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Metamagical Themas: Sanity and Survival

3 essays by AI researcher Douglas Hofstadter exploring cooperation/game theory/‘superrationality’ in the context of the failure of political coordination to prevent global nuclear war

The following 3 essays were prepared from pages 737–780 of an ebook of Metamagical Themas: Questing for the Essence of Mind and Pattern (1985) by Douglas Hofstadter, an anthology of articles & essays primarily published in Scientific American “between January 1981 and July 1983”. (I omit one entry in “Sanity and Survival”, the essay “The Tumult of Inner Voices, or, What is the Meaning of the Word ‘I’?”, which is unconnected to the other entries on cooperation/decision theory/nuclear war.) All hyperlinks are my insertion.

They are interesting for introducing the idea of ‘superrationality’ in game theory, an attempt to devise a decision theory/algorithm for agents which can reach global utility maxima on problems like the prisoner’s dilemma even in the absence of coercion or communication which has partially inspired later decision theories like UDT or TDT, linking decision theory to cooperation (eg. Oesterheld2017) & existential risks (specifically, nuclear warfare), and one networking project.

Note: to hide apparatus like the links, you can use reader-mode ().

Sanity and Survival

by Douglas Hofstadter


Illustration of an abstract font for the Latin alphabet.

Illustration of an abstract font for the Latin alphabet.


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?

Dilemmas for Superrational Thinkers, Leading Up to a Luring Lottery


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


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.

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


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.


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

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


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”


“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


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 an 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


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


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.


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.


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


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


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 an 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 + 3_p_ − _p_2, which grows monotonically as p increases 0 → 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


p maximizes the likelihood of exactly one person getting the go-ahead. It turns out that it is p = 1⁄20, or more generally, p = 1⁄N 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 1⁄e, 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 1/20_e_, 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 $1,000,000 / N, 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)!

Post Scriptum

The emotions churned up by the Prisoner’s Dilemma are among the strongest I have ever encountered, and for good reason. Not only is it a wonderful intellectual puzzle, akin to some of the most famous paradoxes of all time, but also it captures in a powerful and pithy way the essence of a myriad deep and disturbing situations that we are familiar with from life. Some are choices we make every day; others are the kind of agonizing choices that we all occasionally muse about but hope the world will never make us face.

My friend Bob Wolf, a mathematician whose specialty is logic, adamantly advocated choosing D in the case of the letters I sent out. To defend his choice, he began by saying that it was clearly “a paradox with no rational solution”, and thus there was no way to know what people would do. Then he said, “Therefore, I will choose D. I do better that way than any other way.” I protested strenuously: “How dare you say ‘therefore’ when you’ve just gotten through describing this situation as a paradox and claiming there is no rational answer? How dare you say logic is forcing an answer down your throat, when the premise of your ‘logic’ is that there is no logical answer?” I never got what I considered a satisfactory answer from Bob, although neither of us could budge the other. However, I did finally get some insight into Bob’s vision when he, pushed hard by my probing, invented a situation with a new twist to it, which I call “Wolf’s Dilemma”.

Imagine that twenty people are selected from your high school graduation class, you among them. You don’t know which others have been selected, and you are told they are scattered all over the country. All you know is that they are all connected to a central computer. Each of you is in a little cubicle, seated on a chair and facing one button on an otherwise blank wall. You are given ten minutes to decide whether or not to push your button. At the end of that time, a light will go on for ten seconds, and while it is on, you may


either push or refrain from pushing. All the responses will then go to the central computer, and one minute later, they will result in consequences. Fortunately, the consequences can only be good. If you pushed your button, you will get $100, no strings attached, emerging from a small slot below the button. If nobody pushed their button, then everybody will get $1,000. But if there was even a single button-pusher, the refrainers will get nothing at all.

Bob asked me what I would do. Unhesitatingly, I said, “Of course I would not push the button. It’s obvious!” To my amazement, though, Bob said he’d push the button with no qualms. I said, “What if you knew your co-players were all logicians?” He said that would make no difference to him. Whereas I gave credit to everybody for being able to see that it was to everyone’s advantage to refrain, Bob did not. Or at least he expected that there is enough “flakiness” in people that he would prefer not to rely on the rationality of nineteen other people. But of course in assuming the flakiness of others, he would be his own best example—ruining everyone else’s chances of getting $1,000.

What bothered me about Wolf’s Dilemma was what I have come to call reverberant doubt. Suppose you are wondering what to do. At first it’s obvious that everybody should avoid pushing their button. But you do realize that among twenty people, there might be one who is slightly hesitant and who might waver a bit. This fact is enough to worry you a tiny bit, and thus to make you waver, ever so slightly. But suddenly you realize that if you are wavering, even just a tiny bit, then most likely everyone is wavering a tiny bit. And that’s considerably worse than what you’d thought at first—namely, that just one person might be wavering. Uh-oh! Now that you can imagine that everybody is at least contemplating pushing their button, the situation seems a lot more serious. In fact, now it seems quite probable that at least one person will push their button. But if that’s the case, then pushing your own button seems the only sensible thing to do. As you catch yourself thinking this thought, you realize it must be the same as everyone else’s thought. At this point, it becomes plausible that the majority of participants—possibly even all—will push their button! This clinches it for you, and so you decide to push yours.

Isn’t this an amazing and disturbing slide from certain restraint to certain pushing? It is a cascade, a stampede, in which the tiniest flicker of a doubt has become amplified into the gravest avalanche of doubt. That’s what I mean by “reverberant doubt”. And one of the annoying things about it is that the brighter you are, the more quickly and clearly you see what there is to fear. A bunch of amiable slowpokes might well be more likely to unanimously refrain and get the big payoff than a bunch of razor-sharp logicians who all think perversely recursively reverberantly. It’s that “smartness” to see that initial flicker of a doubt that triggers the whole avalanche and sends rationality a-tumblin’ into—the abyss. So, dear reader . . . if you push that button in front of you, do you thereby lose $900 or do you thereby gain $100?


Wolf’s Dilemma is not the same as the Prisoner’s Dilemma. In the Prisoner’s Dilemma, pressure towards defection springs from hope for asymmetry (ie. hope that the other player might be dumber than you and thus make the opposite choice) whereas in Wolf’s Dilemma, pressure towards button-pushing springs from fear of asymmetry (ie. fear that the other player might be dumber than you and thus make the opposite choice). This difference shows up clearly in the games’ payoff matrices for the two-person case (compare Figure 30-lb with Figure 29-1c). In the Prisoner’s Dilemma, the temptation T is greater than the reward R (5 > 3), whereas in Wolf’s Dilemma, R is greater than T (1,000 > 100).

Bob Wolf’s choice in his own dilemma revealed to me something about his basic assessment of people and their reliability (or lack thereof). Since his adamant decision to be a button-pusher even in this case stunned me, I decided to explore that cynicism a bit more, and came up with this modified Wolf’s Dilemma.

Imagine, as before, that twenty people have been selected from your high school graduation class, and are escorted to small cubicles with one button on the wall. This time, however, each of you is strapped into a chair, and a device containing a revolver is attached to your head. Like it or not, you are now going to play Russian roulette, the odds of your death to be determined by your choice. For anybody who pushes their button, the odds of survival will be set at 90%—only one chance in ten of dying. Not too bad, but given that there are twenty of you, it means that almost certainly one or two of you will die, possibly more. And what happens to the refrainers? It all depends on how many of them there are. Let’s say there are N refrainers. For each one of them, their chance of being shot will be one in N2. For instance, if five people don’t push, each of them will have only a 1⁄25 chance of dying. If ten people refrain, they will each get a 99% chance of survival. The bad cases are, of course, when nearly everybody pushes their button (“playing it safe”, so to speak), leaving the refrainers in a tiny minority of three, two, or even one. If you’re the sole refrainer, it’s curtains for you—one chance in one of your death. Bye-bye! For two refrainers, it’s one chance in four for each one. That means there’s nearly a 50% chance that at least one of the two will perish.

