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<h2 class="hd hd-2 unit-title">Maximizing Expected Value</h2>
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<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"> In the previous section I tried to give you an intuitive sense of why the Newcomb case might seem puzzling. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">Now I will give you some mathematical machinery that will help make the puzzle a little more precise. I will introduce you to <em>decision theory.</em></span></p>
<p><span style="font-family: 'book antiqua', palatino;">Suppose that you have different options to choose from. According to the most straightforward form of decision theory, you should make your decision in accordance with the following principle:</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Principle of Expected Value Maximization</strong></span><br /><span style="font-family: 'book antiqua', palatino;">Choose an option whose expected value is at least as high as that of any rival option.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">The best way of getting a handle on the notion of expected value is to consider an example, and we’ll do so shortly. But the intuitive idea is that the expected value of an option is a measure of how desirable the world is expected to be, on the assumption that you choose that option.</span></p>
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<h2 class="hd hd-2 unit-title">An Example</h2>
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<p><span style="font-family: 'book antiqua', palatino;">Suppose you’re wondering what to do tonight. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">You have two <strong>options</strong>: <em>go out for drinks</em> and <em>study</em>. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">There are two relevant <strong>states of the world</strong>: <em>easy exam</em> and <em>hard exam</em>. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">With two options and two relevant states of the world, your choice has four possible <strong>outcomes</strong>, which are represented as the empty cells in the following matrix:</span></p>
<table>
<thead>
<tr class="header"><th align="center"></th><th align="center"><span style="font-family: 'book antiqua', palatino;">easy exam</span></th><th align="center"><span style="font-family: 'book antiqua', palatino;">hard exam</span></th></tr>
</thead>
<tbody>
<tr class="odd">
<td align="center"><span style="font-family: 'book antiqua', palatino;">drinks</span></td>
<td align="center"></td>
<td align="center"></td>
</tr>
<tr class="even">
<td align="center"><span style="font-family: 'book antiqua', palatino;">study</span></td>
<td align="center"></td>
<td align="center"></td>
</tr>
</tbody>
</table>
<p></p>
<p><span style="font-family: 'book antiqua', palatino;">How desirable is each of these outcomes? </span></p>
<p><span style="font-family: 'book antiqua', palatino;">On the one hand, you’ll have fun if you go out for drinks, and be bored if you stay home and study. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">On the other hand, you can pass an easy exam without studying, but you’ll need to study to pass a hard exam.</span></p>
<table>
<thead>
<tr class="header"><th align="center"></th><th align="center"><span style="font-family: 'book antiqua', palatino;">easy exam</span></th><th align="center"><span style="font-family: 'book antiqua', palatino;">hard exam</span></th></tr>
</thead>
<tbody>
<tr class="odd">
<td align="center"><span style="font-family: 'book antiqua', palatino;">drinks</span></td>
<td align="center"><span style="font-family: 'book antiqua', palatino;">fun + pass</span></td>
<td align="center"><span style="font-family: 'book antiqua', palatino;">fun + fail</span></td>
</tr>
<tr class="even">
<td align="center"><span style="font-family: 'book antiqua', palatino;">study</span></td>
<td align="center"><span style="font-family: 'book antiqua', palatino;">bored + pass</span></td>
<td align="center"><span style="font-family: 'book antiqua', palatino;">bored + pass</span></td>
</tr>
</tbody>
</table>
<p><span style="font-family: 'book antiqua', palatino;">Let us put numerical values on the desirability of our outcomes. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Suppose you think that failing the exam would be twice as bad as passing the exam would be good. So if we assign an arbitrary value of 20 to passing the exam, we should assign a value of <span class="math inline">\(-40\)</span> to failing the exam. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Going out for drinks would be almost as good as passing the exam, so we’ll assign it a value of <span class="math inline">\(15\)</span>. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Studying, in contrast, would be only somewhat painful, so we’ll assign it a value of <span class="math inline">\(-2\)</span>. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">This allows us to assign a value to each of our four possible outcomes:</span></p>
