2_Decisions_Risk.ipynb

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    "Decisions Under Risk\n",
    "======================\n",
    "\n",
    "Shane Steinert-Threlkeld  \n",
    "https://www.shane.st  \n",
    "S.N.M.Steinert-Threlkeld AT uva DOT nl"
   ]
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    "Recap\n",
    "-----\n",
    "\n",
    "* Decisions under ignorance: true state unknown, but so are the probabilities\n",
    "* Many choice rules:\n",
    "    - avoid dominance, maximin, leximin, maximax, optimism-pessimism, minimax regret\n",
    "* Every rule seemed to have pros and cons.  Is there a single _correct_ (normatively) choice rule?"
   ]
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    "Outline\n",
    "------\n",
    "\n",
    "* Decisions under risk: probabilities known\n",
    "* Choice rule: maximize _expected_ utility\n",
    "* Justifying the MEU principle\n",
    "    - What is utility?\n",
    "    - What are the probabilities?\n",
    "* Evidential versus Causal Decision Theory\n",
    "* Problems for MEU"
   ]
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   "source": [
    "Ride to Work Revisited\n",
    "----------\n",
    "\n",
    "| &#160; | rain | no rain |\n",
    "| ----- | ----- | ----- |\n",
    "| take clothes | 1 | 1 |\n",
    "| leave clothes | 0 | 2 |"
   ]
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   "source": [
    "Ride to Work Revisited\n",
    "----------\n",
    "\n",
    "| &#160; | rain ($0.7$) | no rain ($0.3$) |\n",
    "| ----- | ----- | ----- |\n",
    "| take clothes | 1 | 1 |\n",
    "| leave clothes | 0 | 2 |"
   ]
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   "source": [
    "(NB: we now assume that these utilities are _cardinal_, representing my actual values, not merely the ordering of my preferences over the outcomes.)"
   ]
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   "source": [
    "from collections import namedtuple\n",
    "\n",
    "decision_problem = namedtuple(\n",
    "    'decision_problem',\n",
    "    ('states', 'actions', 'utilities', 'probabilities'),\n",
    "    defaults=[None]\n",
    ")"
   ]
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   "source": [
    "Maximize Expected Utility\n",
    "---------\n",
    "\n",
    "* Observation: for each $a$, $u( \\cdot , a)$ is a _random variable_ with domain $S$ and range $\\mathbb{R}$.\n",
    "* For full generality, we assume that the function $p$ in the agent's decision problem works as follows: for each action $a$, $p_a$ is a probability distribution over the states $S$.  Intuitively, this assigns probabilities to statements of the form \"if $a$, then $s$\".  Much more on this later."
   ]
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   "source": [
    "$$\\text{MEU}(D) = \\text{argmax}_a \\mathbb{E}_{p_a} u(\\cdot , a)$$"
   ]
  },
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   "source": [
    "Intuitively: we weight the utilities of the action in each outcome by how probable that outcome is."
   ]
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   "source": [
    "MEU in Python\n",
    "------------"
   ]
  },
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   "execution_count": 3,
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   "source": [
    "import numpy as np\n",
    "\n",
    "ride_to_work = decision_problem(\n",
    "    ('rain', 'no_rain'),\n",
    "    ('take_clothes', 'leave_clothes'),\n",
    "    np.array([\n",
    "        [1, 1],\n",
    "        [0, 2]\n",
    "    ]),\n",
    "    np.array([\n",
    "        [0.7, 0.3],\n",
    "        [0.7, 0.3]\n",
    "    ])\n",
    ")"
   ]
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  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
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   "source": [
    "def maximize_expected_utility(decision):\n",
    "    weighted_utilities = decision.utilities * decision.probabilities  # element-wise multiplication\n",
    "    expected_utilities = np.sum(weighted_utilities, axis=1)\n",
    "    max_eu_act_idxs = np.where(expected_utilities == np.amax(expected_utilities))\n",
    "    return list(np.array(decision.actions)[max_eu_act_idxs])"
   ]
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   "cell_type": "code",
   "execution_count": 5,
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     "data": {
      "text/plain": [
       "['take_clothes']"
      ]
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   "source": [
    "maximize_expected_utility(ride_to_work)"
   ]
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   "source": [
    "Where `maximin` and `maximax` differed on the ignorance version of this problem because of encoding \"pessimistic\" and \"optimistic\" outlooks, `meu` relies only on the agent's degrees of belief about the outcomes."
