Social Norms and Game Theory: Harmony or Discord? Cédric Paternotte and Jonathan Grose

Transcription

1 The British Journal for the Philosophy of Science Advance Access published December 21, 2012 Brit. J. Phil. Sci. 0 (2012), 1 37 Social Norms and Game Theory: Harmony or Discord? ABSTRACT Recent years have witnessed an increased number of game-theoretic approaches to social norms, which apparently share some common vocabulary and methods. We describe three major approaches of this kind (due to Binmore, Bicchieri, and Gintis), before comparing them systematically on five crucial themes: generality of the solution, preference transformation, punishment, epistemic conditions, and type of explanation. This allows us to show that these theories are, by and large, less compatible than they seem. We then argue that those three theories struggle to account for three phenomena pertaining to social norms (namely context dependence, conflicting norms, and self-evidence), with which any complete game-theoretic account should in principle be able to deal. 1 Introduction 2 Accounts 2.1 Binmore: Social norms as equilibrium selection solvers 2.2 Bicchieri: Social norms as conditional behavioural rules 2.3 Gintis: Social norms as choreographers 3 Comparing the Accounts 3.1 Games, cues, and generality of solutions 3.2 Norms and other-regarding preferences Other-regarding preferences Artificial utility functions 3.3 The role of sanctions 3.4 Getting the epistemic conditions right Gintis Bicchieri 3.5 Proximate and ultimate accounts 3.6 Taking stock 4 Challenges 4.1 The challenge of context-dependent behaviour Context-dependence: Varying the game ß The Author Published by Oxford University Press on behalf of British Society for the Philosophy of Science. All rights reserved. doi: /bjps/axs024 For Permissions, please journals.permissions@oup.com

2 2 4.2 Conflicting norms 4.3 Self-evidence 5 Conclusion 1 Introduction Social norms are ubiquitous in our daily lives, from holding a door open for someone and letting them pass to refraining from theft; from driving on the correct side of the road to the vagaries of fashion. When it comes to the economics literature, most prominent are norms of what constitutes fair division of resources. Norms appear to be complex phenomena that play a number of roles. Firstly, they allow us to make our behaviour fit with others ; that is, they allow us to coordinate our actions via conformity or adopting a particular role in an interaction. Secondly, they sometimes provide a different type of motivation, prompting us to act in ways that seem contrary to our personal interests. One may feel compelled to be polite, to follow the rules of etiquette, or to refrain from stealing even when the urge to do otherwise is present. In other words, social norms seem to have normative force that can dominate individual interests. Thirdly, and this role feeds into the first two, expectations about others play a role in indicating an appropriate course of action in a given social situation. What counts as appropriate depends both on my expectations of others behaviour and on my expectations of their expectations of me. It has been claimed that social norms are distinct from both mere conventions or habits, and legal or moral rules (Elster [1989], p. 100) on the basis that these other norms need not appeal to the social level, although, as we will see, not all theorists see any distinction here. Social norms are thus phenomena at the intersection of individual and collective behaviour. This article examines and criticizes a recent trend in social norms analysis: formal attempts, made by economically-minded philosophers and philosophicallyminded economists, to model norms by using similar game-theoretic tools. The use of game theory has recently reinvigorated the debate on social norms. Whereas in the past, theorists and philosophers focused on defining what social norms are, the emphasis has now shifted on their explanation the more concrete problem of why and when people follow them. Consequently, game theory, the extension of decision theory to situations of interaction, has been called to the rescue, and it seems that a new, unified research program has emerged. Participants in such a project share the view that game theory is our best tool with which to analyse social norms, and that they are best understood as equilibria, that is, situations in which agents actions mutually justify each other. Both of these points stem from a tradition dating back to

3 Social Norms and Game Theory 3 pioneering work by Thomas Schelling ([1960]) and David Lewis ([1969]). Indeed, it is now widely acknowledged that Hume emphasized the importance of equilibria on conventions more than two hundred and fifty years ago. Theorists embracing the game theory and social norms project share in common the aim of influencing more traditional sociological approaches to norms and promoting the use of formal models. Several questions naturally arise: can the existing game-theoretical approaches of social norms be considered as part of a general theory? Are they united by more than the very generic objectives sketched above? In particular, do they complement or even paraphrase each other? In other words, are we witnessing the dawn of a booming new approach to social norms that would clarify them by trading literary vagueness for formal, quantitative precision? Although we are mostly sympathetic towards game-theoretic approaches and think they could live up to their promises, one aim of this article is to demonstrate that their surface-level similarities obscure fundamental disagreements about how norms should be modelled. This matters because practitioners of the game theoretic approach differ between themselves on what constitutes a social norm cannot help its acceptance across disciplines. We begin by assessing the theories of three of the most vocal advocates of the game-theoretic approach namely Cristina Bicchieri ([1993], [2006], [2008]), Ken Binmore ([1998], [2005], [2010]) and Herb Gintis ([2009], [2010]), chosen because they are representatives of the variety of existing accounts. Indeed, it is enough to show that their theories are irreducible to each other in order to suggest that there is no such thing as a unified game theoretic take on social norms. 1 A second aim of the article is to assess what is currently lacking in all game-theoretic approaches. Consequently, we turn later to three problems with which any game-theoretic theory of social norms should be equipped to deal, arguing that all of these accounts appear unsatisfactory with regard to at least one of these problems and that the explanatory power of game-theoretic approaches remains limited. Section 2 presents the three authors accounts of social norms and within that section we also draw attention to the most important divergence between them. That disagreement is whether or not social norms should be modelled as affecting individuals utilities. Section 3 considers this in more detail and four other potential clashes. First we attempt to diffuse an apparent difference 1 Note that these authors consider themselves to be part of a common tradition, as witnessed by the very first sentences of Gintis ([2010]) and Binmore ([2006]). However, this does not prevent them from criticizing each other s accounts: Rather than confront the barbarians beneath our walls, we spend our time accusing each other of factual inaccuracies and obscure heresies (Binmore [2006], p. 82). However, the common points and differences we highlight have hardly been discussed so far; see the introduction to Section 3 below.

