{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# A (difficult) example (B.4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-05T11:02:15.440147Z",
     "start_time": "2021-02-05T11:02:10.063803Z"
    },
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "from fractions import Fraction\n",
    "import sympy as sp\n",
    "from poisson_approval import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-05T11:02:15.448394Z",
     "start_time": "2021-02-05T11:02:15.442804Z"
    },
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "tau = TauVector({'a': Fraction(3, 20), 'b': Fraction(9, 20), 'ac': Fraction(1, 20), 'bc': Fraction(7, 20)},\n",
    "                symbolic=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-05T11:02:16.464324Z",
     "start_time": "2021-02-05T11:02:15.450482Z"
    },
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{1}{5}$"
      ],
      "text/plain": [
       "-1/5"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.pivot_weak_ab.mu"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-05T11:02:16.532574Z",
     "start_time": "2021-02-05T11:02:16.468665Z"
    },
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{1}{5}$"
      ],
      "text/plain": [
       "-1/5"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.pivot_weak_bc.mu"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-05T11:02:17.465997Z",
     "start_time": "2021-02-05T11:02:16.535486Z"
    },
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{7}{10} + \\frac{\\sqrt{21}}{10}$"
      ],
      "text/plain": [
       "-7/10 + sqrt(21)/10"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.pivot_weak_ac.mu"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-05T11:02:17.478962Z",
     "start_time": "2021-02-05T11:02:17.469986Z"
    },
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{7}{10} + \\frac{\\sqrt{21}}{10}$"
      ],
      "text/plain": [
       "-7/10 + sqrt(21)/10"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.trio.mu"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-05T11:02:17.498909Z",
     "start_time": "2021-02-05T11:02:17.481953Z"
    },
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.241742430504416"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "float(tau.trio.mu)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\delta_{ab}(\\tau)$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-05T11:02:17.543789Z",
     "start_time": "2021-02-05T11:02:17.503895Z"
    },
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{1}{8}$"
      ],
      "text/plain": [
       "1/8"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.score_ab_in_duo_ab - tau.score_c_in_duo_ab"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\delta_{bc}(\\tau)$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-05T11:02:17.601633Z",
     "start_time": "2021-02-05T11:02:17.546781Z"
    },
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{1}{5}$"
      ],
      "text/plain": [
       "1/5"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.score_bc_in_duo_bc - tau.score_a_in_duo_bc"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-05T11:02:17.612604Z",
     "start_time": "2021-02-05T11:02:17.605084Z"
    },
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.pivot_weak_ab.psi_ac"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-05T11:02:17.623575Z",
     "start_time": "2021-02-05T11:02:17.615596Z"
    },
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.pivot_weak_bc.psi_a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-05T11:02:17.633548Z",
     "start_time": "2021-02-05T11:02:17.625569Z"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{1}{3}$"
      ],
      "text/plain": [
       "1/3"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.pivot_weak_bc.psi_b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-05T11:02:17.651501Z",
     "start_time": "2021-02-05T11:02:17.635543Z"
    },
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "exp(- n/5 - log(n)/2 - log(8*pi/5)/2 + o(1))"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.duo_ab.asymptotic"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-05T11:02:17.669452Z",
     "start_time": "2021-02-05T11:02:17.655489Z"
    },
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "exp(- n/5 - log(n)/2 - log(3*pi/5)/2 + o(1))"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau.duo_bc.asymptotic"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-05T11:02:18.962992Z",
     "start_time": "2021-02-05T11:02:17.671447Z"
    },
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{27 \\sqrt{2}}{16 \\sqrt{3} + 27 \\sqrt{2}}$"
      ],
      "text/plain": [
       "27*sqrt(2)/(16*sqrt(3) + 27*sqrt(2))"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "utility_threshold_program = tau.d_ranking_best_response['abc'].utility_threshold\n",
    "utility_threshold_program"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-05T11:02:18.977020Z",
     "start_time": "2021-02-05T11:02:18.964985Z"
    },
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{9 \\sqrt{3}}{8 \\sqrt{2} + 9 \\sqrt{3}}$"
      ],
      "text/plain": [
       "9*sqrt(3)/(8*sqrt(2) + 9*sqrt(3))"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "utility_threshold_theoretical = 9 * sp.sqrt(3) / (9 * sp.sqrt(3) + 8 * sp.sqrt(2))\n",
    "utility_threshold_theoretical"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Check that the value found by the program is actually the same as the theoretical value:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-05T11:02:19.489593Z",
     "start_time": "2021-02-05T11:02:18.978948Z"
    },
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.simplify(utility_threshold_program - utility_threshold_theoretical)"
   ]
  }
 ],
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