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Peter Vanderschraaf, Strategic Justice: Convention and Problems of Balancing Divergent Interests

Robert Sugden
p. 169-177
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Peter Vanderschraaf, Strategic Justice: Convention and Problems of Balancing Divergent Interests, Oxford: Oxford University Press, 2019, 416 pages, ISBN 978-019983219-4

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Crédits : Oxford University Press

1For twenty years, Peter Vanderschraaf has been writing important papers about conventions. His new book, Strategic Justice, synthesises this body of work and displays his ability to write both as a scholarly philosopher and as a rigorous game theorist.

2As a preface to this review, I must declare an interest. I am the author of a book, The Economics of Rights, Cooperation and Welfare (1986), which deals with many of the same topics as Strategic Justice, drawing on the same classic sources. Vanderschraaf is generous in acknowledging my book as an influence on his work, while often arguing to positions that are different from mine. Perhaps prey to a natural bias of authors, I still prefer my original position on many of the points on which we differ.

3Vanderschraaf declares his objective on the first page:

[H]ere is the thesis: Justice is convention. I believe that this justice-as-convention thesis, properly developed, provides the most cogent characterization of the general theory of justice as conceived as a system of rules for mutual benefit, or justice as mutual advantage… In this book I have made my own attempt at such a proper development. (xi)

4He locates his work in the historic tradition of Hugo Grotius, Thomas Hobbes, John Locke, Samuel Pufendorf and (he says “especially”) David Hume, and in the more recent tradition of game-theoretic philosophy developed by writers such as Ken Binmore, David Gauthier, David Lewis, Brian Skyrms and me. In comparison with these other recent writers, however, the question Vanderschraaf sets out to answer is more abstract. If I have understood him correctly, his primary concern is not to formulate any specific principles of justice or to explain how such principles are in fact liable to emerge in human societies. The question he sets out to answer is: “Is a satisfactory general justice-as-convention theory possible?” (xiii). Or, as he puts it towards the end of the book, he wants to characterize the concept of justice as mutual advantage—to identify the “necessary features of any account of justice as mutual advantage” (273). Although Vanderschraaf explores many aspects of conventions along the way, the book is structured around this characterisation project. My review will be structured around that too.

5In his first chapter, Vanderschraaf sets the scene by describing five “dilemmas of interaction,” taken from philosophical and historical sources, which he then reformulates in terms of noncooperative game theory. His premise is that game theory is “the formal theory of all social interactions,” and so “provides a vehicle for philosophers to analyse justice more precisely and rigorously than in times past” (2-3). Two of these dilemmas are taken from Hume—the story of the two farmers who need each another’s help to harvest their crops, and the story of the villagers who need to work together to drain a meadow. Rousseau’s (and later Lewis’s) story of the choice between participating with another person in a deer hunt and hunting hares on one’s own provides a third dilemma. Also featured are Richard Braithwaite’s story of the pianist and the trumpet player, each of whom needs a period in which he can practice, and an episode from ancient Greek history, the siege of the Melians’ city by the Athenians. Vanderschraaf’s reconstructions of these examples are effective in illustrating the range of game theory, but I am uneasy about the confidence with which he asserts that formal game theory, in the state it happens to be now, is the theory of all social interactions, and about the presumption that this theory is always superior to the analyses of times past. Vandeschraaf’s reading of classic texts sometimes overlooks arguments which, although sound on their own terms, do not fit easily into the conceptual framework of standard game theory.

6Vanderschraaf then develops his account of the concept of convention. His aim is “to present an account of convention rich enough and precise enough to provide the foundation for a satisfactory conventionalist analysis of justice” (45). Notice again how he is looking for general characterisations of concepts, rather than theories that can explain concrete phenomena in the world. Vanderschraaf treats the accounts of convention in Hume’s Treatise of Human Nature (1739-1740) and in David Lewis’s Convention (1969) as seminal contributions on which he can build a more formal analysis.