4 4 regarding cues that yield correlated equilibria. Secondly, we return to the role of utility transformations with regard to the concept of a behavioural norm. Thirdly, we look at differing approaches to social sanctioning. Fourthly, we compare the epistemic conditions required by the theories. Finally, we contrast an evolutionary approach, with its focus on repeated interactions, with rational choice theory-based approaches that stress the epistemic requirements for norm adherence, arguing that this distinction underpins some appearances of disagreement between the theories. As a further means of comparison, Section 4 raises three issues that a theory of norms should be able to accommodate. The first is how they account for context-dependent behaviour. Secondly, we look at the problem of what to do when norms conflict. None of our authors deals with this scenario and it is unclear how it would be accommodated. Finally, we examine whether these accounts can accommodate the intuition that recognizing the appropriateness of a norm in a set of circumstances itself provides a reason to follow the norm. Overall, all accounts are found lacking with regard to these problems, although not necessarily for the same reasons; in other words, they are all inadequate although in different ways. However, we do not try to adjudicate between these accounts. Rather, our main point is that independently of the idiosyncratic problems each of these accounts have, it seems it is difficult to imagine how a general game-theoretic account of social norms could be built; moreover, even if it could, this general account would have to confront several conceptual challenges. 2 Accounts In this section we describe the three accounts of social norms. This is not a detailed exegesis; we only aim to be precise enough so as to be able later to locate their points of agreement and of disagreement. Among other things, we pinpoint the roles played by preferences and epistemic considerations such as expectations and knowledge. 2.1 Binmore: Social norms as equilibrium selection solvers Ken Binmore disagrees with some of the characterization of norms presented in the introduction. For instance, for him, moral or legal norms are a subset of social norms. Similarly, the term social convention is not substantively different from a social norm, the two being used interchangeably (Binmore [2010], p. 6). 2 Basically, a norm is a means by which individuals solve the 2 Binmore s book-length presentations go beyond the analysis of norms and attempt to overhaul normative political philosophy. We draw in part on this earlier work but also on recent articles focusing specifically on norms (such as Binmore [2008], [2010]).

5 Social Norms and Game Theory 5 equilibrium selection problem in a coordination game with multiple Nash equilibria (ibid, p. 4). Binmore is particularly interested in equilibrium selection because he thinks that by far the most important strategic interactions between people are infinitely repeated games (Binmore [1998], p. 10). The folk theorem 3 then proves that any outcome in the cooperative payoff region of the stage game that pays each player at least their security level is an equilibrium of the repeated game (Myerson [1991], 7.5). Only some of these equilibria are Pareto optimal but, even if we restrict ourselves to these, the players find themselves with an equilibrium selection problem to solve and this is where Binmore s norms come in. They explain the salience of Schelling s focal-points (Schelling [1960], Chapter 3) in coordination games. Behaviour in accordance with a norm is elicited by an environmental cue that forms part of the framing of the interaction and, for Binmore, this cuing is done in a non-calculating, habitual fashion. So, how do members of a society hit upon a norm? Binmore s answer is that they follow an evolutionary process of trial and improvement. Since his approach is rooted in evolutionary game theory, he does not require players to have any expectations or beliefs about other players actions, beliefs, and expectations. To illustrate: conventions can sometimes be sustained without anyone knowing anything at all in the formal sense required by current theories of knowledge. However, this isn t to say that what the players in a game may or may not know doesn t matter, but that knowledge issues are secondary (Binmore [2008], p. 13). Presumably, when he comments that a society s pool of common knowledge its culture, provides the informational input that individual citizens need to coordinate on equilibria (Binmore [1994], p. 140), Binmore means common knowledge in a looser, non-technical sense. He stresses that there are many more norms than those of which we are typically aware, and that we tend to notice usually smooth-running coordination devices only when they fail to work. J. McKenzie Alexander s ([2007]) modelling of the evolution of, among other things, fairness norms in a structured environment exhibits a similar focus on an a-rational evolutionary process but one lacking the insistence on the primacy of repeated games. H. Peyton Young also characterizes norms in a manner similar to Binmore and adopts an evolutionary approach. His models address the evolutionary stability of norms to stochastic shocks and are technical elaborations of the norm as equilibrium approach in one particular direction (Peyton Young [1998]). Hence, we focus on Binmore as a proponent of this approach to norms. 3 In infinitely repeated games, Nash equilibria cannot be listed or exhaustively described. The folk theorem sidesteps this problem by rather characterizing the payoffs to which an equilibrium can possibly lead. See Osborne and Rubinstein ([1994], Chapter 8).

