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In the four chapters of this concluding section, themes of the previous section are carried further and brought into contact with common social dilemmas and, eventually, the current world situation. On a small scale, we are constantly faced with dilemmas like the Prisoner’s Dilemma, where personal greed conflicts with social gain. For any two persons, the dilemma is virtually identical. What would be sane behavior in such situations? For true sanity, the key element is that each individual must be able to recognize both that the dilemma is symmetric and that the other individuals facing it are equally able. Such individuals—individuals who will cooperate with one another despite all temptations toward crude egoism—are more than just rational; they are superrational, or for short, sane. But there are dilemmas and “egos” on a suprahuman level as well. We live in a world filled with opposing belief systems so similar as to be nearly interchangeable, yet whose adherents are blind to that symmetry. This description applies not only to myriad small, conflicts in the world but also to the colossally blockheaded opposition of the United States and the Soviet Union. Yet the recognition of symmetry—in short, the sanity—has not yet come. In fact, the insanity seems only to grow, rather than be supplanted by sanity. What has an intelligent species like our own done to get itself into this horrible dilemma? What can it do to get itself out? Are we all helpless as we watch this spectacle unfold, or does the answer lie, for each one of us, in recognition of our own typicality, and in small steps taken on an individual level toward sanity?

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June, 1983

And then one fine day, out of the blue, you get a letter from S.N. Platonia, well-known Oklahoma oil trillionaire, mentioning that twenty leading rational thinkers have been selected to participate in a little game. “You are one of them!” it says. “Each of you has a chance at winning one billion dollars, put up by the Platonia Institute for the Study of Human Irrationality. Here’s how. If you wish, you may send a telegram with just your name on it to the Platonia Institute in downtown Frogville, Oklahoma (pop. 2). You may reverse the charges. If you reply within 48 hours, the billion is yours—unless there are two or more replies, in which case the prize is awarded to no one. And if no one replies, nothing will be awarded to anyone.”

You have no way of knowing who the other nineteen participants are; indeed, in its letter, the Platonia Institute states that the entire offer will be rescinded if it is detected that any attempt whatsoever has been made by any participant to discover the identity of, or to establish contact with, any other participant. Moreover, it is a condition that the winner (if there is one) must agree in writing not to share the prize money with any other participant at any time in the future. This is to squelch any thoughts of cooperation, either before or after the prize is given out.

The brutal fact is that no one will know what anyone else is doing. Clearly, everyone will want that billion. Clearly, everyone will realize that if their name is not submitted, they have no chance at all. Does this mean that twenty telegrams will arrive in Frogville, showing that even possessing transcendent levels of rationality—as you of course do—is of no help in such an excruciating situation?

This is the “Platonia Dilemma”, a little scenario I thought up recently in trying to get a better handle on the Prisoner’s Dilemma, of which I wrote

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last month. The Prisoner’s Dilemma can be formulated in terms resembling this dilemma, as follows. Imagine that you receive a letter from the Platonia Institute telling you that you and just one other anonymous leading rational thinker have been selected for a modest cash giveaway. As before, both of you are requested to reply by telegram within 48 hours to the Platonia Institute, charges reversed. Your telegram is to contain, aside from your name, just the word “cooperate” or the word “defect”. If two “cooperate”s are received, both of you will get $3. If two “defect”s are received, you both will get $1. If one of each is received, then the cooperator gets nothing and the defector gets $5.

What choice would you make? It would be nice if you both cooperated, so you’d each get $3, but doesn’t it seem a little unlikely? After all, who wants to get suckered by a nasty, low-down, rotten defector who gets $5 for being sneaky? Certainly not you! So you’d probably decide not to cooperate.

It seems a regrettable but necessary choice. Of course, both of you, reasoning alike, come to the same conclusion. So you’ll both defect, and that way get a mere dollar apiece. And yet—if you’d just both been willing to risk a bit, you could have gotten $3 apiece. What a pity!

