",
      "  0.16666667  0.16666667 -0.        ]\n"
     ]
    }
   ],
   "source": [
    "# Compute P matrix\n",
    "A = nx.adjacency_matrix(transition_graph) # -> return type is scipy.sparse.csr_matrix\n",
    "A = A.toarray() # convert A to a numpy array\n",
    "degrees = np.sum(A,axis=1)\n",
    "D = np.diag(degrees)\n",
    "P = np.linalg.inv(D) @ A\n",
    "\n",
    "# Compute invariant distribution\n",
    "values,vectors = np.linalg.eig(P.T)\n",
    "index = np.argmax(values.real)\n",
    "pi = vectors[:,index].real\n",
    "pi = pi/np.sum(pi)\n",
    "print(\"pi=\", pi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Continuous time best response dynamics\n",
    "In this section we present the continuous time best response dynamics, which is the continuous time analog of the process presented in the previous section.\n",
    "\n",
    "\n",
    "Consider a game $\\left(\\mathcal V, \\mathcal A, \\{u_i\\}_{i \\in \\mathcal V}\\right)$. \n",
    "\n",
    "The **continuous-time asynchronous best response dynamics** is a continuous time Markov chain $X(t)$ with state space $\\mathcal X= \\mathcal A^{\\mathcal V}$ coinciding with the configuration space of the game and transition rate matrix $\\Lambda$ as follows: \n",
    "\n",
    "$\\Lambda_{xy} = 0$ for every two configurations $x, y \\in \\mathcal X$ that differ in more\n",
    "than one entry, and\n",
    "\n",
    "$$\n",
    "\\Lambda_{xy} = \\begin{cases}\n",
    "|B_i(x_{-i})|^{-1} \\quad \\text{if} \\quad y_i \\in B_i(x_{-i}) \\\\\n",
    "0 \\quad \\text{if} \\quad y_i \\notin B_i(x_{-i})\n",
    "\\end{cases}\n",
    "$$\n",
    "for every two configurations $x, y \\in \\mathcal X$ differing in entry $i$ only, i.e., such that\n",
    "$x_i \\neq y_i$ and $x_{-i} = y_{-i}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Continuous time BRD for network games\n",
    "\n",
    "Network games are games where players are associated to nodes of a graph $\\mathcal G$ that describes the interactions among them. In particular, for each player $i\\in\\mathcal V$ and for each couple of configurations $x,y\\in\\mathcal A^{\\mathcal V}$ such that $x_j=y_j$ for each $j\\in\\ N_i\\cup\\{i\\}$ it holds that\n",
    "$$u_i(x)=u_i(y)$$\n",
    "This means that the utility of each player only depends on the actions of its neighbors in the graph $\\mathcal G$.\n",
    "\n",
    "\n",
    "Given an unweighted undirected graph $\\mathcal G=(\\mathcal V,\\mathcal E,W)$ we can construct a network game on it by setting the players’ utilities to coincide with the weighted sum of the payoffs of the same symmetric two-player game played by the player simultaneously with her neighbors.\n",
    "\n",
    "Specifically, for a symmetric two-player game with action set $\\mathcal A$ and utility function $\\varphi(x_1, x_2)$ we define the network game $(\\mathcal V, \\mathcal A, \\{u_i\\}_{i \\in \\mathcal V} )$ by setting the utility of every player $i \\in \\mathcal V$ as\n",
    "$$\n",
    "u_i(x) = \\sum_{j \\in \\mathcal V} W_{ij}\\varphi(x_i,x_j)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise - continuous time BRD for network coordination game\n",
    "A binary coordination game is a symmetric $2 \\times 2$-game with action set $\\mathcal A = \\{0,1\\}$ with payoff matrix $\\varphi$\n",
    "\n",
    "|   | 0   | 1   |\n",
    "|---|-----|-----|\n",
    "| **0** | a,a | d,c |\n",
    "| **1** | c,d | b,b |\n",
    "\n",
    "where $a > c$ and $b > d$. The inequalities above imply that the best response for each player is to copy the action of the other player.\n",
