1.1 simultaneous-move complete information - discrete strategies.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Import necessary packages."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "# import all functions from python labrary: sympy\n",
    "from sympy import *\n",
    "# import python library: matplotlib (for nice plots)\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "# set sympy output to print pretty math expressions\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Declare variables we are going to use as 'symbols'. We will add more along the way, if necessary."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {},
   "outputs": [],
   "source": [
    "### NOTE: it is important here to further declare these variables to be nonzero and positive (otherwise sympy cannot find solution for the FOCs)\n",
    "q1, q2, c1, c2, a, b, pi1, pi2, q1hat, q2hat = symbols('q1 q2 c1 c2 a b \\pi_1 \\pi_2 \\hat{q}_1 \\hat{q}_2', nonzero=True, positive=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 1. Setup---a coordination game with numerical payoffs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## payoff matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We set up a simple payoff matrix below. \n",
    "\n",
    "To tell markdown this is a table, we must \n",
    "* use pipes `|` to mark the separation between columns\n",
    "* use `---` in each cell on the second row to indicate the row above is the title row\n",
    "* make sure each row has the same number of columns\n",
    "\n",
    "|              |       | player 2 |       |\n",
    "| --------     | ----- | -------- | ----- |\n",
    "|              |       | left     | right |\n",
    "| **player 1** | left  | (1,1)    | (0,0) |\n",
    "|              | right | (0,0)    | (2,2) |"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The pure-strategy N.E."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Denote the pure strategy by player 1 and 2 by $a_1$ and $a_2$, respectively.\n",
    "\n",
    "Denote the beliefs about the opponent's play by $\\hat{a}_1$ and $\\hat{a}_2$.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Q:** Show that (L,L) or (left, left) is a NE."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**A:** \n",
    "\n",
    "* Given $\\hat{a}_2 = L$, $a_1=L$ is a best response.\n",
    "* Given the correct belief that $\\hat{a}_1 = L$, $a_2=L$ is a best response.\n",
    "* Lastly, the initial belief that $\\hat{a}_2 = L$ is consistent with player 2's best response $a_2=L$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Q:** Show that (R,R) or (right, right) is a NE."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**A:** \n",
    "\n",
    "* Given $\\hat{a}_2 = R$, $a_1=R$ is a best response.\n",
    "* Given the correct belief that $\\hat{a}_1 = R$, $a_2=R$ is a best response.\n",
    "* Lastly, the initial belief that $\\hat{a}_2 = R$ is consistent with player 2's best response $a_2=R$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The mixed-strategy N.E."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Denote player 1 and 2's probability of playing left by $p_1$ and $p_2$ respectively."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Characterize payoffs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Under these strategies, the *expected* payoffs of each player are\n",
    "\\begin{align}\n",
    "    Ev_1 &= \\underbrace{p_1 p_2 \\cdot 1}_{(L,L)} + \\underbrace{(1-p_1) p_2 \\cdot 0}_{(R,L)} + \\underbrace{p_1 (1-p_2) \\cdot 0}_{(L,R)} + \\underbrace{(1-p_1) (1-p_2) \\cdot 2}_{(R,R)} \\\\\n",
    "                &= p_1 p_2 + 2(1-p_1) (1-p_2)\n",
    "\\end{align}\n",
    "\n",
    "Similarly,\n",
    "\\begin{align}\n",
    "    Ev_2 &= \\underbrace{p_1 p_2 \\cdot 1}_{(L,L)} + \\underbrace{(1-p_1) p_2 \\cdot 0}_{(R,L)} + \\underbrace{p_1 (1-p_2) \\cdot 0}_{(L,R)} + \\underbrace{(1-p_1) (1-p_2) \\cdot 2}_{(R,R)} \\\\\n",
    "                &= p_1 p_2 + 2(1-p_1) (1-p_2)\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Derive best responses"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Player 1's best response is to choose their strategy $p_1$ to maximize the payoff $E v_1$ given their belief of $p_2$, denoted by $\\hat{p}_2$. The maximization problem is therefore\n",
    "$$\\max_{p_1} E v_1 = p_1 p_2 + 2(1-p_1) (1-p_2)$$\n",
    "\n",
    "The first-order derivative is\n",
    "\\begin{align}\n",