It was my discomfort with this seemingly logical analysis of the “one-round Prisoner’s Dilemma” that led me to formulate the following letter, which I sent out to twenty friends after having cleared it with Scientific American

Dear X: I am sending this letter out via Special Delivery to twenty of ‘you’ (namely, various friends of mine around the country). I am proposing to all of you a one-round Prisoner’s Dilemma game, the payoffs to be monetary (provided by Scientific American). It’s very simple. Here is how it goes. Each of you is to give me a single letter: ‘C’ or ‘D’, standing for ‘cooperate’ or ‘defect’. This will be used as your move in a Prisoner’s Dilemma with each of the nineteen other players. The payoff matrix I am using for the Prisoner’s Dilemma is given in the diagram [see Figure 29-1c]. Figure 29-1. The Prisoner’s Dilemma. Thus if everyone sends in ‘C’, everyone will get $57, while if everyone sends in ‘D’, everyone will get $19. You can’t lose! And of course, anyone who sends in ‘D’ will get at least as much as everyone else will. If, for example, 11 people send in ‘C’ and 9 send in ‘D’, then the 11 C-ers will get $3 apiece from each of the other C-ers (making $30), and zero from the D-ers. So C-ers will get $30 each. The D-ers, by contrast, will pick up $5 apiece from each of the C-ers, making $55, and $1 from each of the other D-ers, making $8, for a grand total of $63. No matter what the distribution is, D-ers always do better than C-ers. Of course, the more C-ers there are, the better everyone will do! By the way, I should make it clear that in making your choice, you should not aim to be the winner, but simply to get as much money for yourself as possible. Thus you should be happier to get $30 (say, as a result of saying ‘C’ along with 10 others, even though the 9 D-sayers get more than you) than to get $19 (by [pg741] saying ‘D’ along with everybody else, so nobody ‘beats’ you). Furthermore, you are not supposed to think that at some subsequent time you will meet with and be able to share the goods with your co-participants. You are not aiming at maximizing the total number of dollars Scientific American shells out, only at maximizing the number that come to you! Of course, your hope is to be the unique defector, thus really cleaning up: with 19 C-ers, you’ll get $95 and they’ll each get 18 times $3, namely $54. But why am I doing the multiplication or any of this figuring for you? You’re very bright. So are all of you! All about equally bright, I’d say, in fact. So all you need to do is tell me your choice. I want all answers by telephone (call collect, please) the day you receive this letter. It is to be understood (it almost goes without saying, but not quite) that you are not to try to get in touch with and consult with others who you guess have been asked to participate. In fact, please consult with no one at all. The purpose is to see what people will do on their own, in isolation. Finally, I would very much appreciate a short statement to go along with your choice, telling me why you made this particular choice. Yours…. P. S.—By the way, it may be helpful for you to imagine a related situation, the same as the present one except that you are told that all the other players have already submitted their choice (say, a week ago), and so you are the last. Now what do you do? Do you submit ‘D’, knowing full well that their answers are already committed to paper? Now suppose that, immediately after having submitted your ‘D’ (or your ‘C’) in that circumstance, you are informed that, in fact, the others really haven’t submitted their answers yet, but that they are all doing it today. Would you retract your answer? Or what if you knew (or at least were told) that you were the first person being asked for an answer? And-one last thing to ponder-what would you do if the payoff matrix looked as shown in Figure 30-la ?

FIGURE 30-1. In (a), a modification of Figure 29-1(c). Here, the incentive to defect seems considerably stronger. In (b), the payoff matrix for a [Bob] Wolf’s Dilemma situation involving just two participants. Compare it to that in Figure 29-1(c).

Two game-theory payoff matrixes for variants on the Prisoner’s Dilemma by Hofstadter

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I wish to stress that this situation is not an iterated Prisoner’s Dilemma (discussed in last month’s column). It is a one-shot, multi-person Prisoner’s Dilemma. There is no possibility of learning, over time, anything about how the others are inclined to play. Therefore all lessons described last month are inapplicable here, since they depend on the situation’s being iterated. All that each recipient of my letter could go on was the thought, “There are nineteen people out there, somewhat like me, all in the same boat, all grappling with the same issues as I am.” In other words, there was nothing to rely on except pure reason.

