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
computer-science, experiments, philosophy, sociology
by: Douglas Hofstadter 2012-04-162019-04-01 finished certainty: log importance: 10

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 , an anthol­ogy of arti­cles & essays pri­mar­ily pub­lished in “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 ‘’ in , an attempt to devise a deci­sion theory/algorithm for agents which can reach global util­ity max­ima on prob­lems like the 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 ) & (specifi­cal­ly, ), and one net­work­ing project.

Sanity and Survival

by Dou­glas Hof­s­tadter


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.

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


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.”

(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 .” 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.”

, 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.

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 a unique set of images and asso­ci­a­tions to each word and con­cept, and it is the set of those images and asso­ci­a­tions that will col­lec­tively cre­ate, in that per­son’s mind, a set of men­tal pres­sures like the set of pres­sures inside the earth in an earth­quake zone. When peo­ple decide, you find out how all those pres­sures push­ing in differ­ent direc­tions add up, like a set of force vec­tors push­ing in var­i­ous direc­tions and with strengths influ­enced by pri­vate or unmea­sur­able fac­tors. The assump­tion that it is valid to extrap­o­late has to be based on the idea that every­body is alike inside, only with some­what differ­ent weights attached to cer­tain notions.

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

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

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

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


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 “”.

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 , 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 .

A renor­mal­ized elec­tron’s style of inter­act­ing with, say, a renor­mal­ized pho­ton takes into account that the pho­ton’s quan­tum-me­chan­i­cal struc­ture includes “” 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. 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.”

, 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 a uni­ver­sal choice: C or D? That’s all.

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

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


p max­i­mizes the like­li­hood of exactly one per­son get­ting the go-a­head. It turns out that it is , or more gen­er­al­ly, 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 , which is just under 37%. With twenty super­ra­tional play­ers all throw­ing dice, the chance that you will come up the big win­ner is very close to , 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 $, 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 (i.e., 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 (i.e., 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 , 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.


, an evo­lu­tion­ary biol­o­gist, wrote a famous arti­cle about this type of phe­nom­e­non, called “”. 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 (): 1
  • 10100 (a ): 9
  • (a ): 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 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” [] 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 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 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 “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.

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 (Chap­ter 24) to physi­cist , 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 , 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, -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 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 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 , 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 20%
2 mins 10%
3 mins 7%
4 mins 5%
5 mins 4%
7 mins 3%
10 mins: 2%
12 mins 1.5%
20 mins 1%

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

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’s 1969 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 , 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.