Clearly the crossover line is between three and four refrainers. If you have a reasonable degree of confidence that at least three other people will hold back, you should definitely do so yourself. The only problem is, they’re all making their decisions on the basis of trying to guess how many people will refrain, too! It’s terribly circular, and you hardly know where to start. Many people, sensing this, just give up, and decide to push their button. (Actually, of course, how do I know? I’ve never seen people in such a situation—but it seems that way from evidence of real-life situations resembling this, and of course from how people respond to a mere description of this situation,


where they aren’t really faced with any dire consequences at all. Still, I tend to believe them, by and large.) Calling such a decision “playing it safe” is quite ironic, because if only everybody “played it dangerous”, they’d have a chance of only one in 400 of dying! So I ask you: Which way is safe, and which way dangerous? It seems to me that this Wolf Trap epitomizes the phrase “We have nothing to fear but fear itself.”

Variations on Wolf’s Dilemma include some even more frightening and unstable scenarios. For instance, suppose the conditions are that each button-pusher has a 50% chance of survival, but if there is unanimous refraining from pushing the button, everyone’s life will be spared—and as before, if anyone pushes their button, all refrainers will die. You can play around with the number of participants, the survival chance, and so on. Each such variation reveals a new facet of grimness. These visions are truly horrific, yet all are just allegorical renditions of ordinary life’s decisions, day in, day out.

I had originally intended to close the column with the following paragraph, but was dissuaded from it by friends and editors:

I am sorry to say that I am simply inundated with letters from well-meaning readers, and I have discovered, to my regret, that I can barely find time to read all those letters, let alone answer them. I have been racking my brains for months trying to come up with some strategy for dealing with all this correspondence, but frankly I have not found a good solution yet. Therefore, I thought I would appeal to the collective genius of you-all out there. If you can think of some way for me to ease the burden of my correspondence, please send your idea to me. I shall be most grateful.

Irrationality Is the Square Root of All Evil


September, 1983

The Luring Lottery, proposed in my June column, created quite a stir. Let me remind you that it was open to anyone; all you had to do was submit a postcard with a clearly specified positive integer on it telling how many entries you wished to make. This integer was to be, in effect, your “weight” in the final drawing, so that if you wrote “100”, your name would be 100 times more likely to be drawn than that of someone who wrote ‘I’. The only catch was that the cash value of the prize was inversely proportional to the sum of all the weights received by June 30. Specifically, the prize to be awarded was $1,000,000 / N, where N is the sum of all the weights sent in.

The Luring Lottery was set up as an exercise in cooperation versus defection. The basic question for each potential entrant was: “Should I restrain myself and submit a small number of entries, or should I ‘go for it’ and submit a large number? That is, should I cooperate, or should I defect?” Whereas in previous examples of cooperation versus defection there was a clear-cut dividing line between cooperators and defectors, here it seems there is a continuum of possible answers, hence of “degree of cooperation”. Clearly one can be an extreme cooperator and voluntarily submit nothing, thus in effect cutting off one’s nose to spite one’s face. Equally clearly, one can be an extreme defector and submit a giant number of entries, hoping to swamp everyone else out but destroying the prize in so doing. However, there remains a lot of middle ground between these two extremes. What about someone who submits two entries, or one? What about someone who throws a six-sided die to decide whether or not to send in a single entry? Or a million-sided die?

Before I go further, it would be good for me to present my generalized and non-mathematical sense of these terms “cooperation” and “defection”. As a child, you undoubtedly often encountered adults who admonished you


for walking on the grass or for making noise, saying “Tut, tut, tut just think if everyone did that!” This is the quintessential argument used against the defector, and serves to define the concept:

A defection is an action such that, if everyone did it, things would clearly be worse (for everyone) than if everyone refrained from doing it, and yet which tempts everyone, since if only one individual (or a sufficiently small number) did it while others refrained, life would be sweeter for that individual (or select group).

Cooperation, of course, is the other side of the coin: the act of resisting temptation. However, it need not be the case that cooperation is passive while defection is active; often it is the exact opposite: The cooperative option may be to participate industriously in some activity, while defection is to lay back and accept the sweet things that result for everybody from the cooperators’ hard work. Typical examples of defection are:

  • loudly wafting your music through the entire neighborhood on a fine summer’s day;

  • not worrying about speeding through a four-way stop sign, figuring that the people going in the crosswise direction will stop anyway;

  • not being concerned about driving a car everywhere, figuring that there’s no point in making a sacrifice when other people will just continue to guzzle gas anyway;

  • not worrying about conserving water in a drought, figuring “Everyone else will”;

  • not voting in a crucial election and excusing yourself by saying “One vote can’t make any difference”;

  • not worrying about having ten children in a period of population explosion, leaving it to other people to curb their reproduction;

  • not devoting any time or energy to pressing global issues such as the arms race, famine, pollution, diminishing resources, and so on, saying “Oh, of course I’m very concerned—but there’s nothing one person can do.”

When there are large numbers of people involved, people don’t realize that their own seemingly highly idiosyncratic decisions are likely to be quite typical and are likely to be recreated many times over, on a grand scale; thus, what each couple feels to be their own isolated and private decision (conscious or unconscious) about how many children to have turns into a population explosion. Similarly, “individual” decisions about the futility of working actively toward the good of humanity amount to a giant trend of apathy, and this multiplied apathy translates into insanity at the group level. In a word, apathy at the individual level translates into insanity at the mass level.


Garrett Hardin, an evolutionary biologist, wrote a famous article about this type of phenomenon, called “The Tragedy of the Commons”. His view was that there are two types of rationality: one (I’ll call it the “local” type) that strives for the good of the individual, the other (the “global” type) that strives for the good of the group; and that these two types of rationality are in an inevitable and eternal conflict. I would agree with his assessment, provided the individuals are unaware of their joint plight but are simply blindly carrying out their actions as if in isolation.

However, if they are fully aware of their joint situation, and yet in the face of it they blithely continue to act as if their situation were not a communal one, then I maintain that they are acting totally irrationally. In other words, with an enlightened citizenry, “local” rationality is not rational, period. It is damaging not just to the group, but to the individual. For example, people who defected in the One-Shot Prisoner’s Dilemma situation I described in June did worse than if all had cooperated.

This was the central point of my June column, in which I wrote about renormalized rationality, or superrationality. Once you know you are a typical member of a class of individuals, you must act as if your own individual actions were to be multiplied many-fold, because they inevitably will be. In effect, to sample yourself is to sample the field, and if you fail to do what you wish the rest would do, you will be very disappointed by the rest as well. Thus it pays a lot to reflect carefully about one’s situation in the world before defecting, that is, jumping to do the naively selfish act. You had better be prepared for a lot of other people copping out as well, and offering the same flimsy excuse.

People strongly resist seeing themselves as parts of statistical phenomena, and understandably so, because it seems to undermine their sense of free will and individuality. Yet how true it is that each of our “unique” thoughts is mirrored a million times over in the minds of strangers! Nowhere was this better illustrated than in the response to the Luring Lottery. It is hard to know precisely what constitutes the “field”, in this case. It was declared universally open, to readers and nonreaders alike. However, we would be safe in assuming that few nonreaders ever became aware of it, so let’s start with the circulation of Scientific American, which is about a million. Most of them, however, probably did no more than glance over my June column, if that; and of the ones who did more than that (let’s say 100,000), still only a fraction—maybe one in ten—read it carefully from start to finish. I would thus estimate that there were perhaps 10,000 people motivated enough to read it carefully and to ponder the issues seriously. In any case, I’ll take this figure as the population of the “field”.

In my June column, I spelled out plainly, for all to see, the superrational argument that applies to the Platonia Dilemma, for rolling an N-sided die and entering only if it came up on the proper side. Here, a similar argument goes through. In the Platonia Dilemma, where more than one entry is fatal to all, the ideal die turned out to have N faces, where N is the number of


players—hence, with 10,000 players, a 10,000-sided die. In the Luring Lottery, the consequences aren’t so drastic if more than one entry is submitted. Thus, the ideal number of faces on the die turns out to be about 2⁄3 as many—in the case of 10,000 players, a 6,667-sided die would do admirably. Giving the die fewer than 10,000 sides of course slightly increases each player’s chance of sending in one entry. This is to make it quite likely that at least one entry will arrive!