    "\n",
    "It is a potential game with potential $\\phi$\n",
    "\n",
    "|   | 0   | 1   |\n",
    "|---|-----|-----|\n",
    "| **0** | a-c | 0 |\n",
    "| **1** | 0 | b-d |\n",
    "\n",
    "Consider the **network coordination game**, obtained combining binary coordination games on the link of the following undirected graph\n",
    "\n",
    "![network](network.png)\n",
    "\n",
    "Recall that such network game is potential with potential function $\\Phi$\n",
    "$$\n",
    "\\Phi(x) = \\frac{1}{2} \\sum_{i,j} W_{ij} \\phi(x_i,x_j) .\n",
    "$$\n",
    "\n",
    "1. Simulate the dynamics when the coordination game has the same utility value for coordination of action 0 and action 1 respectively, i.e., $a=b=1$ and $c=d=0$. Run the simulation a few times and plot the potential function as a function of time. What do you observe?\n",
    "\n",
    "**Hint**: you need to fix the best response function to deal with more than 2 players. To do this it is useful to define a Python function to represent the utility functions of the network game. You can find such function in the solution to the next exercise."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "# TO DO"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Remark**: You should able to check the following fact.\n",
    "For potential games, the best response dynamics converges to the set of Nash equilibria. More precisely, it converges to Nash equilibria that are **local maxima of the potential function**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Network games and Noisy Best Response dynamics\n",
    "Consider the graph in Figure below, where a network coordination game takes place.\n",
    "![network](network.png)\n",
    "\n",
    "We consider a **continuous time asyncronous noisy best response  dynamics** with inverse noise parameter $\\eta=2$, i.e., a continuous time Markov chain on the configuration space $\\mathcal X$ (the set of all game's configurations) with transition rates $\\Lambda_{xy} = 0$ for every two configurations $x, y \\in \\mathcal X$ that differ in more\n",
    "than one entry, and\n",
    "\n",
    "$$\n",
    "\\Lambda_{xy} = \\frac{\\exp^{\\eta u_i(y_i,x_{-i})}}{\\sum_{a \\in \\mathcal A} \\exp^{\\eta u_i(a,x_{-i})} }\n",
    "$$\n",
    "for every two configurations $x, y \\in \\mathcal X$ differing in entry $i$ only, i.e., such that\n",
    "$x_i \\neq y_i$ and $x_{-i} = y_{-i}$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1.  Simulate the dynamics when the coordination game has the same utility value for coordination of action 0 and action 1 respectively, i.e., $a=b=1$ and $c=d=0$. Run the simulation a few times and plot the potential function as a function of time. What do you observe?\n",
    "2. Simulate the dynamics when  instead $a=1$, $b=\\frac{1}{2}$ and $c=d=0$. Run the simulation a few times and plot the potential function as a function of time. What do you observe?\n",
    "3. If the simulation continues for a long time, will the probability for different configurations converge towards some specific values? In that case, how can these be computed?\n",
    "4. Compute the ratio between the probability that after a long time (i.e., in stationarity) all nodes choose action 1 and the probability that all nodes choose action 0, both for case **1** and case **2**. Does this seem to agree with your simulations?\n",
    "5. Compute the ratio between the probability that after a long time (i.e., in stationarity) one node chooses action 1 and the rest choose action 0, and the probability that all nodes choose action 0, both for case **1** and case **2**.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solution\n",