   ]
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   "source": [
    "Irrelevant Alternatives are Irrelevant\n",
    "----------"
   ]
  },
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   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
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   "outputs": [],
   "source": [
    "day_out = decision_problem(\n",
    "    ('sun', 'sprinkle', 'thunderstorm'),\n",
    "    ('play football', 'watch movie'),\n",
    "    np.array([\n",
    "        [15, 2, 6],\n",
    "        [8, 10, 9]\n",
    "    ]), \n",
    "    np.array([\n",
    "        [0.2, 0.5, 0.3],\n",
    "        [0.2, 0.5, 0.3]\n",
    "    ])\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "['watch movie']"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "maximize_expected_utility(day_out)"
   ]
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   "cell_type": "markdown",
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     "slide_type": "fragment"
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   "source": [
    "This is the same verdict as `minimax_regret`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
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   "outputs": [],
   "source": [
    "day_out_redux = decision_problem(\n",
    "    ('sun', 'sprinkle', 'thunderstorm'),\n",
    "    ('play football', 'watch movie', 'picnic'),\n",
    "    np.array([\n",
    "        [15, 2, 6],\n",
    "        [8, 10, 9],\n",
    "        [20, 5, 0]\n",
    "    ]), \n",
    "    np.array([\n",
    "        [0.2, 0.5, 0.3],\n",
    "        [0.2, 0.5, 0.3],\n",
    "        [0.2, 0.5, 0.3]\n",
    "    ])\n",
    ")"
   ]
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   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "['watch movie']"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "maximize_expected_utility(day_out_redux)"
   ]
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   "cell_type": "markdown",
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     "slide_type": "fragment"
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   "source": [
    "Adding the third option does not cause `meu` to favor football over the movie, contra `minimax_regret`.\n",
    "\n",
    "Exercise: explain why this holds generally."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "slide_type": "slide"
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   },
   "source": [
    "Justifying the Expected Utility Principle\n",
    "----------\n",
    "\n",
    "1. It's intuitive, and provides good results in a wide range of decision problems."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "slide_type": "fragment"
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   "source": [
    "2. Long-run arguments: you will be better off in the long run if you follow it."
   ]
  },
  {
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   "source": [
    "3. Grounding in rational preferences: if your preferences are rational, it is _as if_ you maximize expected utility."
   ]
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   "source": [
    "Long-run argument\n",
    "------\n",
    "\n",
    "The (strong) law of large numbers: if $X_1 , X_2 , ...$ is an infinite sequence of independent and identically distributed (i.i.d.) random variables, then $$\\text{Pr}\\left( \\lim_{n \\to \\infty} \\frac{1}{n}\\sum_{i=1}^n X_i = \\mathbb{E}(X) \\right) = 1$$"
   ]
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   "source": [
    "Applied to the random variable $u(\\cdot , a)$ in a decision problem $D$, this result says: if you repeatedly make a choice in the same decision problem, your average return will almost surely be the expected utility.\n",
    "\n",
    "So: you should try to maximize that quantity!"
   ]
  },
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     "slide_type": "subslide"
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   "source": [
    "### Some objections:\n",
    "\n",
    "* Keynes: \"In the long run, we are all dead.\"\n",
    "* While gambles arguably can, any important decisions (whether to take a job offer, or move to a new city, or...) simply cannot be repeated."
   ]
  },
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   "source": [
    "Grounding in Rational Preferences\n",
    "------\n",
    "\n",
    "* Main idea: _derive_ the MEU principle from other foundations.\n",
    "* Typically: place constraints on what it means for a set of preferences to be rational.\n",
    "* Prove a _representation theorem_: if an agent has rational preferences, they can be represented as acting in accord with MEU."