With 6,667 faces on the die, each superrational player’s chance of winning is not quite 1 in 10,000, but more like 1 in 13,000; this is because there is about a 22% chance that no one’s die will land right, so no one will send in any entry at all, and no one will win. But if you give the die still fewer faces—say 3,000—the expected size of the pot gets considerably smaller, since the expected number of entrants grows. And if you give it more faces—say 20,000—then you run a considerable risk of having no entries at all. So there’s a trade-off whose ideal solution can be calculated without too much trouble, and 6,667 faces turns out to be about optimal. With that many faces, the expected value of the pot is maximal: nearly $520,000—not to be sneered at.

Now this means that had everyone followed my example in the June column, I would probably have received a total of one or two postcards with ‘1’ written on them, and one of those lucky people would have gotten a huge sum of money! But do you think that is what happened? Of course not! Instead, I was inundated with postcards and letters from all over the world—over 2,000 of them. What was the breakdown of entries? I have exhibited part of it in a table, below:

  • 1: 1,133

  • 2: 31

  • 3: 16

  • 4: 8

  • 5: 16

  • 6: 0

  • 7: 9

  • 8: 1

  • 9: 1

  • 10: 49

  • 100: 61

  • 1,000: 46

  • 1,000,000: 33

  • 1,000,000,000:

  • 11

  • 602,300,000,000,000,000,000,000 (Avogadro’s number): 1

  • 10100 (a googol): 9

  • 1010100 (a googolplex): 14


Curiously, many if not most of the people who submitted just one entry patted themselves on the back for being “cooperators”. Hogwash! The real cooperators were those among the 10,000 or so avid readers who calculated the proper number of faces of the die, used a random-number table or something equivalent, and then—most likely—rolled themselves out. A few people wrote to tell me they had rolled themselves out in this way. I appreciated hearing from them. It is conceivable, just barely, that among the thousand-plus entries of ‘1’ there was one that came from a lucky superrational cooperator—but I doubt it. The people who simply withdrew without throwing a die I would characterize as well—meaning but a bit lazy, not true cooperators—something like people who simply contribute money to a political cause but then don’t want to be bothered any longer about it. It’s the lazy way of claiming cooperation.

By the way, I haven’t by any means finished with my score chart. However, it is a bit disheartening to try to relate what happened. Basically, it is this. Dozens and dozens of readers strained their hardest to come up with inconceivably large numbers. Some filled their whole postcard with tiny ’9’s, others filled their card with rows of exclamation points, thus creating iterated factorials of gigantic sizes, and so on. A handful of people carried this game much further, recognizing that the optimal solution avoids all pattern (to see why, read Gregory Chaitin’s article “Randomness and Mathematical Proof”), and consists simply of a “dense pack” of definitions built on definitions, followed by one final line in which the “fanciest” of the definitions is applied to a relatively small number such as 2, or better yet, 9.

I received, as I say, a few such entries. Some of them exploited such powerful concepts of mathematical logic and set theory that to evaluate which one was the largest, became a very serious problem, and in fact it is not even clear that I, or for that matter anyone else, would be able to determine which is the largest integer submitted. I was strongly reminded of the lunacy and pointlessness of the current arms race, in which two sides vie against each other to produce arsenals so huge that not even teams of experts can meaningfully say which one is larger—and meanwhile, all this monumental effort is to the detriment of everyone.

Did I find this amusing? Somewhat, of course. But at the same time, I found it disturbing and disappointing. Not that I hadn’t expected it. Indeed, it was precisely what I had expected, and it was one reason I was so sure the Luring Lottery would be no risk for the magazine.

This short-sighted race for “first place” reveals the way in which people in a huge crowd erroneously consider their own fancies to be totally unique. I suspect that nearly everyone who submitted a number above 1,000,000 actually believed they were going to be the only one to do so. Many of those who submitted numbers such as a googolplex, or a ‘9’ followed by


thousands of factorial signs, explicitly indicated that they were pretty sure that they were going to “win”. And then those people who pulled out all the stops and sent in definitions that would boggle most mathematicians were very sure they were going to win. As it turns out, I don’t know who won, and it doesn’t matter, since the prize is zero to such a good approximation that even God wouldn’t know the difference. Well, what conclusion do I draw from all this? None too serious, but I do hope that it will give my readers pause for thought next time they face a “cooperate-or-defect” decision, which will likely happen within minutes for each of you, since we face such decisions many times each day. Some of them are small, but some will have monumental repercussions. The globe’s future is in your hands—and yes, I mean you (as well as every other reader of this column).

And with this perhaps sobering conclusion, I would like to draw my term as a columnist for Scientific American to a close. It has been a valuable and beneficial opportunity for me. I have enjoyed having a platform from which to express my ideas and concerns, I have—at least sometimes—enjoyed receiving the huge shipments of mail forwarded to me from New York several times a month, and I have certainly been happy to make new friends through this channel. I won’t miss the monthly deadline, but I will undoubtedly come across ideas, from time to time, that would have made perfect “Metamagical Themas”. I will be keeping them in mind, and maybe at some future time will write a similar set of essays.

But for now, it is time for me to move on to other territory: I look forward to a return to my professional work, and to a more private life. Good-bye, and best wishes to you and to all other readers of this magazine, this issue, this copy, this piece, this page, this column, this paragraph, this sentence, and, last but not least, this “this”.

Post Scriptum

What do you do when in a crushingly cold winter, you hear over the radio that there is a severe natural gas shortage in your part of the country, and everyone is requested to turn their thermostat down to 60 degrees? There’s no way anyone will know if you’ve complied or not. Why shouldn’t you toast in your house and let all the rest of the people cut down their consumption? After all, what you do surely can’t affect what anyone else does.

This is a typical “tragedy of the commons” situation. A common resource has reached the point of saturation or exhaustion, and the questions for each individual now are: “How shall I behave? Am I typical? How does a”


“lone person’s action affect the big picture?” Garrett Hardin’s article “The Tragedy of the Commons” [WP] frames the scene in terms of grazing land shared by a number of herders. Each one is tempted to increase their own number of animals even when the land is being used beyond its optimum capacity, because the individual gain outweighs the individual loss, even though in the long run, that decision, multiplied throughout the population of herders, will destroy the land totally.

The real reason behind Hardin’s article was to talk about the population explosion and to stress the need for rational global planning—in fact, for coercive techniques similar to parking tickets and jail sentences. His idea is that families should be allowed to have many children (and thus to use a large share of the common resources) but that they should be penalized by society in the same way as society “allows” someone to rob a bank and then applies sanctions to those who have made that choice. In an era when resources are running out in a way humanity has never had to face heretofore, new kinds of social arrangements and expectations must be imposed, Hardin feels, by society as a whole. He is a dire pessimist about any kind of superrational cooperation, emphasizing that cooperators in the birth-control game will breed themselves right out of the population. A perfect illustration of why this is so is the man I heard about recently: he secretly had ten wives and by them had sired something like 35 children by the time he was 30. With genes of that sort proliferating wildly, there is little hope for the more modest breeders among us to gain the upper hand. Hardin puts it bluntly: “Conscience is self-eliminating.” He goes even further and says:

The argument has here been stated in the context of the population problem, but it applies equally well to any instance in which society appeals to an individual exploiting a commons to restrain himself for the general good—by means of his conscience. To make such an appeal is to set up a selective system that works toward the elimination of conscience from the race.

An even more pessimistic vision of the future is proffered us by one Walter Bradford Ellis, a hypothetical speaker representing the views of his inventor, Louis Pascal, in a hypothetical speech:

The United States—indeed the whole earth—is fast running out of the resources it depends on for its existence. Well before the last of the world’s supplies of oil and natural gas are exhausted early in the next century, shortages of these and other substances will have brought about the collapse of our whole economy and, indeed, of our whole technology. And without the wonders of modern technology, America will be left a grossly overpopulated, utterly impoverished, helpless, dying land. Thus I foresee a whole world full of wretched, starving people with no hope of escape, for the only countries which could have aided them will soon be no better off than the rest. And thus unless we are saved from this future by the blessing of a nuclear war or a truly lethal


pestilence, I see stretching off into eternity a world of indescribable suffering and hopelessness. It is a vision of truly unspeakable horror mitigated only by the fact that try as I might I could not possibly concoct a creature more deserving of such a fate.