    "#### Step 1\n",
    "> Simulate the dynamics when the coordination game has the same utility value for coordination of action 0 and action 1 respectively, i.e., $a=b=1$ and $c=d=0$. Run the simulation a few times and plot the potential function as a function of time. What do you observe?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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fNhYMKVbZ5PlKlp2jRKJAwyr+RsmzzlHLgZ4j7nUbPQ37mu37vHy5JCkhac073p8HAEyR7TZ2nPhEbYc/0pDyv+r1OS/b6GnYGw4c7XHbTj5S7Z1q2H/M03MAgEmy2cZ0R7sOvfxjjfjStSoafX6vz3vZRk8/oHQ01e7ly7usXP2inrjlr315LgCIWvnN39OwS/r/BdjpdKcOvfq/pMKkzp5+d7+PO5pqy+v1PQ17abE/P7g6f+5NWrrmUV+eCwCi9ner3tVL733c5+fS6bSa1v1vdZxo1ti5jyhR2P+Olhbn97udPV2KqTy3VEOTvZ8i3dmhdHur1NkhpTuVbm9VurOjj2c4/VNXleeN9HIYAGCU/rZRkg5veEptTXs1ds7DKijqPyPgZRs93cd+6HiLvvL4m72uJTW/tVJHflvb42NnfWWeyibP7/UcQ5MF+t13/9bXTgIARKm/bWw/8hd9tPxOqbBIiYLCro+f/bV7NOKyqT0e62UbPV1LGTNiqKouLdfGDw72uK2nbPL8Pkf8TImENHV8OaMOwCn9bWPyrLH6wn9/ddCv97qNnn9A6Z7qChUnCwd/YB+Kk4VaVE2mF4B77qmuUFL5XRDxuo2eh/2K88v00MxKlRTl9lSnewiV5AQAOOn3df+o1O//j4YWJnL6Oj+20ZfbWjKxmrALZgBgmnQ6rccee0zPPvustm7cqN/9pcDOumPG+/uatWzLLm3+sFEJnb7BPiPTHJ46vlyLqis4UwfgnHQ6re9+97tat26dXn/9dX3uc5+TFP42+jrsGU3HW7TmnX1q2H9MK1e/qPlzb1LleSM150p/f0sIAJiio6NDCxcu1I4dO7Ru3TqNHj2612PC2sZAhr3HCyQSCvglACBSra2tuu2229TU1KSXXnpJI0aMGPRrgtzGyH+ZNQDY7OTJk7rhhhvU2tqqV199NatRDxrDDgB5am5uVk1Njc455xytXr1axcXFUR+SJIYdAPJy8OBBTZ06VVdddZV+8YtfKJn0p53lB4YdAHK0Z88eTZ48WTfeeKOWLl2qggKzptSsowEAwzU0NGjy5Mm655579P3vf1+JRG4/gBQGc753AADDvfPOO5o1a5Yee+wxLViwIOrD6RfDDgBZqK+v15w5c/TMM8/oxhtvjPpwBsSlGAAYRF1dnebMmaPa2lrjR11i2AFgQLW1tbrrrrv0yiuv6Nprr436cLLCpRgA6MfTTz+txYsX64033tDll18e9eFkjWEHgDN0LzTW19froosuivqQcsKwA0A33QuNb731Vleh0SYMOwB8qnuhcevWrX0WGm3AsAOAehYaN23aZETMK1/cFQMg9kwsNHrBsAOItUyhcezYsUYVGr1g2AHEVqbQOHHiRP3yl780qtDoBcMOIJa6FxqfeOIJ4wqNXrjzbwIAWbKh0OiFG993AECWbCk0esGwA4gNmwqNXnApBkAs2FZo9IJhB+A8GwuNXnApBoDTbC00esGwA3CS7YVGLxh2AM5xodDoBcMOwCmuFBq9YNgBOMOlQqMX3BUDwAmuFRq9YNgBWM/FQqMXDDsAq7laaPSCYQdgLZcLjV7wXwGAlVwvNHrB9ywArBOHQqMXDDsAq8Sl0OgFl2IAWCNOhUYvGHYAVohbodELLsUAMF4cC41eMOwAjBXnQqMXDDsAI8W90OgFww7AOBQavWHYARglU2g8dOiQ3njjDY0cOTLqQ7IOd8UAMEb3QmNdXR2jnieGHYARKDT6h2EHEDkKjf5i2AFEikKj//gvCCAyFBqDwfc7ACJBoTE4DDuA0FFoDBaXYgCEikJj8Bh2AKGh0BgOLsUACAWFxvAw7AACRaExfAw7gMBQaIwGww4gEBQao8OwA/AdhcZocVcMAF9RaIweww7ANxQazcCwA/AFhUZzMOwAPKPQaBb+6wPwhEKjefheCUDeKDSaiWEHkBcKjebiUgyAnFFoNBvDDiAnFBrNx6UYAFmj0GgHhh3AoCg02oVhBzAgCo32YdgB9ItCo50YdgB9am1t1a233qqmpiYKjZbhrhgAvZw4cULXX3+92traKDRaiGEH0ENzc7NmzJihc845h0KjpRh2AF0oNLqBYQcgiUKjS/g/B4BCo2P4PguIOQqN7mHYgRij0OgmLsUAMUWh0V0MOxBDFBrdxqUYIGYoNLqPYQdigkJjfDDsQAxQaIwXhh1wHIXG+GHYAYdRaIwn7ooBHEWhMb4YdsBBFBrjjWEHHEOhEQw74BAKjZAYdsAZFBqRwfdogAMoNKI7hh2wHIVGnIlLMYDFKDSiLww7YCkKjegPl2IAC1FoxEAYdsAiFBqRDYYdsASFRmSLYQcsQKERuWDYAcNRaESuuCsGMBiFRuSDYQcMRaER+WLYAQNRaIQXDDtgGAqN8Io/MYBBKDTCD3x/BxiCQiP8wrADBsgUGn/2s5/p61//etSHA8sx7EDE6urqdMcdd6i2tpaYF3zBNXYgQhQaEQTO2IGIUGhEUBh2IGQUGhE0hh0IEYVGhIFhB0JCoRFhYdiBEFBoRJi4KwYIGIVGhI1hBwJEoRFRYNiBgFBoRFQYdiAAFBoRJf60AT6j0Iio8b0h4CMKjTABww74hEIjTMGwAz6g0AiTcI0d8IhCI0zDGTvgAYVGmIhhB/KQKTSuWLFCW7du1cUXXxz1IQFdGHYgR90Ljdu2baPQCOMw7EAOKDTCBgw7kCUKjbAFd8UAWaDQCJsw7MAgKDTCNgw7MAAKjbARww70I1NovOGGGyg0wiqcfgB9aGhoUE1Njb7zne/o29/+dtSHA+SEYQfOQKERtmPYgW4oNMIFDDvwKQqNcAXvBgGi0Ai3cMaO2KPQCNcw7IgtCo1wFcOOWKLQCJcx7IgdCo1wHcOOWKHQiDjgrhjEBoVGxAXDjlig0Ig4YdjhvIMHD6q6uppCI2KDYYfTMoXGG2+8kUIjYoNTFziLQiPiimGHkyg0Is4YdjiHQiPijmGHUyg0Arx5CodQaARO44wdTli+fLmWLFlCoREQww7LUWgEemPYYS0KjUDfGHZYiUIj0D+GHdah0AgMjLtiYJVMobG1tZVCI9APhh3