<table>
<thead>
<tr class="header"><th align="center"></th><th align="center"><span style="font-family: 'book antiqua', palatino;">easy exam</span></th><th align="center"><span style="font-family: 'book antiqua', palatino;">hard exam</span></th></tr>
</thead>
<tbody>
<tr class="odd">
<td align="center"><span style="font-family: 'book antiqua', palatino;">drinks</span></td>
<td align="center"><span class="math inline" style="font-family: 'book antiqua', palatino;">\(15+20\)</span></td>
<td align="center"><span class="math inline" style="font-family: 'book antiqua', palatino;">\(15 + (- 40)\)</span></td>
</tr>
<tr class="even">
<td align="center"><span style="font-family: 'book antiqua', palatino;">study</span></td>
<td align="center"><span class="math inline" style="font-family: 'book antiqua', palatino;">\(-2 + 20\)</span></td>
<td align="center"><span class="math inline" style="font-family: 'book antiqua', palatino;">\(-2 + 20\)</span></td>
</tr>
</tbody>
</table>
<p></p>
<p><span style="font-family: 'book antiqua', palatino;">The next thing we need to so is associate a <em>probability</em> with each of the two states of the world that’s relevant to our outcome: easy exam, and hard exam. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Let us suppose that the probability of an easy exam is <span class="math inline">\(80\%\)</span>, and therefore that the probability of a hard exam is <span class="math inline">\(20\%\)</span>.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">We’ve now assigned a value to each outcome, and a probability to each relevant state of the world. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">We can use this to assign <strong>expected values</strong> to each of the two options you can choose from: going out for drinks and studying. We do so by taking the <em>weighted average</em> of the values of the different possible outcomes, with weights given by the probabilities of the relevant states of the world. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">The expected value of going out for drinks <span class="math inline">\((D)\)</span> is: <span class="math display">\[EV(D) \quad = \quad \underbrace{35}_{\tiny \text{value of drinking and easy exam }} \ \cdot \ \underbrace{0.8}_{\tiny \text{probability of easy exam}} \quad + \quad \underbrace{-25}_{\tiny \text{value of drinking and hard exam}} \ \cdot \ \underbrace{0.2}_{\tiny \text{probability of hard exam}} \quad = \quad 23\]</span> Similarly, the expected value of studying <span class="math inline">\((S)\)</span> is: <span class="math display">\[EV(S) \quad = \quad \underbrace{18}_{\tiny \text{value of studying and easy exam}} \ \cdot \ \underbrace{0.8}_{\tiny \text{probability of easy exam}} \quad + \quad \underbrace{18}_{\tiny \text{value of studying and hard exam}} \ \cdot \ \underbrace{0.2}_{\tiny \text{probability of hard exam}} \quad = \quad 18\]</span> We can now look to the Principle of Expected Value Maximization to decide what to do. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">The Principle tells us that we ought to choose an option whose expected value is at least as high as that of any rival. Since <span class="math inline">\(EV(D) > EV(S)\)</span>, this means that you ought to go out for drinks, rather than study.</span></p>
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<h2 class="hd hd-2 unit-title">An evil teacher</h2>
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<p><span style="font-family: 'book antiqua', palatino;">An important feature of our example is that the probability of a given state of the world (easy exam or hard exam) is independent of which option you select (drinking or studying). </span></p>
<p><span style="font-family: 'book antiqua', palatino;">But this needn’t be true in general. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Suppose, for example, that you have an evil teacher who knows whether you go out for drinks, and is more likely to assign a hard exam if you do. Whereas the probability of an easy exam <em>given that you go out for drinks</em> is <span class="math inline">\(30\%\)</span>, the probability of an easy exam <em>given that you study</em> is <span class="math inline">\(90\%\)</span>. The resulting probabilities might be represented as follows:</span></p>