Whew! The circularity of the final thought reminds me of an idea I once had: that it will be just as well if humanity destroys itself in a nuclear holocaust, because civilizations that destroy themselves are barbaric and stupid, and who would want to have one of them around, polluting the universe?

Pascal’s thoughts, expressed in his [1978] article “Human Tragedy and Natural Selection” and in his [1980] rejoinder to an [1980] article by two critics called “The Loving Parent Meets the Selfish Gene” (which is where Ellis’ speech is printed), are strikingly reminiscent of the thoughts of his earlier namesake Blaise, who in an unexpected use of his own calculus of probabilities managed to convince himself that the best possible way to spend his life was in devotion to a God who he wasn’t sure (and couldn’t be sure) existed. In fact, Pascal felt, even if the chances of God’s existence were one in a million, faith in that God would pay off in the end, because the potential rewards (or punishments) if Heaven and Hell exist are infinite, and all earthly rewards and punishments, no matter how great, are still finite. The favored behavior is to be a believer, Pascal “calculated”—regardless of what you do believe. Thus Blaise Pascal devoted his brilliant mind to theology.

Louis Pascal, following in his forebear’s mindsteps, has opted to devote his life to the world’s population problem. And he can produce mathematical arguments to show why you should, too. To my mind, there is no question that such arguments have considerable force. There are always points to nitpick over, but in essence, thinkers like Hardin and Pascal and Anne and Paul Ehrlich and many others have recognized and internalized the novelty of the human situation at this moment in history: the moment when humanity has to grapple with dwindling resources and overwhelmingly huge weapons systems. Not many people are willing to wrestle with this beast, and consequently the burden falls all the more heavily on those few who are.

It has disturbed me how vehemently and staunchly my clear-headed friends have been able to defend their decisions to defect. They seem to be able to digest my argument about superrationality, to mull it over, to begrudge some curious kind of validity to it, but ultimately to feel on a gut level that it is wrong, and to reject it. This has led me to consider the notion that my faith in the superrational argument might be similar to a self-fulfilling prophecy or self-supporting claim, something like being absolutely convinced beyond a shadow of a doubt that the Henkin sentence “This sentence is true” actually must be true—when, of course, it is equally defensible to believe it to be false. The sentence is undecidable; its truth


value is stable, whichever way you wish it to go (in this way, it is the diametric opposite of the Epimenides sentence “This sentence is false”, whose truth value flips faster than the tip of a happy pup’s tail). One difference, though, between the Prisoner’s Dilemma and oddball self-referential sentences is that whereas your beliefs about such sentences’ truth values usually have inconsequential consequences, with the Prisoner’s Dilemma, it’s quite another matter.

I sometimes wonder whether there haven’t been many civilizations Out There, in our galaxy and beyond, that have already dealt with just these types of gigantic social problems—Prisoner’s Dilemmas, Tragedies of the Commons, and so forth. Most likely some would have survived, some would have perished. And it occurs to me that perhaps the ultimate difference in those societies may have been the survival of the meme that, in effect, asserts the logical, rational validity of cooperation in a one-shot Prisoner’s Dilemma. In a way, this would be the opposite thesis to Hardin’s. It would say that lack of conscience is self-eliminating—provided you wait long enough that natural selection can act at the level of entire societies.

Perhaps on some planets, Type I societies have evolved, while on others, Type II societies have evolved. By definition, members of Type I societies believe in the rationality of lone, uncoerced, one-shot cooperation (when faced with members of Type I societies), whereas members of Type II societies reject the rationality of lone, uncoerced, one-shot cooperation, irrespective of who they are facing. (Notice the tricky circularity of the definition of Type I societies. Yet it is not a vacuous definition!) Both types of society find their respective answer to be obvious—they just happen to find opposite answers. Who knows—we might even happen to have some Type I societies here on earth. I cannot help but wonder how things would turn out if my little one-shot Prisoner’s Dilemma experiment were carried out in Japan instead of the U.S. In any case, the vital question is: Which type of society survives, in the long run?

It could be that the one-shot Prisoner’s Dilemma situations that I have described are undecidable propositions within the logic that we humans have developed so far, and that new axioms can be added, like the parallel postulate in geometry, or Godel sentences (and related ones) in mathematical logic. (Take a look at Figure 31-1, and see what kind of logic will extract those two poor devils from their one-shot dilemma.) Those civilizations to which cooperation appears axiomatic—Type I societies—wind up surviving, I would venture to guess, whereas those to which defection appears axiomatic—Type II societies—wind up perishing. This suggestion may seem all wet to you, but watch those superpowers building those bombs, more and more of them every day, helplessly trapped in a rising spiral, and think about it. Evolution is a merciless pruner of ill logic.

Most philosophers and logicians are convinced that truths of logic are “analytic” and a priori; they do not like to think that such basic ideas are grounded in mundane, arbitrary things like survival. They might admit that


“The problem is how to turn loose without letting go.” FIGURE 31-1. One powerful metaphor for the absurdity we have collectively dug ourselves into. The symmetry of the situation is acutely portrayed in this cartoon drawn by Bill Mauldin in 1960. Note that if either person releases his rope, thus chopping of his counterpart’s head, that person’s hand will go limp, thus releasing his rope and causing the other blade to fall and chop of the head of the instigator. That idea is a centerpiece of our current nuclear deterrence strategy: Even if we are wiped of the globe, our trUSty missiles will still wreak divine revenge on the evil empire of Satanic Uglies who dared do harm to US.

“The problem is how to turn loose without letting go.” FIGURE 31-1. One powerful metaphor for the absurdity we have collectively dug ourselves into. The symmetry of the situation is acutely portrayed in this cartoon drawn by Bill Mauldin in 1960. Note that if either person releases his rope, thus chopping of his counterpart’s head, that person’s hand will go limp, thus releasing his rope and causing the other blade to fall and chop of the head of the instigator. That idea is a centerpiece of our current nuclear deterrence strategy: Even if we are wiped of the globe, our trUSty missiles will still wreak divine revenge on the evil empire of Satanic Uglies who dared do harm to US.

natural selection tends to favor good logic—but they would certainly hate the suggestion that natural selection defines good logic! Yet truth and survival value are all tangled together, and civilizations that survive certainly have glimpsed higher truths than those that perish. When you argue with someone whose ideas you are sure are wrong but who dances an infuriatingly inconsistent yet self-consistent verbal dance in front of you, your one solace is that something in life may yet change this person’s mind, even though your own best logic is helpless to do so. Ultimately, beliefs have to be grounded in experience, whether that experience is the organism’s or its ancestors’ or its peer group’s. (That’s what Chapter 5, particularly its


P.S., was all about.) My feeling is that the concept of superrationality is one whose truth will come to dominate among intelligent beings in the universe simply because its adherents will survive certain kinds of situations where its opponents will perish. Let’s wait a few spins of the galaxy and see. After all, healthy logic is whatever remains after evolution’s merciless pruning.

I was describing the Copycat project (Chapter 24) to physicist Victor Weisskopf, and I gave him our canonical example: “If abc goes to abd, what does xyz go to?” After we had discussed various possible answers and settled on wyz as the most compelling for reasons of symmetry, he surprised me by saying this: “You know, the root of the world’s deepest problems is the tragic inability on the part of the world’s leaders to see such basic symmetries. For instance, that the U.S. is to the S.U. what the S.U. is to the U.S.—that is too much for them to accept.” Oh, but how could Weisskopf be so silly? After all, we’re not trying to export communism to the entire world!

Logician Raymond Smullyan, who first heard about the Prisoner’s Dilemma from me and who was absolutely delighted by it, also surprised me, but in a different way: He vehemently insisted on the correctness of defection in a one-shot situation no matter who might be on the other side, including his twin or his clone! (He did waver about his mirror image.) But just as I was giving up on him as a lost cause, he conceded this much to me: “I suspect, Doug, that this problem is a lot knottier than you or I suspect.” Indeed, I suspect so, Raymond.

The Tale of Happiton


June, 1983

Happiton was a happy little town. It had 20,000 inhabitants, give or take 7, and they were productive citizens who mowed their lawns quite regularly. Folks in Happiton were pretty healthy. They had a life expectancy of 75 years or so, and lots of them lived to ripe old ages. Down at the town square, there was a nice big courthouse with all sorts of relics from WW II and monuments to various heroes and whatnot. People were proud, and had the right to be proud, of Happiton.