6 6 2.2 Bicchieri: Social norms as conditional behavioural rules Cristina Bicchieri s analysis of social norms is in many regards different from Binmore s. For Bicchieri, a norm is a behavioral rule known to apply in a situation and to which an individual, i, prefers to conform. The preference for conformity is conditional on the beliefs that: 1) sufficient other individuals will conform; and 2) sufficient other individuals expect i to conform (and in some cases may sanction non-conformity) (Bicchieri [2008], p. 196, Appendix 9.1). Preference for conformity comes about via a transformation of i s utility function, which converts 4 a mixed-motive game (see below) to a coordination game in which all (or, more accurately, enough) players prefer to conform. Once the players utilities have been transformed, mutual norm compliance becomes an equilibrium of the transformed game. Let us develop a few points. Social norms do not arise in coordination problems (that is, in situations with multiple equilibria), but rather in what Bicchieri calls mixed-motive situations: cases where there are conflicts of interests between agents, as well as the potential for a mutual benefit. Bicchieri does not provide any more precise definition of mixed-motive situations. So such situations may contain one equilibrium, multiple equilibria, or none at all. 5 The only additional condition is that she excludes one specific kind of situation: the problem that a social norm is solving in the first place is never a coordination problem. (Bicchieri [2006], p. 34; her emphasis). 6 When a social norm exists, it changes the agents utility functions so as to transform the mixed-motive situation into a coordination problem. Social norms can only appear when the agents utility functions have specific configurations, but they are also triggered by specific configurations of beliefs. For Bicchieri, a social norm is merely a rule; more precisely, a function that specifies the strategies with which an agent should answer to a given combination of the other agents strategies. A norm tells us what we should do, depending on what others do. However, there is more. A social norm is a rule of action that exists whenever an agent expects others to follow it and believes that others expect him to follow it as well. In other words, expectations about others and beliefs about their expectations are the elements that By this (and by the expressions payoff/utility/preference transformation ), we only mean that authors resort to mathematical functions that go from a game s material payoffs to players utilities. In particular we do not mean that a game is literally transformed into another, which would imply that the former has been replaced. In other words, we only mean that players do not always reason from the games material payoffs, which is an uncontroversial claim. We will only consider finite games and pure strategies. It is a standard result that a finite game always contains a Nash equilibrium, though not necessarily one that prescribes pure strategies; and we know of no natural norm that prescribes to randomize among strategies, or leads to probabilistic expectations about others strategies. A game can contain multiple equilibria without being a coordination game. In a coordination game, equilibria are such that all players are worse off if any of them deviates, whereas in a usual equilibrium, the condition is that only the deviating player is worse off.

7 Social Norms and Game Theory 7 will trigger the manifestation of a social norm. A norm thus is defined as a rule that produces a change in preferences, after having been triggered by (at least) an initial configuration of preferences (modelled as a mixed-motive game), an expectation about people s behaviour, and a belief about other people s expectations. When the norm is triggered, it modifies the agents utility for every result of the game as a function of the highest loss caused by anyone s deviations from the norm. My being sensitive to the norm would make me consider as less valuable the consequences of some actions that do not follow it (whether these actions are mine or others ), in proportion to the highest loss that this action may entail. 7 The triggering of a social norm is automatic rather than deliberative. It is situational cues that focus the agent s attention on one specific norm rather than another. One advantage of Bicchieri s account is that it makes the relations between norms and other social concepts explicit. For example, conventions are defined as descriptive norms: to follow them, it is enough to have expectations about other people s behaviour without considering their expectations about oneself. 8 Moreover, this is because they only apply in situations that contain a coordination problem. This makes sense, since having a strong belief about the actions of others can only motivate you to act if all actions are part of an equilibrium. 9 One last interesting feature of Bicchieri s account is that she provides reasons why an agent may comply to a norm or, equivalently, why an agent s preferences may be modified when a social norm is present. These reasons for norm compliance are the following: agents may be motivated by fear of punishment; by the desire to please others; or by recognizing that the normative expectations of others are acceptable. These do not have the same status. Fear of punishment is already implicit in the definition, since beliefs that others may punish me are explicitly mentioned. As for the desire to please others or the acceptance of their expectations, they constitute reasons to modify my More precisely, the transformation takes the form of a reduction in i s utility equal to the maximum (across all other players) payoff loss due to deviations from the norm. This loss is weighted by a parameter, k 0, that represents i s sensitivity to the relevant norm (Bicchieri [2006], p.52, p. 199). More on this later. Other concepts can also be situated with regard to social norms. Moral rules are similar to social norms except that they do not depend on expectations at all. Collective habits correspond to cases where observation of behaviour produces expectations about others but do not trigger a preference change. To take one of Bicchieri s examples, it can be a collective habit that people wear coats during the winter. My choosing to wear a coat is independent from what others do I wear a coat because it is cold. I can still expect them to wear coats, though it does not influence my choice. So tinkering with some parts of the definition of social norms allows us to readily retrieve definitions of other concepts. Again: it cannot justify the choice of the equilibrium strategy, but only motivate it. For more, some common knowledge would be necessary; see Section 3.4 on this topic.

