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<h2 class="hd hd-2 unit-title">The Demon's Game</h2>
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<p><span style="font-family: 'book antiqua', palatino;">Infinite sequences can cause trouble for rational decision making. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">In this lecture I’d like to describe a couple of paradoxes that bring that out. I learned about them from philosophers Frank Arntzenius, Adam Elga and John Hawthorne.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">Imagine an infinite group of people <span class="math inline">\(P_1,P_2,P_3,\ldots\)</span>. An evil demon suggests that they play a game. “I’m going to ask each of you to say <em>aye</em> or <em>nay</em>", he says. “If only finitely many amongst you say <em>aye</em>, there will be prizes for everyone: each of you will receive as many gold coins as there are people who said <em>aye</em>. Should infinitely many of you say <em>aye</em>, however, nobody will receive anything."</span></p>
<p><span style="font-family: 'book antiqua', palatino;">If the members of our group were in a position to agree on a strategy beforehand, they could end up with as many gold coins as they wanted. All they would need to do is pick a number <span class="math inline">\(k\)</span>, and commit to a plan whereby persons <span class="math inline">\(P_1,\ldots, P_k\)</span> answer <em>aye</em>, and everyone else answers <em>nay</em>. As long as everyone sticks to the plan, every member of the group will end up with <span class="math inline">\(k\)</span> gold coins.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">But the demon is no fool. He knows that he could bankrupt himself if he allowed the group to agree on a strategy beforehand. So he isolates each member of the group as soon as the rules have been announced. As a result, each person must decide whether to say <em>aye</em> or <em>nay</em> without having any information about the decisions of other members of the group.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">With such precautions in place, the evil demon has nothing to fear. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">To see this, imagine yourself as a member of the group. You are isolated from your colleagues, and you are pondering your answer. Should you say <em>aye</em> or should you say <em>nay</em>? You know that your decision can have no effect on other people’s decisions, and reason as follows:</span></p>
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<p><span style="font-family: 'book antiqua', palatino;">If infinitely many of my colleagues answer <em>aye</em> nobody will get any coins, regardless of what I decide to do. So my decision can only make a difference to the outcome on the assumption that at most finitely many of my colleagues answer <em>aye</em>. But in that case I should definitely answer <em>aye</em>. For doing so will result in an additional gold coin for everyone, including myself! (If I were to answer <em>nay</em>, on the other hand, I wouldn't be helping anyone.)<br /></span></p>
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<p><span style="font-family: 'book antiqua', palatino;">So: answering <em>aye</em> couldn’t possibly make things worse, and could very well make things better. The rational thing to do is therefore to answer <em>aye</em>!</span></p>
<p><span style="font-family: 'book antiqua', palatino;">Of course, other members of the group are in exactly the same situation as you. So what is rational for you is also rational for them. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">That is why the demon has nothing to fear. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">As long as every member of the group behaves rationally, everyone will answer <em>aye</em>, and the demon won’t have to cough up any money. (It goes without saying that the demon could lose a lot of money if only finitely many members of the group are fully rational. But the demon was careful to avoid selecting such a group for his game: nothing gives him more pleasure than torturing fully rational people.)</span></p>
<p><span style="font-family: 'book antiqua', palatino;">Our group is in a position to avail itself of as much money as it likes. And yet we can predict in advance that it will fail to do so, even though every member of the group behaves rationally. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">What has gone wrong?</span></p>
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<h2 class="hd hd-2 unit-title">Assessment: the Demon's Game</h2>
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<h3><span style="font-family: 'book antiqua', palatino;">The Devil’s Game</span><br /><span style="font-family: 'book antiqua', palatino;">Paradox Grade: 4</span></h3>