On the top of the courthouse, there was a big bell that boomed every hour on the hour, and you could hear it far and wide-even as far out as Shady Oaks Drive, way out nearly in the countryside.

One day at noon, a few people standing near the courthouse noticed that right after the noon bell rang, there was a funny little sound coming from up in the belfry. And for the next few days, folks noticed that this scratching sound was occurring after every hour. So on Wednesday, Curt Dempster climbed up into the belfry and took a look. To his surprise, he found a crazy kind of contraption rigged up to the bell. There was this mechanical hand, sort of a robot arm, and next to it were five weird-looking dice that it could throw into a little pan. They all had twenty sides on them, but instead of being numbered 1 through 20, they were just numbered 0 through 9, but with each digit appearing on two opposite sides. There was also a TV camera that pointed at the pan and it seemed to be attached to a microcomputer or something. That’s all Curt could figure out. But then he noticed that on top of the computer, there was a neat little envelope marked “To the friendly folks of Happiton”. Curt decided that he’d take it downstairs and open it in the presence of his friend the mayor, Janice Fleener. He found Janice easily enough, told her about what he’d found, and then they opened the envelope. How neatly it was written! It said this:


Grotto 19, Hades
June 20, 1983

Dear folks of Happiton,

I’ve got some bad news and some good news for you. The bad first. You know your bell that rings every hour on the hour? Well, I’ve set it up so that each time it rings, there is exactly one chance in a hundred thousand-that is, 1/100,000-that a Very Bad Thing will occur. The way I determine if that Bad Thing will occur is, I have this robot arm fling its five dice and see if they all land with ‘7’ on top. Most of the time, they won’t. But if they do-and the odds are exactly 1 in 100,000-then great clouds of an unimaginably revolting smelling yellow-green gas called “Retchgoo” will come oozing up from a dense network of underground pipes that I’ve recently installed underneath Happiton, and everyone will die an awful, writhing, agonizing death. Well, that’s the bad news.

Now the good news! You all can prevent the Bad Thing from happening, if you send me a bunch of postcards. You see, I happen to like postcards a whole lot (especially postcards of Happiton), but to tell the truth, it doesn’t really much matter what they’re of. I just love postcards! Thing is, they have to be written personally-not typed, and especially not computer-printed or anything phony like that. The more cards, the better. So how about sending me some postcards-batches, bunches, boxes of them?

Here’s the deal. I reckon a typical postcard takes you about 4 minutes to write. Now suppose just one person in all of Happiton spends 4 minutes one day writing me, so the next day, I get one postcard. Well, then, I’ll do you all a favor: I’ll slow the courthouse clock down a bit, for a day. (I realize this is an inconvenience, since a lot of you tell time by the clock, but believe me, it’s a lot more inconvenient to die an agonizing, writhing death from the evil-smelling, yellow-green Retchgoo.) As I was saying, I’ll slow the clock down for one day, and by how much? By a factor of 1.000011. Okay, I know that doesn’t sound too exciting, but just think if all 20,000 of you send me a card! For each card I get that day, I’ll toss in a slow-up factor of 1.00001, the next day. That means that by sending me 20,000 postcards a day, you all, working together, can get the clock to slow down by a factor of 1.00001 to the 20,000th power, which is just a shade over 1.2, meaning it will ring every 72 minutes.

All right, I hear you saying, “72 minutes is just barely over an hour!” So I offer you more! Say that one day I get 160,000 postcards (heavenly!). Well then, the very next day I’ll show my gratitude by slowing your clock down, all day long, midnight to midnight, by 1.00001 to the 160,000th power, and that ain’t chickenfeed. In fact, it’s about 5, and that means the clock will ring only every 5 hours, meaning those sinister dice will only get rolled about 5 times (instead of the usual 24). Obviously, it’s better for both of us that way. You have to bear in mind that I don’t have any personal interest in seeing that awful Retchgoo come rushing and gushing up out of those pipes and causing every last one of you to perish in grotesque, mouth-foaming, twitching convulsions. All I care about is getting postcards! And to send me 160,000 a day wouldn’t cost you folks that much effort, being that it’s just 8 postcards a day just about a half hour a day for each of you, the way I reckon it.


So my deal is pretty simple. On any given day, I’ll make the clock go off once every X hours, where X is given by this simple formula:

X = 1.00001N

Here, N is the number of postcards I received the previous day. If N is 20,000, then X will be 1.2, so the bell would ring 20 times per day, instead of 24. If N is 160,000, then X jumps way up to about 5, so the clock would slow way down just under 5 rings per day. If I get no postcards, then the clock will ring once an hour, just as it does now. The formula reflects that, since if N is 0, X will be 1. You can work out other figures yourself. Just think how much safer and securer you’d all feel knowing that your courthouse clock was ticking away so slowly!

I’m looking forward with great enthusiasm to hearing from you all.

Sincerely yours,
Demon #3127

The letter was signed with beautiful medieval-looking flourishes, in an unusual shade of deep red … ink?

“Bunch of hogwash!” spluttered Curt. “Let’s go up there and chuck the whole mess down onto the street and see how far it bounces.” While he was saying this, Janice noticed that there was a smaller note clipped onto the back of the last sheet, and turned it over to read it. It said this:

P. S.—It’s really not advisable to try to dismantle my little set-up up there in the belfry: I’ve got a hair trigger linked to the gas pipes, and if anyone tries to dismantle it, pssssst! Sorry.

Janice Fleener and Curt Dempster could hardly believe their eyes. What gall! They got straight on the -phone to the Police Department, and talked to Officer Curran. He sounded poppin’ mad when they told him what they’d found, and said he’d do something about it right quick. So he hightailed it over to the courthouse and ran up those stairs two at a time, and when he reached the top, a-huffin’ and a-puffin’, he swung open the belfry door and took a look. To tell the truth, he was a bit ginger in his inspection, because one thing Officer Curran had learned in his many years of police experience is that an ounce of prevention is worth a pound of cure. So he cautiously looked over the strange contraption, and then he turned around and quite carefully shut the door behind him and went down. He called up the town sewer department and asked them if they could check out whether there was anything funny going on with the pipes underground.

Well, the long and the short of it is that they verified everything in the Demon’s letter, and by the time they had done so, the clock had struck five more times and those five dice had rolled five more times. Janice Fleener had in fact had her thirteen-year-old daughter Samantha go up and sit in a


wicker chair right next to the microcomputer and watch the robot arm throw those dice. According to Samantha, an occasional 7 had turned up now and then, but never had two 7’s shown up together, let alone 7’s on all five of the weird-looking dice!

The next day, the Happiton Eagle-Telephone came out with a front-page story telling all about the peculiar goings-on. This caused quite a commotion. People everywhere were talking about it, from Lidden’s Burger Stop to Bixbee’s Druggery. It was truly the talk of the town.

When Doc Hazelthorn, the best pediatrician this side of the Cornyawl River, walked into Ernie’s Barbershop, corner of Cherry and Second, the atmosphere was more somber than usual. “Whatcha gonna do, Doc?” said big Ernie, the jovial barber, as he was clipping the few remaining hairs on old Doc’s pate. Doc (who was also head of the Happiton City Council) said the news had come as quite a shock to him and his family. Red Dulkins, sitting in the next chair over from Doc, said he felt the same way. And then the two gentlemen waiting to get their hair cut both added their words of agreement. Ernie, summing it up, said the whole town seemed quite upset. As Ernie removed the white smock from Doc’s lap and shook the hairs off it, Doc said that he had just decided to bring the matter up first thing at the next City Council meeting, Tuesday evening. “Sounds like a good idea, Doc!” said Ernie. Then Doc told Ernie he couldn’t make the usual golf date this weekend, because some friends of his had invited him to go fishing out at Lazy Lake, and Doc just couldn’t resist.