8 8 preference. Then, the former may still be unconscious or automatic; but agents have to be aware of the latter. 2.3 Gintis: Social norms as choreographers Herb Gintis has argued recently that social norms play the role of a choreographer that leads players to play according to Nash equilibria of games enriched by a signalling system. In doing so, he presents himself explicitly as extending the work of Binmore and Bicchieri (Gintis [2010], p. 251). His toy example is of drivers attempting to coordinate their behaviour at an intersection. The norm as choreographer takes the form of traffic light signals according to which the driver presented with Go goes and the other, presented with a Stop signal, stops. Gintis does not specify a kind of situation in which social norms may appear. Indeed, they could be triggered by any kind of situation, whether it is a coordination game (as in the previous example), a mixed-motive game, or yet another type. In general, for Gintis a social norm is a function which associates with any state of the world (i.e. possible signals) a set of strategies for every agent. A social norm indicates which strategy every agent should choose depending on the observed signal, in such a way that it is rational for every agent to follow the suggestion knowing that the others are following it too. Note that though what the norm itself prescribes is publicly known, it is not necessary that the signal be so; different agents may receive different private information. For instance, they may perceive perceive the signal more or less accurately. What must be common, though, are the agents prior expectations about the signals. This common prior allows them to calculate the expected utility of every strategy and to know that the others make similar calculations. Gintis attempts to distance himself from Binmore by stressing that the norm, relying as it does on signals, is part of a game larger than the original game. The situation in which drivers obey the signals represents a Nash equilibrium of the larger game, G +, but it is not a Nash equilibrium of the base game (Gintis [2010], p. 253, 263). Like Bicchieri, Gintis s focus is on one-shot interactions and he too is interested in modelling the knowledge and rationality constraints that must be met for a player to follow the norm. Having offered this characterization of a norm, Gintis goes on to cite an extra feature. In the traffic case, it is in each player s interest to follow the norm, but Gintis argues that rational individuals may value certain moral virtues so that they voluntarily conform to a social norm in a situation where a perfectly self-regarding and amoral agent would not (Gintis [2010], p. 261). In these cases, norms may induce other-regarding preferences. Individuals then exhibit an a-normative predisposition towards conforming, (Gintis [2010], p. 261), where a is a payoff transformation (merely a constant that is added to the

9 Social Norms and Game Theory 9 payoffs of the relevant action). This change is private, that is, unknown to the other agents. In other words, agents have quantitative normative predispositions: tendencies to follow the norm as long as payoffs from the rival strategies are not too much better. As a consequence, Gintis appears to credit norms with two roles. The first is a Binmore-style coordination device, but outside the repeated game scenario, and the second is something that players have a social preference towards. Why is the latter role needed? This is because Gintis (unlike Bicchieri and Binmore) does not characterize the situations in which norms can arise there are no constraints on the agents payoff structure. However, nothing guarantees the existence of an adequate correlated equilibrium in any given game. So increasing the payoffs for the action that the norm prescribes can make the choice of this action rational. 3 Comparing the Accounts Three main formal accounts of social norms have been introduced. We now focus on several topics that run through them and, more generally, through the game-theoretic literature on social norms, which will help us compare and classify them. The list of points of comparison is by no means exhaustive. However, these themes are so central to social norms that finding that the accounts disagree on some of them is sufficient to cast doubt on their compatibility. We emphasize that unclarity on these themes pervade the entire game-theoretic discourse on social norms, not only these three particular accounts but general points can be made in a less abstract way by considering specific theories. However, and importantly, that different accounts, such as those we discuss, focus on different aspects of social norms does not automatically mean that they are fully incompatible. As the following sections illustrate, apparent disagreements can be superficial whereas more fundamental ones can be obscured. Note that these authors have already discussed and criticized each other s accounts in print Games, cues, and generality of solutions The first difference in the three accounts concerns the type of situation they can be applied to. When do social norms arise in the first place? It may seem 10 For instance, Gintis ([2006]) has criticized Binmore s heavy reliance on the folk theorem, which the latter answered (Binmore [2006]); Gintis ([2010]) and also commented on Bicchieri s account. However, for the most part, the topics they focus on and the arguments they raise concern the empirical relevance of their theories; in contrast, we are interested in their conceptual compatibility, which has hardly, if ever, been discussed in the literature so far.