I had much fun preparing this letter, deciding who to send it out to, anticipating the responses, and then receiving them. It was amusing to me, for instance, to send Special Delivery letters to two friends I was seeing every day, without forewarning them. It was also amusing to send identical letters to a wife and husband at the same address.

Before I reveal the results, I invite you to think how you would play in such a contest. I would particularly like you to take seriously the assertion “everyone is very bright”. In fact, let me expand on that idea, since I felt that people perhaps did not really understand what I meant by it. Thus please consider the letter to contain the following clarifying paragraph:

All of you are very rational people. Therefore, I hardly need to tell you that you are to make what you consider to be your maximally rational choice. In particular, feelings of morality, guilt, vague malaise, and so on, are to be disregarded. Reasoning alone (of course including reasoning about the others’ reasoning) should be the basis of your decision. And please always remember that everyone is being told this (including this!)!

I was hoping for—and expecting—a particular outcome to this experiment. As I received the replies by phone over the next several days, I jotted down notes so that I had a record of what impelled various people to choose as they did. The result was not what I had expected—in fact, my friends “faked me out” considerably. We got into heated arguments about the “rational” thing to do, and everyone expressed much interest in the whole question.

I would like to quote to you some of the feelings expressed by my friends caught in this deliciously tricky situation. David Policansky opened his call tersely by saying, “Okay, Hofstadter, give me the $19!” Then he presented this argument for defecting: “What you’re asking us to do, in effect, is to press one of two buttons, knowing nothing except that if we press button D, we’ll get more than if we press button C. Therefore D is better. That is the essence of my argument. I defect.”

Martin Gardner (yes, I asked Martin to participate) vividly expressed the emotional turmoil he and many others went through. “Horrible dilemma”, he said. “I really don’t know what to do about it. If I wanted to maximize”

[pg743]

“my money, I would choose D and expect that others would also; to maximize my satisfactions, I’d choose C, and hope other people would do the same (by the Kantian imperative). I don’t know, though, how one should behave rationally. You get into endless regresses: ‘If they all do X, then I should do Y, but then they’ll anticipate that and do Z, and so . . .’ You get trapped in an endless whirlpool. It’s like Newcomb’s paradox.” So saying, Martin defected, with a sigh of regret.

In a way echoing Martin’s feelings of confusion, Chris Morgan said, “More by intuition than by anything else, I’m coming to the conclusion that there’s no way to deal with the paradoxes inherent in this situation. So I’ve decided to flip a coin, because I can’t anticipate what the others are going to do. I think—but can’t know—that they’re all going to negate each other.” So, while on the phone, Chris flipped a coin and “chose” to cooperate.

Sidney Nagel was very displeased with his conclusion. He expressed great regret: “I actually couldn’t sleep last night because I was thinking about it. I wanted to be a cooperator, but I couldn’t find any way of justifying it. The way I figured it, what I do isn’t going to affect what anybody else does. I might as well consider that everything else is already fixed, in which case the best I can do for myself is to play a D.”

Bob Axelrod, whose work proves the superiority of cooperative strategies in the iterated Prisoner’s Dilemma, saw no reason whatsoever to cooperate in a one-shot game, and defected without any compunctions.

Dorothy Denning was brief: “I figure, if I defect, then I always do at least as well as I would have if I had cooperated. So I defect.” She was one of the people who faked me out. Her husband, Peter, cooperated. I had predicted the reverse.

By now, you have probably been counting. So far, I’ve mentioned five D’s and two C’s. Suppose you had been me, and you’d gotten roughly a third of the calls, and they were 5-2 in favor of defection. Would you dare to extrapolate these statistics to roughly 14-6? How in the world can seven individuals’ choices have anything to do with thirteen other individuals’ choices? As Sidney Nagel said, certainly one choice can’t influence another (unless you believe in some kind of telepathic transmission, a possibility we shall discount here). So what justification might there be for extrapolating these results?

