The Prisoner's Dilemma

There are thousands of Web-Sites about this, and a huge amount of relevant published books and articles, as well as a substantial and important literature on Game Theory in general. I just want you now to think for yourselves whether we can learn from the dilemma, and if so, what we can learn. You can start from a short article of mine (F.C.T. Moore, `The Martyr's Dilemma', Analysis, 45:1, January 1985, pp. 29-33), reproduced below. [If you want to look at a somewhat wider discussion, you can borrow a copy of my article 'Taking the sting out of the Prisoner's Dilemma' (Philosophical Quarterly, 44: 175, April 1994, pp. 223-234) from the General Office of the Department of Philosophy.] The article in the Stanford Encyclopedia of Philosophy is not too bad, but I find it rather heavy (if you use this link to it, you'll have to use the BACK button to get back here).

It is sometimes stated or implied that the Prisoner's Dilemma arises only for rationally self-interested behaviour. Hillel Steiner, for example, has said (`Prisoner's Dilemma as an insoluble problem, Mind, XCI (1982), pp. 285-6) that the dilemma depends upon the statement that `all persons pursue rationally self-interested strategies'. Commentators on Steiner's article appear to have accepted this widespread view (see Gordon, D., `Is the Prisoner's Dilemma an insoluble problem ?', and Porter, J.P., `Relevant Interest and the Prisoner's Dilemma, Mind, XCIII (1984), pp. 98-102). Thus the question of the rationality of altruism may be held to be settled, or begged, by the dilemma. Indeed, in the usual matrix (for two players A and B, and two choices C and D), where T>R>P>S (the latter, of course, being a preference ordering), C is commonly described as the `altruistic' (but irrational) choice.

			B
		C		D
	C	R,R		S,T
A
	D	T,S		P,P

In fact, moralizing readings of the Prisoner's Dilemma can be quite elaborate: thus C is read "co-operate", and R is the "reward" obtained by A and B for their co-operation; D is read as "deviation" or "defection", and P is the "punishment" both receive for their wickedness; in the case of asymmetrical choices, the player gaining T is described as a "traitor", while the loser (who takes S) is a "saint" or even a "sucker". David Lewis has claimed it to be "rat"ional to "rat" (in `Prisoner's Dilemma is a Newcomb problem', Philosophy and Public Affairs, 83 (1979), pp. 235-40), while Anatol Rapaport in an early article (`Formal games as probing tools for investigating behaviour motivated by trust and suspicion', Journal of Conflict Resolution, VII (1963), pp. 570-9) described a player's repeated co-operation in iterated games (in spite of defection by the other player), as a "martyr run".

Various authors have questioned such readings of the Prisoner's Dilemma, including recently M.C. Geach (Analysis, 44:1, January 1984, pp. 46-8: "a similar sort of case can arise even when people agree about the desire effect, and are not selfish.). Similarly, J.W.N. Watkins has maintained, in `Self Interest and Morality' (Körner, S. (ed.), Practical Reason, Oxford: Blackwell, 1974, pp. 67-77), that the dilemma arises for moral agents who give equal weight to their own and the other's preferences. And Derek Parfit in `Is commonses morality self-defeating ?' (Journal of Philosophy 76:10, 1979, pp. 533-45), applies Prisoner's Dilemma reasoning to cases where the parties are pursuing different goals which each flow from the same moral theory, in order to construct an argument that any such theory is self-defeating. My claim is more general, that Prisoner's Dilemma arises for any actions whatsoever in whose case the outocme depends upon the interplay of different actions: it is thus a problem for the theory of social action, and does not in any way suppose or show the irrationality of altruism.

Note also recent work on applications of Game Theory to the evolution of behaviour in Maynard Smith, J., Evolution and the Theory of Games (Cambridge: Cambridge University Press, 1982), and `Game Theory and the Evolution of Behavior' (in Behavioral and Brain Sciences 7 (1984), pp. 95-125, including pp. 114-115 the commentary by Rapoport `Game Theory without Rationality'). These applications assume the possibility of different strategies, or "behavioural phenotypes", being present in a population (whether because some individuals are genetically determined to do x, and others to do y, or because an individual is genetically determined to do x and y with certain probabilities, or because an individual tends to do x and y with certain probabilities as a result of learning). It is then possible to calculate whether there is an "evolutionarily stable strategy" (pure or mixed) by interpreting payoff in terms of Darwinian fitness, i.e. in terms of the numbers of offspring produced by organisms with various strategies. Efforts have been made using this approach to offer an empirical explanation of the phenomenon of altruism. But it is more relevant to the present argument that the approach itself theoretically allows any behavioural tendencies to occur. We might, for example, have entire populations of doves, or martyrs. The empirical questions are: which behavioural phenotypes actually occur, and which have survival value.