<table>
<thead>
<tr class="header"><th align="center"></th><th align="center"><span style="font-family: 'book antiqua', palatino;">easy exam</span></th><th align="center"><span style="font-family: 'book antiqua', palatino;">hard exam</span></th></tr>
</thead>
<tbody>
<tr class="odd">
<td align="center"><span style="font-family: 'book antiqua', palatino;">drinks</span></td>
<td align="center"><span class="math inline" style="font-family: 'book antiqua', palatino;">\(30\%\)</span></td>
<td align="center"><span class="math inline" style="font-family: 'book antiqua', palatino;">\(70\%\)</span></td>
</tr>
<tr class="even">
<td align="center"><span style="font-family: 'book antiqua', palatino;">study</span></td>
<td align="center"><span class="math inline" style="font-family: 'book antiqua', palatino;">\(90\%\)</span></td>
<td align="center"><span class="math inline" style="font-family: 'book antiqua', palatino;">\(10\%\)</span></td>
</tr>
</tbody>
</table>
<p></p>
<p><span style="font-family: 'book antiqua', palatino;">Note that each cell in this matrix corresponds to a <em>conditional probability</em>. For example, the probability in the lower right corner corresponds to the probability of a hard exam, given that you study.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">In the evil teacher scenario, we need to use these conditional probabilities to calculate expected value. For instance, the expected value of going out for drinks <span class="math inline">\((D)\)</span> should be calculated as follows: <span class="math display">\[EV(D) \quad = \quad \underbrace{35}_{\tiny \text{value of drinking with easy exam}} \ \cdot \ \underbrace{0.3}_{\tiny \text{probability of easy exam, given drinking}} \quad + \quad \underbrace{-25}_{\tiny \text{value of drinking with hard exam}} \ \cdot \ \underbrace{0.7}_{\tiny \text{probability of hard exam, given drinking}} \quad = \quad -7\]</span> Similarly, the expected value of studying <span class="math inline">\((S)\)</span> is now calculated as follows: <span class="math display">\[EV(S) \quad = \quad \underbrace{18}_{\tiny \text{value of studying with easy exam}} \ \cdot \ \underbrace{0.9}_{\tiny \text{probability of easy exam}} \quad + \quad \underbrace{18}_{\tiny \text{value of studying with hard exam}} \ \cdot \ \underbrace{0.1}_{\tiny \text{probability of hard exam}} \quad = \quad 18\]</span> Since <span class="math inline">\(EV(S) > EV(D)\)</span>, the Principle of Expected Value Maximization entails that when you have an evil teacher, you ought to study, rather than go out for drinks.</span></p>
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<h2 class="hd hd-2 unit-title">Definition of Expected Value</h2>
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<p><span style="font-family: 'book antiqua', palatino;">Now that you know how expected value works, I can give you a more precise definition. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">The <strong>expected value</strong> of an option <span class="math inline">\(A\)</span> is the weighted average of the value of the outcomes that <span class="math inline">\(A\)</span> might lead to, with weights determined by the probability of the relevant state of affairs, given that you choose <span class="math inline">\(A\)</span>. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Formally: <span class="math display">\[EV(A) = v(A S_1) \cdot p(S_1|{}A) + v(A S_2) \cdot p(S_2|{}A) + \ldots + v(A S_n) \cdot p(S_n|{}A)\]</span> where <span class="math inline">\(S_1, S_2,\ldots S_n\)</span> is any list of (exhaustive and mutually exclusive) states of the world, <span class="math inline">\(v(A S_i)\)</span> is the value of being in a situation in which you’ve chosen <span class="math inline">\(A\)</span> and <span class="math inline">\(S_i\)</span> is the case, and <span class="math inline">\(p(S|A)\)</span> is the probability of <span class="math inline">\(S\)</span>, given that you choose <span class="math inline">\(A\)</span>.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">Here is how to apply this formula in the case of the evil teacher. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Recall that there are two relevant states of affairs: easy exam (<span class="math inline">\(E\)</span>) and hard exam (<span class="math inline">\(H\)</span>). So the expected value of drinking (<span class="math inline">\(D\)</span>) and studying (<span class="math inline">\(S\)</span>) should be calculated as follows:</span></p>