10 10 surprising that several rather detailed analyses disagree on this basic point. However, most of the disagreement appears to be superficial. As we have seen, Binmore takes social norms to be synonymous with conventions. That is, they apply to problems of coordination: problems containing multiple equilibria. 11 Bicchieri s social norms can be triggered in any mixed-motive game, containing conflict of interest and a potential socially beneficial outcome that need not be (and often is not) an equilibrium. There is no constraint on the type of games to which Gintis s analysis applies. So it seems that, by going from Binmore to Bicchieri and then to Gintis, one goes from the most local account to the most general one. However, this view is misguided, for the games they mention do not have the same status. What must be compared are the games faced by the individuals. Binmore s situation of multiple equilibria corresponds to the potentially infinite repetition of a more basic game; it is the latter that models the way in which individuals interact. Binmore does not provide constraints regarding this simpler game, any more than Gintis does. In that regard, the two accounts are equally general. Bicchieri s solution seems more specific since it is limited to mixed-motive games, but their definition is so vague that they seem to correspond to an extremely wide class of games. Demanding that a game contains conflict to some degree can be taken as meaning that different players payoffs are not perfectly correlated. Requiring the presence of a mutual benefit can be translated as requiring that there be a Pareto-dominant payoff profile, 12 or a Pareto-undominated one, or even one which provides a better average payoff, and so on. In other words, as long as there is no clear-cut definition of mutual benefit and of degree of conflict, Bicchieri s account can be viewed as almost general. Contrary to what appearances suggest, the set of games to which the analyses of our three authors can be applied is extremely large, indeed almost the set of all possible games. Now there seems to be no obvious way to assess the generality of Gintis s and of Binmore s solutions. This is because the set of payoffs profiles, respectively attainable by a Nash equilibrium of the infinitely repeated game, 13 and by a correlated equilibrium of the basic game 14 are not directly comparable in particular none is included in the other. No account appears to be able to explain more potential outcomes of the game than the other. 11 For some simple examples of problems of coordination, see the first pages of Lewis [1969]. 12 That is, a payoff profile for which all players are better off than for another payoff profile. 13 Basically, this is equal to, or contained in, the set of enforceable payoffs, or payoffs superior to the minmax ones (Osborne and Rubinstein [1994], pp. 144). For an illustration, see (Binmore [1998], p. 111). 14 The set of correlated equilibria s possible payoffs is a convex hull, contained in the convex hull of the payoff profiles of the game and containing the convex hull of its Nash equilibria payoff profiles (Osborne and Rubinstein [1994], pp. 45 6).

11 Social Norms and Game Theory 11 Still, the way in which Gintis and Binmore build their solutions is actually strikingly similar, though the latter does not agree explicitly on this point. According to his own presentation, Gintis s norms, when acting in their coordinating capacity, do so outside the original game and he highlights this as a departure from Binmore s account. Note that, in outlining Binmore s approach, nothing was said about the way in which norm-guided behaviour is cued and Binmore himself is often silent on this for reasons that will be discussed below. However, a brief comment by Binmore suggests that this part of Gintis s analysis of norms merely explicates an important way in which norms can be cued: The cooperative payoff region of a game, G, is relevant when it is necessary to consider the kind of sharing that is accomplished by deciding who gets what by tossing a coin[...] When such sharing is possible, it makes sense to replace G by a larger game, H, which is identical to G except that it is preceded by a publicly observable random event (Binmore [1998], p. 300). Binmore s game, H, appears to be precisely Gintis s G +, 15 and so it appears that, on this point, the two agree that a choreographer can cue a norm. 16 So Binmore and Gintis both build larger games to cater for the existence of signals (the origin of which is left unexplained). Now there are conflicting reasons why one account should be more general than the other. Since Binmore considers repetitions of the larger game, he can justify any Nash equilibrium of the basic game but also any linear combination of Nash equilibria payoffs, 17 though Gintis cannot. On the other hand, we have seen that Gintis explicitly allows utility transformations of the game (expressing the normative predispositions of the players), so that what is extended is not the original game but a variant of it. This seems more general than Binmore s solution, but Binmore can always answer that the original game, G, already takes those modified utilities into account (see the next section for an analysis of the issue of preferences transformations). All in all, there is no definitive answer as to which of the three accounts is more general, but Bicchieri s and Gintis s accounts appear, on the face of it, to complement Binmore s, in the sense that they may be seen as describing what goes on (signals, utility transformations) before the game is repeated. We will return to this point of view in Section Reminder: G + is the larger game generated by combining the original game and the signals. 16 More precisely, Binmore uses the well-known fact that the correlated equilibria of a game preceded by a publicly observed signal are Nash equilibria of an extension of the game. So by extending the game, Binmore s method of Nash equilibrium selection can be applied to Gintis s games: [... ] if we can prove a folk theorem [...] in a sufficiently large class of games, then we automatically include the case in which the players use a public randomizing device to coordinate their behavior (Binmore [1998], p. 300). 17 Nash equilibria are enforceable, as well as any payoff profile that is a linear combination of their associated payoffs.