Clearly, any such justification would rely on the idea that people are “like” each other in some sense. It would rely on the idea that in complex and tricky decisions like this, people will resort to a cluster of reasons, images, prejudices, and vague notions, some of which will tend to push them one way, others the other way, but whose overall impact will be to push a certain percentage of people toward one alternative, and another percentage of people toward the other. In advance, you can’t hope to predict what those percentages will be, but given a sample of people in the situation, you can

[pg744]

hope that their decisions will be “typical”. Thus the notion that early returns running 5-2 in favor of defection can be extrapolated to a final result of 14-6 (or so) would be based on assuming that the seven people are acting “typically” for people confronted with these conflicting mental pressures.

The snag is that the mental pressures are not completely explicit; they are evoked by, but not totally spelled out by, the wording of the letter. Each person brings a unique set of images and associations to each word and concept, and it is the set of those images and associations that will collectively create, in that person’s mind, a set of mental pressures like the set of pressures inside the earth in an earthquake zone. When people decide, you find out how all those pressures pushing in different directions add up, like a set of force vectors pushing in various directions and with strengths influenced by private or unmeasurable factors. The assumption that it is valid to extrapolate has to be based on the idea that everybody is alike inside, only with somewhat different weights attached to certain notions.

This way, each person’s decision can be likened to a “geophysics experiment” whose goal is to predict where an earthquake will appear. You set up a model of the earth’s crust and you put in data representing your best understanding of the internal pressures. You know that there unfortunately are large uncertainties in your knowledge, so you just have to choose what seem to be “reasonable” values for various variables. Therefore no single run of your simulation will have strong predictive power, but that’s all right. You run it and you get a fault line telling you where the simulated earth shifts. Then you go back and choose other values in the ranges of those variables, and rerun the whole thing. If you do this repeatedly, eventually a pattern will emerge revealing where and how the earth is likely to shift and where it is rock-solid.

This kind of simulation depends on an essential principle of statistics: the idea that when you let variables take on a few sample random values in their ranges, the overall outcome determined by a cluster of such variables will start to emerge after a few trials and soon will give you an accurate model. You don’t need to run your simulation millions of times to see valid trends emerging.

This is clearly the kind of assumption that TV networks make when they predict national election results on the basis of early returns from a few select towns in the East. Certainly they don’t think that free will is any “freer” in the East than in the West—that whatever the East chooses to do, the West will follow suit. It is just that the cluster of emotional and intellectual pressures on voters is much the same all over the nation. Obviously, no individual can be taken as representing the whole nation, but a well-selected group of residents of the East Coast can be assumed to be representative of the whole nation in terms of how much they are “pushed” by the various pressures of the election, so that their choices are likely to show general trends of the larger electorate.

Suppose it turned out that New Hampshire’s Belknap County and

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California’s Modoc County had produced, over many national elections, very similar results. Would it follow that one of the two counties had been exerting some sort of causal influence on the other? Would they have had to be in some sort of eerie cosmic resonance mediated by “sympathetic magic” for this to happen? Certainly not. All it takes is for the electorates of the two counties to be similar; then the pressures that determine how people vote will take over and automatically make the results come out similar. It is no more mysterious than the observation that a Belknap County schoolgirl and a Modoc County schoolboy will get the same answer when asked to divide 507 by 13: the laws of arithmetic are the same the world over, and they operate the same in remote minds without any need for “sympathetic magic”.

This is all elementary common sense; it should be the kind of thing that any well-educated person should understand clearly. And yet emotionally it cannot help but feel a little peculiar since it flies in the face of free will and regards people’s decisions as caused simply by combinations of pressures with unknown values. On the other hand, perhaps that is a better way to look at decisions than to attribute them to “free will”, a philosophically murky notion at best.

This may have seemed like a digression about statistics and the question of individual actions versus group predictability, but as a matter of fact it has plenty to do with the “correct action” to take in the dilemma of my letter. The question we were considering is: To what extent can what a few people do be taken as an indication of what all the people will do? We can sharpen it: To what extent can what one person does be taken as an indication of what all the people will do? The ultimate version of this question, stated in the first person, has a funny twist to it: To what extent does my choice inform me about the choices of the other participants?