<p><span style="font-family: 'book antiqua', palatino;">What I find interesting about this paradox is that it shows that there can be a discontinuity between the rationality of a group and the rationality of its members. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">More precisely:</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Collective Tragedy</strong></span><br /><span style="font-family: 'book antiqua', palatino;">There are situations in which a group will predictably end up in a situation that is suboptimal for every member of the group, even though every member of the group behaves rationally.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">What should we do about Collective Tragedy? </span></p>
<p><span style="font-family: 'book antiqua', palatino;">My own view is that there is nothing to be done. We need to learn how to live with it. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">In some ways, Collective Tragedy reminds me of a famous theorem due to Stanford economist Kenneth Arrow. <a href="https://plato.stanford.edu/entries/arrows-theorem/" target="[object Object]">Arrow's Impossibility Theorem</a> states that, given certain assumptions, there is no way of using the preferences of individual members of a group to determine the "collective preferences" of the group. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Both Collective Tragedy and Arrow’s Theorem illustrate the fact that one can face discontinuities when one tries to derive the properties of a collective from the properties of its individual members.</span></p>
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<h2 class="hd hd-2 unit-title">A one-person version of the game</h2>
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<p><span style="font-family: 'book antiqua', palatino;">There is a variation of the demon’s evil game that can be applied to a single individual. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Imagine that the demon whispers into your ear: "I am about to ask you an infinite sequence of questions \(q_1, q_2, q_3,\ldots\). Each time I ask you a question, you must answer <em>aye</em> or <em>nay.</em> If you answer aye at most finitely many times, you will receive a prize: as many gold coins as <em>aye</em>-answers you gave. If, however, you answer aye infinitely many times, I will see to it that you leave empty-handed."</span></p>
<p><span style="font-family: 'book antiqua', palatino;">Note that in order for this exercise to make sense there is no need to assume that you lead an infinitely long life. We can assume instead that you are able to answer the demon’s questions faster and faster. Suppose, for example, that the second question is asked half an hour after the first, that the third question is asked a quarter of an hour after the second, and so forth. (In general, question \(q_{n+1}\) is asked \(1/2^n\) hours after question \(q_n\).) This entails that the demon will have asked all his questions within the span of an hour. So as long as you’re able to answer each of his questions before he asks the next one, you will have answered all of his questions within the span of an hour. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">If you were able to commit once and for all to a plan about how to answer the demon’s questions, you could end up with as many gold coins as you wanted. Just as in the multi-person case, you could simply select a number \(k\) and commit to answering <em>aye</em> to questions \(q_1,\ldots,q_k\), and no further. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">But will you be able to stick to your plan? </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Imagine that you’ve committed to answering <em>aye</em> to questions \(q_1,\ldots, q_{k}\) and no further. You’ve responded <em>aye</em> to the first \(k\) questions, just as you had planned. And now the demon asks question \(q_{k+1}\). If you respect the original plan, you should answer <em>nay.</em> </span></p>
<p><span style="font-family: 'book antiqua', palatino;">But why respect the original plan? </span></p>
<p><span style="font-family: 'book antiqua', palatino;">It seems clear that a better plan is available to you: answer <em>aye</em> to questions \(q_1,\ldots, q_{k+1}\) and no further. This new plan will bring you all the benefits of the old plan, plus one gold coin!</span></p>
<p><span style="font-family: 'book antiqua', palatino;">The problem, of course, is that this process could be iterated. Each time your current plan commits you to answering <em>nay,</em> you will be tempted to switch plans. And if you switch plans every time, you’ll end up empty-handed, since you will end up answering <em>aye</em> to every single question.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">There are those who possess an unflappable will: people who are able to make a plan and stick to it even if a better plan presents itself further along the line. Sometimes one needs a trick to ensure that one sticks to one’s plans. There is a famous example of this in Homer’s Odyssey. Here is a paraphrase of the relevant passage:</span></p>