Two days after the Demon’s note, the Eagle-Telephone ran a feature article in which many residents of Happiton, some prominent, some not so prominent, voiced their opinions. For instance, eleven-year-old Wally Thurston said he’d gone out and bought up the whole supply of picture postcards at the 88-Cent Store, $14.22 worth of postcards, and he’d already started writing a few. Andrea McKenzie, sophomore at Happiton High, said she was really worried and had had nightmares about the gas, but her parents told her not to worry, things had a way of working out. Andrea said maybe her parents weren’t taking it so seriously because they were a generation older and didn’t have as long to look forward to anyway. She said she was spending an hour each day writing postcards. That came to 15 or 16 cards each day. Hank Hoople, a janitor at Happiton High, sounded rather glum: “It’s all fate. If the bullet has your name on it, it’s going to happen, whether you like it or not.” Many other citizens voiced concern and even alarm about the recent developments.

But some voiced rather different feelings. Ned Furdy, who as far as anyone could tell didn’t do much other than hang around Simpson’s bar all day (and most of the night) and buttonhole anyone he could, said, “Yeah, it’s a problem, all right, but I don’t know nothin’ about gas and statistics and such.”


“It should all be left to the mayor and the Town Council, to take care of. They know what they’re doin’. Meanwhile, eat, drink, and be merry!” And Lulu Smyth, 77-year-old. proprietor of Lulu’s Thread ’N Needles Shop, said “I think it’s all a ruckus in a teapot, in my opinion. Far as I’m concerned, I’m gonna keep on sellin’ thread ‘n needles, and playin’ gin rummy every third Wednesday.”

When Doc Hazelthorn came back from his fishing weekend at Lazy Lake, he had some surprising news to report. “Seems there’s a demon left a similar set-up in the church steeple down in Dwaynesville”, he said. (Dwaynesville was the next town down the road, and the arch-rival of Happiton High in football.) “The Dwaynesville demon isn’t threatening them with gas, but with radioactive water. Takes a little longer to die, but it’s just as bad. And I hear tell there’s a demon with a subterranean volcano up at New Athens.” (New Athens was the larger town twenty miles up the Cornyawl from Dwaynesville, and the regional center of commerce.)

A lot of people were clearly quite alarmed by all this, and there was plenty of arguing on the streets about how it had all happened without anyone knowing. One thing that was pretty universally agreed on was that a commission should be set up as soon as possible, charged from here on out with keeping close tabs on all subterranean activity within the city limits, so that this sort of outrage could never happen again. It appeared probable that Curt Dempster, who was the moving force behind this idea, would be appointed its first head.

Ed Thurston (Wally’s father) proposed to the Jaycees (of which he was a member in good standing) that they donate $1,000 to support a postcard-writing campaign by town kids. But Enoch Swale, owner of Swale’s Pharmacy and the Sleepgood Motel, protested. He had never liked Ed much, and said Ed was proposing it simply because his son would gain status that way. (It was true that Wally had recruited a few kids and that they spent an hour each afternoon after school writing cards. There had been a small article in the paper about it once.) After considerable debate, Ed’s motion was narrowly defeated. Enoch had a lot of friends on the City Council.

Nellie Doobar, the math teacher at High, was about the only one who checked out the Demon’s math. “Seems right to me”, she said to the reporter who called her about it. But this set her to thinking about a few things. In an hour or two, she called back the paper and said, “I figured something out. Right now, the clock is still ringing very close to once every hour. Now there are about 720 hours per month, and so that means there are 720 chances each month for the gas to get out. Since each chance is 1 in 100,000, it turns out that each month, there’s a bit less than a 1-in-100 chance that Happiton will get gassed. At that rate, there’s about 11 chances in 12 that Happiton will make it through each year. That may sound pretty”


“good, but the chances we’ll make it through any 8-year period are almost exactly 50-50, exactly the same as tossing a coin. So we can’t really count on very many years …”

This made big headlines in the next afternoon’s Eagle-Telephone—in fact, even bigger than the plans for the County Fair! Some folks started calling up Mrs. Doobar anonymously and telling her she’d better watch out what she was saying if she didn’t want to wind up with a puffy face or a fat lip. Seems like they couldn’t quite keep it straight that Mrs. Doobar wasn’t the one who’d set the thing up in the first place.

After a few days, though, the nasty calls died down pretty much. Then Mrs. Doobar called up the paper again and told the reporter, “I’ve been calculating a bit more here, and I’ve come up with the following, and they’re facts every last one of them. If all 20,000 of us were to spend half an hour a day writing postcards to the Demon, that would amount to 160,000 postcards a day, and just as the Demon said, the bell would ring pretty near every five hours instead of every hour, and that would mean that the chances of us getting wiped out each month would go down considerable. In fact, there would only be about 1 chance in 700 that we’d go down the tubes in any given month, and only about a chance in 60 that we’d get zapped each year. Now I’d say that’s a darn sight better than 1 chance in 12 per year, which is what it is if we don’t write any postcards (as is more or less the case now, except for Wally Thurston and Andrea McKenzie and a few other kids I heard of). And for every 8-year period, we’d only be running a 13% risk instead of a 50% risk.”

“That sounds pretty good”, said the reporter cheerfully.

“Well,” replied Mrs. Doobar, “it’s not too bad, but we can get a whole lot better by doublin’ the number of postcards.”

“How’s that, Mrs. Doobar?” asked the reporter. “Wouldn’t it just get twice as good?”

“No, you see, it’s an exponential curve,” said Mrs. Doobar, “which means that if you double N, you square X.”

“That’s Greek to me”, quipped the reporter.

“N is the number of postcards and X is the time between rings”, she replied quite patiently. “If we all write a half hour a day, X is 5 hours. But that means that if we all write a whole hour a day, like Andrea McKenzie in my algebra class, X jumps up to 25 hours, meaning that the clock would ring only about once a day, and obviously, that would reduce the danger a lot. Chances are, hundreds of years would pass before five 7’s would turn up together on those infernal dice. Seems to me that under those circumstances, we could pretty much live our lives without worrying about the gas at all. And that’s for writing about an hour a day, each one of us.”

The reporter wanted some more figures detailing how much different amounts of postcard-writing by the populace would pay off, so Mrs. Doobar obliged by going back and doing some more figuring. She figured out that if 10,000 people—half the population of Happiton—did 2 hours a day for


the year, they could get the same result—one ring every 25 hours. If only 5,000 people spent 2 hours a day, or if 10,000 people spent one hour a day, then it would go back to one ring every 5 hours (still a lot safer than one every hour). Or, still another way of looking at it, if just 1250 of them worked full-time (8 hours a day), they could achieve the same thing.

“What about if we all pitch in and do 4 minutes a day, Mrs. Doobar?” asked the reporter.

“Fact is, ’twouldn’t be worth a damn thing! (Pardon my French.)” she replied. “N is 20,000 that way, and even though that sounds pretty big, X works out to be just 1.2, meaning one ring every 1.2 hours, or 72 minutes. That way, we still have about a chance of 1 in 166 every month of getting wiped out, and 1 in 14 every year of getting it. Now that’s real scary, in my book. Writing cards only starts making a noticeable difference at about 15 minutes a day per person.”

By this time, several weeks had passed, and summer was getting into full swing. The County Fair was buzzing with activity, and each evening after folks came home, they could see loads of fireflies flickering around the trees in their yards. Evenings were peaceful and relaxed. Doc’ Hazelthorn was playing golf every weekend, and his scores were getting down into the low 90’s. He was feeling pretty good. Once in a while he remembered the Demon, especially when he walked downtown and passed the courthouse tower, and every so often he would shudder. But he wasn’t sure what he and the City Council could do about it.

The Demon and the gas still made for interesting talk, but were no longer such big news. Mrs. Doobar’s latest revelations made the paper, but. were. relegated this time to the second section, two pages before the comics, right next to the daily horoscope column. Andrea McKenzie read the article avidly, and showed it to a lot of her school friends, but to her surprise, it didn’t seem to stir up much interest in them. At first, her best friend, Kathi Hamilton, a very bright girl who had plans to go to State and major in history, enthusiastically joined Andrea and wrote quite a few cards each day. But after a few days, Kathi’s enthusiasm began to wane.

“What’s the point, Andrea?” Kathi asked. “A handful of postcards from me isn’t going to make. the slightest bit of difference. Didn’t you read Mrs. Doobar’s article? There have got to be 160,000 a day to make a big difference.”

