{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Bayesian Statistics Seminar\n", "\n", "Copyright 2017 Allen Downey\n", "\n", "MIT License: https://opensource.org/licenses/MIT" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true }, "outputs": [], "source": [ "from __future__ import print_function, division\n", "\n", "%matplotlib inline\n", "\n", "import warnings\n", "warnings.filterwarnings('ignore')\n", "\n", "import numpy as np\n", "\n", "import thinkbayes2\n", "import thinkplot" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## The Price is Right problem\n", "\n", "The following function reads data from http://tpirstats.com/" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [], "source": [ "import csv\n", "\n", "def ReadData(filename='showcases.2011.csv'):\n", " \"\"\"Reads a CSV file of data.\n", "\n", " Args:\n", " filename: string filename\n", "\n", " Returns: sequence of (price1 price2 bid1 bid2 diff1 diff2) tuples\n", " \"\"\"\n", " fp = open(filename)\n", " reader = csv.reader(fp)\n", " res = []\n", "\n", " for t in reader:\n", " _heading = t[0]\n", " data = t[1:]\n", " try:\n", " data = [int(x) for x in data]\n", " res.append(data)\n", " except ValueError:\n", " pass\n", "\n", " fp.close()\n", " return list(zip(*res))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The `Player` object encapsulates the information we have about each of the two players." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [], "source": [ "class Player(object):\n", " \"\"\"Represents a player on The Price is Right.\"\"\"\n", "\n", " def __init__(self, prices, bids, diffs):\n", " \"\"\"Construct the Player.\n", "\n", " prices: sequence of prices\n", " bids: sequence of bids\n", " diffs: sequence of underness (negative means over)\n", " \"\"\"\n", " self.pdf_price = thinkbayes2.EstimatedPdf(prices)\n", " self.cdf_diff = thinkbayes2.MakeCdfFromList(diffs)\n", "\n", " mu = 0\n", " sigma = np.std(diffs)\n", " self.pdf_error = thinkbayes2.NormalPdf(mu, sigma)\n", "\n", " def ErrorDensity(self, error):\n", " \"\"\"Density of the given error in the distribution of error.\n", "\n", " error: how much the bid is under the actual price\n", " \"\"\"\n", " return self.pdf_error.Density(error)\n", "\n", " def PmfPrice(self):\n", " \"\"\"Returns a new Pmf of prices.\n", "\n", " A discrete version of the estimated Pdf.\n", " \"\"\"\n", " xs = np.linspace(0, 75000, 101)\n", " return self.pdf_price.MakePmf(xs=xs)\n", "\n", " def CdfDiff(self):\n", " \"\"\"Returns a reference to the Cdf of differences (underness).\n", " \"\"\"\n", " return self.cdf_diff\n", "\n", " def ProbOverbid(self):\n", " \"\"\"Returns the probability this player overbids.\n", " \"\"\"\n", " return self.cdf_diff.Prob(-1)\n", "\n", " def ProbWorseThan(self, diff):\n", " \"\"\"Probability this player's diff is greater than the given diff.\n", "\n", " diff: how much the oppenent is off by (always positive)\n", " \"\"\"\n", " return 1 - self.cdf_diff.Prob(diff)\n", "\n", " def MakeBeliefs(self, guess):\n", " \"\"\"Makes a posterior distribution based on estimated price.\n", "\n", " Sets attributes prior and posterior.\n", "\n", " guess: what the player thinks the showcase is worth \n", " \"\"\"\n", " pmf = self.PmfPrice()\n", " self.prior = Price(pmf, self, label='prior')\n", " self.posterior = self.prior.Copy(label='posterior')\n", " self.posterior.Update(guess)\n", "\n", " def OptimalBid(self, guess, opponent):\n", " \"\"\"Computes the bid that maximizes expected return.