   ]
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   "source": [
    "von Neumann-Morgenstern\n",
    "-----\n",
    "\n",
    "A lottery can be defined as follows:\n",
    "* Any of some basic prizes are a lottery.\n",
    "* If $L_1$ and $L_2$ are lotteries and $p \\in (0, 1)$, then $pL_1/L_2$ is a lottery.\n",
    "    - Flip a coin with weight $p$: if heads, play lottery $L_1$, else $L_2$\n",
    "    \n",
    "(Can generate any finite chance situation in this way.)"
   ]
  },
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   "cell_type": "markdown",
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     "slide_type": "fragment"
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   "source": [
    "Revealing preferences over the basic prizes:\n",
    "\n",
    "| &#160; | 1 |\n",
    "| --- | --- |\n",
    "| $L_1$ | bitterballen |\n",
    "| $L_2$ | kaastengels |"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Suppose you (like me) have $L_1 \\preceq L_2$.  We can get more precise about how strong the preference is:\n",
    "\n",
    "| &#160; | 0.25 | 0.75 |\n",
    "| --- | --- | --- |\n",
    "| $L_1$ | bitterballen | kaastengels |\n",
    "| $L_2$ | chips | chips |"
   ]
  },
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   "source": [
    "vN-M Representation Theorem\n",
    "------"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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   },
   "source": [
    "Axioms governing rational preferences between lotteries:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* Completeness: for all $L_1 , L_2$ either $L_1 \\prec L_2$, $L_1 \\succ L_2$, or $L_1 \\sim L_2$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* Transitivity: if $L_1 \\prec L_2$ and $L_2 \\prec L_3$, then $L_1 \\prec L_3$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* Continuity: if $L_1 \\preceq L_2 \\preceq L_3$, then there is a $p \\in [0, 1]$ such that $pL_1/L_3 \\sim L_2$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* Independence: if $L_1 \\preceq L_2$, then for all $L_3$ and $p \\in [0, 1]$, $pL_1/L_3 \\preceq pL_2/L_3$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "**Theorem:** If an agent's preferences satisfy the four axioms, there is a function $u$ from lotteries to real numbers such that $$L_1 \\prec L_2 \\text{ iff } \\mathbb{E}[u(L_1)] < \\mathbb{E}[u(L_2)]$$\n",
    "Moreover, this $u$ is unique up to a linear transformation."
   ]
  },
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     "slide_type": "slide"
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   },
   "source": [
    "Discussion of vN-M\n",
    "------"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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   },
   "source": [
    "* What justifies the axioms on rational preference?\n",
    "    - The money pump argument for transitivity. Suppose I have the following preferences\n",
    "    \n",
    "        \\begin{align*}\n",
    "        \\text{Heineken} &\\prec \\text{IJwit} \\\\\n",
    "        \\text{IJwit} &\\prec \\text{Chouffe Bok} \\\\\n",
    "        \\text{Chouffe Bok} &\\prec \\text{Heineken}\n",
    "        \\end{align*}\n",
    "\n",
    "        and that I already have a Heineken.  You can offer me to trade it for an IJwit for 25 cents.  Then I would also pay 25 cents to trade the IJwit for a Chouffe Bok.  Finally, I would trade that for a Heineken for 25 cents.\n",
    "        \n",
    "        I'm back to where I started, just poorer!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "* Not all choices are over lotteries!\n",
    "    - See Savage (1954), Jeffrey (1983), and Joyce (1999) for more general representation theorems in the same style."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "What is utility?\n",
    "-----"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
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   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'utility')"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "dollars = np.arange(0, 2, 0.01)\n",
    "linear_utility = dollars\n",
    "plt.plot(dollars, linear_utility)\n",
    "plt.xlabel(\"euros\")\n",
    "plt.ylabel(\"utility\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x1192e2390>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "averse_utility = dollars**0.8\n",
    "seeking_utility = dollars**1.2\n",