“That’s just the point, Kathi!” replied Andrea exasperatedly. “If you and everyone else will just do your part, we’ll reach that number—but you can’t cop out!” Kathi didn’t see the logic, and spent most of her time doing her homework for the summer school course in World History she was taking. After all, how could she get into State if she flunked World History?

Andrea just couldn’t figure out how come Kathi, of all people, so


interested in history and the flow of time and world-events, could not see her own life being touched by such factors, so she asked Kathi, “How do you know there will be any you left to go to State, if you don’t write postcards? Each year, there’s a 1-in-12 chance of you and me and all of us being wiped out! Don’t you even want to work against that? If people would just care, they could change things! An hour a day! Half an hour a day! Fifteen minutes a day!”

“Oh, come on, Andrea!” said Kathi annoyedly, “Be realistic.”

“Darn it all, I’m the one who’s being realistic”, said Andrea. “If you don’t help out, you’re adding to the burden of someone else.”

“For Pete’s sake, Andrea”, Kathi protested angrily, “I’m not adding to anyone else’s burden. Everyone can help out as much as they want, and no one’s obliged to do anything at all. Sure, I’d like it if everyone were helping, but you can see for yourself, practically nobody is. So I’m not going to waste my time. I need to pass World History.”

And sure enough, Andrea had to do no more than listen each hour, right on the hour, to hear that bell ring to realize that nobody was doing much. It once had sounded so pleasant and reassuring, and now it sounded creepy and ominous to her, just like the fireflies and the barbecues. Those fireflies and barbecues really bugged Andrea, because they seemed so normal, so much like any other summer—only this summer was not like any other summer. Yet nobody seemed to realize that. Or, rather, there was an undercurrent that things were not quite as they should be, but nothing was being done …

One Saturday, Mr. Hobbs, the electrician, came around to fix a broken refrigerator at the McKenzies’ house. Andrea talked to him about writing postcards to the Demon. Mr. Hobbs said to her, “No time, no time! Too busy fixin’ air conditioners! In this heat wave, they been breakin’ down all over town. I work a 10-hour day as it is, and now it’s up to 11, 12 hours a day, includin’ weekends. I got no time for postcards, Andrea.” And Andrea .saw that for Mr. Hobbs, it was true. He had a big family and his children went to parochial school, and he had to pay for them all, and …

Andrea’s older sister’s boyfriend, Wayne, was a star halfback at Happiton High. One evening he was over and teased Andrea about her postcards. She asked him, “Why don’t you write any, Wayne?”.

“I’m out lifeguardin’ every day, and the rest of the time I got scrimmages—for the fall season.”

“But you could take some time out just 15 minutes a day—and write a few postcards!” she argued. He just laughed and looked a little fidgety. “I don’t know, Andrea”, he said. “Anyway, me ’n Ellen have got better things to do—huh, Ellen?” Ellen giggled and blushed a little. Then they ran out of the house and jumped into Wayne’s sports car to go bowling at the Happi-Bowl.


Andrea was puzzled by all her friends’ attitudes. She couldn’t understand why everyone had started out so concerned but then their concern had fizzled, as if the problem had gone away. One day when she was walking home from school, she saw old Granny Sparks out watering her garden. Granny, as everyone called her, lived kitty-corner from the McKenzies and was always chatty, so Andrea stopped and asked Granny Sparks what she thought of all this. “Pshaw! Fiddlesticks!” said Granny indignantly. “Now Andrea, don’t you go around believin’ all that malarkey they print in the newspapers! Things are the same here as they always been. I oughta know—I’ve been livin’ here nigh on 85 years!”

Indeed, that was what bothered Andrea. Everything seemed so annoyingly normal. The teenagers with their cruising cars and loud motorcycles. The usual boring horror movies at the Key Theater down on the square across from the courthouse. The band in the park. The parades. And especially, the damn fireflies! Practically nobody seemed moved or affected by what to her seemed the most overwhelming news she’d ever heard. The only other truly sane person she could think of was little Wally Thurston, that eleven-year-old from across town. What a ridiculous irony, that an eleven-year-old was saner than all the adults!

Long about August 1, there was an editorial in the paper that gave Andrea a real lift. It came from out of the blue. It was written by the paper’s chief editor, “Buttons” Brown. He was an old-time journalist from St. Jo, Missouri. His editorial was real short. It went like this:

The Disobedi-Ant

The story of the Disobedi-Ant is very short. It refused to believe that its powerful impulses to play instead of work were anything but unique expressions of its very unique self, and it went its merry way, singing, “What I choose to do has nothing to do with what any-ant else chooses to do! What could be more self-evident?”

Coincidentally enough, so went the reasoning of all its colony-mates. In fact, the same refrain was independently invented by every last ant in the colony, and each ant thought it original. It echoed throughout the colony, even with the same melody.

The colony perished.

Andrea thought this was a terrific allegory, and showed it to all her friends. They mostly liked it, but to her surprise, not one of them started writing postcards. All in all, folks were pretty much back to daily life. After all, nothing much—seemed really to have changed. The weather had turned real hot, and folks congregated around the various swimming pools in town. There were lots of barbecues in the evenings, and, every once in a while somebody’d make a joke or two about the Demon and the postcards. Folks would chuckle and


then change the topic. Mostly, people spent their time doing what they’d always done, and enjoying the blue skies. And mowing their lawns regularly, since they wanted the town to look nice.

Post Scriptum

The atomic bomb has changed everything except our way of thinking. And so we drift helplessly towards unparalleled disaster.

–Albert Einstein

People of every era always feel that their era has the severest problems that people have ever faced. At first this sounds silly. How can every era be the toughest? But it’s not silly. Things can be getting constantly more dangerous and frightful, and that would mean that each new generation truly is facing unprecedentedly serious problems. As for us, we have the problem of extinction on our hands.

Someone once said that our current situation vis-a-vis the Soviet Union is like two people standing knee-deep in a room filled with gasoline. Both hold open matchbooks in their hands. One person is jeering at the other:.. “Ha ha ha! My matchbook is full, and yours is only half full! Ha ha ha!”

The reality of our situation is about that simple. The vast majority of people, however, refuse to let this reality seep into their systems and change their day-to-day behaviors. And thus the validity of Einstein’s gloomy utterance.

I remember many years ago reading an estimate that the famous geneticist George Wald had made about nuclear war. He said he figured there was a 2% chance per year of a nuclear war taking place.1 This amounts to throwing one 50-sided die (or a couple of seven-sided dice) once a year, and hoping that it doesn’t come up on the bad side. How Wald arrived at his figure of 2% per year, I don’t know. But it was vivid. The figure has stuck with me for a couple of decades. I tend to think that the chances are greater nowadays than they were back then: maybe about 5% per year. But who can say?

The Bulletin of the Atomic Scientists features a clock on its cover. This clock doesn’t tick, it just hovers. It hovers near midnight, sometimes getting closer, sometimes receding a bit. Right now, it’s at three minutes to midnight. Back at the signing of SALT I, it was at twelve minutes before midnight. The closest it ever came was two minutes before midnight, and I think that was at the time of the Cuban missile crisis.

The purpose of the clock is to symbolize the current danger of a nuclear


holocaust. It’s a little like those “Danger of Fire Today” signs that Smokey the Bear holds up for you as you enter a national forest in the summer. It is a subjective estimate, made by the magazine’s board of directors. Now what is the meaning of “danger”, if not probability of disaster per unit time? Surely, the more dangerous a place or situation, the faster you want to get out of it, for, just that reason. Therefore, it seemed to me that the Bulletin’s number of minutes before midnight, B, was really a coded way of expressing a Wald number, W—a probability of nuclear war per year. And so I decided to make a subjective table, matching up the values of B that I knew about with my own best estimates of W. After a bit of experimentation, I came up with the following table:

Bulletin Clock (minutes before midnight)

Wald’s percentage (probability per year)

1 min


2 mins


3 mins


4 mins


5 mins


7 mins


10 mins:


12 mins


20 mins


A fairly accurate summary of this subjective correspondence is given by the following simple equation:

W = 20⁄B

This estimates for you the holocaust danger per orbit of the earth, as a function of the current setting of the Bulletin’s clock.