\n", " \n", " guess: what the player thinks the showcase is worth \n", " opponent: Player\n", "\n", " Returns: (optimal bid, expected gain)\n", " \"\"\"\n", " self.MakeBeliefs(guess)\n", " calc = GainCalculator(self, opponent)\n", " bids, gains = calc.ExpectedGains()\n", " gain, bid = max(zip(gains, bids))\n", " return bid, gain" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The following function reads combines data from two years and makes the `Player` objects." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def MakePlayers():\n", " \"\"\"Reads data and makes player objects.\"\"\"\n", " data = ReadData(filename='showcases.2011.csv')\n", " data += ReadData(filename='showcases.2012.csv')\n", "\n", " cols = zip(*data)\n", " price1, price2, bid1, bid2, diff1, diff2 = cols\n", "\n", " player1 = Player(price1, bid1, diff1)\n", " player2 = Player(price2, bid2, diff2)\n", "\n", " return player1, player2\n", "\n", "player1, player2 = MakePlayers()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here's what the distribution of actual prices looks like for the two showcases." ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { 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9auRgTCwAyy6JVoet37qfSSOiHRTWVWrvMKVOV68j3UTEIyKfAn4HfAV4DHgh\n0YGp5KqxB0daU7nYDffDBk+PsFilRfmcM6UMsAZuVR84jMOu9K2sbaO+1Ze84JQagHprvH8KWAWc\nCzxqjDnfGPMvxpiqfolOJU1NvZcgDsLiwON2kelxUjiI2xquWTQrsr1izQ4mFEXH62zQ6jClTktv\nJZY7gElY07msEhGv/WgREW/iw1PJUlPvxW/37fC4XYwZlonI4O27MW/6GEaPtHqE+fwBgt5oMtF2\nFqVOT29tLA5jTE7MI9d+5BhjcvsrSNX/qo8145OuxJI2qMav9ERE+JvL50b2d2zdjdgzGlXWtlHX\notVhSvVVb1Vh6SLyNRH5mYjcIyKDZxCDOiljDDUNLdESS5pr0Dbcx7po7gRGFFg9wjraO3CHoslE\nq8OU6rveqsJ+A5wHbAGuAX6U8IhU0tU3tREKhfHhwuV04HQ6hkRicbmc3HDZ7Mh+zaHDGGOVWnTu\nMKX6rrfEMt0Yc4cx5hfALcAl/RCTSrKaei8hhKA48bhduJwSWRd+sFu6cCo5WekABFuaafK2A7D/\nWLtWhynVR70llsiiFMYYXa91iKiuaz6u4X5UfgYu58Bfg6UvPO40rrV7iDkxtDc20LV6hDbiK9U3\nvS70FdsTDJitvcIGv5q6Fnwx7SuDdfzKyVx9yUzSPdasxqGWJhrtUosOllSqb3rrFebs1hPMpb3C\nBr/qei/+IdQjrLvsTA9XXzwDgEz8VB9rBgwH6tqp9XYmNzilBoChUb+hTktNnRc/1uqKHrdrUK3B\n0lfXLzmHNJcTJwY6WmlusRLKukqtDlOqN5pY1Amq62JLLC7KhliJBSAvJ4MrLpwGQJbx2YueGa0O\nU6oPNLGo47R3+GloD2AQRKA4P2NQrcFyOm68bA5Op4Ms/LR1dNLS5uNwQwdHmzqSHZpSKU0TizrO\ncVO5pLmGXPtKrOEF2SyZPwUHhkzjjyzVrNVhSp2aJhZ1nOo67/FTuQyBgZGn8jeXz8UhQrbx4W3r\npK3Dx7rKhsjASaXUiRKeWERkmYjsFJEKEXn4JOc8JiK7RWSziMyxj5WJyDsisk1EtojI/YmOVZ04\n+WTpEGy4jzVyeC4XzJ1AJj4cGKrrvBxt6qSqUavDlDqZhCYWEXEAPwOuAmYAt4nI1G7nXA1MMMZM\nAu4FHrefCgIPGmNmYC2R/JXu16r4swZHRnuEjR7iiQXgU5fPwQFkGh+N3nY6fQGtDlPqFBJdYpkP\n7DbGHDARMqULAAAgAElEQVTGBIBngRu7nXMj8BSAMWYNkCcixcaYamPMZvt4K7ADKE1wvEPeodoW\ngmIllqz0NIpz05McUfKNLR3O3GmjyTbWlC419V6tDlPqFBKdWEqBQzH7hzkxOXQ/p6r7OSIyFpgD\nrIl7hOo4B+paI9vlw7NwOAbvGiyn428un0sGfhwmTF1TK1UNbeyva092WEqlpJRvvBeRbOBPwFft\nkotKkFAoTE2zP7I/aVReEqNJLdMnlDClvIgs/BhjTXuzdq+OaVGqJ4keoFAFjInZL7OPdT9ndE/n\n2Ou//An4rTHmxVPd6JFHHolsL168mMWLF59pzEPWscZWOu32lTSXk3FF2UmOKHWICJ+64lw+fuJt\nWiSdY40tfFRxjE/PL9NSnRoQVqxYwYoVK/rlXpLIemIRcQK7gKXAUWAtcJsxZkfMOdcAXzHGXCsi\nC4GfGGMW2s89BdQZYx7s5T5G67vP3se7DnPPzz/EJy6yMzw8+Q+LmFKSk+ywUoYxhvu/9xwf1TkJ\niYPRIwv49zvOY1qpTpunBh4RwRiTkE9FCa0KM8aEgPuAN4BtwLPGmB0icq+I3GOfsxzYJyJ7gF8A\nfw8gIhcBnwMuE5FNIrJRRJYlMt6hrqqmCb/dcO/xaI+w7kSE65fMJhtr3rDaei+r9tQlOSqlUk/C\n5+owxrwGTOl27Bfd9u/r4boPwa6XUf1i95EmDNYHmMIsN5meoTmVy6ksOn8Sv3ppPc1+8AVCvPVx\nFXdcNBa3K+WbK5XqN/rXoCL21rREtscOz0piJKnL407