W6F5oXLNmDYVGoB8MO6xAoRHIHsMO41FoBHLDaQ+MRqERyB3DDmNRaATyw7DDSBQagfwx7DBOXV2dbr/9dtXW1mratGlRHw5gHd6FglEyhcZXX32VUQfyxBk7jEGhEfAHw47IUWgE/MWwI1IUGgH/MeyIDIVGIBgMOyJBoREIDnfFIHQUGoFgMewIFYVGIHgMO0JDoREIB8OOUFBoBMLDKRMCR6ERCBfDjkBRaATCx7AjMBQagWgw7AgEhUYgOryDBd9RaASixRk7fEWhEYgeww5fUGgEzMGww7NMobGuro5CI2AAhh2edC801tfXU2gEDMCwI28UGgEzcVcM8kKhETAXw46cUWgEzMawIycUGgHzMezIGoVGwA6cbiErFBoBezDsGFSm0PjDH/5Qt99+e9SHA2AQDDsGRKERsA/Djn5RaATsxLtf6BOFRsBenLGjFwqNgN0YdnSh0Ai4gWGHJAqNgEsYdlBoBBzDsMcchUbAPdwVE2MUGgE3MewxRaERcBfDHkMUGgG3MewxQ6ERcB+najFCoRGIB4Y9Jig0AvHBsMcAhUYgXhh2x1FoBOKHd84cRqERiCfO2B1FoRGIL4bdMRQaATDsDqHQCEBi2J1BoRFABsPuAAqNALrjrhjLUWgEcCaG3WIUGgH0hWG3VKbQeOWVV1JoBNADw26h7oXGJ598kkIjgB44zbMMhUYAg2HYLUKhEUA2GHZLUGgEkC2G3QIUGgHkgnfdDFdbW6s777xTr7zyCqMOICucsRuse6HxS1/6UtSHA8ASDLuBKDQC8IJhNwyFRgBeMewG6ejo0N13363333+fQiOAvDHshqDQCMAv3BVjAAqNAPzEsEeMQiMAvzHsEaLQCCAIDHtEKDQCCAqniBGg0AggSAx7yLZv367Zs2dTaAQQGIY9RBQaAYSBYQ8JhUYAYeEduxBQaAQQJs7YA0ahEUDYGPaAUGgEEBWGPQAUGgFEiWH3GYVGAFFj2H1EoRGACbgrxicUGgGYgmH3AYVGACZh2D2i0AjANAy7BxQaAZiI08s8UWgEYCqGPQ8UGgGYjGHPEYVGAKZj2HNAoRGADXi3L0sUGgHYgjP2LFBoBGAThn0AFBoB2Ihh7weFRgC2Ytj7kCk07tixg0IjAOsw7GfoXmjctGkTMS8A1uGumG4oNAJwAcP+KQqNAFzBsItCIwC3xH7YKTQCcE2sT00pNAJwUWyHnUIjAFfFctgpNAJwWeyGnUIjANfF6p1CCo0A4iA2Z+wUGgHEhfPDTqERQNw4PewUGgHEkbPDTqERQFw5OewUGgHEmXN3xVBoBBB3Tg07hUYAcGjYKTQCwGlODDuFRgD4jPWntRQaAaAnq4edQiMA9GbtsFNoBIC+WTnsFBoBoH/WvctIoREABmbVGfvy5cu1ePFiCo0AMAArhr17obG+vp5CIwAMwPhhp9AIALkxetgpNAJA7owddgqNAJAfI++KodAIAPkzbtgpNAKAN0YNO4VGAPDOmGGn0AgA/jDilJhCIwD4J/Jhp9AIAP4qfOSRRx7x+0kPHW/R87/foxf+aY/+NT1WH6XLtLvppC4cM1zDhnz2d0l9fb1uuOEGPf3005o3b57fhwEARsl2G71KpNPptF9PtmNvs57asktbdzZKklraO7s+V5wsUFpS9fhyLaqq0L73f0uhEUAs5LKNV5xf5vn1fBv2F97erSXrGpRq79BAz5hISEmldWLbr/Xij+7TpEmT/Hh5ADBSLttYnCzUQzMrdeukCzy9pi+3npw+8A90qm3gA5ekdFpqSyc0csoC7dK5frw8ABgp12081dahJes+0Atv7/b0up7P2HfsbdYtK97WqbaOHh8/9MqPldq9Q51tKRUOH6XSSTdr5BUzejympKhQq745SRPGef/WAwBM0t82ZrQd/kgf//xbGl75FY257oEen/O6jZ7P2J/askup9t4HXjpprj6/8Bf6q/tXa+yc76m5/tdqObCrx2NS7R1atmVXr68FANv1t40Zh19/WkPPu6TPz3ndRk/Dfuh4i7bubOzzW4wh5V9QIln06T8llFBC7Z/s7/GYdFra/GGjmo63eDkMADDKQNsoSSf+ZasKioer+AtX9Pl5r9voadjXbN834OebNizTn398sz5ecbcKR5ytkouv6vWYhKQ17wz8PABgk4G2sbPlpJrfWqlRf3vXgM/hZRs93TjZcOBoj9t2zjR6xiKdPf2/quWjBqX+/CclCot6PSbV3qmG/ce8HAYAGGWgbWyu/7VGXFGjZGn5gM/hZRs9DfvRVPugj0kUFKr4/Mt04p8369i761R61fW9HrNy9Yt64pa/9nIoAGCM8pu/p2GXXN3r460H/1WpPTt03h1PZvU8R1Nteb2+p2EvLc7hyzs7e11jz5g/9yYtXfOol0MBAGP83ap39dJ7H/f6eOrPf1L7kYPat+wOSVK6NSWlO7X/0Lf7HPvS4t5XObLhadgrzy3V0OSBXt9ydJxoVmrPDpVU/I0SySFK7X5PJz7YqjHX/X2v5yhOFqjyPH6RBgB39LeNI/79DA3/4pSufz76f19U+5GDOnvGPb2ew8s2ehr2ORPHaekbO3t/IpHQsXfXq2nDMindqeRZYzXq2v+iYZf2/inTtKQ5V47zchgAYJT+trGgqFgq+uyXByWKipVIDlHhsLN6PdbLNnoa9jEjhqrq0nJt/OBgj9t6CoedpXPnPzbo1ycS0tTx5Ro9YqiXwwAAo/S3jWcqmzy/z4973UbPP6B0T3WFipOFeX1tcbJQi6orvB4CABgnym30POxXnF+mh2ZWqqQot6cqKSrQQzMryQkAcFKU2+hLADhTIgu7YAYAJotqG33tsb+/r1nLtuzS5g8bldDpG+wzMs3hqePLtai6gjN1ALER9jb6OuwZTcdbtOadfWrYf0xHU20qLS5S5XkjNefKcbxRCiC2wtrGQIYdABAdX37RBgDAHAw7ADiGYQcAxzDsAOAYhh0AHMOwA4BjGHYAcAzDDgCOYdgBwDEMOwA4hmEHAMcw7ADgGIYdABzDsEgjvDcAAAAgSURBVAOAYxh2AHAMww4AjmHYAcAxDDsAOIZhBwDH/H9xJR7GQl37mgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 360x504 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import networkx as nx\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import scipy as sp\n",
    "\n",
    "# Construct the undirected graph as shown in the picture\n",
    "G = nx.Graph()\n",
    "G.add_nodes_from(range(5))\n",
    "G.add_edges_from([\n",
    "    (0,1),(0,2),\n",
    "    (1,2),(1,3),\n",
    "    (2,3),(2,4),\n",
    "    (3,4)\n",
    "])\n",