    "plt.plot(dollars, averse_utility, label=\"risk averse\")\n",
    "plt.plot(dollars, linear_utility, label=\"risk neutral\")\n",
    "plt.plot(dollars, seeking_utility, label=\"risk seeking\")\n",
    "plt.xlabel(\"euros\")\n",
    "plt.ylabel(\"utility\")\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "What is utility? Kahneman and Tversky's Prospect Theory\n",
    "------"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x119332278>]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "gains = np.arange(0, 0.5, 0.01)\n",
    "losses = np.arange(-0.5, 0, 0.01)\n",
    "\n",
    "alpha = 0.7\n",
    "lambda_param = 2.25\n",
    "\n",
    "plt.plot(gains, gains**alpha)\n",
    "plt.plot(losses, -lambda_param*(-losses)**alpha)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "What are the probabilities?\n",
    "------"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* Classical: \"The theory of chance consists in reducing all the events of the same kind to a certain number of cases equally possible\" (Laplace)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* Logical: measures _evidential support_ between evidence and hypothesis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* Frequentist: probabilities measure the frequency with which certain events occur"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* Subjective: they are an agent's _degree of belief_\n",
    "    \n",
    "    This is the notion that we are using in this class."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Evidential vs. Causal Decision Theory\n",
    "------"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* What are the probabilities?\n",
    "    - Subjective degrees of belief\n",
    "    \n",
    "* Still open: _which_ probabilities?\n",
    "    - How do we specify what $p_a$ is for each act in a decision problem?\n",
    "    - It turns out that a lot hinges on the answer to this question.\n",
    "    - EDT and CDT are distinguished exactly by how they answer it."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Newcomb's Problem\n",
    "-------\n",
    "\n",
    "Before you are two boxes, one transparent and one opaque.  Inside the transparent box is 1,000 euros.  You can choose to receive the contents of both boxes (\"two-box\") or only the contents of the opaque box (\"one-box\").  The opaque box was filled yesterday, its contents yesterday by a very good predictor.  If they predicted that you will two-box, the opaque box is empty.  If they predicted that you will one-box, the opaque box has 1,000,000 euros in it.  \n",
    "\n",
    "So: two-box or one-box?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "| &#160; | predicted one-box | predicted two-box |\n",
    "| --- | --- | --- |\n",
    "| one-box | 1,000,000 | 0 |\n",
    "| two-box | 1,001,000 | 1,000 |"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* Two-box! It strictly dominates one-boxing.  Plus, the contents of the opaque box are already fixed."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* One-box! That makes it much more likely that I'll get a million euros."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Two Views\n",
    "-----"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* Evidential Decision Theory: $$p_a(s) := p(s | a)$$\n",
    "\n",
    "    This is sometimes called the \"news value\" of an action: what evidence about the state does it provide.\n",
    "    \n",
    "    Intuitively: $p(\\text{predict one-box} | \\text{one-box})$ is very high, so you should one-box."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* Causal Decision Theory: $$p_a(s) := p(a > s)$$\n",
    "\n",
    "    where \"$a > s$\" is the _counter-factual_ conditional \"If I had done $a$, then $s$ would have obtained.\"\n",
    "    \n",
    "    A concrete proposal: $p(a > s) := p(s | \\text{do}(a))$.\n",
    "    \n",
    "    Intuitively: $p(\\text{predict one-box} | \\text{do}(\\text{one-box})) = p(\\text{predict one-box})$, so you should two-box."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Implementing the Two Views\n",
    "-----\n",
    "\n",
    "* By way of concrete implementation, we will make the two views more precise.\n",
    "* Using [Pyro](http://pyro.ai/), a universal _probabilistic programming langauge_ (PPL)\n",
    "    - Allows free mixing of stochastic primitives with native Python\n",