7The basic idea in Vanderschraaf’s account is that a convention corresponds with one of two or more correlated equilibria in a noncooperative game. A correlated equilibrium is essentially the same as a Nash equilibrium, but with the additional assumption that all the players in the relevant game observe the same realisation of some random device and so are able to coordinate their strategies if they so choose. For example, consider a Battle of the Sexes game in which Arthur and Betty both want to meet at a restaurant, but Arthur would prefer the Italian restaurant and Betty would prefer the Vietnamese. If they both choose the strategy ‘Italian if the sun is shining, Vietnamese if not’ (IV), this is a correlated equilibrium. If Alice and Betty have common knowledge that this strategy and its opposite (VI) are correlated equilibria, and have common knowledge that each of them expects the other to follow IV, then IV is the incumbent convention for them (81). Vanderschraaf extends this theoretical analysis to games that are played repeatedly (and not necessarily anonymously) by players drawn from a larger population. In doing so, he makes technical advances beyond Lewis’s account. But some of Hume’s and Lewis’s most important ideas are lost in the reduction of “convention” to “correlated equilibrium.”

8If orthodox game theory really were the theory of social interaction, Vanderschraaf would be entitled to assume that all coherent ideas about social interaction can be expressed in terms of that theory. But the truth is that game theory in general, and orthodox game theory still more so, is just a model of social interaction; like all models, it leaves out many features of the world that it tries to represent. Of course, there is a generic sense in which Hume and Lewis are game theorists: indeed, they are among the all-time greats of the field. But their deviations from the current orthodoxy are not all errors or eccentricities. For example, they both recognise that rational play in games can depend on common understandings about how strategies are labelled—the idea that Thomas Schelling develops in The Strategy of Conflict (1960). The conceptual framework of orthodox game theory does not recognise labels. In defining convention as correlated equilibrium, Vanderschraaf is committing himself to this particular framework. I think that this commitment sometimes leads him astray, but I will leave this issue aside.

9In Hume’s and Lewis’s accounts, a convention is mutually beneficial. “On balancing the account” (as Hume puts it), each individual gains from everyone’s conformity to a convention (Treatise, 497). Lewis makes it part of the definition of a convention that, conditional on her own conformity, each individual prefers that other individuals conform too (Convention, 36-42). Not all correlated equilibria have this property. For example, consider a Hawk-Dove game in which two individuals are in conflict over the possession of some valuable resource. The ‘Hawk’ strategy is to demand the whole resource and fight if necessary; the ‘Dove’ strategy is to propose an equal split but to accept nothing if the other player threatens to fight. Hawk is the best response to Dove, but Dove is the best response to Hawk. Suppose that every game is played between a man and a woman, and that it is common knowledge that men are expected to play Hawk and women to play Dove. This is a correlated equilibrium but not a Lewisian convention: conditional on playing Dove, women would prefer men to play Dove rather than Hawk.

10Mutual benefit is important for Hume’s and Lewis’s answers to a question that Vanderschraaf does not address. In Hume’s words, the question is “why we annex the idea of virtue to justice, and of vice to injustice” (Treatise, 498). Hume treats this as an empirical question, and gives an answer based on hypotheses about human psychology. It is crucial for this argument that the rules of justice are mutually beneficial. Lewis offers a similar explanation of why conventions are also norms—that is, regularities to which people believe they ought to conform (Convention, 97). In contrast, Vanderschraaf’s conventions are merely regularities to which each individual prefers to conform, conditional on other people’s conformity. When, later in the book, Vanderschraaf discusses conventions of justice, he treats these as norms, but he does not look for any psychological explanation of their (perceived) normativity.

