ETHICS AND GAME THEORY: THE PRISONER'S DILEMMA By R. L. C u n n i n g h a m U n i v e r s i t y of San F r a n c i s c o

After first saying a word about g a m e theory, I shall describe the Prisoner's D i l e m m a and then go on to consider critically s o m e recent discussions of the dilemma by s o m e of those w h o take a "moral point of view". In the course of m y remarks I shall s h o w that the Prisoner's D i l e m m a is not, properly speaking, a dilemma (or even a paradox); that neither the generalization argument nor Rousseauean "enforced collusion" offers help in solving this "dilemma"; and that s o m e analyses of suggested analogues to the Prisoner's D i l e m m a are failures. The term 'game theory' is generally used to refer to a branch of mathematics dealing with decision-making processes through attempting to simulate mathematically a situation of decision-making so as to discover the most rational decision. (I) (The term 'game theory' might, but should not, be taken to suggest that the scope of the discipline is limited to socially unimportant conflicts such as parlor g a m e s. ) As to the overall relevance of g a m e theory to ethics, I shall say no more here than to remark that I agree with ~. R. Lucas w h e n he wrote: The theory of G a m e s is never likely to provide a calculus of Morals; but it m a y well provide models on which to sharpen our logical teeth and develop our moral sense. (2)

The Prisoner's D i l e m m a is a 2-person, non-zero-sum, noncooperative g a m e first devised b y A . W . Tucker in %he early 1950's, and is usually given the following interpretation: T w o persons are thought to be jointly guilty of a serious crime, but the evidence is not adequate to convict them of that crime. The district attorney

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separates them and asks each if he would like to confess, saying that if both confess he will r e c o m m e n d a somewhat lighter sentence, e.g.,6 years in prison, than the usual one, e.g., i0 years in prison; but that if both decide not to confess, both will be prosecuted on a minor charge (of which there is conclusive evidence) and each will get 2 years in prison. O n the other hand, if one confesses and the other does not, the squealer will get only 1 year in prison and the other will get the full I0 years. The following 2-by-2 matrix diagrams the options: Jack Don't Squeal

Squeal

Don't Squeal I0 Willy i0

Squeal

Figure I Prisoner' s Dilemma

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(The lower number in each box gives the years in prison for Willy, the upper number gives the years in prison for Jack) (3) For W i l l y t h e b o t t o m b o x e s are p r e f e r a b l e to t h e t o p b o x e s no m a t t e r w h a t Jack d o e s : if Jack d o e s not c o n f e s s or s q u e a l , it w o u l d be i n W i l l y ' s i n t e r e s t to s q u e a l (a s i n g l e y e a r i s p r e f e r a b l e t o two y e a r s ) ; if Jack d o e s s q u e a l , it w o u l d a g a i n be i n W i l l y ' s i n t e r e s t to s q u e a l (6 y e a r s are p r e f e r a b l e to 10). For Jack t h e r i g h t - s i d e b o x e s are p r e f e r a b l e to the l e f t - s i d e b o x e s : 1 i s p r e f e r a b l e to 2 y e a r s , a n d 6 is p r e f e r a b l e to 10 y e a r s . So if e a c h d o e s the r a t i o n a l l y s e l f i n t e r e s t e d t h i n g , e a c h w i l l s q u e a l a n d b o t h end up i n t h e b o t t o m r i g h t - h a n d box ( 6 , 6 ) . (Note t h a t e v e n if t h e P r i s o n e r s are put t o g e t h e r for a few m i n u t e s a n d p e r m i t t e d to c o n s p i r e , t h i s w i l l not c h a n g e m a t t e r s a s long a s no b i n d i n g a g r e e m e n t c a n be made a n d t h e y s q u e a l s e p a r a t e l y a n d e a c h is i n t e r e s t e d o n l y i n g e t t i n g t h e s h o r t e s t s e n t e n c e for h i m s e l f . But s h o u l d t h e r e b e c o m m u n i c a t i o n a m o n g the p r i s o n e r s s u f f i c i e n t to d e v e l o p f r i e n d s h i p so t h a t t h e i n d i v i d u a l p r i s o n e r b e c o m e s c o n c e r n e d w i t h t h e w e l l b e i n g of h i s f e l l o w , t h e r e s u l t i n g c h o i c e might be d i f f e r e n t . And e x p e r i m e n t a l s e q u e n t i a l r u n s of the P r i s o n e r ' s D i l e m m a g i v e e v i d e n c e t h a t i n d i v i d u a l s t e n d to a d o p t c o o p e r a t i v e b e h a v i o r e v e n i n the a b s e n c e of e x p l i c i t c o m m u n i c a t i o n . ) (4)

