jdossgollin/Ellsberg.ipynb

## Ellsberg.ipynb
{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy import stats\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "OK so to recap we have 30 red balls in the urn, and 60 balls which are either black or yellow.\n",
    "Let's take this to the continuous space for now and apply a beta distribution to the probability of a red ball, i.e.: $$p(\\text{black}) \\sim \\text{Beta}(\\alpha, \\beta)$$\n",
    "where $\\alpha$ and $\\beta$ are hyperparameters that we have no information whatsoever on.\n",
    "However, since we have no information, it probably makes sense to assume that the distribution is symmetric around 0.5, ie we have no reason to suspect more red than yellow balls or vice versa.\n",
    "(From the properties of the beta distribution, this implies $\\alpha=\\beta$.\n",
    "We can change these numbers and it doesn't change the conclusions, but let's put $\\alpha=\\beta=1.25$, which looks like this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "alpha = 1.25 # this is arbitrary\n",
    "beta = alpha # if beta=alpha you have symmetry\n",
    "x = np.linspace(0, 1, 500)\n",
    "p = stats.beta(alpha, beta).pdf(x)\n",
    "plt.plot(x, p)\n",
    "plt.xlabel(\"Fraction of black balls\")\n",
    "plt.ylabel(\"Probability Density\")\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "OK, back to the 1-urn problem.\n",
    "We have a choice between Gamble A: earn 100 dollars to draw a red ball, and Gamble B: earn 100 dollars to draw a black ball.\n",
    "If we take Gamble A, we have a $\\frac{1}{3}$ chance of winning 100 dollars.\n",
    "If we take Gamble B, we have *approximately* a probability $p \\times 60$ of winning 100 dollars.\n",
    "I say approximately because, strictly speaking we should be using a binomial distribution.\n",
    "Here's a histogram of expected payouts using Gamble B:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Expected payout is 30.0914\n"
     ]
    }
   ],
   "source": [
    "n_sim = 10000\n",
    "black_obs = np.nan * np.ones(n_sim)\n",
    "for i in np.arange(n_sim):\n",
    "    prob_i = np.random.beta(a=alpha,b=beta, size=1)\n",
    "    black_obs[i] = np.random.binomial(n=60, p=prob_i, size=1)\n",
    "plt.hist(black_obs, density=True, edgecolor=\"black\")\n",
    "plt.title(\"Payouts of Gamble B\")\n",
    "plt.yticks([])\n",
    "plt.xlabel(\"Dollar Payout\")\n",
    "plt.ylabel(\"Payout likelihood\")\n",
    "plt.show()\n",
    "print(\"Expected payout is {}\".format(black_obs.mean()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see that even though the expected payout is essentially 30 dollars (slightly off because I only used 10000 simulations), there is a big spread of payouts.\n",
    "If I have *any* risk aversion, I will choose the certain bet, always.\n",
    "This holds for any value of $\\alpha$, as long as $\\beta=\\alpha$, ie I have no prior reason to suspect that there are \n",
    "more yellow or black balls (and certainly the problem gives me no information to this effect!)\n",
    "\n",
    "The same issue holds with the second problem between Gamble C (100 dollars for red OR yellow) and Gamble D (100 dollars for black OR yellow).\n",
    "Again, Gamble C has expected payout of 60 dollars with a lot of spread (if $p=0$ then you have a 2 in 3 chance of a payout and if $p=1$ you have a 1 in 3 chance), while Gamble D has an expected payout of 60 dollars with no spread.\n",
    "If you have risk aversion, you'll strictly prefer the sure bet, which is Gamble C"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So either\n",
    "* the paradox isn't actually a paradox if you frame it in this way (i.e., as a Bayesian putting distributions on probabilities)\n",
    "* I misunderstood the question\n",
    "* I typed in something wrong"
   ]
  }
 ],
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