    "    \\frac{\\partial E v_1}{\\partial p_1} &= \\hat{p}_2 - 2(1-\\hat{p}_2) \\\\\n",
    "    &= 3\\hat{p}_2 - 2\n",
    "    \\quad\n",
    "    \\begin{cases}\n",
    "        > 0  & { \\text{ if } } \\hat{p}_2 > \\frac{2}{3} \\\\\n",
    "        = 0  & { \\text{ if } } \\hat{p}_2 = \\frac{2}{3} \\\\\n",
    "        < 0  & { \\text{ if } } \\hat{p}_2 < \\frac{2}{3}\n",
    "    \\end{cases}\n",
    "\\end{align}\n",
    "\n",
    "Recall that $p_1$ is a probability, thus $p_1 \\in [0,1]$. Therefore, the best response of $p_1$ is\n",
    "\\begin{align}\n",
    "p_1^*(\\hat{p}_2) = \n",
    "    \\begin{cases}\n",
    "        1  & { \\text{ if } } \\hat{p}_2 > \\frac{2}{3} \\\\\n",
    "        \\in [0,1]  & { \\text{ if } } \\hat{p}_2 = \\frac{2}{3} \\\\\n",
    "        0  & { \\text{ if } } \\hat{p}_2 < \\frac{2}{3}\n",
    "    \\end{cases}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Player 2's best response is to choose their strategy $p_2$ to maximize the payoff $E v_2$ given their belief of $p_1$, denoted by $\\hat{p}_1$. The maximization problem is therefore\n",
    "$$\\max_{p_2} E v_2 = p_1 p_2 + 2(1-p_1) (1-p_2)$$\n",
    "\n",
    "The first-order derivative is\n",
    "\\begin{align}\n",
    "    \\frac{\\partial E v_2}{\\partial p_2} &= \\hat{p}_1 - 2(1-\\hat{p}_1) \\\\\n",
    "    &= 3\\hat{p}_1 - 2\n",
    "    \\quad\n",
    "    \\begin{cases}\n",
    "        > 0  & { \\text{ if } } \\hat{p}_1 > \\frac{2}{3} \\\\\n",
    "        = 0  & { \\text{ if } } \\hat{p}_1 = \\frac{2}{3} \\\\\n",
    "        < 0  & { \\text{ if } } \\hat{p}_1 < \\frac{2}{3}\n",
    "    \\end{cases}\n",
    "\\end{align}\n",
    "\n",
    "Recall that $p_2$ is a probability, thus $p_2 \\in [0,1]$. Therefore, the best response of $p_2$ is\n",
    "\\begin{align}\n",
    "p_2^*(\\hat{p}_1) = \n",
    "    \\begin{cases}\n",
    "        1  & { \\text{ if } } \\hat{p}_1 > \\frac{2}{3} \\\\\n",
    "        \\in [0,1]  & { \\text{ if } } \\hat{p}_1 = \\frac{2}{3} \\\\\n",
    "        0  & { \\text{ if } } \\hat{p}_1 < \\frac{2}{3}\n",
    "    \\end{cases}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plotting best responses"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "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": [
    "# p1 as a best response to p2\n",
    "plt.plot([0, 2/3, 2/3, 1],[0, 0, 1, 1 ], color='salmon')\n",
    "plt.xlabel(\"$\\hat{p}_2$\")\n",
    "plt.ylabel(\"$p_1$\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "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": [
    "# p2 as a best response to p1 (inversed)\n",
    "plt.plot([0, 0, 1, 1 ],[0, 2/3, 2/3, 1], color='blue')\n",
    "plt.xlabel(\"$p_2$\")\n",
    "plt.ylabel(\"$\\hat{p}_1$\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "scrolled": true
   },
   "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": [
    "# plot intersection of best responses under correct beliefs: $p_1 = \\hat{p}_1$, $p_2 = \\hat{p}_2$\n",
    "plt.plot([0, 2/3, 2/3, 1],[0, 0, 1, 1 ], color='salmon')\n",
    "plt.plot([0, 0, 1, 1 ],[0, 2/3, 2/3, 1], color='blue')\n",
    "plt.xlabel(\"$p_2$\")\n",
    "plt.ylabel(\"$p_1$\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From the graph, we can see three NE, two pure-strategy NE\n",
    "* $(p_1=0, p_2=0)$, i.e., (L,L)\n",
    "* $(p_1=1, p_2=1)$, i.e., (R,R)\n",
    "\n",
    "and a mixed-strategy NE\n",
    "* $(p_1=2/3, p_2=2/3)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculating mixed-strategy best responses using sympy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Import necessary packages."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# import all functions from python labrary: sympy\n",
    "from sympy import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Declare variables we are going to use as 'symbols'. We will add more along the way, if necessary."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "### NOTE: it is important here to further declare these variables to be nonzero and positive (otherwise sympy cannot find solution for the FOCs)\n",
    "p1, p2, p1hat, p2hat = symbols('p1 p2 \\hat{p}_1 \\hat{p}_2', positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "## define player payoffs in each outcome\n",
    "# payoffs in outcome (L,L)\n",
    "v1LL = 1\n",
    "v2LL = 1\n",
    "# payoffs in outcome (L,R)\n",
    "v1LR = 0\n",
    "v2LR = 0\n",
    "# payoffs in outcome (R,L)\n",