jxosmk2ZCAByqaWbLoeYkR6VUatHEoiIO\nN0RHk0/UtpWTuvrSmeSINaalpd3HqxsPJjkipVKLJhYFWA3TtS2RlaiZPmZYEqNJbcPys1kyY2Rk\n/72tR2n36crdSnXRxKIAaGhuoyNs/To4HcLkUdrT6VRuu+IcPMZKxHXNbby79WiSI1IqdWhiUQDs\nrWokJNavQ6bHRZFO5XJKE8uLmFpk9ZozBv744d4kR6RU6tDEogDYdjA6qeKIbDciOuivN7cvmYJg\njZ/afriJww1tSY5IqdSgiUUBsLvaG9nW8St9s+S8iRS4rN5h/kCIZ9/fk+SIlEoNmlgUAAfrop+2\nx4/UqVz6wp3m4orZoyL7b26uIhTWGSCU0sSiAKhu9kW2p5UVJDGSgeXzV87CacIA1DR38OEObcRX\nShOLwhcI0dQZsvcMM8uHJzWegaSsOJ/pI6NVh0+vqEhiNEqlBk0sit1HmwmGrE/d6RKmZLgOjjwd\nn71kUmR74/5Gmlp9pzhbqcFPE4ti6/7ouiLDMl3aI+w0Xb1wEnlp1rY/GOa37+w49QVKDXKaWBS7\njjRFtkfl6/iV0+V0Olg6KzoSf/mmI7pssRrSNLEo9h+L9gjTxb3OzBeunIHDbsQ/2tTB6oraJEek\nVPJoYhnijDEcbeqM7E8u1eWIz8SY4nymFEVLe797Vxvx1dCliWWIO9bio63TmvPKacKMG5mf5IgG\nrs9cNCGyvWFfA/Ut2oivhiZNLEPcwfp2fH5rZl43QUpGaInlTF1zwSRyHFa37U5/kD+s1PnD1NCk\niWWI21vTgj9ovRlmEKSoULsanymPO41Lp0bHAL2y8VCkG7dSQ4kmliFux6Foj7CinDRcLl0N+mzc\nsXQ6Lnt1ySP1bazeXZfkiJTqf5pYhjBjDHtiliMuH649ws7W1HHFlOdaf1ZhY/jNu7u167EacjSx\nDGH1rf7IKHGHCTN+pC7udbZEhJsvGBeZTn9nVSPbq7y9XKXU4KKJZQg7UNdOp8/qEeYhSNlInXwy\nHq66YAp5pgOAljYfz685mOSIlOpfmliGsIP17XTaa7VrYomfYfnZLBxXECm1rKmopbK2NclRKdV/\nNLEMYfvr2uj02yUWE6S0SMewxMuyCyaRbaxqxoamNl79uDrJESnVfzSxDFHGGPZWt0S6w2a7DMPy\ns5Ic1eCx8JzxjHD6AejwB/hoVy1HmzqSHJVS/UMTyxDV2Bagzmu90TlMmPKiXJ3VOI7SPWksmjOG\nTLvUUtfYyl836iJgamjQxDJEHahr69Zwr9Vg8bZ4/hQKTDsA9c1trNlbz4GYJaCVGqw0sQxRVo+w\nmIb7Yk0s8TZr0ihKct1kGR/BUBhvawfPr6tKdlhKJZwmliFqf10bHXaJxW2ClBZrj7B4czgcXDJv\nEoWmDTDUN7Wxvcqr41rUoKeJZQgyxlglFn+0Kqy0SCefTITF8yfjJkSu6aSppZ1QKMyf1h7W0fhq\nUNPEMgQ1tQfwdgTw+YM4MLgJ6azGCVI+ahhjS4dTYNoxxtDobeNgfTtr9jb0frFSA5QmliEotrTi\nNgFGFOTgcaclOarBa/H5k3ERJs90UN9kNd7/ae1h2u02LqUGG00sQ9D+Y9ojrD9dPG8iDhHyTTvt\nbR34/AGa2gP8eb025KvBSRPLELS3tjXSIyzdBCkt0ob7RCrIzWTOtNE4MQw3rZFSy3s7j7GnRqd6\nUYOPJpYhJhw2VB5XYgkwShvuE27R+ZMByMJHsLUZMBgDT63cr4uBqUFHE8sQU9XYgS8QptMXwGVC\nuAhTqmNYEm7+rLFkpLsRIL2tHp+d2I80dvLaJzqPmBpcEp5YRGSZiOwUkQoRefgk5zwmIrtFZLOI\nzI05/qSI1IjIJ4mOc6jYW9sKGDr9ATwEEdAxLP3AnebignPGA5BGmCJHdN6wv246yr5jOiJfDR4J\nTSwi4gB+BlwFzABuE5Gp3c65GphgjJkE3Av8PObp/7GvVXGyp6aVQDBEKGxINwEy093k52QkO6wh\nYfH8yZHto5X7GTPM+rmHwoZfvVtJpz+UrNCUiqtEl1jmA7uNMQeMMQHgWeDGbufcCDwFYIxZA+SJ\nSLG9vxJoTHCMQ0plbVu04Z4ApcX5OvlkP5k+oYRR9nghnz/ArHxDepr1J1jr9fG7jw4kMzyl4ibR\niaUUOBSzf9g+dqpzqno4R8WBtyNArddHhy+AYKwR91oN1m9EhKUXTIvsr928mzsvHhvZX72ngVW7\n65MQmVLx5Up2APHyyCOPRLYXL17M4sWLkxZLqqqsterxOzr9eIzVvqKTT/avJfOn8PQrawmFwlTs\nr2FkJlw4aRgf2Qnldx8dYMywTEoLtXpSxdeKFStYsWJFv9wr0YmlChgTs19mH+t+zuhezulVbGJR\nPdtrL4/b3uEnHatX0vjRI5IZ0pCTl5PB/FnjWLV5LwBvrdrB7dctZG9NKzVeH75AmJ+9uYdv3TiN\n7PRB87lPpYDuH7gfffTRhN0r0VVh64CJIlIuIm7gVuClbue8BNwJICILgSZjTE3M82I/1FnaW9OK\nMYZ2nx+PsRNL2fAkRzX0XHFhtDpsxdoKHGL48hUT8djtLcdafPzq3UrCYZ2oUg1MCU0sxpgQcB/w\nBrANeNYYs0NE7hWRe+xzlgP7RGQP8Avgy13Xi8jTwEfAZBE5KCJ3JzLewSwYCrO/rp1OXwBjIJ0g\nwwuyyclKT3ZoQ87syaUUFeYA0NbhY/XHlZQWZPC3l46LnLOtysvz6w8nK0SlzkrCy9rGmNeAKd2O\n/aLb/n0nufb2BIY2pBxu6MAfDNPe6Y8MjNTSSnJ0NeI/88paAF7/cDuXnjeZeeMKuG5OCS9vtpYw\nfv2TGkryM7h4sv4/qYFFR94PEV3tK20x7SvjNLEkzWULpuBwWH9+Oyur2bXPGn1/47xRzB4dnWLn\ntysPsOOILgymBhZNLEPE7mq74b7TT7qxxrFow33yFOZlcel5kyL7L7y1GbBKM/csGU9ZYXTw5M/f\n3kt1U2dS4lTqTGhiGQLCYWN/6jV2jzA/oA33yXbT0jmR7XVb93Oo2hoLnO52cv+Vk8jLtNbIafeF\neOyN3Xg7AkmJU6nTpYllCDhQ306bL0SnL4iEQ9ZSudkZFORmJju0IW30yALOnzk2sv/iO5sj24XZ\nbu67YiJpTqtDZK3Xx8/e3IM/qDMhq9SniWUI2FFl1dG3d/rJwI8AE0YP16lcUkBsqeX99bupa4yu\nzzJuRBb3LBlP139TZW0bT6zQbsgq9WliGQK2H4kmlkxjVYONK9VqsFQwdfxIpo0vASAUCvPyiuMn\n8p47toDPLoiOH964v4k/rtVuyCq