<p><span class="math display" style="font-family: 'book antiqua', palatino;">\[\begin{array}{ccccccccccc} EV(D) &= &v(D E) &\cdot &p(E|D) &+ &v(DH) &\cdot &p(H|D) \\ &= &35 &\cdot &0.3 &+ &(-25) &\cdot &0.7 &= &-7\\ \ \\ EV(S) &= &v(S E) &\cdot &p(E|S) &+ &v(SH) &\cdot &p(H|S) \\ &= &18 &\cdot &0.9 &+ &18 &\cdot &0.1 &= &18 \end{array}\]</span></p>
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<span style="font-family: 'book antiqua', palatino;">A fair coin will be tossed, and you must choose between the following two bets:</span>
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<table>
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<span class="math inline">\(B_1\)</span>
</th>
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<span class="math inline">\(B_2\)</span>
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<td align="left">$1000 if Heads; <span class="math inline">\(-\$200\)</span> if Tails. </td>
<td align="left"> $100 Heads; $50 if Tails.</td>
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<span style="font-family: 'book antiqua', palatino;">(Assume the degree to which you value a given outcome corresponds to the amount of money you end up with. So, for example, you assign value 1000 to an outcome in which you receive $1000, and value <span class="math inline">\(-200\)</span> to an outcome in which you pay $200.)</span>
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<span style="font-family: 'book antiqua', palatino;">What is the expected value of accepting <span class="math inline">\(B_1\)</span>?</span>
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<span style="font-family: 'book antiqua', palatino;">What is the expected value of accepting <span class="math inline">\(B_2\)</span>?</span>
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<span style="font-family: 'book antiqua', palatino;">Which of the two bets should you accept, according to the Principle of Expected Value Maximization?</span>
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<h2 class="hd hd-2 unit-title">Back to Newcomb</h2>
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<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"><span style="font-size: 12pt;"> What does the Principle of Expected Value Maximization say we should do in a Newcomb situation? </span></span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"><span style="font-size: 12pt;">Recall that you can choose between the following two options:</span></span></p>
<table>
<thead>
<tr class="header"><th align="left"><span class="math inline" style="font-family: 'book antiqua', palatino;">\(1B\)</span></th><th align="left"><span class="math inline" style="font-family: 'book antiqua', palatino;">\(2B\)</span></th></tr>
</thead>
<tbody>
<tr class="odd">
<td align="left"><span style="font-family: 'book antiqua', palatino;">One-Box</span></td>
<td align="left"><span style="font-family: 'book antiqua', palatino;">Two-Box</span></td>
</tr>
</tbody>
</table>
<p></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"><span style="font-size: 12pt;">Recall, moreover, that there are two relevant states of the world:</span></span></p>
<table>
<thead>
<tr class="header"><th align="left"><span class="math inline" style="font-family: 'book antiqua', palatino;">\(F\)</span></th><th align="left"><span class="math inline" style="font-family: 'book antiqua', palatino;">\(E\)</span></th></tr>
</thead>
<tbody>
<tr class="odd">
<td align="left"><span style="font-family: 'book antiqua', palatino;">The large box is full</span></td>
<td align="left"><span style="font-family: 'book antiqua', palatino;">The large box is empty</span></td>
</tr>
</tbody>
</table>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"><span style="font-size: 12pt;">This means that there are four possible outcomes, depending on whether you One-Box or Two-Box and on whether the large box is full or empty. </span></span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"><span style="font-size: 12pt;">The value of each outcome is as follows:</span></span></p>
<table>
<thead>
<tr class="header"><th align="center"></th><th align="center"><span style="font-family: 'book antiqua', palatino;">Large box full (<span class="math inline">\(F\)</span>)</span></th><th align="center"><span style="font-family: 'book antiqua', palatino;">Large box empty (<span class="math inline">\(E\)</span>)</span></th></tr>
</thead>
<tbody>
<tr class="odd">
<td align="center"><span style="font-family: 'book antiqua', palatino;">One-Box (<span class="math inline">\(1B\)</span>)</span></td>