    "\n",
    "plt.figure(figsize=(5,7))\n",
    "pos = {3: (0, 0), 4: (1, 0), 1: (0, 1), 2: (1, 1), 0: (0.5, 1.5)}\n",
    "nx.draw(G, pos=pos, with_labels=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Define the network coordination game\n",
    "\n",
    "# number of nodes in G = number of players in the game\n",
    "n_players = len(G)\n",
    "# utility values\n",
    "a = 1\n",
    "b = 1\n",
    "c = 0\n",
    "d = 0\n",
    "# utility matrix for the 2x2 coordination game\n",
    "phi = np.array([[a,d],\n",
    "                [c,b]])\n",
    "# the potential function of the 2x2 coordination game\n",
    "pot = np.array([[a-c,0],\n",
    "                [0,b-d]])\n",
    "# inverse noise parameter\n",
    "eta = 2\n",
    "# available actions\n",
    "actions = [0,1]\n",
    "n_actions = len(actions)\n",
    "# adjacency matrix\n",
    "W = nx.convert_matrix.to_numpy_matrix(G)\n",
    "\n",
    "def utility(player, x, phi):\n",
    "    result = 0\n",
    "    for other_player in G.neighbors(player):\n",
    "         result += phi[x[player], x[other_player]]\n",
    "    return result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Define the noisy best response dynamics\n",
    "\n",
    "# Initialize transition rates matrix\n",
    "n_config = n_actions**n_players\n",
    "Lambda = sp.sparse.lil_matrix((n_config,n_config))\n",
    "\n",
    "# Number of actions for each player\n",
    "n_states =tuple(n_actions for _ in range(n_players))\n",
    "\n",
    "# Fill transition rates matrix\n",
    "for x_id in range(n_config):\n",
    "    x = np.unravel_index(x_id,shape=n_states)\n",
    "    x = np.array(x)\n",
    "    for player in range(n_players):\n",
    "        # compute utilities gained by `player` for each of its possible actions\n",
    "        # while the other players are in the current configuration x\n",
    "        utilities = np.zeros(n_actions)\n",
    "        for action in actions:\n",
    "            y = np.array(x)\n",
    "            y[player] = action\n",
    "            utilities[action] = utility(player,y, phi)\n",
    "        exp_utilities = np.exp(eta*utilities)\n",
    "        for action in actions:\n",
    "            if action == x[player]:\n",
    "                continue\n",
    "            y = np.array(x)\n",
    "            y[player] = action\n",
    "            y_id = np.ravel_multi_index(tuple(y), dims = n_states)\n",
    "            Lambda[x_id, y_id] += exp_utilities[action] / np.sum(exp_utilities)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Simulate the continuous time Markov chain with 2nd approach: local clocks\n",
    "\n",
    "w = np.sum(Lambda, axis=1)\n",
    "w = np.array(w.T)[0]\n",
    "for x, weight in enumerate(w):\n",
    "    if weight == 0:\n",
    "        Lambda[x,x] = 1\n",
    "        w[x] = 1\n",
    "D = np.diag(w)\n",
    "P = np.linalg.inv(D) @ Lambda\n",
    "\n",
    "# number of iterations\n",
    "n_steps = 100\n",
    "\n",
    "states = np.zeros(n_steps, dtype=int)\n",
    "# initial configuration\n",
    "x = np.random.choice(actions, size = n_players)\n",
    "x_id = np.ravel_multi_index(tuple(x), dims = n_states)\n",
    "states[0] = x_id\n",
    "\n",
    "transition_times = np.zeros(n_steps)\n",
    "t_next = -np.log(np.random.rand())/w[0]\n",
    "\n",
    "P_cum = np.cumsum(P, axis=1)\n",
    "\n",
    "for i in range(1,n_steps):\n",
    "    states[i] = np.argwhere(P_cum[states[i-1]] > np.random.rand())[0]\n",
    "    transition_times[i] = transition_times[i-1] + t_next\n",