You might feel that each person is completely unique and therefore that no one can be relied on as a predictor of how other people will act, especially in an intensely dilemmatic situation. There is more to the story, however. I tried to engineer the situation so that everyone would have the same image of the situation. In the dead center of that image was supposed to be the notion that everyone in the situation was using reasoning alone—including reasoning about the reasoning—to come to an answer.

Now, if reasoning dictates an answer, then everyone should independently come to that answer (just as the Belknap County schoolgirl and the Modoc County schoolboy would independently get 39 as their answer to the division problem). Seeing this fact is itself the critical step in the reasoning toward the correct answer, but unfortunately it eluded nearly everyone to whom I sent the letter. (That is why I came to wish I had included in the letter a paragraph stressing the rationality of the players.) Once you realize

[pg746]

this fact, then it dawns on you that either all rational players will choose D or all rational players will choose C. This is the crux.

Any number of ideal rational thinkers faced with the same situation and undergoing similar throes of reasoning agony will necessarily come up with the identical answer eventually, so long as reasoning alone is the ultimate justification for their conclusion. Otherwise reasoning would be subjective, not objective as arithmetic is. A conclusion reached by reasoning would be a matter of preference, not of necessity. Now some people may believe this of reasoning, but rational thinkers understand that a valid argument must be universally compelling, otherwise it is simply not a valid argument.

If you’ll grant this, then you are 90%of the way. All you need ask now is, “Since we are all going to submit the same letter, which one would be more logical? That is, which world is better for the individual rational thinker: one with all C’s or one with all D’s?” The answer is immediate: “I get $57 if we all cooperate, $19 if we all defect. Clearly I prefer $57, hence cooperating is preferred by this particular rational thinker. Since I am typical, cooperating must be preferred by all rational thinkers. So I’ll cooperate.” Another way of stating it, making it sound weirder, is this: “If I choose C, then everyone will choose C, so I’ll get $57. If I choose D, then everyone will choose D, so I’ll get $19. I’d rather have $57 than $19, so I’ll choose C. Then everyone will, and I’ll get $57.”

To many people, this sounds like a belief in voodoo or sympathetic magic, a vision of a universe permeated by tenuous threads of synchronicity, conveying thoughts from mind to mind like pneumatic tubes carrying messages across Paris, and making people resonate to a secret harmony. Nothing could be further from the truth. This solution depends in no way on telepathy or bizarre forms of causality. It’s just that the statement “I’ll choose C and then everyone will”, though entirely correct, is somewhat misleadingly phrased. It involves the word “choice”, which is incompatible with the compelling quality of logic. Schoolchildren do not choose what 507 divided by 13 is; they figure it out. Analogously, my letter really did not allow choice; it demanded reasoning. Thus, a better way to phrase the “voodoo” statement would be this: “If reasoning guides me to say C, then, as I am no different from anyone else as far as rational thinking is concerned, it will guide everyone to say C.”

The corresponding foray into the opposite world (“If I choose D, then everyone will choose D”) can be understood more clearly by likening it to a musing done by the Belknap County schoolgirl before she divides: “Hmm, I’d guess that 13 into 507 is about 49—maybe 39. I see I’ll have to calculate it out. But I know in advance that if I find out that it’s 49, then sure as shootin’, that Modoc County kid will write down 49 on his paper as well; and if I get 39 as my answer, then so will he.” No secret transmissions are involved; all that is needed is the universality and uniformity of arithmetic.

[pg747]

Likewise, the argument “Whatever I do, so will everyone else do” is simply a statement of faith that reasoning is universal, at least among rational thinkers, not an endorsement of any mystical kind of causality.