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Rapoport a n d C h a m m a h , i n t h e i r r e c e n t b o o k P r i s o n e r ' s D i l e m m a (5), write as follows: The w i d e i n t e r e s t i n P r i s o n e r ' s D i l e m m a a m o n g b o t h p s y c h o l o g i s t s a n d game t h e o r e t i c i a n s is i n i t s s t a t u s of a g e n u i n e d i l e m m a or p a r a d o x . It s e e m s to us t h a t t h i s p a r a d o x is of t h e sort t h a t a p p e a r e d at t i m e s o n the i n t e l l e c t u a l h o r i z o n a s h a r b i n g e r s ~ic--] of i m p o r t a n t s c i e n t i f i c a n d p h i l o s o p h i c a l r e c o n s t r u c t i o n s . Such h a s b e e n the A c h i l l e s ' p a r a d o x d i s c o v e r e d by the E l e a t i c s . Both p a r a d o x e s w e r e p r e c u r s o r s of m o d e r n m a t h e m a t i c a l t h e o r i e s of t h e c o n t i n u u m . Such h a v e a l s o b e e n the l o g i c a l p a r a d o x e s w i t h s e l f n e g a t i n g p r o p o s i t i o n s a s k e r n e l s , t h e p r e c u r s o r s of t h e m o d e r n c r i t i q u e s of the f o u n d a t i o n of m a t h e m a t i c s . The p a r a d o x e s of r e l a t i v i t y a n d of q u a n t u m p h y s i c s a l s o i n i t i a t e d a f a r - r e a c h i n g p h i l o s o p h i c a l r e c o n s t r u c t i o n . (p. 1t) O n e might define 'dilemma' rather loosely as follows: a state of affairs in which a choice is d e m a n d e d but the alternatives appear to be equally bad or undesirable. (6) Is the Prisoner's D i l e m m a a d i l e m m a ? It is hard to see that it is: confessing (squealing) strictly dominates not confessing (not squealing): the consequences of confessing are better for either, no matter what the other fellow does, than are the consequences of not confessing. N o w Rapoport and C h a m m a h write that the Prisoner's D i l e m m a is an "example of a g a m e which has no satisfactory solution. That is to say, whatever choice is r e c o m m e n d e d by 'rational considerations', has something wrong with it in spite of the fact that nothing remains u n k n o w n about the situation. In other words, the chooser cannot do better by finding out more. H e n c e the dilemma." This s e e m s confused. There is nothing wrong about the rational choice for any prisoner, given the conditions of the game, though of course both prisoners might like to play s o m e other sort of game. The source of confusion, I think, is this: if one compares a situation in which both act "rationally" with a situation in which both act "irrationally" (both choose not to squeal), the latter situation is preferable to the players; this is confused with the Prisoner's D i l e m m a where no joint choice is possible. It remains true that any individual choice is better if "rational," worse if "irrational, " no matter what the other prisoner does. The prisoners will end up in the upper left-hand box (2,2) if either a) they are playing the g a m e cooperatively with an enforceable contract and each chooses rationally; or if b) the two players choose irrationally; or if c) the two players choose rationally but both wish to minimize total pain and each evaluates any one year in prison for either as equivalent to any other year. But of course in the first case, there is no Prisoner's Dilemma; in the second, each is ignorant

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of t h e c h o i c e w h i c h w o u l d m a x i m i z e h i s own u t i l i t y ; and i n t h e t h i r d c a s e we h a v e a c h o i c e m a d e b y t w o u t i l i t a r i a n s w h o s e h e d o n i c c a l c u l u s is not t h e o n e u s u a l l y a t t r i b u t e d t o t h e p r i s o n e r s . (One might n o t e t h a t e v e n if o n e p r i s o n e r w a s a u t i l i t a r i a n who had no k n o w l e d g e of t h e " m o r a l s " of t h e o t h e r , he w o u l d c h o o s e not to c o n f e s s no m a t t e r w h a t t h e o t h e r did so l o n g a s he o p e r a t e d on t h e a s s u m p t i o n t h a t t h e d i s u t i l i t y of y e a r s in p r i s o n for e i t h e r p r i s o n e r i s e f f e c t i v e l y l i n e a r and a s s u m e d t h a t t h e d i s u t i l i t y of o n e y e a r in p r i s o n for e i t h e r p r i s o n e r i s e q u a l to t h a t for t h e o t h e r p r i s o n e r . ) (7) A n d one cannot, I think, defend the claim that the Prisoner's D i l e m m a is a paradox, a n assertion seemingly contradictory or opp o s e d to c o m m o n sense. Rapoport and C h a m m a h write:

In . . . t h e P r i s o n e r ' s D i l e m m a , t h e r a t i o n a l c h o i c e of s t r a t e g y by b o t h p l a y e r s l e a d s to a n o u t c o m e w h i c h i s w o r s e for b o t h t h a n if they had c h o s e n their s t r a t e g y ' i r r a t i o n a l l y ' . ['But a g a i n , p a r e n t h e t i c a l l y , a n y i n d i v i d u a l r a t i o n a l c h o i c e i s p r e f e r a b l e to a n y i r r a t i o n a l c h o i c e . ~ The p a r a d o x r e m a i n s u n r e s o l v e d a s l o n g a s w e i n s i s t on a d h e r i n g to t h e c o n c e p t of r a t i o n a l i t y w h i c h m a k e s p e r f e c t s e n s e in z e r o - s u m g a m e s ['[-hose g a m e s in w h i c h t h e g a i n of one p l a y e r from a n y c h o i c e r e s u l t s in a n e q u i v a l e n t l o s s to t h e othe-Y] but w h i c h m a k e s q u e s t i o n a b l e s e n s e in n o n - z e r o - s u m g a m e s . T h u s , t h e p a r a d o x f o r c e s a r e e x a m i n a t i o n of our c o n c e p t of r a t i o n a l decision. (p. 13, op. c i t , ) I think it obvious that there are Prisoner's D i l e m m a a paradox than A n d I fail to see that in a 2-person choice is s o m e h o w contraindicated

no better grounds for calling the there are for calling it a dilemma. non-zero s u m g a m e , the "rational" or has less prescriptive power, (8)

