VI.-GAMES, JUSTICE AND THE GENERAL WILL

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IN this paper, we argue that a useful sense may be given'to the "General Will" and the " Common Good " by reference to the theory of non-zero-sum non-cooperative games, and we suggest some possible implications of this for the nQtion of social justice.

I

We begin by looking at the two-person, non-zero-sum, noncooperative game known as the " prisoner's dilemma ".1 Two persons are thought to be jointly guilty of a serious crime, but the evidence is not adequate to convict them at a trial. The district attorney tells the prisoners that he will take them separately and ask them whether they would like to confess, though of course they need not. If both of them confess, they will be prosecuted, but he will recommend a lighter sentence than is usual for such a crime, say 6 years of imprisonment rather than 10 years. If neither confesses, the attorney will put them up only for a minor charge of illegally possessing a weapon, of which there is conclusive evidence, and they can expect to get 2 years each. If, however, one confesses and the other does not, the one who confesses receives lenient treatment for providing evidence to the state and gets only 1 year, and the one who does not receives the full punishment of 10 years. From an egoistic point of view, the strategy of confession for either of the prisoners " strictly dominates " over the strategy of non-confession, i.e. no matter what the other prisoner is assumed to do it is always better for this prisoner to confess. If, for example, this prisoner thinks that the other one is going to confess, then by confessing himself he gets only 6 years rather than the full 10 years which he will receive if he does not confess. If, on the other hand, he assumes that the other prisoner will not confess, then by confessing he gets only 1 year and by not confessing he receives 2 years. So no matter what this prisoner assumes about the other's behaviour, it is always in his interest to confess. The same, of course, holds for the other prisoner. Therefore, assuming egoism, both will confess and receive 6 years each, but if neither confessed, they would have received only 2 years 1 First devised by A. W. Tucker, and discussed in R. Duncan Luce and Howard Raiffa, Games and Decisions (New York, 1957), pp. 94-102. 554 This content downloaded from 13.236.81.23 on Thu, 06 Apr 2023 12:05:18 UTC All use subject to https://about.jstor.org/terms each. So what seems rational from an individualistic point of view, i.e. each taking the other person's actions as given, produces a relatively worse overall result. To clarify the picture for ready reference, we reproduce below in a summary form the consequence of alternative policies for the two prisoners. The rows represent Prisoner l's policies and the columns Prisoner 2's policies. We can find the overall result by looking at the item that belongs to the appropriate row and, the appropriate column. The first figure within each pair of brackets represents the consequences for Prisoner 1 and the second those for Prisoner 2, e.g. (10, 1) represents 10 years for the first and 1 year for the second. As is clear, for Prisoner 1 the second row is better than the first, no matter which column he is in, and for Prisoner 2 the second column is better than the first, no matter which row he is in. So the self-seeking of the two will lead them to (6, 6), as a consequence of confession by each, whereas they would have been both better off at (2, 2), the consequence of neither confessing.'

It is important to recognize that even if the two prisoners are allowed to talk to each other and to conspire, this will not affect the outcome. Since both prisoners are self-seekers, and since the confession takes place separately, it will be in the interest of each to break the. contract not to confess which they might have made; and it will be in the interest of each to do this quite irrespective of whether each assumes that the other prisoner will also break his contract or not.

This conflict between what seems individually better and what seems to produce the best over-all result contains, in our view, 1 Repetitions of this game over time, under certain circumstances, may take the two to the mutually beneficial solution, but this is not always the case (Luce and Raiffa, pp. 97-102). Besides, such pure repetitions are normally not possible in social games, with which we shall be concerned in the rest of the paper. In fact, in the case of the " prisoner's dilemma " itself, pure repetitions will be difficult and many repetitions unlikely.