    "    t_next = -np.log(np.random.rand())/w[states[i]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Compute and plot the evolution of the potential along the simulated \n",
    "# path of the chain\n",
    "\n",
    "# store the potential value at each step of the simulation\n",
    "potentials = np.zeros(n_steps)\n",
    "for step in range(n_steps):\n",
    "    x_id = states[step]\n",
    "    x = np.array(np.unravel_index(x_id, shape = n_states))\n",
    "    for i, j in G.edges:\n",
    "        potentials[step] += 0.5*pot[x[i],x[j]]\n",
    "\n",
    "plt.step(transition_times, potentials, 'bo-', where=\"post\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 648x864 with 25 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# To better understand the process, visualize the \n",
    "# evolution of the game's configuration in time\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Plot the evolution of the game's configurations\n",
    "fig = plt.figure(figsize=(9,12))\n",
    "for t in range(0,min(n_steps,25)):\n",
    "    plt.subplot(5,5,t+1)\n",
    "    x_id = states[t]\n",
    "    x = np.array(np.unravel_index(x_id, shape = n_states))\n",
    "    nx.draw(G,\n",
    "        pos = pos,\n",
    "        with_labels=True,\n",
    "        nodelist=np.argwhere(x==0).T[0].tolist(),\n",
    "        node_color = 'r')\n",
    "    nx.draw(G,\n",
    "        pos = pos,\n",
    "        with_labels=True,\n",
    "        nodelist=np.argwhere(x==1).T[0].tolist(),\n",
    "        node_color = 'b')\n",
    "    plt.title('jump step = {0}'.format(t+1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see that the potential during most of the\n",
    "time is at its maximum value, corresponding to all nodes choosing the\n",
    "same action. By repeating the simulation, it seems like all nodes tend\n",
    "to end up in the same state after long time, independently of the initialization. The asymptotic state consists equally often in that all nodes\n",
    "choose action 0 as that all nodes choose action 1.\n",
    "\n",
    "We know from the theory that the noisy best response dynamics operates a selection among the NE and concentrates, in the small noise limit, most of the probability of its invariant distribution in the subset of those Nash equilibria that correspond to the set of potential maximizers $$\\arg \\max_{x \\in \\mathcal X} \\Phi(x)$$ With the current choice for $a,b,c,d$ the maximum value of the potential $\\Phi$ is realized by both the all-one and all-zero configurations.\n",
    "\n",
    "In this example we have a fixed noisy parameter $\\eta = 2$ but we are still able to observe that the dynamics spends most part of the time in global maxima of the potential.\n",
    "Try to change the noise parameter and see how the dynamics behaves."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Step 2\n",
    "> Simulate the dynamics when  instead $a=1$, $b=\\frac{1}{2}$ and $c=d=0$. Run the simulation a few times and plot the potential function as a function of time. What do you observe?\n",
    "\n",
    "The same code as previously can be reused, by only changing the parameter $b$. The simulations and the evolution of the potential values along it behave very similarly, with the difference that, among the states in which all nodes choose the same action, state $(0,\\ldots,0)$ is much more frequent than $(1,\\ldots,1)$. Even if all nodes are initialized to choose action 1, with high probability after a while they will all choose action 0 instead. This happens since with the current choice of $a,b,c,d$ the global maximum of the potential $\\Phi$ is realized by the all-zero configuration."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "# TO DO"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Step 3\n",