1aWIZ5HzBEHuqoyPuM7p6hI3WxJIqbro8Wmp5beU2ahtajnv+\n8pnFXDGzOLL/5tYa3t5Wg1KpShPLILe7upVg2ACGUGcHLqw6+vFl2iMsVcybPoaxdgkyEAzxP3/+\n8IRzPj2/jLljo/O6Pbv6EBv26cTfKjVpYhnkttvtK/5ACFfQWlwqM91N8bCcZIalYogIX7rl4sj+\n2i372bDt+Cn0HQ7hS4vHM74oCwBj4IkVleytaUWpVKOJZZDrSiztHX4yTHRgpDbcp5ap40eyZEF0\ngoonn/8QfyB43Dlul4P7rphIUa4HgEDI8Ngbu3WMi0o5mlgGMW9HgMMNVimlwxfTvqIN9ynp89cv\nJCvDSho19V7+/NamE87JzUjja8smRWY+bvOF+MnrFTrGRaUUTSyDWFc3YwDxdeDA6qY6cYy2r6Si\nvJwMPnfd/Mj+n9/cxLY9R044ryg3nfuvjI5xqWvx85PXduvSxiplaGIZxLbZiSUcDtPpbYocnzlZ\nF+hMVVdcOI0J9lQ7oVCYf3/ydY7UNp1w3viibP7PZRMiY1wO1rfz83f2EgzpAEqVfJpYBqlgKMzm\ng9YbUkubD0/IB8DokkIdcZ/CHA4HD/3tleRmW3OFtbb7+N4vltPSdmI7yjnl+Xz+ovLI/rbDXv73\ng/0YowMoVXJpYhmktlV5afdZVSMhvw8PVkPwOZPLkhmW6oOiwhy+ec8y0lxOAKrrvPzfX73WY3K5\ndOoIrp9bEtlfvaeB51Yf0uSikkoTyyC1rrIhsi0dLZElOGdP0WqwgWBSeTFf/fzSyP6ufdX88w+f\nZ9/huhPOveHcUVw6Ndoh461ttSz/uLpf4lSqJ5pYBiFfMMSmA1Y1WCAYwtdovRk5HA5mTByVzNDU\nabhgzni+cNOFkf3ahha+8eMXeGf1zuNKJCLCHReWM29cQeTYC+ureHd7bb/Gq1QXTSyD0JZDzfgC\nViOuKxzEjVUlNmVsMemetGSGpk7T9Utm89DdV0b+3wLBEP/1zAq+9dMX2VkZLZU4HMIXF49j2qjo\nwNfff3SQD3Yd6/eYldLEMgit3RutBssIdWg12AB3wZzx/PtDN1NWHC2R7NpXzbd++hf+7VevsWPv\nUYwxpDkdfOXyiYwbkRU576mVB1i1uz4ZYashTAZDI5+ImMHwfcRDhz/Eg7/fTCBkzQ/mP1BBm9ea\n1PB7X72JqeNHJjdAdcY6fQGeeWUdr67cSqhbt+JxZcO59tJZXHTuBAJh+NHyCg7WtwMgAl9aPJ75\nEwqTEbZKUSKCMSYhU3BoYhlkVu2u58n39gFQmOFg79q1AKR70vjNv34Bl93TSA1cNfVenn5lLSs3\n7Dnhucx0NxfPm8jCuRP58yeNVDVaMy+IwJ0Xl3PJFB0cqyyaWHqhiSXqJ69VsPWwNTByfHaYDas2\nAnD+zLF8/UvLkhmairMDR+p5ecUWPtiwm0DwxFH3w4bl4s0aicOTTmZ6GiB8en4ZV83WUqvSxNIr\nTSyWI40d/P/Pb4vsj/TVsqPiIAB/+6mLuHbRrGSFphLI29rBW6t28vbqHVTXeY97LoRwVPIwaenk\n5WSQm53Op+aP4faLxupEpEOcJpZeaGKx/Pq9fXxkN9ROGpHB+ndXErZ/Lj/7/26jZEReMsNTCWaM\nYUdlNW+v3snqjyvp9FkTU4YQqiWXTnFHzi3JFK6cms/0ccVMHlvMmJJCnE7tyzOUaGLphSYWqG/1\n8c0/bCVkr4d+Tn6QFR9sBmDmpFE8et8NyQxP9TN/IMimHYf4cNNeNmw7QLsvQI3k0i6eyDkeE2Sk\nacZFmDSXk8lji5k1uZTZk0uZOKZIE80gp4mlF5pY4JlVB3l7mzUgblJxFnvWraehuQ2AB79wBRfN\nnZDM8FQSBYMhKg7UsnH7QV7cVM2epiBdfy5OE2aEaSEL/3HXZGV4uHDueBadN5mp40dqtdkgpIml\nF0M9sXg7Ajz87Cd2F2O4fEIWf/zLewDkZmfwq0fv0N5gKuLtrdU8+e5evG0+2jqsR7q/lWGmtceB\nbUWFOVy7aBaXXzBNB9gOIppYejHUE8tf1lfx8uajAIwZlolU72PzzkMA/M3SOdxxw8JkhqdS0K6j\nLfzq3Uqa2q12mEAwiDMcpMzt4+jBI5HSbqzsTA/LLpnJdYtmkZOV3t8hqzjTxNKLoZxY6lt9fOf5\nbXTaU7h85ryR/PJ/Xqbrp/Ff376dkcNzkxegSlmtnUGeWrmfjfuPX+9lbnk+54508/H2fXy4cS/t\nncdXk2Wmu7lx6RyuWzRLSzADmCaWXgzVxGKM4UevVrDziDWyvjjPw7T0dl6wl7SdM3U03/77a5MZ\nokpxxhhWVtTx3OpDkQ8nAC6ncPHk4SydPpyPt+7npXc/pqb++K7MeTkZfOaq87j8gqla1ToAaWLp\nxVBNLG9ureG51VaVlwh88eIyfvrEK/j8VvXGP//dVSyYPS6ZIaoBoqnNz5/WHWb1nobjjjsdwoIJ\nhSyaMpwjh6v5w2vrOXKs+bhzRg7P5bZr5nPRuRO0kX8A0cTSi6GYWKoaO/iXv2wnaDfYX3POSDat\n2sD2vVZbS2lRPj/++me0y6g6LXtqWnlu9SH2HTuxjWV0YQYXThpGR0Mdf3170wntMGNKCrn5inO5\ncO54HA79vUt1mlh6MdQSS2tnkB8t38WhBmseqDHDMpmdG+C3L60CwCHCvz5wE5PKi5MZphqgjDHs\nONLCy5uOUFHdesLzIjB+RBbhtma2bN6FvzM6gzZYJZjrF5/DJedNJCvDc8L1KjVoYunFUEosjW1+\nfvxaBUcarWVqXU7hSxeX8oPH/xqZL+qWK8/ltmvnJzNMNUjsrWnlvZ3HWFfZEOnOHisUCtPqbaG+\nupa0YCceArgII0Cay8nCc8Zz6XmTmDlpFO40V/9/A+qkNLH0YqgkluqmTn78WgX1rdFeOtfPHsE7\nb63mcE0jAOWjhvHv//gpbUxVcdXuC7JmbwPrKhvYXdNK9z+3YChETX0LtfVeJBTEQxA3QdzGWmgu\nwwkzJ5Ywa3Ip40ePYHzZ8JTvsmyMoTMQpqUzQGtnkJbOIO2+EO3+IJ2BMIFgGF8wTChsELFqCkQg\nPc1JhttJepqD3Iw0CrLc5GemkZPuSqk2qAGdWERkGfATrEXFnjTGfL+Hcx4DrgbagC8YYzb39Vr7\nvEGdWPzBMO9sr2X55qO0+61SidMhXDQmgzffXB3pDup0OvjBQzdTPmpYMsNVg5y3I8Cm/U18fLCJ\nXdUtkdVKwSrB1De1UtfU1q2bssFlwrgI47QfuRlpDMvNYER+FsPzMxmWl8mwvCyG52eRl52Ow+FA\nBPryzhf7hi2RfwADBgiFDWFjCIYMobDBHwrjC4TwBcJ0BkK0+0O0+YK0+UJ2EgnQ0hmMtGHGg9vl\noCjXQ1Guh+K8dEblZzCqIJ2R+el4kvBBcMAmFhFxABXAUuAIsA641RizM+acq4H7jDHXisgC4KfG\nmIV9uTbmNVI+saxYsYLFixef1jXejgAb9jWy/OOjNLYFIsdDwSATM/1s2rA9Ml7F6XRw3+2LufS8\nyf0eZzJonPF1pnEGQ2H21rax44iXyto29ta2RhJNe6ef+sZWvK2ddPgDvbzS8UTAnebCk+bC43aR\n7knD43bRtH8LY2aej0NSu3PA0V0bKJkyr0/nisCIHA+lBRmUFmRQkp9OSUEGxXmehCacRCaWRFd6\nzgd2G2MOAIjIs8CNQGxyuBF4CsAYs0ZE8kSkGBjXh2sHjFP94RpjaPOFONbio6a5k6NNnWyv8to9\ncwz+QIiOzgAdvgCdbW14vDV4CUauH5afxT//7VVMLC9KaJypROOMrzON0+V0MKUkhyklOQCEw4Yj\nTR0cqu/gUEM7hxs6qGnupLqxnebWDto6/LR3+Gn3+U+oTotlDPj8QXz+oFWPYate8x51jpG4XU48\nbhdut5V83GlOXC4nbpf11eV04HAIfSvvnJzb5SAn3UVOhossj4sst4tMj1XV5XY5SHM6cDkEg/V3\nHAwb/IEwv121k4UTr8DbEaSp3U9TWyBS29DT91rr9VHr9bHpwPGDVQuy0hiR42FEroei3HQun1GE\nJy31q7kTnVhKgUMx+4exkk1v55T28dqIW763/KwCjTfTbXv7+7v5+LvLCRsIY/0BhoBgCPx2TYIx\nVnHdGAiFw4RC1sNgTRZYYNrIpfO4P5WZk0bx4F1XkJeT0V/fmlIn5XAIZYWZlBVmcgHRKtlgKEx9\nq5/GNj/eziBNbX6O1rdS29hGXXM7jS0dtLb5aOnw09rhw2e/CXfPPS5CZBg/BCAYgGAbtJ8kFhEh\nzeXA5XTgcjhwuhw4RXA5xf4KaSI4HeB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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "thinkplot.Pdf(player1.PmfPrice(), label='Player 1')\n", "thinkplot.Pdf(player2.PmfPrice(), label='Player 2')\n", "thinkplot.Config(xlabel='Showcase price', ylabel='PMF', loc='upper right')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "And here's the CDF of the difference between the actual price and the players bid (positive difference means the actual price was higher; negative difference means the bid was too high)." ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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10foG7N+6qH51RSwxikj5MHeP78nNTgeq3X1uuP9VwKOFdDO7A3jU3ZeH+y8C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\ngYiIyIAogYiIyID8f4tUvRYg3sNcAAAAAElFTkSuQmCC\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "thinkplot.Cdf(player1.CdfDiff(), label='Player 1')\n", "thinkplot.Cdf(player2.CdfDiff(), label='Player 2')\n", "thinkplot.Config(xlabel='Showcase price', ylabel='PDF')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Median differences:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "(3902, 3136)" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "cdf1 = player1.CdfDiff()\n", "cdf2 = player2.CdfDiff()\n", "cdf1.Percentile(50), cdf2.Percentile(50)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We need an error model for the contestants. Ideally, we would have data on what the contestants thought, rather than what they bid. Since we don't have that, we have to make some assumptions." ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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DqQfmrdzBmMHWlGrqLuiTlKouqPwC1gPrqxyrSTsgcLnO3c6x2pQJVjdTVfc7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\nRGSjiGwWkceqKfOCiOSJyCo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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "thinkplot.Pdf(player1.pdf_error, label='Player 1')\n", "thinkplot.Pdf(player2.pdf_error, label='Player 2')\n", "thinkplot.Config(xlabel='Error', ylabel='PDF')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Exercise 1:** What is the probability of overbidding for each player?" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# Solution goes here" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## The Bayesian part\n", "\n", "Here's a class that encapsulates the Bayesian update.