In s o m e r e c e n t d i s c u s s i o n s of t h e P r i s o n e r ' s D i l e m m a by p h i l o s o p h e r s , n a m e l y t h o s e by W o l f f , T h o m p s o n , and H e l d (9), s t a t e m e n t s a r e m a d e i m p l y i n g t h a t t h e p r o b l e m is o n e of m u t u a l t r u s t . But (as w i l l be p o i n t e d out b y G o r d o n T u l l o c k in a f o r t h c o m i n g a r t i c l e i n E t h i c s ) t h i s i s of c o u r s e not so: o n e p r i s o n e r m i g h t b e p e r f e c t l y s u r e t h a t t h e o t h e r w i l l not c o n f e s s w i t h o u t m a k i n g it u n d e s i r a b l e for t h e f i r s t to c o n f e s s , But if o n e w e r e to go on to s a y , a s T u l l o c k d o e s , t h a t the p r o b l e m r a i s e d by the P r i s o n e r ' s D i l e m m a i s t h e " s o c i a l i r r a t i o n a l i t y " of t h e i n d i v i d u a l l y r a t i o n a l d e c i s i o n s of t h e p r i s o n e r s , t h e n o n e h a s to e x p l a i n w h y , in t h i s l i t t l e s o c i e t y of t h e t w o p r i s o n e r s , two i n d i v i d u a l l y r a t i o n a l d e c i s i o n s l e a d to s o c i a l i r r a t i o n a l i t y a l t h o u g h t w o i n d i v i d u a l l y i r r a t i o n a l d e c i s i o n s l e a d to s o c i a l r a t i o n ality. I t h i n k t h e s h o r t a n s w e r to t h i s q u e s t i o n i s t h a t t h e g a m e t h e

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prisoners are asked to play is one which leads to social irrationality so far as the prisoners' little society is concerned but presumably leads to social rationality so far as the larger society, civil society, is concerned. I shall return to this point later.

II t n a r e c e n t a r t i c l e i n t h e R e v i e w o f P o l i t i c s , R. F . H o p k i n s suggests that we apply the "generalization argument" to the Prisoner's D i l e m m a . (10) Popularly phrased in the form: "But suppose everyb o d y d i d t h e s a m e . T'', t h e g e n e r a l i z a t i o n argument calls for appealing to the consequences of everybody's doing x as the criterion of the rightness of my doing x (where x stands for an identifiable kind of act). For example, if I tell you that I don't think I'll bother voting in the next election, you might use the generalization argument to try to show me that I would be wrong in not voting. You say to me: "But suppose everybody did the same, suppose everybody decided not to vote; then the results would be disastrous; and so it would be wrong of you not to vote. " Very briefly and loosely, the answer to one who tells me that my not voting is immoral -- suppose everybody did the same! -- is that I know that not everybody will abstain from voting: and that since there is no causal nexus between my not voting and the disastrous consequences of everybody's not voting, that is, since the disastrous consequences will or will not occur whether I vote or not, then it is n o t i m m o r a l of m e n o t t o v o t e , a n d , i f v o t i n g i s m o r e t h a n s l i g h t l y troublesome, it m i g h t e v e n b e i r r a t i o n a l o f m e t o v o t e i n a n e l e c t i o n in which the chance that my voting will have important political consequences is infinitesimal. (11)

I-low do w e apply the generalization argument to the prisoner's d i l e m m a ? Here "doing the same thing" could refer either to both prisoners' confessing or to both prisoners' not-confessing, and it is clear that if both were to confess the consequences would be worse than if both were not to confess, for (6,6) is worse than (2,2). The matrix has been simplified by effectively eliminating all options where the prisoners do different things; and so w e no longer have the Prisoner's Dilemma. An adherent of the generalization argument must on moral grounds choose the course of action which, if chosen by everybody, would have the best consequences: here the choice that would have the best consequences is the choice not to confess. Since the generalization argument is far from being universally accepted as true or sound, and since someone, e.g., an egoist,

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might surely object to characterizing the choice not to confess as the moral choice, Hopkins attempts to support his position by illustrating the "apparent conflict" between prudence and morality by another example. " I n t h e s i t u a t i o n of a n o p e n m a r k e t w i t h m a n y s m a l l p r o d u c e r s w h o s e p r o d u c t i v e c a p a c i t y e x c e e d s d e m a n d , w h e a t f a r m e r s for i n s t a n c e , it i s in t h e i n t e r e s t s of a l l t h e p r o d u c e r s t o l i m i t p r o d u c t i o n . N e v e r t h e l e s s , t h e a c t i o n s of a n y o n e w h e a t f a r m e r c a n n o t a f f e c t t h e p r i c e , so no m a t t e r w h a t t h e o t h e r f a r m e r s d e c i d e it i s to t h e a d v a n t a g e o f e a c h i n d i v i d u a l f a r m e r t o r a i s e a s much w h e a t a s he c a n . . . . H e n c e , e a c h f a r m e r , s e e k i n g t o m a x i m i z e h i s p e r s o n a l i n t e r e s t s , e n s u r e s t h e g e n e r a l r u i n of w h e a t f a r m e r s . Of c o u r s e , t h i s s t r a t e g y , w h i c h a g a m e t h e o r e t i c a l s i m u l a t i o n w o u l d p r e s c r i b e for t h e f a r m e r , i s not in h i s r e a l i n t e r e s t s . " (p. 4 9 5 - 6 ) (It is not precisely true that "each farmer ... ensures the general ruin of wheat farmers" -- this is not in any one farmer's power -it is rather the actions of all, or most, of the farmers that "ensures the general ruin". Further, the claim that "this strategy . .. is not in his real interests" is wrong: in the circumstances described there is no better "strategy" available, given the costs of colluding. (12)) "Were the farmer, on the other hand, prone to reason ethically, such steps [~ctions of the federal government with respect to price supports and acreage limitation~ might not be necessary. Using the generalization argument, for instance, each farmer could see that bad consequences For whom--]? would result if all raised as m u c h wheat as they could, and hence, would conclude that he ought to follow self-imposed limitations. " (p. 496) F o l l o w i n g s e l f - i m p o s e d l i m i t a t i o n s i s the m o r a l t h i n g t o d o . But, u n l e s s a l l or m o s t f o l l o w s u c h l i m i t a t i o n s , a n y f a r m e r who did " t h e m o r a l t h i n g " w o u l d be a f o o l . H o p k i n s s e e s t h i s , and b a c k s off from s a y i n g t h a t t h e man who a c t s r a t i o n a l l y or p r u d e n t l y and d o e s not f o l l o w s e l f - i m p o s e d l i m i t a t i o n s is a c t i n g i m m o r a l l y . For, a c c o r d i n g t o H o p k i n s , w e a r e r e a l l y in a " s t a t e of n a t u r e " s i t u a t i o n (an i m p l i c i t r e f e r e n c e t o H o b b e s ' u s e of t h a t t e r m w h e n t h e r e e x i s t s no c i v i l s o c i e t y and so no m o r a l i t y ) . If t h e f a r m e r s f o l l o w no l i m i t a t i o n s t h e n t h e m a r k e t w i l l c o l l a p s e and t h e g o v e r n m e n t w i l l e n t e r t h e g a m e t o i m p o s e a c r e a g e l i m i t a t i o n s and o f f e r p r i c e - s u p p o r t s . So t h e m o r a l t h i n g for e a c h f a r m e r t o do w o u l d be t o a p p l y t h e g e n e r a l i z a t i o n a r g u m e n t , but if m o s t do n o t , t h e n t h e n e x t - b e s t t h i n g

