logitdyn_jl.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Logit Response Dynamics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Yuya Furusawa**\n",
    "\n",
    "*Graduate School of Economics, University of Tokyo*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This notebook demonstrates how to study logit best response dynamics model using the logitdyn Julia library."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Plots.PlotlyBackend()"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using GameTheory\n",
    "using Plots\n",
    "plotly()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Logit Response Dynamics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Logit response dynamics model is one of the learning models.\n",
    "There is much literature that studies the learning in games, for example, fictitious play, (myopic) best response dynamics, Kandori-Mailath-Rob, sampling best response models. Please refer these notes:\n",
    "\n",
    "- fictitious play\n",
    "- best response dynamics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The learning model with random mistakes, including KMR model, is criticised by the absense of the foundation of mistakes.\n",
    "\n",
    "In contrast, **Logit response dynamics**, which also incorporates the random mistakes, is based on a theory about the origin of mistakes."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $\\Gamma = (I, {(S_i, u_i)}_{i \\in I})$ be a finite normal form game.\n",
    "- $I=\\{1,\\ldots,N\\}$ : player set\n",
    "- $S_i$ : strategies set ($S$ : pure strategy profile)\n",
    "- $u_i$ : payoff function\n",
    "\n",
    "Denote $s=(s_i, s_{-i}) \\in S$ and $u_i(s_i, s_{-i})=u_i(s)$.\n",
    "\n",
    "Players behave as myopic best replier, but have mistakes in thier decisions.\n",
    "A player is chosen to update her action given other players' actions every period (**asynchronous learning**)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When player $i$ is chosen, she will update her action to $s_i$ given actions $s_{-i}$.\n",
    "The probability of choosing action $s_i$ is given by **logit choice function** :\n",
    "$$\n",
    "    p_i(s_i, s_{-i}) = \\frac{\\exp(\\beta u_i(s_i, s_{-i}))}{\\sum_{s_i' \\in S_i} \\exp(\\beta u_i(s_i', s_{-i}))}\n",
    "$$\n",
    "where $0 < \\beta < \\infty$.\n",
    "\n",
    "$\\beta$ can be interpreted as an inverse measure of the level of noise in players' decisions.\n",
    "As $\\beta$ becomes larger, the size of noise becomes smaller.\n",
    "From the microeconomic point of view, this modeling is based on a random-utility model, decision thoretic foundation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Logit response dynamics can converge to the set of Nash equilibria in the potential game.\n",
    "First, we introduce the definition of potential game.\n",
    "\n",
    "**Definition(potential game)**\n",
    "\n",
    "The $\\Gamma$ is a *potential game* if there exists a function $\\rho:S \\rightarrow \\mathbb{R}$ such that for all $i \\in I$, $s_i, s_i' \\in S_i$, $s_{-i} \\in S_{-i}$,\n",
    "$$\n",
    "    u_i(s_i, s_{-i}) - u_i(s_i', s_{-i}) = \\rho(s_i, s_{-i}) - \\rho(s_i', s_{-i})\n",
    "$$\n",
    "\n",
    "$\\rho$ is called the potential."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following game is the potential game:\n",
    "\n",
    "1$\\backslash$2 | $A$           | $B$ \n",
    " ------------- |---------------| ---------\n",
    " $A$           | 4, 4          |   0, 3   \n",
    " $B$           | 3, 0          |   2, 2     \n",
    " \n",
    "To see this, the potential $\\rho$ can be defined\n",
    "\n",
    "1$\\backslash$2 | $A$        | $B$ \n",
    " ------------- |------------| ---------\n",
    " $A$           | 1          |   0  \n",
    " $B$           | 0          |   2     \n",
    "\n",
    "(Note every $2 \\times 2$ symmetric game is the potential game)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Theorem**\n",
    "\n",
    "In the potential game, the set of stable states of logit response dynamics is equal to the set of maximizers of  𝜌 .\n",
    "\n",
    "We will check this later."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## `logitdyn.jl` usage"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To create LogitDynamics class instance, you need the played game and the parameter $\\beta$.\n",
    "The game has to be NormalFormGame instance.\n",
    "As argued before, $\\beta$ is the level of noise in the players' decisions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LogitDynamics{2, Int64, Float64}((2×2 Player{2, Int64}:\n",
       " 4  0\n",
       " 3  2, 2×2 Player{2, Int64}:\n",
       " 4  0\n",
       " 3  2), (2, 2), 2.0, Array[[1.0 1.1353352832366128; 0.01831563888873418 1.0183156388887342], [1.0 1.1353352832366128; 0.01831563888873418 1.0183156388887342]])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "payoff = [4 0;\n",
    "          3 2]\n",
    "game = NormalFormGame(payoff)\n",
    "beta = 2.0\n",
    "ldy = LogitDynamics(game, beta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Vector{Array}:\n",
       " [1.0 1.1353352832366128; 0.01831563888873418 1.0183156388887342]\n",
       " [1.0 1.1353352832366128; 0.01831563888873418 1.0183156388887342]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ldy.choice_probs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### `play` method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`play` method returns a new action profile in the model.\n",
    "You can specify the initial action profile.\n",
    "In addition, you can proceed the dynamics multiple times by specifying num_reps, which is set to 1 by default."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Vector{Int64}:\n",
       " 1\n",
       " 1"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "init_actions = (1, 1)\n",