11Vanderschraaf introduces mutual benefit through the medium of the “Circumstances of Justice.” This is another idea that derives from Hume. In the Treatise, Hume says: “’tis only from the selfishness and confin’d generosity of men, along with the scanty provision nature has made for his wants, that justice derives its origin” (495). Expanding on this idea in his Enquiries (1777), he says:

Reverse, in any considerable circumstance, the condition of men: Produce extreme abundance or extreme necessity: Implant in the human breast perfect moderation and humanity, or perfect rapaciousness and malice: By rendering justice totally useless, you totally destroy its essence, and suspend its obligation on mankind. (188)

12The Circumstances of Justice play a significant role in John Rawls’s Theory of Justice (1971, 126-130). Vanderschraaf calls Rawls’s formulation of this idea the “Standard Account” in modern philosophy, claiming that it differs from Hume’s. But it seems to me that Rawls’s essential idea—that individuals’ interests are not perfectly aligned, but there are opportunities for mutually beneficial cooperation—is much the same as Hume’s. According to Rawls, the Circumstances of Justice are “the normal conditions under which human cooperation is both possible and necessary.”

13Vanderschraaf claims to identify a fatal flaw in the Standard Account. He argues at some length that the Circumstances of Justice apply to the people in Hobbes’s State of Nature. That seems obviously true. From this, Vanderschraaf concludes: “How could people be in conditions in which justice is ‘both possible and necessary’ if they are also doomed to a war of all against all that precludes justice?... This shows that the Standard Account circumstances are not sufficient conditions for justice” (86). I do not understand this objection. The obvious answer to Vanderschraaf’s question is that Hobbes’s State of Nature is a society in which, as a matter of empirical fact, people’s behaviour is not constrained by rules of justice. Each person acts in his own interests, in the light of correct beliefs about other people’s behaviour. The results are dire. Everyone would be better off if all were constrained by rules of justice. Hobbes tells them how to achieve this result—by agreeing to appoint a sovereign. So justice is possible. To use Hume’s word, justice would certainly be useful to these people. It is also necessary in the ordinary sense of the word: it is something that the inhabitants of the State of Nature desperately need. Arguably, it is necessary in another sense too. In the world as it really is, a society that had the features of the State of Nature would probably fail to reproduce itself: it would collapse in the face of natural calamities that require collective action, or be destroyed or taken over by better-organised societies. What more is Vanderschraaf asking for?

14Working within the framework of orthodox game theory, Vanderschraaf presents his own definition of the “Generic Circumstances of Justice” as a property of a game. Consider any given game. As a first step, Vanderschraaf defines the game’s baseline outcomes. This definition is complicated and its rationale is not fully explained, but the intuitive idea is that each baseline outcome corresponds with some “‘free-for-all’ strategy profile that results if each party tries to gain at the expense of the others” (112). I take it that Vanderschraaf is using “free-for-all” in the sense that might be applied to a drunken brawl. A baseline outcome is something that might happen if people acted without any restraint at all, not even with the restraints of prudence and caution that Hobbes attributes to the inhabitants of the State of Nature. A strategy profile creates a cooperative surplus if it leads to an outcome that everyone prefers to every baseline outcome—in other words, if it is unambiguously better for everyone than free-for-all interaction. This is Vanderschraaf’s concept of mutual benefit. Vanderschraaf defines a game as satisfying the Generic Circumstances of Justice if there are at least two distinct strategy profiles such that (i) each of these profiles is a correlated equilibrium, (ii) each of these profiles generates a cooperative surplus, and (iii) players’ preferences over these profiles are not perfectly aligned.

15For Vanderschraaf, the Circumstances of Justice are not (as they are for Hume and Rawls) the normal conditions of human life that give justice a point; they are the basic features of his characterisation of justice as mutual advantage. Interpreting these conditions, he says: “The Generic Circumstances of Justice summarize the structure of any social interaction that could be regulated by a norm, or set of norms, such that one who violates any of these norms is liable to be accused of committing an injustice;” they are “background conditions that effectively set certain formal constraints upon norms of justice” (110-111; 116). The implication is that there are three such constraints. First (which I will call J1), a norm of justice must specify a correlated equilibrium. Thus, each individual must have a preference-based reason to follow the norm, provided that others do so too. Second (J2), if everyone follows the norm, each individual must do better than in any free-for-all baseline outcome. Third (J3), there must be at least two ways of satisfying the first two conditions, such that individuals have conflicting preferences between them.