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<p><span style="font-family: 'book antiqua', palatino;">So enchanting were the songs of the sirens, that no sailor could resist them. Bewitched, the sailor would follow the music into the sea, and drown. Odysseus wanted to hear the song of the sirens, but he didn’t want to die. So he ordered his men to bind him to the mast of his ship, and made clear that he was not to be released until the deadly sirens had been left behind. He then ordered his men cover their own ears with wax. When the sirens began to sing, it seemed to Odysseus that his original plan ought to be replaced with a different one: the plan of following the sirens’ songs. But his men forced him to stick to the original plan, and refused to release him until they were safely out of the sirens’ reach.</span></p>
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<p><span style="font-family: 'book antiqua', palatino;">There is an interesting analogy between Odysseus’ predicament and the demon’s game. In both cases, things are set up so that one is tempted to abandon one’s original plan midway. In the case of Odysseus, the new plan will only seem like a good idea to someone who has fallen under the sirens’ spell. When it comes to the demon’s game, however, everyone should agree that switching plans is the rational thing to do. For even before the start of the game, everyone should agree that answering <em>aye</em> to the first \(k+1\) questions (and <em>nay</em> to the rest) is better than answering <em>aye</em> to the first \(k\) questions (and <em>nay</em> to the rest).</span></p>
<p><span style="font-family: 'book antiqua', palatino;">You will only be able to beat the evil demon if you’re able to make decisions on the basis of plans that can be seen to be inferior from the start, and then stick to those plans. You’ll have to somehow find a way of "tying yourself to the mast". But, for most people, that is not easy to do.</span></p>
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<h2 class="hd hd-2 unit-title">Assessment: one-person version of the game</h2>
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<h3>A One-Person Version of the Devil’s Game<br />Paradox Grade: 4</h3>
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<p><span style="font-family: 'book antiqua', palatino; font-size: 12pt;"> The reason I like this paradox, and the reason I think it deserves a higher paradoxicality grade than its multi-person counterpart, is that it suggests that an individual person might be thought of as a kind of collective—a collective consisting of her different "time slices"— and that collectives consisting of time slices are just as subject to Collective Tragedy as collectives consisting of ordinary individuals. </span></p>
<p><span style="font-family: 'book antiqua', palatino; font-size: 12pt;">In other words: there are scenarios in which you will predictably end up in a suboptimal situation, even though each of your time-slices behaves rationally. </span></p>
<p><span style="font-family: 'book antiqua', palatino; font-size: 12pt;">How should we respond to this result? </span></p>
<p><span style="font-family: 'book antiqua', palatino; font-size: 12pt;">As before, my own view is that there’s nothing to be done. We need to learn to live with it. If your decisions are the decisions of your time slices, you may end up doing things that are predictably suboptimal, even though you never behave irrationally.</span></p>
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<span style="font-family: 'book antiqua', palatino;">The Devil offers you a deal: spend a day in Hell, and you will get two in Heaven.</span>
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<span style="font-family: 'book antiqua', palatino;">You value a day in Heaven just as much as you disvalue a day in Hell. And, you value two days in Heaven twice as much as one day in Heaven. Also, you do not value immediate pleasures over future pleasures, or disvalue immediate pains over future pains.</span>
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<span style="font-family: 'book antiqua', palatino;">Given your values, would your present time-slice act rationally if it accepted the Devil's deal?</span>
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<span style="font-family: 'book antiqua', palatino;">Yes, it would act rationally.</span>
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<span style="font-family: 'book antiqua', palatino;">No, it wouldn't.</span>
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<span style="font-family: 'book antiqua', palatino;">Suppose that you do accept the deal above.</span>
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<span style="font-family: 'book antiqua', palatino;">Towards the end of your horrible day in Hell, the Devil sidles up to your telemarketing cubicle and offers you another deal: stay in Hell tomorrow and you get two more days in Heaven, for a total of four days in Heaven. (You value four days in Heaven twice as much as you value two days in Heaven.)</span>
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<p>
<span style="font-family: 'book antiqua', palatino;">Given your values, would your current time-slice act rationally if it accepted this deal? (As you consider your answer, note that by iterating this strategy, the Devil could get you to spend an eternity in Hell&#8230;)</span>
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