{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Games\n",
    "### EC297\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Nash Equilibrium \n",
    "\n",
    "A set of strategies for all players that are all _best response_. \n",
    "\n",
    "The idea is that such set of strategies is stable as noboy has a reason to\n",
    "unilaterally change\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Do people play NE?\n",
    "\n",
    "1. Epistemic -> people reason their way to equilibrium.\n",
    "\n",
    "1. Evolution -> Groups of people mutually learn their way to\n",
    "equilibrium.\n",
    "\n",
    "*We are going to analyze these ideas in detail*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Methods\n",
    "\n",
    "Experimental Econ. test game theory in the lab using several distinct\n",
    "timing and matching protocols.\n",
    "\n",
    "- **One shot:** Ss play game once, get paid. \n",
    "\n",
    " * Probably, useful for reasoning.\n",
    "\n",
    "\n",
    "- **Random matching** _or strangers matching_: Ss play a number of rounds\n",
    "but each round are randomly and anonymously: matched with someone\n",
    "new.\n",
    "\n",
    " * Learning\n",
    "\n",
    "\n",
    "- **Repeated rematching** or _partners matching_: Ss are matched with the\n",
    "some counterparts for a number of rounds. \n",
    "\n",
    " * Reputations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# PD Game\n",
    "\n",
    "- Two players (row and column) meet and pick either C or D.\n",
    "\n",
    "- Thus, the action set is {C,D}\n",
    "\n",
    "\n",
    "\n",
    "|     | C   | D   |\n",
    "|-----|:-------:|:-------:|\n",
    "| C   |10,10| 5,20|\n",
    "| D   |20,5| 8,8|\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# PD Game\n",
    "\n",
    "- The equilibrium is (D,D)!\n",
    "\n",
    "\n",
    "|     | C   | D   |\n",
    "|-----|:-------:|:-------:|\n",
    "| C   |10,10| 5,__20__|\n",
    "| D   |__20__,5| __8__,__8__|\n",
    "\n",
    "- Note that they can improve their payoffs by playing (C,C).\n",
    "\n",
    "- The 20 is usually called *the temptation payoff*, and 10 is the\n",
    "reward payoff.\n",
    "\n",
    "**Experimental Results: **\n",
    "\n",
    "  - Modest cooperation (C) in one shot games with low-temptation and high reward.\n",
    "\n",
    "  - Modest initial cooperation and decay in strangers.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Finitely Repeated PD\n",
    "\n",
    "- Think about backward induction (aka BI): NE should be the same.\n",
    "\n",
    "- But, data shows early cooperation and eventual defection.\n",
    "\n",
    "- Why?\n",
    "\n",
    "  * **Reputation** Suppose that there is small prob. that your\n",
    "      counterparty cooperates until defection, then it can be rational\n",
    "      \n",
    "      to _pretend_ to be this type of player in early rounds and defect later.\n",
    "  * $\\epsilon$ **-equilibrium:** When the rewards of C are high, Ss are\n",
    "      willing to accept minor deviations from BR, causing them to\n",
    "      cooperate. Note that benefits of cooperation get higher, the\n",
    "      larger the game is repeated!\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\infty$ Repeated PD\n",
    "\n",
    "|     | C   | D   |\n",
    "|-----|:-------:|:-------:|\n",
    "| C   |2,2| 0,3|\n",
    "| D   |3,0| 1,1|\n",
    "\n",
    "- We analyze these games with _trigger strategies_\n",
    "\n",
    "- Cooperation\n",
    "\n",
    "$$\n",
    "2 + 2\\delta + 2 \\delta^2 + \\dots = \\frac{2}{1-\\delta}\n",
    "$$\n",
    "\n",
