{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "import collections\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def roll_n_sided_dice(n=6):\n",
    "    \"\"\"\n",
    "    Roll a dice with n sides.\n",
    "    \"\"\"\n",
    "    return np.random.randint(1, n + 1)\n",
    "\n",
    "class MoranProcess:\n",
    "    \"\"\"\n",
    "    A class for a moran process with a population of \n",
    "    size N=3 using the standard Hawk-Dove Game:\n",
    "    \n",
    "    A = \n",
    "       [0, 3]\n",
    "       [1, 2]\n",
    "\n",
    "    Note that this is a simulation corresponding to an \n",
    "    in class activity where students roll dice.\n",
    "    \"\"\"\n",
    "    def __init__(self, number_of_hawks=1, seed=None):\n",
    "        \n",
    "        if seed is not None:\n",
    "            np.random.seed(seed)\n",
    "            \n",
    "        self.number_of_hawks = number_of_hawks\n",
    "        self.number_of_doves = 3 - number_of_hawks\n",
    "        \n",
    "        self.dice_and_values_for_hawk_birth = {1: (6, {1, 2, 3}), 2: (4, {1, 2, 3})}\n",
    "        self.dice_and_values_for_hawk_death = {1: (6, {1, 2}), 2: (6, {1, 2, 3, 4})}\n",
    "\n",
    "        self.history = [(self.number_of_hawks, self.number_of_doves)]\n",
    "        \n",
    "    def step(self):\n",
    "        \"\"\"\n",
    "        Select a hawk or a dove for birth.\n",
    "        Select a hawk or a dove for death.\n",
    "        \n",
    "        Update history and states.\n",
    "        \"\"\"      \n",
    "        birth_dice, birth_values = self.dice_and_values_for_hawk_birth[self.number_of_hawks]\n",
    "        death_dice, death_values = self.dice_and_values_for_hawk_death[self.number_of_hawks]\n",
    "\n",
    "        select_hawk_for_birth = self.roll_dice_for_selection(dice=birth_dice, values=birth_values)\n",
    "        select_hawk_for_death = self.roll_dice_for_selection(dice=death_dice, values=death_values)\n",
    "       \n",
    "        if select_hawk_for_birth:\n",
    "            self.number_of_hawks += 1\n",
    "        else:\n",
    "            self.number_of_doves += 1\n",
    "            \n",
    "        if select_hawk_for_death:\n",
    "            self.number_of_hawks -= 1\n",
    "        else:\n",
    "            self.number_of_doves -= 1\n",
    "        \n",
    "        self.history.append((self.number_of_hawks, self.number_of_doves))\n",
    "\n",
    "             \n",
    "    def roll_dice_for_selection(self, dice, values):\n",
    "        \"\"\"\n",
    "        Given a dice and values return if the random roll is in the values.\n",
    "        \"\"\"\n",
    "        return roll_n_sided_dice(n=dice) in values\n",
    "        \n",
    "    def simulate(self):\n",
    "        \"\"\"\n",
    "        Run the entire simulation: repeatedly step through \n",
    "        until the number of hawks is either 0 or 3.\n",
    "        \"\"\"\n",
    "        while self.number_of_hawks in [1, 2]:\n",
    "            self.step()\n",
    "        return self.number_of_hawks\n",
    "    \n",
    "    def __len__(self):\n",
    "        return len(self.history)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "repetitions = 10 ** 7\n",
    "end_states = []\n",
    "path_lengths = []\n",
    "for seed in range(repetitions):\n",
    "    mp = MoranProcess(seed=seed)\n",
    "    end_states.append(mp.simulate())\n",
    "    path_lengths.append(len(mp))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5452601"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "counts = collections.Counter(end_states)\n",
    "counts[3] / repetitions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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KOZmYmVkxJxMzMyvmZGJmZsWcTMzMrJiTiZmZFXMyMTOzYk4mZmZWzMnEzMyKOZmYmVkx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\nzMnEzMyKOZmYmVkxJxMzMyv2/wHFqBfsXIOnNwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa09eeb24e0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(path_lengths, bins=20)\n",
    "plt.title(\"Path lengths: Mean={:0.2f}, 95%={:0.2f},  80%={:0.2f}\".format(np.mean(path_lengths), \n",
    "                                                                         np.percentile(path_lengths, 95),\n",
    "                                                                         np.percentile(path_lengths, 80)));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that this is not how we would do this in practice:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.4545,  0.    ,  0.    ,  0.5455])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P = np.array([[1, 0, 0, 0],\n",
    "              [1 / 6, 1 / 2, 1 / 3, 0],\n",
    "              [0, 1 / 6, 7 / 12, 1 / 4],\n",
    "              [0, 0, 0, 1]])\n",
    "np.round(np.linalg.matrix_power(P, 100)[1], 4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def theoretic_fixation(N, game, i=1):\n",
    "    \"\"\"\n",
    "    Calculate x_i\n",
    "    \"\"\"\n",
    "    f_ones = np.array([(game[0, 0] * (i - 1) + game[0, 1] * (N - i)) / (N - 1) for i in range(1, N)])\n",
    "    f_twos = np.array([(game[1, 0] * i + game[1, 1] * (N - i - 1)) / (N - 1) for i in range(1, N)])\n",
    "    gammas = f_twos / f_ones\n",
    "    return (1 + np.sum(np.cumprod(gammas[:i-1]))) / (1 + np.sum(np.cumprod(gammas)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.54545454545454553"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "theoretic_fixation(N=3, game=np.array([[0, 3], [1, 2]]), i=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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MZnEOsKmAuMzMrIK6j1aKiL2SFgKrgQ7gqojYKGkJ0B0RK4GPSpoD7AWeBs6rd1xmZlaZ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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa09ede1f60>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Ns = range(2, 20 + 1)\n",
    "fixations = [theoretic_fixation(N=n, game=np.array([[0, 3], [1, 2]]), i=1)\n",
    "             for n in Ns]\n",
    "plt.scatter(Ns, fixations)\n",
    "plt.xlabel(\"$N$\")\n",
    "plt.ylabel(\"$x_1$\")\n",
    "plt.title(\"Theoretic fixation probability\");"
   ]
  }
 ],
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