This content downloaded from 13.236.81.23 on Thu, 06 Apr 2023 12:05:18 UTC All use subject to https://about.jstor.org/terms the essence of Rousseau's distinction between the " will of all" and the " general will ". The " general will" of the prisoners, we can say, is to avoid confession, but each person's " particular will " is to confess. Since, in the absence of successful collusion their self-seeking will take them to a situation worse for both, what is needed is an enforceable contract between them." They would both be ready to appoint an agent who would see to it that neither of them confessed. In the absence of sanction (or, we might even say, of a Sovereign), each prisoner may be driven by rational self-seeking to break the contract which is to the common advantage of both. This gives an immediate and plausible sense to Rousseau's notion of the members of a society being " forced to be free ", and can be claimed to correspond with what Rousseau says in the Social Contract about the general will. It is not our purpose to offer yet another reinterpretation of Rousseau as either totalitarian or liberal; but it seems worthwhile to point out how Rousseau's notion can be given a valid sense by reference to the model of the prisoner's dilemma. The purpose of the general will, says Rousseau,' is the good of all, or common good: the general will " always tends to the public advantage ". What makes the general will general is not the number of citizens involved, but " the common interest by which they are united ". The divergence between the general will and the will of all arises because the individual's personal interest " may dictate a line of action quite other than that demanded by the interest of all ". The general will, though remaining unalterable, becomes subordinated to the encroachment of individual wills when " each, separating his interest from the interest of all, sees that such separation cannot be complete, yet the part he plays in the general damage 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 who seeks to gain an advantage by breaking the contract. The general will is general not only in its origins but its objects, and is " applicable to all as well as operated by all ". It tends always to equality: and all citizens being equal by virtue of the contract, " none has the right to demand that another should do what he does not do himself ". It is, finally, the legislator who must persuade the people of their true interest: the average man " finds it difficult to see what benefit he is likely to derive from the ceaseless privations which good laws will impose upon him ", so that. the good 1 The quotations which follow are taken from the translation in Social Contract (ed. Barker), World's Classics edition (London, 1947) .

This content downloaded from 13.236.81.23 on Thu, 06 Apr 2023 12:05:18 UTC All use subject to https://about.jstor.org/terms legislator is the legislator who will lead, those " to whom mere mortal prudence would ever be a stumbling block " (or who would, in other words, pursue what might well be their rational individual strategy, but at the cost of their real benefit). These remarks of Rousseau's seem to us to make it possible for an interpretation to be given,to the general will on the model of the prisoner's dilemma. This interpretation requires, however, to be distinguished from the interpretation of the general will given by Arrow in Social Choice and Individual Values. According to Arrow, " The idealistic doctrine may then be summed up by saying that each individual has two orderings, one which governs him in his everyday actions and one which would be relevant under some ideal conditions and is in some sense truer than the first ordering ".1 Our interpretation, by contrast, does not require us to impute to each person more than a single set of orderings. On our view, each person has (as in Rousseau) a single and consistent aim. The conflict between the will of all and the general will arises not because theindividual must be required to change his preference orderings, but because of the difference between the outcome of individual strategy and of enforced collusion which arises under the conditions of the non-cooperative, non-zero-sum game. We thus further assume that a substantive sense may be given to the common good (again, as in Rousseau). This is not to say that the general will enjoins a set of specific practices on specific persons. Rousseau makes clear at several points that this is not so. It does, however, mean more than the sense to which, it has been argued by Benn and Peters, the common good must be restricted, namely that all claims should have been impartially considered before legislation is passed.2 The limit which must be imposed on the meaning of the common good is rather that it loses its meaning where there is a conflict of " real " interests even in Rousseau's sense. That is to say, the common good may be taken to be substantively embodied in what the general will wiLls: but the general will does not will anything which requires that any person should be (in terms of his own preference ordering) the long-term loser, although it may, of course, require him to forego the pursuit of an individual advantage which, without enforced collusion, would leave him in the end worse off. We may say, if you like, that the general will always fulfils the conditions of Pareto optimality, although it is not suggested thereby that Pareto optimality is a sufficient criterion of justice. Whether or how far it may be useful to say that the general will wills justice will be considered in the second section of this paper.

There is, however, a difficulty which may arise evei where Pareto optimality is satisfied. Even where a set of social arrangements may be envisaged whereby all individuals would benefit, there may be a choice where one class of persons would be better off under one possible set of arrangements and another class under another. Consider, for example, a country where some people have cars with a left-hand drive and others with a right-hand drive, so that the former would prefer a rule of driving on the right and the latter of driving on the left. Which law does the general will enjoin ? It will be in the interest of both the two classes to have either of the two laws in preference to none at all, but one class will prefer one law and the other class another. We are here faced with a choice of cooperative solutions to a non-zero-sum game, and there does not appear to be any criterion by which one solution better embodies the general will (or, we may say, is more just) than the other. In other words, the appeal to the general will appears to yield weak but not strong orderings of aggregated preferences.