16J3 is puzzling. In effect, it says that any norm of justice must also be a convention. But why? One might argue that if a norm of justice is to have a point, there must be some credible state of affairs in which people fail to conform to it. But, in Vanderschraaf’s theory, there clearly is such a state of affairs—free-for-all interaction. If there is a unique correlated equilibrium that everyone prefers to every free-for-all outcome, there is surely room for a norm that directs people to that equilibrium. Conversely, if one is creating a theory only for rational agents, and if one makes the questionable assumption that rational individuals always reach some correlated equilibrium, then the baseline should be a correlated equilibrium, not a free-for-all. (Hobbes’s State of Nature seems more like a Nash equilibrium than a free-for-all.)

17One might also ask whether J2 is strong enough to capture the idea of justice as mutual advantage. Take my example of the Hawk-Dove convention according to which men always play Hawk and women always play Dove. If everyone follows this convention, everyone does better than in what Vanderschraaf would classify as an unrestrained free-for-all. Nevertheless, men act just as if they were unrestrained; men and women both benefit from the women’s restraint. This is not mutual advantage.

18Vanderschraaf recognises this problem, but does not want to say, as Gauthier does in Morals by Agreement (1986), that a person can have a legitimate claim on a cooperative surplus only if she has contributed to its production (Gauthier goes further, requiring that each individual’s share of a cooperative surplus is proportional to her contribution). Vanderschraaf raises what he calls the “Vulnerability Objection” to any theory—Hume’s as well as Gauthier’s—that takes “mutual advantage” literally. Vanderschraaf defines members of society as vulnerable if they are “unable through their own efforts to contribute to the cooperative surplus” (280). Given Vanderschraaf’s definition of “cooperative surplus,” a person is vulnerable only if there is no way by which she could benefit others by restraining herself from trying (however imprudently—we are talking about free-for-all interaction) to benefit herself. Vanderschraaf’s examples of vulnerable people include very young children, the seriously ill and the demented old. He takes it to be uncontroversial that if a putative theory of justice as mutual advantage “denies the vulnerable any of its advantages,” then that theory is “no theory of justice at all.” It is (he says) a commonly held considered judgement that “the vulnerable are not to be denied the benefits of justice merely because they are vulnerable” (281; 283).

19Vanderschraaf presents an interesting model of a norm that satisfies his three conditions and protects the vulnerable. The game to which the norm applies is a variant of the “Mutual-Aid Game” discussed in my Economics of Rights, Cooperation and Welfare. The Mutual-Aid Game is played repeatedly within a community. In each period, each individual has a small probability of needing assistance; each other individual can choose whether to contribute to providing that assistance. I show that there is a self-enforcing norm by which assistance is provided to anyone who is both in need and in “good standing.” An individual achieves and retains good standing by contributing when the norm requires this. This norm satisfies J1-J3. It is also mutually beneficial: everyone is both a contributor and a potential beneficiary. Vanderschraaf’s insight is that if a not-too-large number of individuals (the vulnerable) were always in need and never able to contribute, these individuals could be deemed to have permanent good standing and the aid-giving norm would still satisfy J1-J3.

20In the final chapter, Vanderschraaf completes his characterisation project by stating what he claims are “necessary conditions for justice as mutual advantage.” This chapter has a rather strange structure. The first five of its six substantive sections build naturally on previous arguments in the book, and end with a coherent characterisation of justice as mutual advantage in terms of four conditions (M1-M4). Then, in the final section, Vanderschraaf points out that this characterisation imposes only very weak constraints on the rules of justice. Many of the rules of justice that fit this characterisation would allow extreme social inequalities. There are, he says, “too many equilibria;” a theory of justice needs “some criteria for narrowing down the set of equilibria … so that only the genuinely just equilibria remain” (308). Vanderschraaf’s readers might reasonably feel let down. They have followed him over more than three hundred pages of intricate argument about the implications of his favoured philosophical conception of justice. Now they are told that what has been missing all along is a criterion of “genuine” justice. Out of the blue, Vanderschraaf proposes an additional condition (M5 or “Baseline Consistency”) that he “believe[s] does the job” (311). His entire discussion of this criterion takes up less than seven pages, and provides no real explanation of why the criterion is one of genuine justice, or how it is compatible with the concept of justice as mutual advantage. I will treat this final section as an afterthought and concentrate on what seems to me to be the main line of Vanderschraaf’s argument.