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to do is to act prudently, to count the situation as a Hobbesian "state of nature" situation and to grow all the wheat he can. Then government will change the nature of the g a m e by its coercive acreage limitations and price-supports. There's something wrong here. Is it not quite odd that w e usually find fault with an industry dominated by price-searchers (monopolists, oligopolists, etc.) and are ready (in such a context) to write paeans of praise for the purely competitive market, where each seller is a price-taker -- that is, sells too small a share of the total product to affect its price significantly -- pointing out h o w the c o n s u m e r is thus benefitted (for competitive equilibrium is Pareto optimal for the inclusive group of both producers and consumers); and yet, w h e n w e look at things from the point of view of the individual seller in a "perfectly competitive market", find our hearts going out to him in sympathy, even to the extent of sometimes being willing to guarantee that the market will not be competitive (to the disadvantage of the consumer, of course). O n the one hand w e act, by anti-trust laws, etc. to ensure that individual sellers will not be in a position to affect the price significantly and so to exploit others for in general, market organization works efficiently only to the extent that the Prisoner's D i l e m m a applies to each producer sub-group; on the other, w e act to protect sellers from the consequences of "perfect competition", modifying the rules so that exploitation is possible. Let us ask two questions: would the bulk of wheat farmers ever generally do the "moral thing" -- voluntarily impose limitations on themselves? The answer, given the impossibly high costs of organizing and maintaining so large a "ring" of colluders, is: No. N o farmer acting independently will have any incentive to modify his o w n behavior either in the short or long-run despite his recognition that joint action in restricting supply would increase total profits. The second question is this: w h o would be benefitted and w h o hurt by such a "moral" collusion? It is clear that collusion would benefit those farmers w h o would benefit from a stable market (but hurt those farmers w h o would have gained a larger share of the market once the less efficient farmers were driven out of the market) and would hurt the consumers -- giving them wheat at higher prices than otherwise. And if it is reasonable for farmers to collude -- in the n a m e of morality! -- then it would be reasonable for all other producers to collude: butchers, bakers, candle-stick makers, doctors, lawyers, and merchant chiefs. In short, every group which could be hurt by competition a m o n g themselves is implicitly encouraged by Hopkins to collude -- on moral grounds.

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Hopkins appears to believe that the farmers' dilemma is really analogous to the prisoners' dilemma, which would m e a n that maximizing the production of wheat always strictly dominates producing the Pareto optimal quantity; and he then goes on to m a k e policy recommendations based on this assumption. H e appears not only to fail to see the relevance of the fact that m a n y Prisoner's D i l e m m a g a m e s are being played simultaneously, but also to have failed to take sufficiently into account important characteristics of commodity production. H e appears to forget that the g a m e of wheat farming is an iterated game, with all the consequences this entails; that the risks involved in farming can be reallocated or "laid off" on others willing to take or share these risks through selling short, "hedging", using futures markets and all the other institutions of a mature capitalist e c o n o m y which facilitate reallocating risk; that wheat farmers are not normally limited to producing nothing but wheat; that even in price-takers' markets (or pure competition), there are variations in the speed with which individuals m a k e adjustments, and that in such markets there would tend to be a convergence towards a solution which would maximize the utility of various factors of production. That Hopkins has failed to see the bad consequences of government restrictive action is clear. Consider the paragraph with which he ends his essay. Finally, it m a y be suggested that g a m e theory m a y be thoroughly analogous, in an ideal manner, to the generalization principle in ethics. For in a perfect or ideal society, g a m e matrices, as a tool for calculating strategy, would reflect situations in which it would never be prudent to act immorally. ~ust as in our example of the wheat farmer in which government action can alter each individual's g a m e matrix so that avoiding bad consequences for all is in the interest of each, in the ideal society all situations would be structured in this manner. (p. 500; m y italics) This is a glorious example of myopic social policy but one nicely consistent with a vision of a Platonic "closed society. " The longrun consequences for the farmers and for society are ignored -nothing is said about the bad consequences of attenuating private property rights or about the function of the open market in allocating inefficient farmers to jobs where they are more productive and so worth more. Nothing is said about the consequences for consumers, w h o must not only pay higher prices for wheat but also pay taxes to keep the market up (for the farmer has proved pretty skillful in evading government regulations). W e have done in the United States precisely what Hopkins suggests is the "ideal" thing to do, and yet it is clear that w e have dismally failed to do the job of protecting

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the farmer (who is counted as a special case because his job entails playing a cooperative game with a sometimes uncooperative nature -at least this is the only conceivable justification for protecting farmers more than for protecting hula-hoop manufacturers).