The problem, therefore, in such cases as these is to settle in advance the rules of a fair game by which the choice of cooperative solutions to a non-zero-sum non-cooperative game may be made. In the example offered, it could be claimed that the general will might require an appeal to the majority principle (with, perhaps, an agreement in the case of a tie to abide by the toss of a coin). Everyone could be assumed to be ready to accept a general method of resolving deadlocks which would be preferable to the pursuit by everyone of his atomistic rational strategy. The implementation of the law enacted on this basis would be consistent with the general will. In such a case, it could be said not only that the general will wills the common interest (either of two alternative laws being demonstrable as in the common interest in preference to no enforced collusion at all), but also that the general will wills the just resolution of conflicting interests. This formulation, however, directly raises the relation of the general will to the notion of justice, and the limits of the scope of the general will when applied to problems traditionally associated with the concept of social justice. It is to these questions that we turn in the following section.

This content downloaded from 13.236.81.23 on Thu, 06 Apr 2023 12:05:18 UTC All use subject to https://about.jstor.org/terms II For this purpose it will be convenient to start from the contractual model of justice as fairness as developed by John Rawls.' It is the notion of the contract which, on Rawls's model as on Rousseau's, must link any notion of the general will to that of justice, since it is by virtue of the contract that all persons are to be viewed as equally subject to the general will, and it is by reference to the contract that assessment is made of the justice of demanding that the individual -should surrender the right to act purely according to his atomistic strategy. As used by Rawls, the contract implicit in justice as fairness is not, of course, a historical contract. But it is a contract in that all parties seeking a " just " aggregation or resolution of their individual interests are to be envisaged as having made, in Rawls's phrase, a "firm commitment in advance ". This commitment is not on specific practices, but on " principles of appraisal ' mutually acknowledged by " free persons who have no authority over each other " ; and it is according to these principles that their competing claims are to be settled. This means, in effect, that every member of a society might as well have been party to a fictitious contract to abide by decisions of social policy to the extent that he would expect his own claims to be vindicated when in accordance with the principles agreed in advance. That this is the essence of justice, and that the social contract (in a modified form) does provide the model for it, seems to us to be adequately demonstrated by Rawls. It remains to consider the relation of the general will as we see it to this conception of justice.

On the model of the prisoner's dilemma, both prisoners partially share a common preference ordering-both, that is, prefer (2, 2) to (6, 6), since both wish to minimize their joint and equal sentences. Extending this, therefore, we must say that the model only applies to cases where social aims are up to a point unanimous.2 However, even where aims may be said to conflict in that people may have conflicting preferences between cooperative solutions to a non-zero-sum non-cooperative game, but where each cooperative solution is better than no cooperation at all, we have seen that the general will may will a solution acceptable 1 John Rawls, " Justice as Fairness ", Philosophical Review, lxvii (1958), 164-194. This content downloaded from 13.236.81.23 on Thu, 06 Apr 2023 12:05:18 UTC All use subject to https://about.jstor.org/terms to both parties. The real difficulty comes when we are confronted with a case where cooperation cannot make everybody better off. Take, for example, zero-sum social games-games, that is, where the gain of one person or group must be the loss of another. By our sense of the general will, a person cannot here be " forced to be free ", except where his own long-run preference o'rdering can be shown to be better realized by his acceptance of the loss in this particular case. What this means is that when the rules of a zero-sum game are to be settled then the general will may be described as enjoining the rules of a game of " fair " division to the extent that such rules can be shown to be in the long-term or CC real " interest of all the players relative to the outcome which would result without such rules. A " fair " zero-sum game, to accord with the requirements of our sense of the general will, must in this sense accord with the preference orderings of all the players. What will it mean, therefore, to say that (as Rousseau seems sometimes to imply) the general will wills social justice ? It is clear that on our analysis the justification of either enforced collusion or of " fair " rules for zero-sum social games will cover fewer cases than could be appealed to by Rawlsian justice. If justice is interpreted in Rawls's sense-and a case could perhaps be made for thus interpreting Rousseau's " true principle of equity "-then justice corresponds to such solutions to zero-sum games as accord with principles to which the players, had they met before the game under conditions of primordial equality, would have jointly agreed to. Our view of the " general will ", which rather follows Rousseau's emphasis on common interests than his implication of common principles, does not offer any way of establishing principles by which some players must accept to be losers except in so far as the acceptance of rules entailing loss accords with the players' long-term interest or preference. Interest does not mean here that terms of " fair " contracts depend purely upon the threat-advantage of individual players -an objection effectively raised by Rawls against Braithwaite's Theory of Games as a Toolfor the Moral Philosopher. But it does mean that we do not extend the general will so far as to allow persons to be " forced to be free " by the criterion of any principle to which-they could be supposed, if rational, to have been prepared to assent from the state of nature.