This analysis shows why you should cooperate even when the opaque envelopes containing the other players’ answers are right there on the table in front of you. Faced so concretely with this unalterable set of C’s and D’s, you might think, “Whatever they have done, I am better off playing D than playing C—for certainly what I now choose can have no retroactive effect on .what they chose. So I defect.” Such a thought, however, assumes that the logic that now drives you to playing D has no connection or relation to the logic that earlier drove them to their decisions. But if you accept what was stated in the letter, then you must conclude that the decision you now make will be mirrored by the plays in the envelopes before you. If logic now coerces you to play D, it has already coerced the others to do the same, and for the same reasons; and conversely, if logic coerces you to play C, it has also already coerced the others to do that.

Imagine a pile of envelopes on your desk, all containing other people’s answers to the arithmetic problem, “What is 507 divided by 13?” Having hurriedly calculated your answer, you are about to seal a sheet saying “49” inside your envelope, when at the last moment you decide to check it. You discover your error, and change the ‘4’ to a ‘3’. Do you at that moment envision all the answers inside the other envelopes suddenly pivoting on their heels and switching from “49” to “39”? Of course not! You simply recognize that what is changing is your image of the contents of those envelopes, not the contents themselves. You used to think there were many “49”s. You now think there are many “39”s. However, it doesn’t follow that there was a moment in between, at which you thought, “They’re all switching from ‘49’ to ‘39’!” In fact, you’d be crazy to think that.

It’s similar with D’s and C’s. If at first you’re inclined to play one way but on careful consideration you switch to the other way, the other players obviously won’t retroactively or synchronistically follow you—but if you give them credit for being able to see the logic you’ve seen, you have to assume that their answers are what yours is. In short, you aren’t going to be able to undercut them; you are simply “in cahoots” with them, like it or not! Either all D’s, or all C’s. Take your pick.

Actually, saying “Take your pick” is 100% misleading. It’s not as if you could merely “pick”, and then other people—even in the past—would magically follow suit! The point is that since you are going to be “choosing” by using what you believe to be compelling logic, if you truly respect your logic’s compelling quality, you would have to believe that others would buy it as well, which means that you are certainly not “just picking”. In fact, the more convinced you are of what you are playing, the more certain you should be that others will also play (or have already played) the same way, and for the same reasons. This holds whether you play C or D, and it is the real core of the solution. Instead of being a paradox, it’s a self-reinforcing solution: a benign circle of logic.

[pg748]

If this still sounds like retrograde causality to you, consider this little tale, which may help make it all make more sense. Suppose you and Jane are classical music lovers. Over the years, you have discovered that you have incredibly similar tastes in music—a remarkable coincidence! Now one day you find out that two concerts are being given simultaneously in the town where you live. Both of them sound excellent to you, but Concert A simply cannot be missed, whereas Concert B is a strong temptation that you’ll have to resist. Still, you’re extremely curious about Concert B, because it features Zilenko Buznani, a violinist you’ve always heard amazing things about.

At first, you’re disappointed, but then a flash crosses your mind: “Maybe I can at least get a first-hand report about Zilenko Buznani’s playing from Jane. Since she and I hear everything through virtually the same ears, it would be almost as good as my going if she would go.” This is comforting for a moment, until it occurs to you that something is wrong here. For the same reasons as you do, Jane will insist on hearing Concert A. After all, she loves music in the same way as you do—that’s precisely why you wish she would tell you about Concert B! The more you feel Jane’s taste is the same as yours, the more you wish she would go to the other concert, so that you could know what it was like to have gone to it. But the more her taste is the same is yours, the less she will want to go to it!

The two of you are tied together by a bond of common taste. And if it turns out that you are different enough in taste to disagree about which concert is better, then that will tend to make you lose interest in what she might report, since you no longer can trust her opinion as that of someone who hears music “through your ears”. In other words, hoping she’ll choose Concert B is pointless, since it undermines your reasons for caring which concert she chooses!

The analogy is clear, I hope. Choosing D undermines your reasons for doing so. To the extent that all of you really are rational thinkers, you really will think in the same tracks. And my letter was supposed to establish beyond doubt the notion that you are all “in synch”; that is, to ensure that you can depend on the others’ thoughts to be rational, which is all you need.