\n", "\n", "`Likelihood` computes the probability of the player's guess, given the actual price." ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [], "source": [ "class Price(thinkbayes2.Suite):\n", " \"\"\"Represents hypotheses about the price of a showcase.\"\"\"\n", "\n", " def __init__(self, pmf, player, label=None):\n", " \"\"\"Constructs the suite.\n", "\n", " pmf: prior distribution of price\n", " player: Player object\n", " label: string\n", " \"\"\"\n", " thinkbayes2.Suite.__init__(self, pmf, label=label)\n", " self.player = player\n", "\n", " def Likelihood(self, data, hypo):\n", " \"\"\"Computes the likelihood of the data under the hypothesis.\n", "\n", " hypo: actual price\n", " data: the contestant's guess\n", " \"\"\"\n", " price = hypo\n", " guess = data\n", "\n", " error = price - guess\n", " like = self.player.ErrorDensity(error)\n", "\n", " return like\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "`MakeBeliefs` uses `PmfPrice` as a prior and updates with a given guess. Let's suppose Player 1 thinks the first showcase is worth $20000:" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "(30299.488817891368, 25096.411816206251)" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "player1.MakeBeliefs(20000)\n", "player1.prior.Mean(), player1.posterior.Mean()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The posterior mean is between the prior mean and the guess.\n", "\n", "Now we can plot the prior and posterior:" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def PlotBeliefs(player):\n", " \"\"\"Plots prior and posterior beliefs.\n", "\n", " player: Player object\n", " \"\"\"\n", " thinkplot.Pdfs([player.prior, player.posterior])\n", " thinkplot.Config(xlabel='Price ($)', ylabel='PMF')" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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NgP+x4xGpoHN1WZMPPc6mxvKurfnp5kWziYx0Zouf7F4SC0DRPF+/SvGGfRij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\n+iqs++wTIydlBz22QIiOjGC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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "PlotBeliefs(player1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "And here's what it looks like for Player 2, assuming that the initial guess is $40000." ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "(31034.710418731363, 35641.216869713542)" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "player2.MakeBeliefs(40000)\n", "player2.prior.Mean(), player2.posterior.Mean()" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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qO+hze4gZQ5BU6nwM/sC9adMmx8oa9rfaGOM2xvzBGPNxYBVwAigWkY2jfP5u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\nFYGm1k6effXAmJ/pcknA2mE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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "PlotBeliefs(player2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Exercise 2:** Run these two updates again with some different guesses to see how the posterior distribution depends on the guess." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Decision analysis\n", "\n", "Now that we know what the players should believe, let's see what they should do.\n", "\n", "`GainCalculator` encapsulates the process for computing expected gain for each potential bid.\n", "\n", "Reading from bottom to top:\n", "\n", "* `ProbWin` computes the probability of winning, given how much you are off by.