Note that if the government is to control production it must control the producers and limit their freedom. Would, for example, a m a n w h o wanted to b e c o m e a wheat farmer be able to get a wheat allotment? Hopkins' ideal society would be one in which one person would have a wheat allotment, another an allotment to dig so m a n y tons of coal, another an allotment to cut so m a n y heads of hair, another an allotment to teach logic to so m a n y college students, etc. If one were to wonder h o w different such an "ideal'' society is from our present one in which there are already effective cartels of physicians, of hairdressers, of street-car conductors, etc., the answer is that the principles w e n o w see exemplified would be extended (on moral grounds !) over the whole range of production instead of only over part, as at present. It i s a m i s t a k e t o a s s u m e t h a t t h e r e i s a l w a y s (at l e a s t ) o n e g o o d solution to a problem. The farmer's dilemma has no easy solution. Nor would an easy solution -- indeed any solution -- be in the general interest. Fortunately our basic social and economic institutions are such that a man has a choice between entering a relatively risky occupation like farming or hula-hoop manufacturing, and a relatively secure occupation like teaching. A wheat farmer who cannot derive satisfactory income, monetary and psychic, in the relatively risky game of farming is free to leave the farm for another occupation, one that promises more security. There is no reason to seek to eliminate the risk attached to certain occupations, given the cost to others of doing so, and certainly no moral reason -- though of course most people seem to expect other people to bear part of the cost when things go sour. If a m a n w e r e b o u n d b y l a w t o b e a w h e a t f a r m e r , if he were in effect a serf, if he had no choice in adjusting his taste for risk to an occupation which satisfied that taste, then he might well deserve money from other people's pockets when things go bad; but not otherwise.

III

Another author, Anatol Rapoport, offers a s o m e w h a t (but only somewhat) different solution to the Prisoner's Dilemma. (13) H e suggests that underlying assumptions are chosen arbitrarily. Instead of using as a basis of calculation " W h e r e a m I better off?", each Prisoner might well start with " M y partner F!] is like m e and so is likely to

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a c t l i k e me. If I c o n c l u d e t h a t t s h o u l d c o n f e s s , h e w i l l p r o b a b l y a l s o c o n c l u d e t h e s a m e ; and if I c o n c l u d e t h a t I s h o u l d not c o n f e s s , h e w i l l p r o b a b l y c o n c l u d e t h e s a m e . But in t h e f i r s t c a s e we w o u l d b o t h l o s e 2 y e a r s ; in t h e s e c o n d c a s e w e w o u l d b o t h l o s e 6 y e a r s . T h i s i n d i c a t e s t h a t I p e r s o n a l l y b e n e f i t by not c o n f e s s i n g . " N o w w h e n Rapoport says that w e choose our underlying assumptions arbitrarily, supporting this by the assertion that m a n y such a s s u m p tions, though seemingly absurd, are useful in science, and that what counts is not the absurdity of the assumption but the consequences of making it, he m o v e s into an area in which w e cannot follow him. It's true that to the extent I k n o w another person is like me, I can predict his choices by asking what I should do in similar circumstances. If I know, for example, that a friend has musical tastes similar to m y own, I can predict which phonograph records he will consider buying for himself at the big record sale at the local e m porium. But this is the case where there is no conflict of interests; and game-theory deals precisely with methods of analyzing conflict. Rapoport believes, in effect, that he has found a w a y of virtually eliminating from consideration the lower-left and upper-right boxes. These give results of choices in which the prisoners do different things. But the fact of the matter is that it is better for each individual to confess no matter what the other does -- if that individual is not at all interested in what happens to the other and if a choice m a d e by one has no causal influence on the choice m a d e by the other. O n e cannot really m o v e from egoism to utilitarianism quite so easily as Rapoport implies. Of course if I can count on the other's good will or concern for "social values" and I k n o w that he can count on mine, I will choose not to confess. If the two prisoners are brothers, or father and son, or, to use Rapoport's term "partners", they will choose a jointly maximizing strategy. The assumption I should m a k e about the other's strategy, and he about mine, is not arbitrary: it depends on w h o w e are. (14) Rapoport has in e f f e c t changed the g a m e from that of the Prisoner's D i l e m m a to a g a m e in which the choosers are friends (remember Aristotle's definition of 'friend': another self~ or partners, to a g a m e played cooperatively -- and a different set of standards of course applies. There's no contradiction in saying that it is rational to confess in one g a m e and rational not to confess in another. In sum, unless I a m interested in the welfare of the other, I shall m a k e an assumption in a Prisoner's D i l e m m a g a m e that what I do the other will do only if making that assumption is in m y interest. Rapoport goes on to defend the prisoners' "assumption of similarity"