    "> If the simulation continues for a long time, will the probability to be in the different configurations converge towards some specific values? In that case, how can these be computed?\n",
    "\n",
    "According to the theory about noisy best response dynamics, this type of dynamics is, for any choice of $\\eta > 0$, an irreducible reversible Markov chain with invariant probability distribution\n",
    "$$\n",
    "\\pi_x = \\frac{e^{\\eta \\Phi(x)}}{Z_\\eta}, \\quad Z_\\eta = \\sum_{y\\in \\mathcal{X}} e^{\\eta \\Phi(y)}.\n",
    "\\tag{1}\n",
    "$$\n",
    "\n",
    "Compute the values of $\\pi$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "# TO DO"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Step 4\n",
    "> Compute the ratio between the probability that after a long time (i.e., in stationarity) all nodes choose action 1 and the probability that all nodes choose action 0, both for case **1** and case **2**. Does this seem to agree with your simulations?\n",
    "\n",
    "According to (1), we have that \n",
    "$$\n",
    "\\frac{\\pi_{(1,1,1,1,1)}}{\\pi_{(0,0,0,0,0)}} = \\frac{e^{\\eta \\Phi(1,1,1,1,1)}}{e^{\\eta \\Phi(0,0,0,0,0)}}.\n",
    "$$\n",
    "We can thus compute the result according to the following code: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Case 1:\n",
      "The ratio between the probability of configuration 'all ones' and configuration 'all zeros' is 1.0\n",
      "Case 2:\n",
      "The ratio between the probability of configuration 'all ones' and configuration 'all zeros' is 0.0301973834223185\n"
     ]
    }
   ],
   "source": [
    "print(\"Case 1:\")\n",
    "# utility values\n",
    "a = 1\n",
    "b = 1\n",
    "c = 0\n",
    "d = 0\n",
    "# the potential function of the 2x2 game\n",
    "pot = np.array([[a-c,0],\n",
    "                [0,b-d]])\n",
    "x_0 = [0 for node in G]\n",
    "potential_0 = 0\n",
    "for i, j in G.edges:\n",
    "    potential_0 += 0.5*pot[x_0[i],x_0[j]]\n",
    "\n",
    "x_1 = [1 for node in G]\n",
    "potential_1 = 0\n",
    "for i, j in G.edges:\n",
    "    potential_1 += 0.5*pot[x_1[i],x_1[j]]\n",
    "\n",
    "probability_ratio_1_over_0 = np.exp(eta*potential_1) / np.exp(eta*potential_0)\n",
    "print(\"The ratio between the probability of configuration 'all ones' and configuration 'all zeros' is\", probability_ratio_1_over_0)\n",
    "\n",
    "print(\"Case 2:\")\n",
    "\n",
    "# utility values\n",
    "a = 1\n",
    "b = 1/2\n",
    "c = 0\n",
    "d = 0\n",
    "# the potential function of the 2x2 game\n",
    "pot = np.array([[a-c,0],\n",
    "                [0,b-d]])\n",
    "\n",
    "x_0 = [0 for node in G]\n",
    "potential_0 = 0\n",
    "for i, j in G.edges:\n",
    "    potential_0 += 0.5*pot[x_0[i],x_0[j]]\n",
    "\n",
    "x_1 = [1 for node in G]\n",
    "potential_1 = 0\n",
    "for i, j in G.edges:\n",
    "    potential_1 += 0.5*pot[x_1[i],x_1[j]]\n",
    "\n",
    "probability_ratio_1_over_0 = np.exp(eta*potential_1) / np.exp(eta*potential_0)\n",
    "print(\"The ratio between the probability of configuration 'all ones' and configuration 'all zeros' is\", probability_ratio_1_over_0)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Check if this result agree with your simulations by estimating $\\pi$ for case **1** and **2** by measuring the time spent in each configuration.\n",
    "\n",