Well, not quite. You need to depend not just on their being rational, but on their depending on everyone else to be rational, and on their depending on everyone to depend on everyone to be rational—and so on. A group of reasoners in this relationship to each other I call superrational. Superrational thinkers, by recursive definition, include in their calculations the fact that they are in a group of superrational thinkers. In this way, they resemble elementary particles that are renormalized.

A renormalized electron’s style of interacting with, say, a renormalized photon takes into account that the photon’s quantum-mechanical structure includes “virtual electrons” and that the electron’s quantum-mechanical structure includes “virtual photons”; moreover it takes into account that all

[pg749]

these virtual particles (themselves renormalized) also interact with one another. An infinite cascade of possibilities ensues but is taken into account in one fell swoop by nature. Similarly, superrationality, or renormalized reasoning, involves seeing all the consequences of the fact that other renormalized reasoners are involved in the same situation-and doing so in a finite swoop rather than succumbing to an infinite regress of reasoning about reasoning about reasoning …

‘C’ is the answer I was hoping to receive from everyone. I was not so optimistic as to believe that literally everyone would arrive at this conclusion, but I expected a majority would—thus my dismay when the early returns strongly favored defecting. As more phone calls came in, I did receive some C’s, but for the wrong reasons. Dan Dennett cooperated, saying, “I would rather be the person who bought the Brooklyn Bridge than the person who sold it. Similarly, I’d feel better spending $3 gained by cooperating than $10 gained by defecting.”

Charles Brenner, who I’d figured to be a sure-fire D, took me by surprise and C’d. When I asked him why, he candidly replied, “Because I don’t want to go on record in an international journal as a defector.” Very well. Know, World, that Charles Brenner is a cooperator!

Many people flirted with the idea that everybody would think “about the same”, but did not take it seriously enough. Scott Buresh confided to me: “It was not an easy choice. I found myself in an oscillation mode: back and forth. I made an assumption: that everybody went through the same mental processes I went through. Now I personally found myself wanting to cooperate roughly one third of the time. Based on that figure and the assumption that I was typical, I figured about one third of the people would cooperate. So I computed how much I stood to make in a field where six or seven people cooperate. It came out that if I were a D, I’d get about three times as much as if I were a C. So I’d have to defect. Water seeks out its own level, and I sank to the lower right-hand corner of the matrix.” At this point, I told Scott that so far, a substantial majority had defected. He reacted swiftly: “Those rats—how can they all defect? It makes me so mad! I’m really disappointed in your friends, Doug.” So was I, when the final results were in: Fourteen people had defected and six had cooperated—exactly what the networks would have predicted! Defectors thus received $43 while cooperators got $15. I wonder what Dorothy’s saying to Peter about now? I bet she’s chuckling and saying, “I told you I’d do better this way, didn’t I?” Ah, me … What can you do with people like that?

A striking aspect of Scott Buresh’s answer is that, in effect, he treated his own brain as a simulation of other people’s brains and ran the simulation enough to get a sense of what a “typical person” would do. This is very

[pg750]

much in the spirit of my letter. Having assessed what the statistics are likely to be, Scott then did a cool-headed calculation to maximize his profit, based on the assumption of six or seven cooperators. Of course, it came out in favor of defecting. In fact, it would have, no matter what the number of cooperators was! Any such calculation will always come out in favor of defecting. As long as you feel your decision is independent of others’ decisions, you should defect. What Scott failed to take into account was that cool-headed calculating people should take into account that cool-headed calculating people should take into account that cool-headed calculating people should take into account that …

This sounds awfully hard to take into account in a finite way, but actually it’s the easiest thing in the world. All it means is that all these heavy-duty rational thinkers are going to see that they are in a symmetric situation, so that whatever reason dictates to one, it will dictate to all. From that point on, the process is very simple. Which is better for an individual if it is a universal choice: C or D? That’s all.