12 Norms and other-regarding preferences Other-regarding preferences The question of other-regarding preferences is a point on which Binmore certainly parts ways with Gintis and Bicchieri. Gintis s account of a-normative predispositions leading to norm compliance begins by positing other-regarding preferences and he also talks of individuals internalising norms (Gintis [2010], p. 261; Gintis [2003]); similarly with Bicchieri s sensitivity to norms factor, k. In both cases, the presence of a norm is linked to utility functions that differ from the game s payoffs. However, Bicchieri and Gintis s other-regarding preferences are of different scope. Gintis s a-normative predisposition expresses an individual s general tendency to follow norms, whereas Bicchieri s norm sensitivity parameter depends on the situation at hand. In other words, Gintis s other-regarding preferences are defined globally, but locally for Bicchieri. 18 Bicchieri defends her version by mentioning that there is much evidence that we are conditional followers of norms (Bicchieri [2010], p. 308). This alone does not suffice to completely dismiss Gintis s account though because the same other-regarding utility may well lead to norm-following behaviour in one situation but not in another (for instance, when the stakes are very different). 19 Nonetheless, so far Bicchieri has found more experimental evidence in favour of her view than Gintis has in favour of his. 20 Binmore does not deny the existence of other-regarding preferences although he is sceptical about the strength with which they tend to be held and how widely they are directed (Binmore [2005], 7.1). Gintis and Bicchieri build other regarding preferences or transformed utility functions into their accounts of norms. In contrast, Binmore insists that all otherregarding preferences be built into the payoff structure of the game before the analysis in terms of norm evolution begins. 21 This methodological insistence reflects a commitment to revealed preference theory, according to which preferences are inferred from choices. Binmore s norms are coordinating devices but they are not allowed to play a role in preference formation unless one 18 This does not mean that for Gintis the agents always have the same preferences, but that for a given agent, the function that links a game s material payoffs to the agent s utility has a fixed form. In contrast, Bicchieri s k parameter is explicitly allowed to vary across different norms. 19 This also does not entail that Gintis cannot account for framing effects; that agents have constant sensitivity to norms once triggered does not prevent different norms to be triggered in different contexts. 20 See for instance the various experiments she mentions and interprets (Bicchieri [2006], Ch. 4; Bicchieri [2010]). 21 Our methodology remains unchanged whether our players are Attila the Hun or St Francis of Assisi. We simply recognize that they have different tastes by writing different numbers in their payoff matrices (Binmore and Shaked [2010], p. 98).

13 Social Norms and Game Theory 13 means preference in a very loose sense. Deviating from his orthodox usage, Binmore notes that, on the idea that people like to play a norm, There is, of course, a tautological sense in which this is true. I like driving on the left in London because I am less likely to have an accident [...]I like driving on the right in New York for the same reason. But all that is being said here is that it is optimal for me to play my part in whatever Nash equilibrium the local social norm selects. (Binmore [2010], p. 5) Bicchieri and Gintis s approach feeds off our intuition that social norms tug on our utilities, but for Binmore that tug is illusory since social preferences would be built in before the norm analysis. More precisely, when it comes to preferences for norms, Binmore s quote refers only to preferences for actions when the actions of others are held fixed. Knowing that most other people drive on the left, I prefer to conform because this is optimal in terms of smooth driving and accident avoidance. In this sense, preference and choice are equivalent: to prefer an action amounts to being disposed to choose it. However, following orthodoxy, Bicchieri, Gintis, and (aside from the above quote) Binmore consider preferences for outcomes (combinations of actions), that is, preferences that motivate the choice of actions but are not conceptually reducible to them. There is virtue in restricting norms in a formal account to just one role in Binmore s case that of coordination even if it fails to encompass some of what we might, like Gintis and Bicchieri, want to cover. The coordinating role is a different phenomenon from the postulation of preference or utility transformations, and it makes sense for the two to be kept apart in formal modelling in order to avoid ambiguity and the risk of authors talking past each other. One specific reason for this is that other-regarding preferences can transform a game from one in which there is a unique Nash equilibrium to one that presents an equilibrium selection problem. In that case, one of the proposed roles for a norm produces the need for a norm in the alternative role Artificial utility functions Another of Binmore s arguments is that as long as the agents choices of actions are not too inconsistent, it will always be possible to concoct a specific utility function that justifies them 22 which does not mean that agents actually reason from this specific utility function. So churning out utility functions in order to account for problematic behaviour may actually not be very interesting and, in fact, could be positively misleading. To borrow an example from 22 For instance: All behaviour can be described in terms of maximising a utility function, provided only that whatever is behind the behavior causes the subject to act consistently (Binmore [2010], p. 12; Binmore and Shaked [2010], Section 3).