The initial difficulty in-fitting the notion of a " game of fair division " even to our limited sense of the general will is that there are occasions when such games as formally defined by game theory are manifestly unfair. This is the difficulty inherent in This content downloaded from 13.236.81.23 on Thu, 06 Apr 2023 12:05:18 UTC All use subject to https://about.jstor.org/terms the attempt to apply the theory of games to the resolution of conflicting claims without (as Rawls shows to be necessary) introducing some further concept of fairness. The difficulty most obviously arises when one player in a game of " fair " division is initially very rich and the other very poor. Nor, unless unanimity already exists, is it possible without a regress to make the rules of the game themselves a part of the players' desired outcome. By Rawlsian justice, we could perhaps settle the matter by establishing the principles by which rational persons would, before knowing if they would be rich or poor, have agreed to decide when a game of " fair " division was in fact appropriate. But this involves the assumption, which we do not ourselves wish to make, that there are no conflicts between the principles of, say, needs and deserts, except those dictated by a vested position.' We can, however, demonstrate that persons can be " forced to be free " in the cases where their preference ordering of social states will be better realized as a result of their willingness not to pursue their atomistic strategy. With the concept of " social justice " which follows from Rawls's model of justice and that of the "general will" as developed above, it becomes possible to make the following statements about their relationships in the context of rules that are used to compel individuals. The set of rules that satisfy the criterion of " social justice " we refer to as S, and the set of rules that conform to an unambiguous " general will" we refer to as G. Our contentions are the following:

(I) G is a sub-set of S. (II) The " complement " of G in S, i.e. the set of rules that belong to S but not to G, may be non-empty. This means that it is possible that some rules may be socially " just" without conforming to an unambiguous " general will ". (III) Because of the possibility of a genuine conflict of principles about who should gain and who should lose, it may be difficult to decide whether a rule belongs to S or not, if it does not belong to G. Statement (I) says that if a particular rule conforms to an unambiguous " general will ", it will pass Rawls's test of " social justice ". If a certain contract will lead the persons concerned to a situation where everyone would be better off than they would 1 See, on this, Brian M. Barry, " Justice and the Common Good "I, Analysis, xxxi (1961) All use subject to https://about.jstor.org/terms be if they followed atomistic strategy, then it will be correct to say that everyone, if rational (as defined by Rawls), will accept the contract in a situation of primordial equality, i.e. even without knowing his own exact position in the game. In the example of the " prisoner's dilemma ", the contract of nonconfession will conform to "social justice " from the point of view of the two prisoners. Breaking this contract, moreover, can be regarded as " unjust ", and if either prisoner tries to break the contract he may be justly " forced to be free ". Statement (II) can be understood by visualizing a case where it is known that everybody concerned would have accepted a rule in a state of primordial equality, even though the rule in practice leads to the greater detriment of somebody than would result if he were to follow his atomistic strategy. If, for example, it were known that everybody would have supported a certain system of progressive taxation in a state where they had absolutely no idea what their own pre-tax income would be, then such a system of taxation would satisfy the Rawlsian requirement of justice, even though there might not be the necessary partial unanimity of interests required for the existence of a " general will ". Statement (III) can be illustrated with the same example. Since there exist genuine conflicts of principles, e.g. between " needs " and" deserts ", it will become difficult beyond a certain point to establish what principles people would have subscribed to in a state of primordial equality. Whether, therefore, a particular person's argument is attributable to his vested interest in the situation in which he finds himself, or whether it is attributable to a principle he would (if rational) have accepted even when he had no such vested interest, may be difficult to resolve. In contrast with this, cases that fit the " general will " pose no special difficulty, since irrespective of a person's vested position, the enforceable contract is to his advantage.

We do not, therefore, wish to restrict the use of the notion of " social justice " to the cases that conform to a " general will ", but we do want to point out that in these cases there is no possibility of ambiguity in the interpretation of " social justice ", unlike the cases where the model of a " general will " cannot be applied.

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Kenneth J. Arrow, Social Choice and Individual Values (New York, 1951), pp. 82-83. 2 S. I. Benn and R. S. Peters, Social Principles and the Democratic State (London, 1959), p. 227.

Cases of this kind vary from such civic rules as not dumping rubbish on the street, to important national economic decisions. For an example of the last, see A. K. Sen, " On Optimizing the Rate of Saving ", Economic Journal, lxxi (1961), esp. pp. 487-489.