    "play(ldy, init_actions)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### `time_series` method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`time_series` method returns a time series of action distributions.\n",
    "You have to specify the length of time series.\n",
    "In addition, you can choose initial action profile from which the dynamics proceeds."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×5 Matrix{Int64}:\n",
       " 1  1  1  1  1\n",
       " 1  1  1  2  1"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ts_length = 5\n",
    "init_actions = (1, 1)\n",
    "time_series(ldy, ts_length, init_actions)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see the convergence of the logit response dynamics in the potential game."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LogitDynamics{2, Int64, Float64}((2×2 Player{2, Int64}:\n",
       " 4  0\n",
       " 3  2, 2×2 Player{2, Int64}:\n",
       " 4  0\n",
       " 3  2), (2, 2), 4.0, Array[[1.0 1.0183156388887342; 0.00033546262790251185 1.0003354626279024], [1.0 1.0183156388887342; 0.00033546262790251185 1.0003354626279024]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "payoff = [4 0; 3 2]\n",
    "game = NormalFormGame(payoff)\n",
    "beta = 4.0\n",
    "ldy = LogitDynamics(game, beta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×1000 Matrix{Int64}:\n",
       " 1  1  1  1  1  1  1  1  1  1  1  1  1  …  2  2  2  2  2  2  2  2  2  2  2  2\n",
       " 1  1  1  1  1  1  1  1  1  1  1  1  1     2  2  2  2  2  2  2  2  2  2  2  2"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ts_length = 1000\n",
    "init_actions = (1, 1)\n",
    "series = time_series(ldy, ts_length, init_actions)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
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     "execution_count": 8,
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   "source": [
    "seriesB_1 = [i==2 ? 1 : 0 for i in series[1,:]]\n",
    "seriesB_2 = [i==2 ? 1 : 0 for i in series[2,:]]\n",
    "freq1 = accumulate(+, seriesB_1) ./ collect(1:ts_length) \n",
    "freq2 = accumulate(+, seriesB_2) ./ collect(1:ts_length)\n",
    "labels=[\"Player 1\" \"Player 2\"]\n",
    "plot([freq1, freq2], label=labels)"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you can see, (B,B) is a stable state of this dynamics, and this state maximizes the potential $\\rho$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, we simulate the logit reponse dynamics in the Young's game.\n",
    "Young's game is defined as follows:\n",
    "\n",
    "1$\\backslash$2 | $A$           | $B$      | $C$\n",
    " ------------- |---------------| ---------| --------\n",
    " $A$           | 6, 6          |   0, 0   |   0, 0\n",
    " $B$           | 5, 5          |   7, 7   |   5, 5\n",
    " $C$           | 0, 0          |   5, 5   |   8, 8"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LogitDynamics{2, Int64, Float64}((3×3 Player{2, Int64}:\n",
       " 6  0  0\n",
       " 5  7  5\n",
       " 0  5  8, 3×3 Player{2, Int64}:\n",
       " 6  0  0\n",
       " 5  7  5\n",
       " 0  5  8), (3, 3), 4.0, Array[[1.0 1.0183156388887342 1.0183156389264856; 6.914400106940203e-13 1.0000000000006914 1.0003354626285939; 1.2664165549094176e-14 6.144212365992376e-6 1.000006144212366], [1.0 1.0183156388887342 1.0183156389264856; 6.914400106940203e-13 1.0000000000006914 1.0003354626285939; 1.2664165549094176e-14 6.144212365992376e-6 1.000006144212366]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "young_game = [6 0 0;\n",
    "              5 7 5;\n",
    "              0 5 8]\n",
    "game = NormalFormGame(young_game)\n",
    "beta = 4.0\n",
    "ldy = LogitDynamics(game, beta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
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       "        \"tickcolor\": \"rgb(0, 0, 0)\",\n",
       "        \"type\": \"-\"\n",
       "    },\n",
       "    \"legend\": {\n",
       "        \"yanchor\": \"auto\",\n",
       "        \"xanchor\": \"auto\",\n",
       "        \"bordercolor\": \"rgba(0, 0, 0, 1.000)\",\n",
       "        \"bgcolor\": \"rgba(255, 255, 255, 1.000)\",\n",
       "        \"borderwidth\": 1,\n",
       "        \"tracegroupgap\": 0,\n",
       "        \"y\": 1.0,\n",
       "        \"font\": {\n",
       "            \"color\": \"rgba(0, 0, 0, 1.000)\",\n",
       "            \"family\": \"sans-serif\",\n",
       "            \"size\": 11\n",
       "        },\n",
       "        \"title\": {\n",
       "            \"font\": {\n",
       "                \"color\": \"rgba(0, 0, 0, 1.000)\",\n",
       "                \"family\": \"sans-serif\",\n",
       "                \"size\": 15\n",
       "            },\n",
       "            \"text\": \"\"\n",
       "        },\n",
       "        \"traceorder\": \"normal\",\n",
       "        \"x\": 1.0\n",
       "    },\n",
       "    \"width\": 600\n",
       "}\n",
       ");\n",
       "    });\n",
       "    </script>\n"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ts_length = 1000\n",
    "init_actions = (1, 1)\n",
    "series = time_series(ldy, ts_length, init_actions)\n",
    "seriesB_1 = [i==2 ? 1 : 0 for i in series[1,:]]\n",
    "seriesB_2 = [i==2 ? 1 : 0 for i in series[2,:]]\n",
    "freq1 = accumulate(+, seriesB_1) ./ collect(1:ts_length) \n",
    "freq2 = accumulate(+, seriesB_2) ./ collect(1:ts_length)\n",
    "labels=[\"Player 1\" \"Player 2\"]\n",
    "plot([freq1, freq2], label=labels)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## References"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Alós-Ferrer, Carlos, and Nick Netzer. \"The logit-response dynamics.\" Games and Economic Behavior 68.2 (2010): 413-427."
   ]
  }
 ],
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