    "- Defect\n",
    "\n",
    "$$\n",
    "3 + \\delta +  \\delta^2 + \\dots = 3+\\frac{\\delta}{1-\\delta}\n",
    "$$\n",
    "\n",
    "- Thus, cooperate if\n",
    "\n",
    "$$\n",
    "\\frac{2}{1-\\delta} \\geq  3+\\frac{\\delta}{1-\\delta} \\quad \\text{or}\n",
    "\\quad \\delta\\geq 1/2\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\infty$ Repeated PD\n",
    "\n",
    "- The (standard) theory predicts cooperation if the discount factor is big enough.\n",
    "\n",
    "- But, how can we run these type of games?\n",
    "\n",
    "  - Assume that I can kill you if one appears after rolling a dice. In\n",
    "    this case, the prob. of survival is 5/6. (due to Roth and Murrigham, 78)\n",
    "\n",
    "- C doesn't emerge when discount rates are too low for it to emerge in standard theory (Dalbo and Frechette, 2011).\n",
    "\n",
    "- Higher $\\delta$, we observe higher C (Dalbo, 2005)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matching pennies\n",
    "|     | R   | L   |\n",
    "|-----|:-------:|:-------:|\n",
    "| U   |10,5| 5,10|\n",
    "| D   |5,10| 10,5|\n",
    "\n",
    "In this case, we cannot find a Pure Nash Equilibrium. Let's try to figure out if other type of equilibrium exists\n",
    "\n",
    "let's call $q$ the prob. of playing R and $p$ the probability of playing p. \n",
    "\n",
    "Therefore, the payoffs for playing U and D are\n",
    "\n",
    "$$\n",
    "\\pi^U = 10\\times q + 5\\times (1-q)\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\pi^D = 5\\times q + 10\\times (1-q)\n",
    "$$\n",
    "\n",
    "Let's substract one from the other, \n",
    "\n",
    "$$\n",
    "\\Delta \\pi = \\pi^U-\\pi^D = 5q - 5(1-q) = 10q-5\n",
    "$$\n",
    "\n",
    "Now, \n",
    "\n",
    "If Player 2 plays $q=1$ or R, then $\\Delta \\pi>0$ or $\\pi^U>\\pi^D$. Thus, Player 1 picks $p=1$ or U \n",
    "\n",
    "If Player 2 plays $q=0$ or L, then $\\Delta \\pi<0$ or $\\pi^U<\\pi^D$. Thus, Player 1 picks $p=1$ or D \n",
    "\n",
    "Note that we can say that Player 1 is indifferent between U and D when $\\pi^D=\\pi^L$. This is true when $q=0$\n",
    "\n",
    "Perhaps, a graph it is more useful to represent the different responses: \n",
    "\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
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VkZOCWQmKKwV/vax+vNaalWCk6PrMrhSshpwUzEpQfJyCk4LVj5OCWQl82mxrCicFsxIU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\ngg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c0dd310>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "### Code to draw a Best Response Function\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from __future__ import print_function\n",
    "from ipywidgets import interact, interactive, fixed, interact_manual\n",
    "import ipywidgets as widgets\n",
    "%matplotlib inline \n",
    "q = np.arange(0,1.01,.01) #The prob. of playing P(right)\n",
    "\n",
    "def plotbr(a,b,c,d,no):\n",
    "    ## a,b,c,d, the payoff matrix\n",
    "    piup = a*(q)+b*(1-q)\n",
    "    pido = c*(q)+d*(1-q)\n",
    "    pile = b*(q)+d*(1-q)\n",
    "    piri = d*q + c*(1-q)\n",
    "    dpi = piup-pido\n",
    "    d2pi = pile-piri\n",
    "    lp = np.copy(dpi)\n",
    "    lq =np.copy(d2pi)\n",
    "    lp[dpi>0]=1\n",
    "    lp[dpi<0]=0\n",
    "    lq[d2pi>0]=1\n",
    "    lq[d2pi<0]=0\n",
    "    if no>0:\n",
    "        plt.title('Figure 1: The best response of Player 1')\n",
    "        plt.plot(q,lp,linewidth=5)\n",
    "    else:\n",
    "        plt.title('Figure 2: The best response of Player 1 and 2')\n",
    "        plt.plot(q,lp,lq,q,linewidth=5)\n",
