{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Ordinal Equilibria: Condorcet Consistency of Approval Voting (4.1, C.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-09T13:45:58.150932Z",
     "start_time": "2021-02-09T13:45:51.721736Z"
    }
   },
   "outputs": [],
   "source": [
    "import random\n",
    "import pandas as pd\n",
    "from poisson_approval import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-09T13:45:58.156916Z",
     "start_time": "2021-02-09T13:45:58.152926Z"
    }
   },
   "outputs": [],
   "source": [
    "N_SAMPLES = 10000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-09T13:45:58.193508Z",
     "start_time": "2021-02-09T13:45:58.160906Z"
    }
   },
   "outputs": [],
   "source": [
    "def has_an_ordinal_equilibrium(profile):\n",
    "    return len(profile.analyzed_strategies_ordinal.equilibria) > 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-09T13:45:58.226765Z",
     "start_time": "2021-02-09T13:45:58.197844Z"
    }
   },
   "outputs": [],
   "source": [
    "def has_an_ordinal_equilibrium_not_electing_the_cw(profile):\n",
    "    return not profile.analyzed_strategies_ordinal.winners_at_equilibrium.issubset(profile.condorcet_winners)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-01-29T11:34:18.252052Z",
     "start_time": "2021-01-29T11:34:18.248038Z"
    }
   },
   "source": [
    "## By Number of Rankings"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-09T13:46:03.460764Z",
     "start_time": "2021-02-09T13:45:58.230823Z"
    }
   },
   "outputs": [
    {
     "data": {
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       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>P(∃ ord eq | ∃ Cond)</th>\n",
       "      <th>P(∃ ord eq not Cond | ∃ Cond)</th>\n",
       "      <th>P(∃ Cond)</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Number of rankings</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.4083</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.2408</td>\n",
       "      <td>0.0378</td>\n",
       "      <td>0.9721</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.2742</td>\n",
       "      <td>0.0175</td>\n",
       "      <td>0.9493</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>0.2687</td>\n",
       "      <td>0.0031</td>\n",
       "      <td>0.9404</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>0.2941</td>\n",
       "      <td>0.0012</td>\n",
       "      <td>0.9367</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                   P(∃ ord eq | ∃ Cond) P(∃ ord eq not Cond | ∃ Cond)  \\\n",
       "Number of rankings                                                      \n",
       "1                                     1                             0   \n",
       "2                                0.4083                             0   \n",
       "3                                0.2408                        0.0378   \n",
       "4                                0.2742                        0.0175   \n",
       "5                                0.2687                        0.0031   \n",
       "6                                0.2941                        0.0012   \n",
       "\n",
       "                   P(∃ Cond)  \n",
       "Number of rankings            \n",
       "1                          1  \n",
       "2                          1  \n",
       "3                     0.9721  \n",
       "4                     0.9493  \n",
       "5                     0.9404  \n",
       "6                     0.9367  "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "table = pd.DataFrame(index=range(1, 7), columns=[\n",
    "    'P(∃ ord eq | ∃ Cond)',\n",
    "    'P(∃ ord eq not Cond | ∃ Cond)',\n",
    "    'P(∃ Cond)'\n",
    "])\n",
    "table.index.name = \"Number of rankings\"\n",
    "for n_rankings in table.index:\n",
    "    def rand_profile():\n",
    "        rankings = random.sample(RANKINGS, n_rankings)\n",
    "        return RandProfileOrdinalUniform(orders=rankings)()\n",
    "    p1, p2 = probability(\n",
    "        factory=rand_profile,\n",
    "        n_samples=N_SAMPLES,\n",
    "        test=(has_an_ordinal_equilibrium, has_an_ordinal_equilibrium_not_electing_the_cw),\n",
    "        conditional_on=is_condorcet\n",
    "    )\n",
    "    table.loc[n_rankings, 'P(∃ ord eq | ∃ Cond)'] = p1\n",
    "    table.loc[n_rankings, 'P(∃ ord eq not Cond | ∃ Cond)'] = p2\n",
    "    p3 = probability(\n",
    "        factory=rand_profile,\n",
    "        n_samples=N_SAMPLES,\n",
    "        test=is_condorcet\n",
    "    )\n",
    "    table.loc[n_rankings, 'P(∃ Cond)'] = p3\n",
    "table"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## By Subset of Rankings"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-09T13:46:12.845707Z",
     "start_time": "2021-02-09T13:46:03.463628Z"
    }
   },
   "outputs": [
    {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>P(∃ ord eq not Cond | ∃ Cond)</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Rankings</th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>('abc', 'bac', 'cab')</th>\n",
       "      <td>0.1299</td>\n",
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       "    <tr>\n",