\n", "\n", "* `Gain` computes how much you'll win, given the actual price.\n", "\n", "* `ExpectedGain` computes how much you'll win, averaged over possible prices, given a bid.\n", "\n", "* `ExpectedGains` computes the expected gain for all bids." ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": true }, "outputs": [], "source": [ "class GainCalculator(object):\n", " \"\"\"Encapsulates computation of expected gain.\"\"\"\n", "\n", " def __init__(self, player, opponent):\n", " \"\"\"Constructs the calculator.\n", "\n", " player: Player\n", " opponent: Player\n", " \"\"\"\n", " self.player = player\n", " self.opponent = opponent\n", "\n", " def ExpectedGains(self, low=0, high=75000, n=101):\n", " \"\"\"Computes expected gains for a range of bids.\n", "\n", " low: low bid\n", " high: high bid\n", " n: number of bids to evaluates\n", "\n", " returns: tuple (sequence of bids, sequence of gains)\n", " \n", " \"\"\"\n", " bids = np.linspace(low, high, n)\n", "\n", " gains = [self.ExpectedGain(bid) for bid in bids]\n", "\n", " return bids, gains\n", "\n", " def ExpectedGain(self, bid):\n", " \"\"\"Computes the expected return of a given bid.\n", "\n", " bid: your bid\n", " \"\"\"\n", " suite = self.player.posterior\n", " total = 0\n", " for price, prob in sorted(suite.Items()):\n", " gain = self.Gain(bid, price)\n", " total += prob * gain\n", " return total\n", "\n", " def Gain(self, bid, price):\n", " \"\"\"Computes the return of a bid, given the actual price.\n", "\n", " bid: number\n", " price: actual price\n", " \"\"\"\n", " # if you overbid, you get nothing\n", " if bid > price:\n", " return 0\n", "\n", " # otherwise compute the probability of winning\n", " diff = price - bid\n", " prob = self.ProbWin(diff)\n", "\n", " # if you are within 250 dollars, you win both showcases\n", " if diff <= 250:\n", " return 2 * price * prob\n", " else:\n", " return price * prob\n", "\n", " def ProbWin(self, diff):\n", " \"\"\"Computes the probability of winning for a given diff.\n", "\n", " diff: how much your bid was off by\n", " \"\"\"\n", " prob = (self.opponent.ProbOverbid() + \n", " self.opponent.ProbWorseThan(diff))\n", " return prob\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "I'll instantiate a `GainCalculator` for each player, providing the other player as an opponent." ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [], "source": [ "calc1 = GainCalculator(player1, player2)\n", "calc2 = GainCalculator(player2, player1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For Player 1, here's the plot of expected gain versus bid." ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Player 1 optimal bid (16668.781463564384, 21000.0)\n" ] }, { "data": { "image/png": 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S2/mTxnD5xyfEGFH8ogMR1pV6F5xzrYEnoDRTtmUXP3n05cTojlFD+vDVGz6e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\nXCw8ATnnnIuFJyDnnHOx8ATknHMuFv8f5Xl5TwaC6dMAAAAASUVORK5CYII=\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "bids, gains = calc1.ExpectedGains()\n", "thinkplot.Plot(bids, gains, label='Player 1')\n", "thinkplot.Config(xlabel='Bid ($)', ylabel='Expected gain ($)')\n", "print('Player 1 optimal bid', max(zip(gains, bids)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Interestingly, in this case the optimal bid is higher than the guess, which is contrary to the assumption that you would always want to round down to avoid going over." ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Player 2 optimal bid (19365.953590885678, 31500.0)\n" ] }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "bids, gains = calc2.ExpectedGains()\n", "thinkplot.Plot(bids, gains, label='Player 2')\n", "thinkplot.Config(xlabel='Bid ($)', ylabel='Expected gain ($)')\n", "print('Player 2 optimal bid', max(zip(gains, bids)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The peak in these curves is pretty steep, which suggests it might be important to get the model right! One of the assumptions we made is that the contestant guesses are unbiased.\n", "\n", "**Exercise 3:** In `Player.__init__`, change the value of `mu` to add some bias to the contestant guesses. What effect does this change have on the results? What if you increase or decrease `sigma`? " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Optimal bidding\n", "\n", "Finally, this function plots the optimal bid for a range of guesses." ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def PlotOptimalBid(playerA, playerB):\n", " \"\"\"Plots optimal bid vs estimated price.\n", " \"\"\"\n", " low, high = 15000, 60000\n", " guesses = np.linspace(low, high, 21)\n", "\n", " res = []\n", " for guess in guesses:\n", " playerA.MakeBeliefs(guess)\n", " mean = playerA.posterior.Mean()\n", " calc = GainCalculator(playerA, playerB)\n", " bids, gains = calc.ExpectedGains()\n", " gain, bid = max(zip(gains, bids))\n", " res.append((guess, mean, gain, bid))\n", "\n", " guesses, means, gains, bids = zip(*res)\n", " \n", " thinkplot.PrePlot(num=3)\n", " thinkplot.Plot([low, high], [low, high], color='gray')\n", " thinkplot.Plot(guesses, means, label='Posterior mean')\n", " thinkplot.Plot(guesses, bids, label='Optimal bid')\n", " thinkplot.Plot(guesses, gains, label='Expected gain')\n", " thinkplot.Config(xlabel='Guessed price ($)', loc='upper left')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here are the results for Player 1." ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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au0Z/NtoXe1Yf5RWVfJdwmG/W76e8wlwV3DAkjvtuH05wgHezzp2cW8wHm8+Q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\nsX6O9Y3AZaq6SUTuBG5Q1btF5CPA61X1UhE5D3i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a74eBN03gt2FMiVjYDhhGI4jIV3Cz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\nAaGtpbAi1qP9AAAAAElFTkSuQmCC\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "player1, player2 = MakePlayers()\n", "PlotOptimalBid(player1, player2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Not surprisingly, the optimal bid is always below the posterior mean.\n", "\n", "But when the guess is low, the optimal bid sometimes exceeds the guess." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here's the same analysis for Player 2." ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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kDgkYNRqn+Oqrr5gwYQJWq5VHHnmEQYMGucyBaNwLb419APy45Wf+9fEaKqpO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\nAAAASUVORK5CYII=\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "player1, player2 = MakePlayers()\n", "PlotOptimalBid(player2, player1)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "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": "2.7.11" } }, "nbformat": 4, "nbformat_minor": 0 }