    "    plt.xlabel('P(right)')\n",
    "    plt.ylabel('P(up)')\n",
    "    plt.ylim(-0.1,1.1)\n",
    "    plt.xlim(-0.1,1.1)\n",
    "    plt.show()\n",
    "    \n",
    "plotbr(10,5,5,10,1)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Figure 1 shows the best choice (aka **best response**) of players 1. Using similar ideas, we can do the graph for both players\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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iFvT19U16kGatGA73KVj1lZkU7gK2yQ3PSePyDgfOB4iIXwEzgFklxmRWmsI+BScFq5gy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\nFMzMrMFJwczMGv4XIP32nWcYIaAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c084610>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plotbr(10,5,5,10,-1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From Figure 2, we realize that both best responses intersect at 0.5. Both players are doing the best when they pick $(p,q)=(.5,5)$. \n",
    "In this case, they are mixing between both choices. We will say that this game have a unique mixed NE. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<link href='http://fonts.googleapis.com/css?family=Alegreya+Sans:100,300,400,500,700,800,900,100italic,300italic,400italic,500italic,700italic,800italic,900italic' rel='stylesheet' type='text/css'>\n",
       "<link href='http://fonts.googleapis.com/css?family=Arvo:400,700,400italic' rel='stylesheet' type='text/css'>\n",
       "<link href='http://fonts.googleapis.com/css?family=PT+Mono' rel='stylesheet' type='text/css'>\n",
       "<link href='http://fonts.googleapis.com/css?family=Shadows+Into+Light' rel='stylesheet' type='text/css'>\n",
       "<link href='http://fonts.googleapis.com/css?family=Nixie+One' rel='stylesheet' type='text/css'>\n",
       "<link href='https://fonts.googleapis.com/css?family=Source+Code+Pro' rel='stylesheet' type='text/css'>\n",
       "<style>\n",
       "\n",
       "@font-face {\n",
       "    font-family: \"Computer Modern\";\n",
       "    src: url('http://mirrors.ctan.org/fonts/cm-unicode/fonts/otf/cmunss.otf');\n",
       "}\n",
       "\n",
       "#notebook_panel { /* main background */\n",
       "    background: rgb(245,245,245);\n",
       "}\n",
       "\n",
       "div.cell { /* set cell width */\n",
       "    width: 750px;\n",
       "}\n",
       "\n",
       "div #notebook { /* centre the content */\n",
       "    background: #fff; /* white background for content */\n",
       "    width: 1000px;\n",
       "    margin: auto;\n",
       "    padding-left: 0em;\n",
       "}\n",
       "\n",
       "#notebook li { /* More space between bullet points */\n",
       "    margin-top:0.8em;\n",
       "}\n",
       "\n",
       "/* draw border around running cells */\n",
       "div.cell.border-box-sizing.code_cell.running { \n",
       "    border: 1px solid #111;\n",
       "}\n",
       "\n",
       "/* Put a solid color box around each cell and its output, visually linking them*/\n",
       "div.cell.code_cell {\n",
       "    background-color: rgb(256,256,256); \n",
       "    border-radius: 0px; \n",
       "    padding: 0.5em;\n",
       "    margin-left:1em;\n",
       "    margin-top: 1em;\n",
       "}\n",
       "\n",
       "div.text_cell_render{\n",
       "    font-family: 'Alegreya Sans' sans-serif;\n",
       "    line-height: 140%;\n",
       "    font-size: 125%;\n",
       "    font-weight: 400;\n",
       "    width:600px;\n",
       "    margin-left:auto;\n",
       "    margin-right:auto;\n",
       "}\n",
       "\n",
       "\n",
       "/* Formatting for header cells */\n",
       ".text_cell_render h1 {\n",