Actually, it’s not quite all, for I’ve swept one possibility under the rug: maybe throwing a die could be better than making a deterministic choice. Like Chris Morgan, you might think the best thing to do is to choose C with probability p and D with probability 1−p. Chris arbitrarily let p be 1⁄2, but it could be any number between 0 and 1, where the two extremes represent Ding and C’ing respectively. What value of p would be chosen by superrational players? It is easy to figure out in a two-person Prisoner’s Dilemma, where you assume that both players use the same value of p. The expected earnings for each, as a function of p, come out to be I+3p−p2, which grows monotonically as p increases from 0 to 1. Therefore, the optimum value of p is 1, meaning certain cooperation. In the case of more players, the computations get more complex but the answer doesn’t change: the expectation is always maximal when p equals 1. Thus this approach confirms the earlier one, which didn’t entertain probabilistic strategies.—Rolling a die to determine what you’ll do didn’t add anything new to the standard Prisoner’s Dilemma, but what about the modified-matrix version I gave in the P. S. to my letter? I’ll let you figure that one out for yourself. And what about the Platonia Dilemma? There, two things are very clear: (1) if you decide not to send a telegram, your chances of winning are zero; (2) if everyone sends a telegram, your chances of winning are zero. If you believe that what you choose will be the same as what everyone else chooses because you are all superrational, then neither of these alternatives is very appealing. With dice, however, a new option presents itself to roll a die with probability p of coming up “good” and then to send in your name if and only if “good” comes up.

Now imagine twenty people all doing this, and figure out what value of

[pg751]

p maximizes the likelihood of exactly one person getting the go-ahead. It turns out that it is p=120, or more generally, p=1N where N is the number of participants. In the limit where N approaches infinity, the chance that exactly one person will get the go-ahead is 1e, which is just under 37%. With twenty superrational players all throwing icosahedral dice, the chance that you will come up the big winner is very close to 120e, which is a little below 2%. That’s not at all bad! Certainly it’s a lot better than 0%.

The objection many people raise is: “What if my roll comes up bad? Then why shouldn’t I send in my name anyway? After all, if I fail to, I’ll have no chance whatsoever of winning. I’m no better off than if I had never rolled my die and had just voluntarily withdrawn!” This objection seems overwhelming at first, but actually it is fallacious, being based on a misrepresentation of the meaning of “making a decision”. A genuine decision to abide by the throw of a die means that you really must abide by the throw of the die; if under certain circumstances you ignore the die and do something else, then you never made the decision you claimed to have made. Your decision is revealed by your actions, not by your words before acting!

If you like the idea of rolling a die but fear that your will power may not be up to resisting the temptation to defect, imagine a third “Policansky button”: this one says ‘R’ for “Roll”, and if you press it, it rolls a die (perhaps simulated) and then instantly and irrevocably either sends your name or does not, depending on which way the die came up. This way you are never allowed to go back on your decision after the die is cast. Pushing that button is making a genuine decision to abide by the roll of a die. It would be easier on any ordinary human to be thus shielded from the temptation, but any superrational player would have no trouble holding back after a bad roll.

This talk of holding back in the face of strong temptation brings me to the climax of this column: the announcement of a Luring Lottery open to all readers and nonreaders of Scientific American. The prize of this lottery is $1,000,000N, where N is the number of entries submitted. Just think: If you are the only entrant (and if you submit only one entry), a cool million is yours! Perhaps, though, you doubt this will come about. It does seem a trifle iffy. If you’d like to increase your chances of winning, you are encouraged to send in multiple entries—no limit! Just send in one postcard per entry. If you send in 100 entries, you’ll have 100 times the chance of some poor slob who sends in just one. Come to think of it, why should you have to send in multiple entries separately? Just send one postcard with your name and address and a positive integer (telling how many entries you’re making) to:

Luring Lottery c/o Scientific American 415 Madison Avenue New York, N.Y. 10017

You will be given the same chance of winning as if you had sent in that number of postcards with ‘1’ written on them. Illegible, incoherent, ill-specified, or incomprehensible entries will be disqualified. Only entries received by midnight June 30, 1983 will be considered. Good luck to you (but certainly not to any-other reader of this column)!