<td align="center"><span style="font-family: 'book antiqua', palatino;">$1,000,000</span></td>
<td align="center"><span style="font-family: 'book antiqua', palatino;">$0</span></td>
</tr>
<tr class="even">
<td align="center"><span style="font-family: 'book antiqua', palatino;">Two-Box (<span class="math inline">\(2B\)</span>)</span></td>
<td align="center"><span style="font-family: 'book antiqua', palatino;">$1,001,000</span></td>
<td align="center"><span style="font-family: 'book antiqua', palatino;">$1,000</span></td>
</tr>
</tbody>
</table>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"><span style="font-size: 12pt;">When the predictor is assumed to be 99% accurate, we work with the following probabilities:</span></span></p>
<table>
<thead>
<tr class="header"><th align="center"></th><th align="center"><span style="font-family: 'book antiqua', palatino;">Large box full (<span class="math inline">\(F\)</span>)</span></th><th align="center"><span style="font-family: 'book antiqua', palatino;">Large box empty (<span class="math inline">\(E\)</span>)</span></th></tr>
</thead>
<tbody>
<tr class="odd">
<td align="center"><span style="font-family: 'book antiqua', palatino;">One-Box (<span class="math inline">\(1B\)</span>)</span></td>
<td align="center"><span class="math inline" style="font-family: 'book antiqua', palatino;">\(99\%\)</span></td>
<td align="center"><span class="math inline" style="font-family: 'book antiqua', palatino;">\(1\%\)</span></td>
</tr>
<tr class="even">
<td align="center"><span style="font-family: 'book antiqua', palatino;">Two-Box (<span class="math inline">\(2B\)</span>)</span></td>
<td align="center"><span class="math inline" style="font-family: 'book antiqua', palatino;">\(1\%\)</span></td>
<td align="center"><span class="math inline" style="font-family: 'book antiqua', palatino;">\(99\%\)</span></td>
</tr>
</tbody>
</table>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"><span style="font-size: 12pt;">As before, each cell in this matrix corresponds to a <em>conditional probability</em>. For example, the probability in the lower right corner corresponds to the probability of finding the large box empty, given that you two-box. </span></span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"><span style="font-size: 12pt;">The expected value of one-boxing and two-boxing can then calculated as follows:</span></span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"><span class="math display">\[\begin{array}{ccccccccccc} EV(1B) &= &v(1B\, F) &\cdot &p(F|1B) &+ &v(1B\,E) &\cdot &p(E|1B) \\ &= &1000000 &\cdot &0.99 &+ &0 &\cdot &0.01 &= &990,000 \end{array}\]</span></span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"><span class="math display">\[\begin{array}{ccccccccccc} EV(2B) &= &v(2B\, F) &\cdot &p(F|2B) &+ &v(2B\,E) &\cdot &p(E|2B)\\ &= &1001000 &\cdot &0.01 &+ &1000 &\cdot &0.99 &= &11,000 \end{array}\]</span></span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">The expected value of one-boxing ($990,000) is much greater than the expected value of two-boxing ($11,000). So according to the Principle of Expected Utility Maximization, we should one-box. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">Is that the right result? </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">We’ll try to tackle that question in the next section.</span></p>
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<span style="font-family: 'book antiqua', palatino;">Assume the predictor has an 80% chance of making the right prediction.</span>
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<span style="font-family: 'book antiqua', palatino;">What is the expected value of one-boxing?</span>
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<span style="font-family: 'book antiqua', palatino;">What is the expected value of two-boxing?</span>
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Problem 2
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<span style="font-family: 'book antiqua', palatino;">How accurate does the predictor need to be in order for the Principle of Expected Value Maximization to entail that one should one-box?</span>
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<p>
<span style="font-family: 'book antiqua', palatino;">In other words: identify the smallest number <span class="math inline">\(x\)</span> such that, as long as the predictor has an accuracy of more than <span class="math inline">\(x\)</span>, the Principle of Expected Value Maximization entails that one should one-box.</span>
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<p>