       "    font-family: 'Nixie One', serif;\n",
       "    font-style:regular;\n",
       "    font-weight: 400;    \n",
       "    font-size: 45pt;\n",
       "    line-height: 100%;\n",
       "    color: rgb(0,51,102);\n",
       "    margin-bottom: 0.5em;\n",
       "    margin-top: 0.5em;\n",
       "    display: block;\n",
       "}\n",
       "\n",
       ".text_cell_render h2 {\n",
       "    font-family: 'Nixie One', serif;\n",
       "    font-weight: 400;\n",
       "    font-size: 30pt;\n",
       "    line-height: 100%;\n",
       "    color: rgb(0,51,102);\n",
       "    margin-bottom: 0.1em;\n",
       "    margin-top: 0.3em;\n",
       "    display: block;\n",
       "}\t\n",
       "\n",
       ".text_cell_render h3 {\n",
       "    font-family: 'Nixie One', serif;\n",
       "    margin-top:16px;\n",
       "    font-size: 22pt;\n",
       "    font-weight: 600;\n",
       "    margin-bottom: 3px;\n",
       "    font-style: regular;\n",
       "    color: rgb(102,102,0);\n",
       "}\n",
       "\n",
       ".text_cell_render h4 {    /*Use this for captions*/\n",
       "    font-family: 'Nixie One', serif;\n",
       "    font-size: 14pt;\n",
       "    text-align: center;\n",
       "    margin-top: 0em;\n",
       "    margin-bottom: 2em;\n",
       "    font-style: regular;\n",
       "}\n",
       "\n",
       ".text_cell_render h5 {  /*Use this for small titles*/\n",
       "    font-family: 'Nixie One', sans-serif;\n",
       "    font-weight: 400;\n",
       "    font-size: 16pt;\n",
       "    color: rgb(163,0,0);\n",
       "    font-style: italic;\n",
       "    margin-bottom: .1em;\n",
       "    margin-top: 0.8em;\n",
       "    display: block;\n",
       "}\n",
       "\n",
       ".text_cell_render h6 { /*use this for copyright note*/\n",
       "    font-family: 'PT Mono', sans-serif;\n",
       "    font-weight: 300;\n",
       "    font-size: 9pt;\n",
       "    line-height: 100%;\n",
       "    color: grey;\n",
       "    margin-bottom: 1px;\n",
       "    margin-top: 1px;\n",
       "}\n",
       "\n",
       ".CodeMirror{\n",
       "    font-family: \"Source Code Pro\";\n",
       "    font-size: 90%;\n",
       "}\n",
       "\n",
       ".alert-box {\n",
       "    padding:10px 10px 10px 36px;\n",
       "    margin:5px;\n",
       "}\n",
       "\n",
       ".success {\n",
       "    color:#666600;\n",
       "    background:rgb(240,242,229);\n",
       "}\n",
       "</style>\n",
       "<script>\n",
       "    MathJax.Hub.Config({\n",
       "                        TeX: {\n",
       "                           extensions: [\"AMSmath.js\"],\n",
       "                           equationNumbers: { autoNumber: \"AMS\", useLabelIds: true}\n",
       "                           },\n",
       "                tex2jax: {\n",
       "                    inlineMath: [ ['$','$'], [\"\\\\(\",\"\\\\)\"] ],\n",
       "                    displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ]\n",
       "                },\n",
       "                displayAlign: 'center', // Change this to 'center' to center equations.\n",
       "                \"HTML-CSS\": {\n",
       "                    styles: {'.MathJax_Display': {\"margin\": 4}}\n",
       "                }\n",
       "        });\n",
       "</script>\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.core.display import HTML\n",
    "def css_styling():\n",
    "    import os\n",
    "    styles = open(os.path.expanduser(\"custom.css\"), \"r\").read()\n",
    "    return HTML(styles)\n",
    "css_styling()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}