THE PRISONER'S DILEMMA

21

by considering an analogous assumption: "The assumption of similarity is indeed the rationale which induce~ the individual citizen to vote. The argument that a single vote 'makes no appreciable difference' is countered with 'Yes, but if everyone thought so, the will of the collective would find no expression'. This is the rationale behind cooperative effort. " But, a s w a s s a i d a b o v e , w h a t i s i m p o r t a n t i s not t h a t a g i v e n i n d i v i d u a l v o t e ; w h a t is i m p o r t a n t i s t h a t not e v e r y o n e (or n e a r l y e v e r y one) a v o i d s v o t i n g . In a n e l e c t i o n i n w h i c h my s i n g l e v o t e m a k e s , it i s v i r t u a l l y c e r t a i n , no p o l i t i c a l d i f f e r e n c e , I h a v e no o b l i g a t i o n to v o t e , for I s h a l l at t h e s a m e t i m e be v i r t u a l l y c e r t a i n e n o u g h othel w i l l v o t e to make t h e i n s t i t u t i o n " w o r k " . Rapoport gives another example: "It is c o n c e i v a b l e t h a t t h e (minimal) d a n g e r i n h e r e n t i n s u b m i t t i n g to v a c c i n a t i o n a c t u a l l y e x c e e d s t h e ( s t i l l more m i n i m a l ) d a n g e r of r e m a i n i n g u n v a c c i n a t e d . But if everyor%.e m i n i m i z e d t h e d a n g e r to h i m s e l f a l o n e , e v e r y o n e w o u l d be s u b j e c t e d to t h e much g r e a t e r d a n g e r of a s m a l l p o x e p i d e m i c . " It i s c l e a r t h a t g e n e r a l l y w h e n a p e r s o n i s i n n o c u l a t e d a g a i n s t a c o m m u n i c a b l e d i s e a s e , he p r o t e c t s not o n l y h i m s e l f , b u t b y r e d u c i n g t h e p o t e n t i a l a g e n t s of c o n t a g i o n , he a l s o b e n e f i t s o t h e r s . On t h e o t h e r h a n d , i m m u n i z a t i o n u s e s up r e s o u r c e s (and may be u n p l e a s a n t or e v e n s l i g h t l y d a n g e r o u s ) a n d so t h e o p t i m a l p r o p o r t i o n of t h e p o p u l a t i o n i m m u n i z e d w i l l not n o r m a l l y be 100%. To i l l u s t r a t e : s u p p o s e a l l but o n e p e r s o n were i n n o c u l a t e d ; t h e n to i n n o c u l a t e t h a t l a s t p e r s o n w o u l d not b e n e f i t o t h e r s w h o , t h e m s e l v e s b e i n g i n n o c u l a t e d , c o u l d not c a t c h t h e d i s e a s e . So h e r e a g a i n , a l t h o u g h o n e c o u l d s a y to t h e l a s t u n i n n o c u l a t e d p e r s o n : "But s u p p o s e e v e r y o n e did t h e s a m e ! " , he c o u l d r e a s o n a b l y a n s w e r , "But e v e r y o n e d i d n ' t , a n d s i n c e t h e r e i s no d a n g e r t h a t a n y o n e e l s e w i l l be h a r m e d I ' l l not be i n n o c u l a t e d . " (15)

IV A c c o r d i n g to R u n c i m a n a n d Sen, i n a r e c e n t a r t i c l e i n M i n d (16), t h e c o n f l i c t b e t w e e n what s e e m s i n d i v i d u a l l y b e t t e r ( c o n f e s s ) a n d w h a t p r o d u c e s the b e s t o v e r - a l l r e s u l t (not c o n f e s s ) c o n t a i n s "the e s s e n c e of R o u s s e a u ' s d i s t i n c t i o n b e t w e e n t h e ' w i l l of a l l ' a n d the ' g e n e r a l w i l l ' . The ' g e n e r a l w i l l ' of t h e p r i s o n e r s , we

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D E C I S I O N PAPERS

can say, is to avoid confession, but each person's 'particular will' is to confess. Since in the absence of enforced collusion their self-seeking will take them to a situation worse for both, what is needed is an enforceable contract b e t w e e n them. They would both be ready to appoint an agent w h o would see to it that neither of t h e m confessed. In the absence of sanction (or, w e might even say, of a Sovereign), each prisoner m a y be driven by rational self-seeking to break the contract which is to the c o m m o n advantage of both. This gives an immediate and plausible sense to Rousseau's notion of the m e m b e r s of a society being 'forced to be free' .... " Runciman and Sen have offered us a model which comes perhaps as close as any model can to "voluntary coercion". If each prisoner sees that his only real alternative to 2 years in prison is one which he values as worse than two years -- he will of course choose the lesser penalty. Our n e w matrix is this:

2 2

2+ 10

10 2+

6

Figure II Rousseau g a m e

6

N o w w e have a g a m e in which each Prisoner will m a k e a "rational" choice which is also better for both jointly. Both would, presumably, prefer to play a R o u s s e a u - g a m e , to choose w h e n the matrix w a s coercive; but of course one could no longer say that the Prisoner's D i l e m m a g a m e w a s being played. And if a m a n were a wheat farmer, he would probably prefer a R o u s s e a u g a m e if his allotment were high enough, if he w a s content with a stable market, and if only wheat farmers were playing this g a m e . But if all Prisoner's D i l e m m a g a m e s were changed into R o u s s e a u g a m e s (by the addition of coercion), the players in any one g a m e would be colluding against all other players playing similar collusive games; and rationality would not lead a person to a choice of a R o u s s e a u - g a m e if a necessary concomitant of his choice w a s all others' choice of R o u s s e a u g a m e s -- he would prefer that he and everyone else play Prisoner's D i l e m m a games. Barbers, to avoid price-cutting and market competition collude against all w h o would like the price of a haircut to be no higher than the competitive rate -- and against all those w h o would like to be barbers,