14 14 Larry Samuelson, 23 take the case of George Akerlof s Nobel prize-winning work on asymmetric information and the market for lemons (Akerlof [1970]). The fact that the price of young but used cars is dramatically lower than that of new cars could be explained by postulating preferences for novelty. If the data were explained away in that way we would miss out on important insights on asymmetric information between buyers and sellers. 24 More generally, it is useless to try to base utility functions on some realistic parameters, as all we can hope for is that our models have a modicum of predictive power. This view appears to dismiss Bicchieri s use of her k i parameter, which represents the agents sensitivity to norms (Bicchieri [2006], p. 52), as well as Gintis s a-normative dispositions, not only because they are proximate explanations, but because they purport to be realistic ones. However, it would be a mistake to reduce Bicchieri and Gintis s analyses to merely relying on fancy utility functions. Binmore s attacks are directed at results such as Fehr and Schmidt s ([1999]), whose sole aim is to accommodate experimental data by way of utility functions. But as we have seen, the gist of Bicchieri and Gintis s approaches lies somewhere else. Gintis only needs social preferences to complement his correlated equilibria solution so that it extends to all types of games. Bicchieri needs social preferences to ensure that mixed-motive games can be transformed into coordination problems, that is, in situations where agents epistemic states give them good reasons to choose the equilibrium. 25 In other words, for Gintis as well as for Bicchieri, utility transformation is only a means to an end, but neither of them emphasizes it as primary, which should make Binmore feel more sympathetic towards their accounts. 3.3 The role of sanctions Punishment is an integral part of the naive concept of social norms. In many contexts, it seems that we follow a norm for instance, when we respect eating manners or refrain from pushing in a queue partly in order to avoid sanctions, be it mild ones such as glances of disapproval. In this section, we show that our three approaches differ in the way in which they model sanctions. Peyton Young ([2008]) mentions three ways in which social norms can be sustained. Firstly, in coordination problems where any change in behaviour harms oneself as well as others, none has any incentive to deviate. Secondly, 23 Given during his talk The Evolution of Decision and Experienced Utility at the Paris EVO-ECO workshop, April 12th, Sugden expresses a similar idea: Invoking preferences for acting morally can become an ad hoc strategy which insulates economic theory from falsification by eliminating its predictive power (Sugden [1998], p. 75). 25 See Section 3.4 for more on the epistemic aspects of their solutions.

15 Social Norms and Game Theory 15 the threat of punishment by others can motivate one to keep abiding by a norm. Thirdly, people can internalize norms, that is, their preferences partially reflect a tendency to follow norms. Binmore focuses on the first option (coordination problems), while Bicchieri and Gintis choose the third (norm internalization). Hence we might expect them to ignore sanctions, and yet their accounts retain repeated mentions of punishment. The issue thus deserves some clarification. Let us start with Binmore. For him, punishment affects the payoffs of a game before one embarks on the analysis in terms of norms. More specifically, punishment is represented within the context of the repeated games that form his Game of Life : The strategies that support such efficient equilibria in repeated games work by requiring a player to punish any deviation by an opponent from the implicitly agreed equilibrium path by withdrawing cooperation (Binmore [2010], p. 8). In other words, punishment is not built into preferences; it forms some of the moves of the (repeated) game, the advantage being that different punishing strategies can be modelled explicitly. More precisely, the punishing actions are those whose combinations can prevent the payoff of a given player exceeding a certain limit, which is lower than what this player would get out of cooperation. If one player does not play his part, other players can switch to these punishing actions for as long as it takes to offset the individual benefit her deviation granted her. Importantly, the original, one-shot game need not have been explicitly built so as to allow punishment; rather, some of its plays are used to carry out punishment in the context of repeated plays. Note that the reason Binmore does not emphasize punishment is not that he considers coordination games. The problem of coordination appears in the global, repeated game, not in the simple game that is repeated. Binmore s account simultaneously relies upon punishment and can stay more or less silent about it because it is already embedded in the concept of a Nash equilibrium in a repeated game. Punishment is thus not constitutive only of norms, but of any social concept that would rely on game theoretic folk theorems. After his toy driving game example, Gintis continues we add to the social norm that [...] there is a sufficiently effective surveillance system that no driver has an incentive to disobey (Gintis [2010], p. 253). Firstly, it appears question begging to introduce exogenous sanctions since their stability cannot simply be assumed. Secondly, with sanctions built into the payoffs, it appears that we no longer require a norm in a coordinating role. If Gintis is correct that the correlating device should be modelled in terms of an expanded game, any sanctions should similarly be placed within a further expanded game and this is what Binmore allows (Binmore [1998], Section 3.3.6). Gintis admits the importance, or at least the ubiquitousness, of sanctions in social norms