    "    - Easy specification of complex generative probabilistic models\n",
    "    - Most importantly: `pyro.condition` and `pyro.do`\n",
    "* Such tools are becoming increasingly useful in AI, so exposure to it will likely help you in the future."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "The Basic Newcomb Model\n",
    "-------"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "import math\n",
    "import torch\n",
    "import pyro\n",
    "import pyro.distributions as dist\n",
    "\n",
    "def newcomb(prob_two_box=0.99, accuracy=0.99):\n",
    "\n",
    "    # 'raw facts' about agent, used by the predictor and causing the act\n",
    "    predisposition = pyro.sample('predisposition',\n",
    "                                 dist.Bernoulli(prob_two_box)).item()\n",
    "\n",
    "    # whether the Bernoulli should be p or (1-p)\n",
    "    predict_act_prob = (\n",
    "        math.sqrt(accuracy)*(predisposition == 1.) +\n",
    "        (1 - math.sqrt(accuracy))*(predisposition == 0.))\n",
    "\n",
    "    act = pyro.sample('act', dist.Bernoulli(predict_act_prob))\n",
    "    prediction = pyro.sample('prediction', dist.Bernoulli(predict_act_prob))\n",
    "\n",
    "    return prediction"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'one_box', 'two_box', 'one_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'one_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'one_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box']\n"
     ]
    }
   ],
   "source": [
    "def name(value):\n",
    "    return 'two_box' if value == 1. else 'one_box'\n",
    "\n",
    "print([name(newcomb().item()) for _ in range(100)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Conditioning and Intervening\n",
    "-------"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "def get_marginal_distribution(model, num_samples):\n",
    "    return pyro.infer.EmpiricalMarginal(\n",
    "                pyro.infer.Importance(model, num_samples=num_samples).run())\n",
    "\n",
    "def conditioned_marginal(model, data, num_samples=2000):\n",
    "    conditioned = pyro.condition(model, data=data)  # conditioning!\n",
    "    return get_marginal_distribution(conditioned, num_samples)\n",
    "\n",
    "def intervened_marginal(model, data, num_samples=2000):\n",
    "    intervened = pyro.do(model, data=data)  # intervening!\n",
    "    return get_marginal_distribution(intervened, num_samples)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['one_box', 'two_box', 'one_box', 'one_box', 'one_box', 'two_box', 'two_box', 'two_box', 'two_box', 'two_box', 'one_box', 'two_box', 'two_box', 'two_box', 'two_box', 'one_box', 'one_box', 'two_box', 'two_box', 'one_box', 'two_box', 'one_box', 'one_box', 'one_box', 'two_box', 'one_box', 'one_box', 'one_box', 'one_box', 'one_box', 'two_box', 'one_box', 'one_box', 'one_box', 'one_box', 'one_box', 'one_box', 'two_box', 'one_box', 'one_box', 'one_box', 'one_box', 'two_box', 'one_box', 'one_box', 'two_box', 'one_box', 'one_box', 'one_box', 'two_box', 'one_box', 'one_box', 'two_box', 'two_box', 'one_box', 'two_box', 'one_box', 'one_box', 'two_box', 'one_box', 'one_box', 'one_box', 'two_box', 'one_box', 'one_box', 'one_box', 'one_box', 'two_box', 'one_box', 'two_box', 'one_box', 'two_box', 'one_box', 'two_box', 'one_box', 'two_box', 'one_box', 'one_box', 'one_box', 'two_box', 'one_box', 'one_box', 'two_box', 'one_box', 'one_box', 'one_box', 'one_box', 'one_box', 'two_box', 'one_box', 'one_box', 'two_box', 'one_box', 'two_box', 'one_box', 'two_box', 'one_box', 'one_box', 'one_box', 'two_box']\n"
     ]
    }
   ],
   "source": [
    "one_box_act = {'act': 0.0}\n",
    "two_box_act = {'act': 1.0}\n",
    "\n",
    "conditioned_one_box = conditioned_marginal(newcomb, data=one_box_act)\n",
    "print([name(conditioned_one_box().item()) for _ in range(100)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "The Evidential Decision Theory Verdict\n",
    "--------"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0.66789787 0.33210298]\n",
      " [0.00506551 0.99493456]]\n"
     ]
    }
   ],
   "source": [
    "conditioned_dists = [\n",
    "    conditioned_marginal(newcomb, data=act)\n",
    "    for act in [one_box_act, two_box_act]\n",
    "]\n",
    "conditioned_probs = np.array([\n",
    "    [math.exp(dist.log_prob(prediction)) for prediction in [0.0, 1.0]]\n",
    "    for dist in conditioned_dists\n",
    "])\n",
