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<h2 class="hd hd-2 unit-title">Chapter Appendix: The Prisoner's Dilemma</h2>
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<p><span style="font-family: 'book antiqua', palatino; font-size: 12pt;"><span style="font-family: 'book antiqua', palatino; font-size: 12pt;"> Newcomb’s Problem is closely connected to a classic problem in Game Theory: the Prisoner’s Dilemma. </span></span></p>
<p><span style="font-family: 'book antiqua', palatino; font-size: 12pt;"><span style="font-family: 'book antiqua', palatino; font-size: 12pt;">Here is the Dilemma. </span></span></p>
<p><span style="font-family: 'book antiqua', palatino; font-size: 12pt;"><span style="font-family: 'book antiqua', palatino; font-size: 12pt;">Imagine you and Jones committed a crime together, and you’ve both been arrested. The police don’t have much evidence against either of you, though. If both of you keep quiet, you’ll each be convicted of a relatively minor offense, and spend only 1,000 days in prison (a payoff of -1,000). But there is an offer on the table. If either of you agrees to sign a statement accusing the other, the defector will be allowed to leave scot-free (a payoff of 0), and the accused will be convicted of a felony offense, and sentenced to 10,000 days in prison (a payoff of -10,000). </span></span></p>
<p><span style="font-family: 'book antiqua', palatino; font-size: 12pt;"><span style="font-family: 'book antiqua', palatino; font-size: 12pt;">Unfortunately, there is a catch. Should <em>both</em> you and Jones defect, you will both be convicted with felony offenses, but given the lesser sentence of 9,000 days in prison because of your cooperation with the police (a payoff of -9,000). </span></span></p>
<p><span style="font-family: 'book antiqua', palatino; font-size: 12pt;"><span style="font-family: 'book antiqua', palatino; font-size: 12pt;">Here is a summary of the payoffs:</span></span></p>
<table>
<thead>
<tr class="header"><th align="left"></th><th align="left">
<p>You defect</p>
</th><th align="left">
<p>You keep quiet</p>
</th></tr>
</thead>
<tbody>
<tr class="odd">
<td align="left">
<p>Jones defects</p>
</td>
<td align="left">
<p>Jones <span class="math inline">\(\rightarrow\)</span> <span class="math inline">\(-\)</span>9,000<br /> You <span class="math inline">\(\rightarrow\)</span> <span class="math inline">\(-\)</span>9,000</p>
</td>
<td align="left">
<p>Jones <span class="math inline">\(\rightarrow\)</span> 0<br /> You <span class="math inline">\(\rightarrow\)</span> <span class="math inline">\(-\)</span>10,000</p>
</td>
</tr>
<tr class="even">
<td align="left">
<p>Jones keeps quiet</p>
</td>
<td align="left">
<p>Jones <span class="math inline">\(\rightarrow\)</span> <span class="math inline">\(-\)</span>10,000<br /> You <span class="math inline">\(\rightarrow\)</span> 0</p>
</td>
<td align="left">
<p>Jones <span class="math inline">\(\rightarrow\)</span> <span class="math inline">\(-\)</span>1,000<br /> You <span class="math inline">\(\rightarrow\)</span> <span class="math inline">\(-\)</span>1,000</p>
</td>
</tr>
</tbody>
</table>
<p><span style="font-family: 'book antiqua', palatino; font-size: 12pt;"><span style="font-family: 'book antiqua', palatino; font-size: 12pt;">Before the offer is made, you and Jones are placed in separate interrogation rooms. You’re then each told of the offer, and each of you is also told that the other has been told of the offer. But, you’re not allowed to communicate with each other. So neither of you can do anything to affect the other’s decision. Finally, assume that all you care about is minimizing your jail time. So, in particular, you do not care about Jones at all, or about any consequences beyond those that have already been mentioned. (No need to worry about Jones exacting revenge, or about feeling guilt, or about acquiring a bad reputation, or anything like that.)</span></span></p>
<p><span style="font-family: 'book antiqua', palatino; font-size: 12pt;"><span style="font-family: 'book antiqua', palatino; font-size: 12pt;"> Should you defect or should you keep quiet?</span></span></p>
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<h2 class="hd hd-2 unit-title">The Standard Answer</h2>
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<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"> Philosophers and economists tend to think that the right thing to do in a Prisoner’s Dilemma is to defect. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">Why? </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">Because defecting <strong>dominates</strong> not defecting, in the sense of Lecture 5.3.1:</span></p>
<ul>
<li><span style="font-family: 'book antiqua', palatino;">If Jones defects, you’ll be better off if you defect than if you don’t.</span></li>
<li><span style="font-family: 'book antiqua', palatino;">If Jones doesn’t defect, you’ll be better off if you defect than if you don’t.</span></li>
</ul>
<p><span style="font-family: 'book antiqua', palatino;">So: regardless of what Jones does, you’ll be better off if you defect.</span></p>
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<span style="font-family: 'book antiqua', palatino;">Suppose that Jones will, in fact, defect. Will you be better off if you defect? (Remember that you are in separate interrogation rooms, so there is nothing you can do to affect Jones's decision.)</span>
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<h2 class="hd hd-2 unit-title">Newcomb's Problem as a Prisoner's Dilemma</h2>
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<p><span style="font-family: 'book antiqua', palatino; font-size: 12pt;"> An interesting observation, due to philosopher David Lewis, is that the Newcomb Problem can be thought of as a special case of the Prisoner’s Dilemma in which you and Jones are similar enough that you’re highly likely to end up making the same decisions, even though you haven’t communicated with one another. </span></p>
<p><span style="font-family: 'book antiqua', palatino; font-size: 12pt;">Suppose, for example, that Jones is your clone. The two of you are molecule for molecule identical, and have grown up in parallel environments. As a result, the probability that you’ll make the same decision in this case is 99%. Now imagine yourself facing 10,000 days of prison. You have two boxes in front of you. The small box contains a voucher, which reduces your prison time by 1,000 days; the large box contains a voucher which will either be counted as reducing your prison time by 9,000 days, or declared void. You are offered two choices: take both boxes or take just the large box. </span></p>
<p><span style="font-family: 'book antiqua', palatino; font-size: 12pt;">How is the value of the voucher in the large box determined? Jones is in a situation analogous to yours. If he decides to two-box, the voucher will be declared void; and if he decides to one-box, it will be counted as giving you a 9,000 day reduction. (Similarly, if you decide to two-box, the voucher in Jones’s large box will declared void, and if you decide to one-box, it will be counted as giving him a 9,000 day reduction.) Since there is no communication between you, your choices are causally independent. But since you are clones, the choices are not probabilistically independent: it is 99% likely that you will both end up making the same choice. </span></p>
<p><span style="font-family: 'book antiqua', palatino; font-size: 12pt;">We have constructed a version of the Prisoners’ Dilemma in which defecting is, in effect, two-boxing and keeping quiet is, in effect, one-boxing. In this sense, the Newcomb Problem is a special case of the Prisoners’ Dilemma.</span></p>
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<span style="font-family: 'book antiqua', palatino;">On the assumption that the probability that Jones and you will make the same decision is <span class="math inline">\(99\%\)</span>, what is the expected value of defecting? (Note: in an outcome where you spend <em>n</em> days in prison, your payoff is <em>-n</em>.)</span>
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<span style="font-family: 'book antiqua', palatino;">On that assumption, what is the expected value of keeping quiet?</span>
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