It is not my aim here to say anything about solutions to the Prisoner's Dilemma, but rather to establish a point about its character. Indeed, one interesting line of solution was developed formally by Howard, N., in `The Theory of Meta-Games' and `The Mathematics of Meta-Games' (in General Systems 11, 1966, pp. 167-200), and explained and elaborated by Pörn, Ingmar, Action Theory and Social Science: some formal models, Dordrecht: Reidel, 1977, pp.98-102. This line of solution involves the introduction of policies in the first-level meta-game (by which a makes his choice of C or D conditional on a's choice of C or D), and then in the second-level meta-game a can make his choice of C or D conditional upon b's choice of policy. If this line of reasoning is accepted, it turns out that a co-operative policy is rationally preferable to a non-co-operative policy (see also Rapoport, A., `Escape from Paradox', Scientific American 217 (1967), pp.50-6).

My aim here is rather to support the view that the Prisoner's Dilemma is morally neutral by arguing that it arises out of interdependent choices under uncertainty, whatever the moral character of the objectives of the agents, and, in particular, independently of whether those objectives are self-interested or altruistic in character. The rational choice may commonly have been as a "selfish" or self-interested one, but this is only because self-interested objectives happend to have been selected in illustrating the dilemma.

It is sufficient proof of this contention to construct another instantiation of the same problem in which the objective is admittedly altruistic.

Three people, A, B and C, are in a dug-out. A had-grenade lands close to and between A and B, but some distance from C. This is how the situation appears to A and B: if A and B both throw themselves away from the grenade, all three will be injured. If A throws himself on the grenade, he will die, and B will be injured, but C will be unscathed. Correspondingly, if B throws himself on the grenade. If both A and B throw themselves on the grenade, both will be seriously injured, but C will be unscathed.

If A is a martyr, all he will be concerned with is the fate of the others: that is, in the case of the four possible outcomes as they affect the three men in the dug-out, we measure the pay-off from A's point of view by ignoring the outcome for himself. Now let us suppose that in fact both A and B are martyrs. Let them accordingly share the following preference ordering over the possible outcomes (for both of them, as explained, the outcome for only two people out of the three in the dug-out is relevant):

1	least bad	1 injured
2	bad	2 injured
3	very bad	1 seriously injured
4	worst	1 dead

What then should A do ? Consider this diagram:

			B
		saves himself		sacrifices himself
	saves himself	2i/2i		1d/1i
A
	sacrifices himself	1i/1d		1si/1si

outcomes as judged by A and B, each ignoring himself (i=injured; d=dead; u=unscathed; si=seriously injured)

Given A's preferences specified above (i.e. 1 least bad, 2 bad, 3 very bad, 4 worst), he faces the following situation. If he sacrifices himself, there are two possible outcomes - 3 very bad (if B does the same), or 1 least bad (if B tries to save his skin); but if he saves his skin, the two possible outcomes are - 4 worst (in case B sacrifices himself) or 2 bad (in case B also saves his skin). By minimax, as a martyr, A should sacrifice himself, since the outcome cannot be worse than 3 very bad, and there is chance of 1, the least bad outcome. He should not save his skin, since though there is a chance of 2 the (merely) bad outcome - which is collectively the best for two martyrs, there is also the chance of 4, the worst outcome, if the other sacrifices himself. Let me emphasize again that if A is a martyr (that is, if his preference ordering of the outcomes is the one already described), then it would be rational for A, using minimax, to sacrifice himself; but if both A and B are martyrs the best result for them collecitvely, i.e. the outcome which would produce the best net result taking both their preferences into account, would be achieved by both saving their skin, for there will then be two injured, which both avgree to be better than one seriously injured, or one dead.

Thus if both A and B are martyrs, they will in 'rationality' both sacrifice themselves, producing an outcome 3, which is very bad (i.e. one seriously injured, from either point of view); but irrational martyrs, both of whom saved their skin, would actually produce the best net result. This can be checked on the following diagram:

			B
		saves himself		sacrifices himself
	saves himself	A B C i i i		A B C i d u
A
	sacrifices himself	A B C d i u		A B C si si u

I conclude that we can readily construct a dilemma formally identical to the Prisoner's Dilemma, but in which the parties have altruistic objectives, in the sense of ignoring their own interests. My dilemma could be called the "Martyr's Dilemma" - except that if my argument here is cogent, the two dilemmas are in fact just the same dilemma in different instantiations. I am uncertain whether the example tells us anything about martyrdom; but it does, I believe, prove that moralizing interpretations of the Prisoner's Dilemma are wrong.