       "      <th>('abc', 'acb', 'bac', 'cab')</th>\n",
       "      <td>0.0590</td>\n",
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       "    <tr>\n",
       "      <th>('abc', 'acb', 'bca', 'cba')</th>\n",
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       "    <tr>\n",
       "      <th>('abc', 'acb', 'bac', 'bca', 'cab')</th>\n",
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       "      <th>('abc', 'acb', 'bac', 'cba')</th>\n",
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       "      <th>('abc', 'acb', 'bac', 'bca', 'cab', 'cba')</th>\n",
       "      <td>0.0012</td>\n",
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       "    <tr>\n",
       "      <th>('abc',)</th>\n",
       "      <td>0.0000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>('abc', 'acb')</th>\n",
       "      <td>0.0000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>('abc', 'bac')</th>\n",
       "      <td>0.0000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>('abc', 'bca')</th>\n",
       "      <td>0.0000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>('abc', 'cba')</th>\n",
       "      <td>0.0000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>('abc', 'acb', 'bac')</th>\n",
       "      <td>0.0000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>('abc', 'acb', 'bca')</th>\n",
       "      <td>0.0000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>('abc', 'bca', 'cab')</th>\n",
       "      <td>0.0000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>('abc', 'acb', 'bac', 'bca')</th>\n",
       "      <td>0.0000</td>\n",
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      ],
      "text/plain": [
       "                                            P(∃ ord eq not Cond | ∃ Cond)\n",
       "Rankings                                                                 \n",
       "('abc', 'bac', 'cab')                                              0.1299\n",
       "('abc', 'acb', 'bac', 'cab')                                       0.0590\n",
       "('abc', 'acb', 'bca', 'cba')                                       0.0094\n",
       "('abc', 'acb', 'bac', 'bca', 'cab')                                0.0046\n",
       "('abc', 'acb', 'bac', 'cba')                                       0.0038\n",
       "('abc', 'acb', 'bac', 'bca', 'cab', 'cba')                         0.0012\n",
       "('abc',)                                                           0.0000\n",
       "('abc', 'acb')                                                     0.0000\n",
       "('abc', 'bac')                                                     0.0000\n",
       "('abc', 'bca')                                                     0.0000\n",
       "('abc', 'cba')                                                     0.0000\n",
       "('abc', 'acb', 'bac')                                              0.0000\n",
       "('abc', 'acb', 'bca')                                              0.0000\n",
       "('abc', 'bca', 'cab')                                              0.0000\n",
       "('abc', 'acb', 'bac', 'bca')                                       0.0000"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "table = pd.DataFrame()\n",
    "table.index.name = 'Rankings'\n",
    "for rankings in SETS_OF_RANKINGS_UP_TO_RELABELLING:\n",
    "    rand_profile = RandProfileOrdinalUniform(orders=rankings)\n",
    "    p = probability(\n",
    "        factory=rand_profile, \n",
    "        n_samples=N_SAMPLES,\n",
    "        test=has_an_ordinal_equilibrium_not_electing_the_cw,\n",
    "        conditional_on=is_condorcet\n",
    "    )\n",
    "    table.loc[str(rankings), 'P(∃ ord eq not Cond | ∃ Cond)'] = p\n",
    "table.sort_values(by='P(∃ ord eq not Cond | ∃ Cond)', ascending=False, inplace=True)\n",
    "table"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Focus on Single-Peaked Profiles"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Over the whole single-peaked domain:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-09T13:46:12.861662Z",
     "start_time": "2021-02-09T13:46:12.850693Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.059"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "table.loc[\"('abc', 'acb', 'bac', 'cab')\", 'P(∃ ord eq not Cond | ∃ Cond)']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When only 3 rankings are present:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-09T13:46:12.897104Z",
     "start_time": "2021-02-09T13:46:12.869834Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.1299"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "table.loc[\"('abc', 'bac', 'cab')\", 'P(∃ ord eq not Cond | ∃ Cond)']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-02-09T13:46:12.910069Z",
     "start_time": "2021-02-09T13:46:12.902091Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "table.loc[\"('abc', 'acb', 'bac')\", 'P(∃ ord eq not Cond | ∃ Cond)']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "N.B.: the above cases actually cover all single-peaked cases, up to relabeling the candidates."
   ]
  }
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