THE PRISONER'S DILEMMA

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t h o s e w h o s e t r a i n i n g a n d t a l e n t w o u l d fit t h e m to b e m o s t p r o d u c t i v e a s b a r b e r s but to whom t h e market i s c l o s e d b y a g u i l d ; but of c o u r s e b a r b e r s do not l i k e t h e f a c t t h a t o t h e r g u i l d s f o r c e t h e m to p a y h i g h e r p r i c e s for g o o d s p r o d u c e d b y m e m b e r s of t h e s e o t h e r g u i l d s . As a p r o d u c e r of x , I w a n t to p l a y a c o l l u s i v e g a m e w i t h my f e l l o w p r o d u c e r s of x ; but a s a c o n s u m e r I d o n ' t w a n t o t h e r p r o d u c e r s , p r o d u c e r s of n o n - x , to p l a y s u c h g a m e s . {In t h e o r i g i n a l P r i s o n e r ' s D i l e m m a , t h e p u b l i c i s p r e s u m a b l y i n t e r e s t e d i n s e e i n g to it t h a t t h e p r i s o n e r s are p u n i s h e d , a n d if the p e n a l t i e s a s s i g n e d i n the o r i g i n a l m a t r i x (Figure I) are " f a i r " , t h e p u b l i c w o u l d be g l a d if b o t h g e t 6 y e a r s , a s s u m i n g b o t h are g u i l t y . If a n e n f o r c e a b l e c o n t r a c t were p o s s i b l e , a s it s o m e t i m e s i s i n p r a c t i c e , w h e n t h e s q u e a l e r who g e t s to p r i s o n i s t r e a t e d h a r s h l y b y t h o s e who d o n ' t l i k e s q u e a l e r s , t h e n "the p u b l i c " w o u l d l o s e . ) In sum, Runciman and Sen have offered no "solution" to the Prisoner's D i l e m m a though they have clearly s h o w n w h y the prisoners would prefer to play a similar g a m e to which the element of coercion to enforce contracts w a s added. With respect to their suggestion that the "general will", w h e n coercively enforced, fulfills the conditions of Pareto optimality, it appears clear that this is true only of the subgroup of prisoners and not of the inclusive group of the general public, of the subgroup of wheat farmers but not of the inclusive group of consumers-producers. One further point relevant to the extent of realistic Rousseau g a m e analogues. Runciman and Sen write: The general will, though remaining unalterable, b e c o m e s subordinated to the encroachment of individual wills w h e n "each, separating his interest from the interest of all, sees that such separation cannot be complete, yet the part he plays in the general d a m a g e seems to him as nothing compared with the exclusive good which he seeks to appropriate" -- an account which fits exactly the case of the prisoner w h o seeks to gain an advantage by breaking the contract. (p. 556) I a m inclined to believe that neither the Prisoner's D i l e m m a nor the Rousseau g a m e best analyzes the rational choice of an individual w h e n the goods to be chosen are various quantities of private and public (or "common") goods. A more satisfactory analysis, would, I have been persuaded (by ~ames Buchanan in a recent article) (17), lead to the conclusion that an expanded 3-by-3 choice matrix would better illustrate the opposing pressures on the individual, some of which lead to cooperation and others of which lead to conflict. Though I a m not sure, it might even be possibly better to illustrate the

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f a r m e r ' s d i l e m m a w i t h a n a p p r o p r i a t e l y e x p a n d e d game m a t r i x , o n e i n w h i c h t h e d o m i n a n c e f e a t u r e s are r e m o v e d . In conclusion, I should like to suggest that the fairly extensive technical literature dealing with the Prisoner's D i l e m m a and related g a m e s might well be found worth studying by one w h o attempts to deal with s o m e central issues in social philosophy, issues, for example, such as the desirable range and allocation of "publicgoods" (together with the related "free rider" problem); the nature of, and incentives to, "social" and "anti-social" behavior in large-number and small-number groups; the desirable range of institutions which foster competition and/or cooperation, together with related problems concerning inter-firm organization (duopoly, oligopoly, etc.), problems concerning federalism and its alternatives, and the like.

NOTES 1. The t h e o r y of g a m e s , R. B. B r a i t h w a i t e t e l l s u s ("The T h e o r y of G a m e s a n d Its R e l e v a n c e to P h i l o s o p h y " , P h i l o s o p h y i n M i d C e n t u r y , F i r e n z e : 1958), w a s d e v e l o p e d to p r o v i d e " e x a c t c r i t e r i a for a p e r s o n ' s d e c i d i n g how he s h o u l d c h o o s e to a c t i n a s i t u a t i o n i n w h i c h the o u t c o m e d e p e n d s not o n l y u p o n h i s own c h o i c e of a c t i o n b u t a l s o u p o n t h e c h o i c e s of a c t i o n of a n u m b e r of o t h e r p e o p l e . Such s i t u a t i o n s a r i s e w h e n e v e r p e o p l e w i t h different interests can only further their individual interests by acting in c o m m o n , and lie at the root of economics and of the other social sciences. " (p. 148) If one takes 'rational decision' to m e a n a decision which maximizes expected utility (a decision which yields the more preferred outcome, a decision in which an individual chooses "more" rather than "less") then clearly one decision can be "more rational" than another if the individual is better informed as to the range of alternatives open to him, if the results of alternative choices of action are more fully k n o w n and predictable, and if separate decisions are consistent with one another. There is evidence that g a m e theory can help an individual in making rational decisions, can help us, in Seriven's words, in "deciding h o w most efficiently to do what w e want to do. " (Ethics, Vol. 68, Oct. '57, p. 62)

2. "Moralists and G a m e s m e n " ,

Philosophy, January 1959, p. ii. C.f. " G a m e s and Aims", Aurel Kolnai, Proceedings of the Aristotelian Society 1966, esp. pp. 120-124.