    "print(conditioned_probs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['one_box']"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "newcomb_edt = decision_problem(\n",
    "    ('predict_one_box', 'predict_two_box'),\n",
    "    ('one_box', 'two_box'),\n",
    "    np.array([\n",
    "        [1000000, 0],\n",
    "        [1001000, 1000]\n",
    "    ]),\n",
    "    conditioned_probs\n",
    ")\n",
    "maximize_expected_utility(newcomb_edt)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "The Causal Decision Theory Verict\n",
    "-------"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0.0125     0.98749982]\n",
      " [0.0135     0.98649971]]\n"
     ]
    }
   ],
   "source": [
    "intervened_dists = [\n",
    "    intervened_marginal(newcomb, data=act)\n",
    "    for act in [one_box_act, two_box_act]\n",
    "]\n",
    "intervened_probs = np.array([\n",
    "    [math.exp(dist.log_prob(prediction)) for prediction in [0.0, 1.0]]\n",
    "    for dist in intervened_dists\n",
    "]) \n",
    "print(intervened_probs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['two_box']"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "newcomb_cdt = decision_problem(\n",
    "    ('predict_one_box', 'predict_two_box'),\n",
    "    ('one_box', 'two_box'),\n",
    "    np.array([\n",
    "        [1000000, 0],\n",
    "        [1001000, 1000]\n",
    "    ]),\n",
    "    intervened_probs\n",
    ")\n",
    "maximize_expected_utility(newcomb_cdt)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Newcomb and Dominance\n",
    "-------\n",
    "\n",
    "* Real-world Newcomb-like problems:\n",
    "    - The Smoking Lesion: if (as believed in the 1950s) a desire to smoke and lung cancer have a common cause, one should not smoke\n",
    "    - A certain kind of _Prisoner's Dilemma_ (next week)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* Whither dominance?  Joyce's (1999) extortionist:\n",
    "    > Suppose you have just parked in a seedy neighborhood when a man approaches and offers to “protect” your car from harm for 10 euros. You recognize this as extortion and have heard that people who refuse “protection” invariably return to find their windshields smashed. Those who pay find their cars intact.  You cannot park anywhere else because you are late for an important meeting.  It costs 400 euros to replace a windshield.  Should you buy “protection”?  Dominance says that you should not. Since you would rather have the extra 10 euros both in the even that your windshield is smashed and in the event that it is not, Dominance tells you not to pay. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Some Problems for MEU\n",
    "------"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Allais' Paradox and the Sure-Thing Principle\n",
    "-------"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Would you prefer a ticket from Gamble 1 or Gamble 2?\n",
    "\n",
    "| &#160; | no. 1 | no. 2-11 | no. 12-100 |\n",
    "| --- | --- | --- | ---|\n",
    "| Gamble 1 | 1,000,000 | 1,000,000 | 1,000,000 |\n",
    "| Gamble 2 | 0 | 5,000,0000 | 1,000,000 |"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Would you prefer a ticket from Gamble 3 or Gamble 4?\n",
    "\n",
    "| &#160; | no. 1 | no. 2-11 | no. 12-100 |\n",
    "| --- | --- | --- | ---|\n",
    "| Gamble 3 | 1,000,000 | 1,000,000 | 0 |\n",
    "| Gamble 4 | 0 | 5,000,0000 | 0 |"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "**Problem**: The judgments $G_1 \\succ G_2$ and $G_4 \\succ G_3$ are inconsistent with maximizing expected utility."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Transformative Decisions\n",
    "--------"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "(See Laurie Paul's recent book _Transformative Experience_.) Consider the following decisions:\n",
    "* whether to try bitterballen for the first time\n",
    "* whether to go to graduate school or not\n",
    "* whether to have your first child"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "| &#160; | actual world |\n",
    "| --- | --- |\n",
    "| try | get bitterballen |\n",
    "| don't try | status quo |"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* What is $u(@, bb)$?\n",
    "* When an experience transforms you, your utilities change.  Which ones should be used to evaluate the choice?"
   ]
  }
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