    "**Hint:** exploit the code for continuous time Markov chains from Lab04, where we kept trace of time intervals spent in each configuration during the simulation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Step 5. \n",
    "> Compute the ratio between the probability that after a long time (i.e., in stationarity) one node chooses action 1 and the rest choose action 0, and the probability that all nodes choose action 0, both for case **1** and case **2**.\n",
    "\n",
    "Using the invariant probability distribution formula (1) again, we can compute the ratio\n",
    "$$\n",
    "\\frac{\\sum_{i \\in \\mathcal V} \\pi_{\\delta(i)}}{\\pi_{(0,0,0,0,0)}}\n",
    "$$\n",
    "where $\\delta(i)_i=1$ and $\\delta(i)_j=0$, $\\forall j \\neq i$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Case 1:\n",
      "The ratio between the probability of configuration 'one one, rest zero' and configuration 'all zeros' is 0.3885603420976875\n",
      "Case 2:\n",
      "The ratio between the probability of configuration 'one one, rest zero' and configuration 'all zeros' is 0.3885603420976875\n"
     ]
    }
   ],
   "source": [
    "print(\"Case 1:\")\n",
    "# utility values\n",
    "a = 1\n",
    "b = 1\n",
    "c = 0\n",
    "d = 0\n",
    "# the potential function of the 2x2 game\n",
    "pot = np.array([[a-c,0],\n",
    "                [0,b-d]])\n",
    "x_0 = [0 for node in G]\n",
    "potential_0 = 0\n",
    "for i, j in G.edges:\n",
    "    potential_0 += 0.5*pot[x_0[i],x_0[j]]\n",
    "\n",
    "exp_potential_1 = 0\n",
    "for player in G:\n",
    "    x_1 = [0 for node in G]\n",
    "    x_1[player] = 1\n",
    "    potential_1 = 0\n",
    "    for i, j in G.edges:\n",
    "        potential_1 += 0.5*pot[x_1[i],x_1[j]]\n",
    "    exp_potential_1 += np.exp(eta*potential_1)\n",
    "\n",
    "# print(exp_potential_1)\n",
    "probability_ratio_1_over_0 = exp_potential_1 / np.exp(eta*potential_0)\n",
    "print(\"The ratio between the probability of configuration 'one one, rest zero' and configuration 'all zeros' is\", probability_ratio_1_over_0)\n",
    "\n",
    "print(\"Case 2:\")\n",
    "\n",
    "# utility values\n",
    "a = 1\n",
    "b = 1/2\n",
    "c = 0\n",
    "d = 0\n",
    "# the potential function of the 2x2 game\n",
    "pot = np.array([[a-c,0],\n",
    "                [0,b-d]])\n",
    "\n",
    "x_0 = [0 for node in G]\n",
    "potential_0 = 0\n",
    "for i, j in G.edges:\n",
    "    potential_0 += 0.5*pot[x_0[i],x_0[j]]\n",
    "\n",
    "exp_potential_1 = 0\n",
    "for player in G:\n",
    "    x_1 = [0 for node in G]\n",
    "    x_1[player] = 1\n",
    "    potential_1 = 0\n",
    "    for i, j in G.edges:\n",
    "        potential_1 += 0.5*pot[x_1[i],x_1[j]]\n",
    "    exp_potential_1 += np.exp(eta*potential_1)\n",
    "\n",
    "# print(exp_potential_1)\n",
    "probability_ratio_1_over_0 = exp_potential_1 / np.exp(eta*potential_0)\n",
    "print(\"The ratio between the probability of configuration 'one one, rest zero' and configuration 'all zeros' is\", probability_ratio_1_over_0)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Even if in case 1 and 2 are different, the ratio \n",
    "$$\n",
    "\\frac{\\sum_{i \\in \\mathcal V} \\pi_{\\delta(i)}}{\\pi_{(0,0,0,0,0)}}\n",
    "$$\n",
    "is the same in the two cases.\n",
    "This happens because the potential value of configurations $\\delta(i)$ and $(0,0,0,0,0)$ does not depends on $b$, as the configurations does not contain any couple of neighboring players playing action $1$."
   ]
  }
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