    "v1RL = 0 \n",
    "v2RL = 0\n",
    "# payoffs in outcome (R,R)\n",
    "v1RR = 2\n",
    "v2RR = 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( p_{1} p_{2} + 2 \\left(1 - p_{1}\\right) \\left(1 - p_{2}\\right), \\  p_{1} p_{2} + 2 \\left(1 - p_{1}\\right) \\left(1 - p_{2}\\right)\\right)$"
      ],
      "text/plain": [
       "(p₁⋅p₂ + 2⋅(1 - p₁)⋅(1 - p₂), p₁⋅p₂ + 2⋅(1 - p₁)⋅(1 - p₂))"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# define player payoffs in sympy (recall p1, p2 are prob. of the players playing L)\n",
    "pi1 = p1*p2*v1LL + (1-p1)*p2*v1RL + p1*(1-p2)*v1LR + (1-p1)*(1-p2)*v1RR\n",
    "pi2 = p1*p2*v2LL + (1-p1)*p2*v2RL + p1*(1-p2)*v2LR + (1-p1)*(1-p2)*v2RR\n",
    "\n",
    "# show defined payoffs pi1 and pi2\n",
    "(pi1,pi2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 3 p_{2} - 2$"
      ],
      "text/plain": [
       "3⋅p₂ - 2"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# find best responses for p1\n",
    "pi1.diff(p1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{2}{3}\\right]$"
      ],
      "text/plain": [
       "[2/3]"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# solve for p2 such that all p1 \\in [0,1] is a best response\n",
    "p2_sol = solve(pi1.diff(p1), p2)\n",
    "# show solution\n",
    "p2_sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{2}{3}$"
      ],
      "text/plain": [
       "2/3"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# retrieve the solution from the list (1st element in list)\n",
    "p2mix = p2_sol[0]\n",
    "p2mix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 3 p_{1} - 2$"
      ],
      "text/plain": [
       "3⋅p₁ - 2"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# find best responses for p2\n",
    "pi2.diff(p2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABsAAAAzCAYAAABmK3MOAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACh0lEQVRYCe2Y71ECMRDFwbEAtATsALUD7EBKUEtw/ARftQMpQexASlA70BLUDvT9bi5MyP3J42T84LAzmWz2su8lu3uQXH86nQ56NTKbzb5qzJZJvrWYe/J+UftM2q2F2jxpkeCBv+hrZ29SrrSaZbPv754I+1IIk30XRg5Dzb0u55+o/2As+2tpy3YWWUl0q34SEKUT6hf1Z2pWVMiZIwBfxBNFwC4pIvJjiUs2FtqnCNIqY0cD2QlxVlwyQN8F2vQ6pIuoJXZztspVgjJirEVYReLuLOEoCCAifKFCK3NSQ2cyAVEYc+3qLgVtGnciE8G9AJfqr5qA6+wbk4mAX4PepkT4bEQmgnP5HMVE0oc0wHJikwmQgjhVnxYEC+CnKytu6bNyCoI8ka8gh1JGsllFYpEJECIIi3ypj8V6x3CwyLTy4xi9q27nrCtB7Lcji6PRWd+FsXPoYsfse6Z37Dt2aNM1t9/23CFrBWgDT5/tCiSNSKfxn4YxWyBhC6q0ofRw5uCcyJgjuXX0BsciEyDgXCICWU86/21P6idqj4DlxA0jwJcC5QgQ5KFUboIh17tk/Btz9KYVIuKVHmy53g0jeTmIwaJdxmeSeEpFd3e25iiisQxco7ixztcetgysnQV/AXOcg+hUjdA+q9myKRkExWmqDCM3z61XY2X1IqHcKZKF9EFlQo3BypnAOIgSwlRCGAltVrJk5ar5VkLIrB00sTpkhIr2KDL6WPhEgVg/WW6BpJeJnoj5NWGnlH+6CBZQEYtMYNwwx2rxCzwUmv0NBGaLjIkiIlRWuJhfJ9mc1Tl1tf1fspAz7sVU1ko0tips5RApKZYeFdghjFRZ/IFz2x83C7wfnDTJVLHq1jYAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{2}{3}\\right]$"
      ],
      "text/plain": [
       "[2/3]"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# solve for p1 such that all p2 \\in [0,1] is a best response\n",
    "p1_sol = solve(pi2.diff(p2), p1)\n",
    "# show solution\n",
    "p1_sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{2}{3}$"
      ],
      "text/plain": [
       "2/3"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# retrieve the solution from the list (1st element in list)\n",
    "p1mix = p1_sol[0]\n",
    "p1mix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "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": [
    "# plot intersection of best responses under correct beliefs: $p_1 = \\hat{p}_1$, $p_2 = \\hat{p}_2$\n",
    "#     note: 1. compare to the previous plot, the indifferent cutoffs can be updated by sympy calculations\n",
    "#           2. this code only applies to coordination games (why?)\n",
    "plt.plot([0, p2mix, p2mix, 1],[0, 0, 1, 1 ], color='salmon')\n",
    "plt.plot([0, 0, 1, 1 ],[0, p1mix, p1mix, 1], color='blue')\n",
    "plt.xlabel(\"$p_2$\")\n",
    "plt.ylabel(\"$p_1$\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Challenge"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q1. The above plotting code is limited to coordination games, try modifying it to adapt to battle-of-sexes games."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q2. The above plotting code is limited to coordination games, try modifying it to adapt flexibly to battle-of-sexes games and coordination games."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}