21It is a premise for Vanderschraaf’s conditions M1-M4 that the members of society are in the Generic Circumstances of Justice. A system of justice as mutual advantage is a set of rules that individuals are normatively required to follow; the rules provide a “remedy” for the “predicament” posed by those Circumstances (274). Three of these conditions (‘M2,’ ‘M3’ and ‘M4’) effectively restate the constraints J1-J3, but are applied only to the non-vulnerable. An additional condition (‘M1’) requires each non-vulnerable individual to restrain her pursuit of her own interests ‘to some extent’ in order to advance the interests of other individuals. This rules out the possibility that some non-vulnerable individuals contribute nothing at all to the cooperative surplus. Thus, these conditions require that the system of justice is minimally mutually advantageous among the non-vulnerable.

22M1-M4 do not require that, if the non-vulnerable follow the rules of justice, then the vulnerable benefit. These conditions merely allow this possibility. As in Vanderschraaf’s variant of the Mutual-Aid Game, a system of justice that is mutually advantageous for the non-vulnerable may also require that benefits are provided for the vulnerable; but equally, it may not. Given what Vanderschraaf has said about the Vulnerability Objection, this might seem surprising. But what more could one expect? If justice is understood solely in terms of mutual advantage, how can anyone have a claim in justice on a cooperative surplus to which they have made no contribution?

23Nevertheless, an advocate of justice as mutual advantage might still say that the vulnerable ought not to be denied the benefits of justice. Think of the Mutual-Aid Game. A society that upholds justice as mutual advantage can create a just institution by which the non-vulnerable find it in their interests to provide assistance to one another. The principal benefit of this institution is that each non-vulnerable person is assured of assistance if and when need arises. It makes perfect moral sense to say that the vulnerable ought not to be excluded from this benefit just because they are unable to contribute; and Vanderschraaf has shown how to adapt the institution so as to achieve this desirable end. This “ought not” is not a dictate of justice as mutual advantage, but it is perhaps (as Hume puts it when discussing an analogous case) a dictate of “the laws of humanity” (Enquiries, 190). Ultimately, what matters is whether the non-vulnerable recognise the normative force of this prohibition, not what moral category it falls into.

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Gauthier, David. 1986. Morals by Agreement. Oxford: Oxford University Press.

Hume, David. [1739-40] 1978. A Treatise of Human Nature. Oxford: Oxford University Press.

Hume, David. [1777] 1975. Enquiries concerning Human Understanding and concerning the Principles of Morals. Oxford: Oxford University Press.

Lewis, David. 1969. Convention: A Philosophical Study. Cambridge, MA: Harvard University Press.

Rawls, John. 1971. A Theory of Justice. Cambridge, MA: Harvard University Press.

Schelling, Thomas. 1960. The Strategy of Conflict. Cambridge, MA: Harvard University Press.

Sugden, Robert. 1986. The Economics of Rights, Cooperation and Welfare. Oxford: Blackwell.

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Robert Sugden, « Peter Vanderschraaf, Strategic Justice: Convention and Problems of Balancing Divergent Interests », Œconomia, 9-1 | 2019, 169-177.

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Robert Sugden, « Peter Vanderschraaf, Strategic Justice: Convention and Problems of Balancing Divergent Interests », Œconomia [En ligne], 9-1 | 2019, mis en ligne le 01 mars 2019, consulté le 27 juin 2019. URL :

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Robert Sugden

University of East Anglia,

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