<span style="font-family: 'book antiqua', palatino;">(As usual, assume that the small box contains $1000 and that the large box contains $1M or nothing.)</span>
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Problem 3
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<span style="font-family: 'book antiqua', palatino;">Is the following true or false?</span>
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<p>
<span style="font-family: 'book antiqua', palatino;">As long as one has a predictor who does better than chance, it is possible to find payoffs that generate a Newcomb Problem.</span>
</p>
<p>
<span style="font-family: 'book antiqua', palatino;">More precisely: for any small-box value <span class="math inline">\(s\)</span> and any positive value <span class="math inline">\(\epsilon\)</span> (<span class="math inline">\(0 &lt; \epsilon \leq 0.5\)</span>), one can find an large-box value <span class="math inline">\(l\)</span> such that when the predictor has an accuracy of <span class="math inline">\(0.5+\epsilon\)</span>, the Principle of Expected Value Maximization entails that one should one-box in a Newcomb Scenario based on <span class="math inline">\(s\)</span> and <span class="math inline">\(l\)</span>.</span>
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Problem 4
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<span style="font-family: 'book antiqua', palatino;">Consider a variant of the Newcomb case that works as follows:</span>
</p>
<blockquote>
<p>
<span style="font-family: 'book antiqua', palatino;">You can choose either the large box or the small box, but not both.</span>
</p>
<p>
<span style="font-family: 'book antiqua', palatino;">If the predictor predicts that you would choose the large box, then she left the large box empty, and placed $100 in the small box.</span>
</p>
<p>
<span style="font-family: 'book antiqua', palatino;">If the predictor predicted that you will choose the small box, then she left the small box empty, and placed $1000 in the large box.</span>
</p>
</blockquote>
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<span style="font-family: 'book antiqua', palatino;">According to the Principle of Expected Value Maximization, which of the two boxes should you choose? (Assume the predictor is not perfectly accurate.)</span>
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<input type="radio" name="input_88dc08a1ef6e45e290561286ffe53970_2_1" id="input_88dc08a1ef6e45e290561286ffe53970_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="88dc08a1ef6e45e290561286ffe53970_2_1-choice_0-label" for="input_88dc08a1ef6e45e290561286ffe53970_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_88dc08a1ef6e45e290561286ffe53970_2_1">
<span style="font-family: 'book antiqua', palatino;">Choose the large box.</span>
</label>
</div>
<div class="field">
<input type="radio" name="input_88dc08a1ef6e45e290561286ffe53970_2_1" id="input_88dc08a1ef6e45e290561286ffe53970_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="88dc08a1ef6e45e290561286ffe53970_2_1-choice_1-label" for="input_88dc08a1ef6e45e290561286ffe53970_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_88dc08a1ef6e45e290561286ffe53970_2_1">
<span style="font-family: 'book antiqua', palatino;">Choose the small box.</span>
</label>
</div>
<div class="field">
<input type="radio" name="input_88dc08a1ef6e45e290561286ffe53970_2_1" id="input_88dc08a1ef6e45e290561286ffe53970_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="88dc08a1ef6e45e290561286ffe53970_2_1-choice_2-label" for="input_88dc08a1ef6e45e290561286ffe53970_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_88dc08a1ef6e45e290561286ffe53970_2_1">
<span style="font-family: 'book antiqua', palatino;">None of the above.</span>
</label>
</div>
<span id="answer_88dc08a1ef6e45e290561286ffe53970_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_88dc08a1ef6e45e290561286ffe53970_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div><div class="solution-span">
<span id="solution_88dc08a1ef6e45e290561286ffe53970_solution_1"/>
</div></div>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 4" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_88dc08a1ef6e45e290561286ffe53970" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_88dc08a1ef6e45e290561286ffe53970">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="88dc08a1ef6e45e290561286ffe53970-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="88dc08a1ef6e45e290561286ffe53970-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="88dc08a1ef6e45e290561286ffe53970-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
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