T H E PRISONER'S D I L E M M A

25

3. Note that the numbers in Figure I are somewhat arbitrary; any numbers will do but only so long as the following relationships prevail: where R stands for the Reward for cooperating (e.g., 2 years), S for the Sucker's payoff (e.g., I0 years), T for the Temptation to get a "free ride", (e.g., 1 year), and P for Punishment (e.g., 6 years), then S , < P ~ R ~ T (e.g., i 0 < 6 < 2 < 1 ) and 2 R ~ S + T (e.g., 4 7 1 0 + I). 4. Cf. the general discussion of the issue in R. D. Luce and H. Raiffa, G a m e s and Decision (New York: 1957); and the experiments of Rapoport and C h a m m a h , Prisoner's Dilemma (Ann Arbor: 1965), which indicate that the proportion of cooperative responses reaches approximately 70% after repeated plays

(Ch. s). 5. University of Michigan Press (Ann Arbor: 1965). 6. The C e n t u r y D i c t i o n a r y (1914) i l l u s t r a t e s : A strong dilemma in a d e s p e r a t e c a s e ! To act with infamy, or quit the place.

(Swift)

7. If the matrix were s l i g h t l y c h a n g e d b y s u b s t i t u t i n g (5,5) in the lower r i g h t - s i d e box for (6,6), the problem for the u t i l i t a r i a n would be c o m p l i c a t e d by the need to e s t i m a t e p r o b a b i l i t i e s , e t c . 8. "Once the limits of 2-person zero-sum games are transcended, game theory, while remaining a powerful tool for analyzing the logical structure of conflicts of interest, loses its prescriptive power. In this realm, strategically rationalizable courses of action are frequently intuitively unacceptable and vice versa, " Rapoport and C h a m m a h , op. cir., p. 23. 9. "Reflections on G a m e Theory and the Nature of Value, " R.P. Wolff Ethics, April 1962, p. 171; " G a m e Theory and 'Social Value' States," G. Thompson, Ethics, October 1964, p. 36; and "Rationality and Social Value in G a m e Theoretical Analysis," V. Held, Ethics, April 1966, p. 215.

I0. " G a m e Theory and Generalization in Ethics", October 1965. II° Cf. m y " W h y Care What Would Happen if Everybody Acted Like M e ?", Memorias del XIII Congres so Internacional de Filosofia, Vol. 7, 1964, and "Is It Necessary or Useful to Randomize?", Analysis, lanuary, 1966. 12. 'Strategic behavior' usually refers to an action expected to exert

26

DECISION PAPERS some influence on the behavior of others, e.g., in an iterated Prisoner's D i l e m m a game, one might adopt a tit-for-tat strategy, playing whatever the other played last time. O n this use of 'strategic behavior', no strategic behavior is available to the farmers. Strategic behavior is unavailable to members of a group w h o s e size is critically large; c . f . J . M . Buchanan, "Ethical Rules, Expected Values, and Large Numbers", Ethics, Vol. 76, No. I, October, 1965.

13. Fights, G a m e s , and Debates (Ann Arbor: 1961), pp. 175 ff. Other interpretations of the Prisoner's D i l e m m a matrix have been suggested by Rapoport and C h a m m a h : " ... the two players can be asked to imagine that they are two firms in competition. Each has a choice of selling its product at one of two price levels. If one firm sells at a high level while the other sells at a low level, the second firm reaps the profit (by winning the market). If both sell at a high level, both profit (though not as m u c h as w h e n competition is eliminated). If both sell at a low level, both lose money. Clearly this situation is isomorphic to the Prisoner's Dilemma. Or, the players can imagine that they are rival power blocs w h o have m a d e a disarmament agreement. The cooperative choice n o w m e a n s keeping the agreement; the defecting choice, breaking it. There is supposedly an advantage accruing to the bloc which breaks the agreement unilaterally, etc." Prisoner's Dilemma, p. 222. 14. Who we are d e p e n d s v e r y o f t e n , a s J a m e s B u c h a n a n h a s p o i n t e d out (Ethical R u l e s , E x p e c t e d V a l u e s , a n d Large N u m b e r s " , E t h i c s O c t o b e r 1965), o n the s i z e of t h e p o p u l a t i o n d e a l t w i t h . The g e n e r a l r u l e of p r e d i c t i o n : t h e l a r g e r t h e p o p u l a t i o n the l e s s l i k e l y t h e y w i l l a c t l i k e f r i e n d s (or a p p l y t h e g e n e r a l i z a t i o n a r g u m e n t ) . If w e ' r e i n t e r a c t i n g w i t h j u s t a few p e o p l e w h o s e i n t e r e s t s are v i s i b l y a f f e c t e d b y a c t i o n s of o u r s , we a c t o n e way; if w e ' r e i n t e r a c t i n g w i t h a g r e a t m a n y p e o p l e who are not v i s i b l y a f f e c t e d b y what we d o , we a c t a n o t h e r w a y . 15. C.f. "Economic Analogues to the Generalization Argument", J. M . Buchanan and G. Tullock, Ethics, July 1964. 16.

" G a m e s , T u s t i c e , a n d t h e G e n e r a l W i l l " , O c t o b e r 1965.

17.

"Cooperation and Conflict in P u b l i c - G o o d s Interaction" (mimeographed, 10/28/66).