16 16 and, thus, has to include them in his model. However, punishment receives no special status; moreover, no formal guarantee is given that payoffs which reflect punishments will always, or more often, lead to a satisfying correlated equilibrium. Consequently, this addition is somewhat artificial or incomplete, and poorly integrated with the signalling/correlated equilibrium core. In contrast, for Binmore punishment also has no special status but is integral to his account. Bicchieri takes punishment as an optional feature of social norms, which can be one of the reasons why agents are motivated to follow a norm: for [some norms], the possibility of sanctions is crucial to induce a preference for conformity (Bicchieri [2006], p. 15). Now as we noted above, Bicchieri s account of norm following involves an expectation that others will sanction violation but these sanctions do not appear in the general utility function that she presents in her model and so their role is somewhat hidden. More precisely, Bicchieri mentions two possible options for punishment to motivate players to follow norms. First, expectation could be a reason why agents have a preference for conformity, that is, it would trigger a preference change. This is what the above quote suggests, but it seems mistaken insofar as punishment should be represented by auxiliary payoffs of the game. This is the second option: fear of punishment in a one-shot setting is nothing other than the expectation of a potential decrease in utility (intentionally caused by the actions of others). This would require transforming the game into a larger, more complex one so as to include punishing actions. Now one may think that Bicchieri actually adopts a third option that consists in directly reflecting the effect of punishment on utility through the preference change itself. After all, the utility transformation she defends changes a mixed-motive game into a coordination problem, in which we have seen that punishment is already embedded implicitly. There is a problem though: Bicchieri s specific utility transformation cannot do this job because the deviant behaviour of any player can equally cause utility to decrease, depending on one unique parameter of norm sensitivity. However, if expected punishment influences one player, then it should be expressed formally by a larger influence of one s own deviations on a utility decrease. In other words, the sensitivity to norms parameter, if it was also taken to represent the effect of expected punishment, should be higher depending on which player failed to follow the norm 26 which is not the case in Bicchieri s model. As a general consequence, all three accounts are unsatisfactory as to the precise role of punishment. Binmore s view is consistent but incomplete; Gintis s is too vague; and no part (formal and non-formal) of Bicchieri s two-folded account leads to a satisfactory answer. 26 This point is also made, although for different reasons, in Hausman [2008].

17 Social Norms and Game Theory Getting the epistemic conditions right Epistemic (and doxastic) aspects play a major role in Bicchieri s and Gintis s accounts of social norms, although for different reasons. For Bicchieri, beliefs about the actions and beliefs of others are what can trigger a social norm. As we will see, this allows her to cater for context sensitivity, for example. Gintis s worry is different: it is because the epistemic conditions that guarantee that agents play according to a Nash equilibrium are too strong that he prefers to refer to correlated equilibria. This section examines whether the epistemic part of these accounts can fulfil the role it is expected to Gintis Two decades ago, the epistemic program started flourishing in game theory. It aimed at providing minimal epistemic conditions that, when met among a group of agents, entail that they will play according to a certain equilibrium (see Aumann and Brandenburger [1995] for a characteristic example). Gintis s argument relies on the fact that epistemic conditions required for a correlated equilibrium are weaker and more realistic than those required for a Nash equilibrium in particular, the latter requires a greater amount of common knowledge among agents. 28 Nonetheless, this line is threatened by Gintis s use of a-normative predispositions, that is, of other-regarding preferences. Since the a-normative predisposition is private, 29 an agent does not necessarily know that the others have such a predisposition, even if she has it herself (and if she does know, still ignores its precise value), so she will not take this into account when she reasons about the optimal choice. As a consequence, the agents do not act according to a correlated equilibrium anymore. If a strategy can only be rational for agents when they have an a-normative predisposition, then without it agents would not act according to the equilibrium. But if an agent ignores that others have such a predisposition, or ignores its value, then she can think that they will not act according to the equilibrium, and thus not follow the norm. For this solution to work, the quantitative normative predispositions, 27 Binmore will not be mentioned since he deems all epistemic conditions secondary. 28 More precisely, a correlated equilibrium can be guaranteed by the existence of a common prior (common probabilistic belief) among players who are Bayesian rational, that is, whose strategy maximizes their expected utility given their beliefs. A mixed (probabilistic) Nash equilibrium is guaranteed by a common prior among rational players whose beliefs are all common knowledge. Consequently, it is easier to guarantee the latter than the former since it is quite rare that the beliefs of human agents are common knowledge among them. See Gintis ([2009], Chapters 7 and 8). 29 [E]ach agent s payoff might consist of a public component that is known to the choreographer and a private component that reflects the idiosyncrasies of the agent and is unknown to the choreographer (Gintis [2010], p. 260). The a-normative predisposition corresponds to this private component.