W and B may not be estimable in any truly scientific way, but there is a definite reality behind them, even if not. so simple as that of N and X in Happiton. Obviously it is not a “random” dice-like process that will determine whether nuclear war erupts in any given year. Nonetheless, it makes good sense to think of it in terms of a probability per year, since what actually does determine history is a lot of things that are in effect random, from the point of view of any less-than-omniscient being. What other people (or countries) do is unpredictable and uncontrollable: it might as well be random.

If tensions get unbearably high in the Middle East or in Central America, that is not something that we could have predicted or forestalled. If some terrorist group manufactures and uses or threatens to use—a nuclear bomb, that is essentially a “random” event. If overpopulation in Asia or starvation


In Africa or crop failures in the Soviet Union or oil gluts or shortages create huge tensions between nations, that is like a random variable, like a throw of dice. Who could have predicted the crazy flareup between Britain and Argentina over the silly Falkland Islands? Who knows where the next hot spot will turn out to be? The global temperature can change as swiftly and capriciously as a bright summer day can turn sultry and menacing—even in Happiton.

It is the vivid imagery behind the Wald number and the Bulletin clock that first got me thinking in terms of the Happiton metaphor. The story was pretty easy to write, once the metaphor had been concocted. I had to work out the mathematics as I went along, but otherwise it flowed easily. It was crucial to me that the numbers in. the allegory seem realistic. The most important numbers were: (1) the chance of devastation per year, which came out about right, as I see it; and (2) the amount of time per day that I think would begin to make a significant difference if devoted by a typical person to some sort of activity geared toward the right ends. In Happiton, that threshold turned out to be about fifteen minutes per day per person. Fifteen minutes a day is just about the amount of time that I think would begin to make a real difference in the real world, but there are two ways that one might draw a distinction between the situation in Happiton and the actual case.

Firstly, some people say that the situation in Happiton is much simpler than that of global competition and potential nuclear war. In Happiton, it’s obvious that writing postcards will do some good, whereas it’s not so obvious (they claim) what kind of action will do any good in the real world. Working hard for a freeze or for a reduction of US-SU tensions might even be harmful, they claim! The situation is so complex that nothing corresponds to the simplistic and sure-fire recipe of writing postcards.

Ah, but there is a big fallacy here. Writing postcards in Happiton is not sure-fire. The gas could still come oozing up at any time. All that changes is the odds. Now in the real world, we must follow our own best estimates, in the absence of perfect information, as to what actions are likely to be positive and what ones to be negative. You can only follow your nose. You can never be sure that any action, no matter how well intended, is going to improve. the situation. That’s just the way life is.

I happen to believe that the odds of a holocaust will be reduced (perhaps by a factor of 1.0000001) by writing to my representatives and senators fairly regularly, by attending local freeze meetings, by contributing to various organizations, by giving lectures here and there on the topic, and by writing articles like this. How can I know that it will do any good? I can’t, of course. And it’s no different in Happiton. The best of intentions can backfire for totally unforeseeable reasons. It might turn out that little Wally Thurston, by moving his pencil in a certain graceful curlicue motion one


afternoon while writing his 1,000th postcard to the Demon, stirs up certain air molecules which, by bouncing and jouncing against other ones helter-skelter, wind up giving that tiny last push to the caroming icosahedral dice atop the belfry, and bang! They all come up ‘7’! Wally, oh Wally, why such folly? Why did you ever write those postcards?

Those who would caution people that it might be counter-productive to work against the arms race—unless they believe one should work for the arms race—are in effect counseling paralysis. But would they do so in other areas of life? You never know if that car trip to the grocery store won’t be the last thing you do in your life. All life is a gamble.

The second distinction between Happiton and reality is this. In Happiton, for fifteen minutes a day to make a noticeable dent, it would have had to be donated by all 20,000 citizens, adults and children. Obviously I do not think that is realistic in our country. The fifteen minutes a day per person that I would like to see spent by real people in this country is limited to adults (or at least people of high-school age), and I don’t even include most adults in this. I cannot realistically hope that everyone will be motivated to become politically active. Perhaps a highly active minority of 5% would be enough. It is amazing how visible and influential an articulate and vocal minority of,that size can be! So, being realistic, I limit ’my desires to an average of fifteen minutes of activity per day for 5% of the adult American population. I sincerely believe that with about this much work, a kind of turning point would be reached—and that at 30 minutes or 60 minutes per day (exactly as in Happiton), truly significant changes in the national mood (and hence in the global danger level) could be effected.

I think I have explained what Happiton was written for. Trigger activity it may not. I’m growing a little more realistic, and I don’t expect much of anything. But I would like to understand human nature. better, to understand what it is that makes us so much like stupid gnats dully buzzing above a freeway, unable to see the onrushing truck, 100 yards down the road, against whose windshield we are about to be smashed.

One last thought: Although to me it seems that nuclear war is the gravest threat before us, I would grant that to other people it might appear otherwise. I don’t care so much what kinds of efforts people invest their time in, as long as they do something. The exact thing that corresponds to the threat to Happiton doesn’t much matter. It could be nuclear weapons, chemical or biological weapons, the population explosion, the U.S.’s ever-deepening involvement in Central America, or even something more contained, like the environmental devastation inside the U.S. What it seems to me is needed is a healthy dose of indignation: a spark, a flame, a fire inside. Until that happens, that courthouse clock’ll be tickin’ away, once every hour, on the hour, until …


Post Post Scriptum

Two magazines are devoted to the prevention of nuclear war. They are: the Bulletin of the Atomic Scientists and Nuclear Times. The Bulletin, founded in 1945, aims to forestall nuclear holocaust by promoting awareness and understanding of the issues involved. It describes itself as “a magazine of science and world affairs”. Its address is: 5801 South Kenwood Avenue, Chicago, Illinois 60637.

Nuclear Times is a more recent arrival, and calls itself “the news magazine of the antinuclear weapons movement”. Its articles are shorter and lighter than those of the Bulletin, but it keeps you up to date on what’s happening all over the country and the world. Its address is: Room 512, 298 Fifth Avenue, New York, New York 10001.

The following organizations are effective and important forces in the attempt to slow down the arms race and to reduce global tensions. Most of them put out excellent literature, which is available in, large quantities at low prices (sometimes free) for distribution. Needless to say, they can always use more members and more funding. Many have local chapters.

The Council for a Livable World
11 Beacon Street Boston,
Massachusetts 02108

711 G Street, S.E. Washington,
D.C. 20003

Nuclear Weapons Freeze Campaign
4144 Lindell Boulevard, Suite 404
St. Louis, Missouri 63108

The Center for Defense Information
303 Capital Gallery West
600 Maryland Avenue, S.W.
Washington, D.C. 20024

Physicians for Social Responsibility,
639 Massachusetts Avenue
Cambridge, Massachusetts 02139

International Physicians for the Prevention of Nuclear War
225 Longwood Avenue, Room 200
Boston, Massachusetts 02115

Union of Concerned Scientists
1384 Massachusetts Avenue
Cambridge, Massachusetts 02238

  1. Possibly this refers to Wald’s1969 speech, “A Generation in Search of a Future”. In it, Wald doesn’t make the estimate himself but attributes it to another Harvard professor:

    A few months ago, Senator Richard Russell, of Georgia, ended a speech in the Senate with the words “If we have to start over again with another Adam and Eve, I want them to be Americans; and I want them on this continent and not in Europe.” That was a United States senator making a patriotic speech. Well, here is a Nobel laureate who thinks that those words are criminally insane.

    How real is the threat of full-scale nuclear war? I have my own very inexpert idea, but, realizing how little I know, and fearful that I may be a little paranoid on this subject, I take every opportunity to ask reputed experts. I asked that question of a distinguished professor of government at Harvard about a month ago. I asked him what sort of odds he would lay on the possibility of full-scale nuclear war within the foreseeable future. “Oh”, he said comfortably, “I think I can give you a pretty good answer to that question. I estimate the probability of full-scale nuclear war, provided that the situation remains about as it is now, at two per cent per year.” Anybody can do the simple calculation that shows that two per cent per year means that the chance of having that full-scale nuclear war by 1990 is about one in three, and by 2000 it is about fifty-fifty.

    I think I know what is bothering the students. I think that what we are up against is a generation that is by no means sure that it has a future.


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