Bayesian Rationality and Social Norms Herbert Gintis∗ November 13, 2007 1 Introduction There is a glaring contrast between economics, which models social interaction as a Nash equilibrium of a game played by rational decision-makers (Mas-Colell, Whinston and Green 1995), and sociology, which models social interaction as the role-playing of individuals guided by social norms (Durkheim 1933[1902], Parsons 1967).1 The incentive compatibility requirements of Nash equilibrium are virtually ignored by sociologists, who argue that humans are prosocial by nature, and gener- ally carry out the duties and obligations associated with the social roles they assume rather than maximizing personal gain. The commonality of beliefs stressed by so- ciologists as key to social efficiency have traditionally been ignored by economists, who typically celebrate the irreducible heterogeneity of rational agents (Becker and Stigler 1977). The remarkable body of empirical evidence supporting the sociological model has made little impression on economists, who favor deriving social norms from the interaction of rational actors. The equally remarkable body of evidence supporting the rational actor model is rejected by sociologists on grounds that the standard sociological model incorporates all the strengths of the rational actor model, and in addition can deal with the socially determined nature of beliefs (Boudon 2003). The discrepancy between these two firmly held positions has been a major factor sustaining theoretical disarray in the behavioral sciences (Gintis 2007). Recent findings suggest, however, that economic and sociological theory can be rendered consistent and mutually supportive. First, epistemic game theory has ∗ Santa Fe Institute and Central European University. I am grateful for the comments of Masahiko Aoki, RobertAumann, Adam Brandenburger, Gerry Mackie,Yusuke Narita, Robert Sugden, Giacomo Sillari, Luca Tumolini, and Peter Vanderschraff. I would like to thank the John D. and Catherine T. MacArthur Foundation for financial support. 1A Bayesian rational agent maximizes expected utility, given a subjective prior over the various possible outcomes (Savage 1954, Anscombe and Aumann 1963, Kreps 1988). 1 demonstrated the centrality of common priors and common knowledge in ensuring that rational agents play a Nash equilibrium (Aumann and Brandenburger 1995, Polak 1999). Second, behavioral game theory has demonstrated that normative behavior need not, and usually does not, violate Bayesian rationality (Fehr and Gächter 2002, Andreoni and Miller 2002, Gintis, Bowles, Boyd and Fehr 2005, Gneezy 2005). Third, Aumann (1987) has shown that the natural equilibrium con- dition for game theory is the correlated equilibrium, and in this paper I will show that the most promising candidate for a correlating device—which epistemic game theory does not provide, even in principle—is the social norm. Finally, epistemic game theory has shown how a commonality of beliefs can be maintained by a set of rational actors facing asymmetric information. The need for such commonality was stressed by Durkheim (1933[1902]), who observed that the division of labor in modern society fosters a heterogeneous citizenry and an ethic of individual autonomy, yet social efficiency requires organic solidarity based on collective representations that harmonize the mental constructs of individuals.2 Never answered in the sociological literature is how a common culture can be maintained by rational actors facing asymmetric information. Aumann (1976) an- swered this question by showing that if rational agents have a common prior and if their posteriors following an event are common knowledge, then their posteriors agree as well.3 This analysis shows that collective representations can be main- tained by rational actors, so the ‘suspension of disbelief’ prima facie required by Durkheim’s organic solidarity is in fact unnecessary. If Bayesian rational agents have common prior probabilities or expectations, the receipt of asymmetric in- formation conserves the commonality of posteriors, provided these are common knowledge. However, game theory fails to supply conditions under which pos- teriors will be common knowledge, and supplies no reasons for the existence of common priors. If game theory could explain collective representations (common priors and common knowledge), even in principle, as an outcome of the interaction of rational agents, it would be plausible to view game theory as the unique foundation for all of the behavioral sciences (economics, sociology, anthropology, social psychology, and political science). While the aim of recent game-theoretic research has been exactly this, I will show that this task cannot be achieved, even in principle. The foundations of behavioral science must therefore include analytical constructs in addition to game theory, and the minimum set of human characteristics needed to 2To the extent that there are ethnic, regional, cultural, and social differences within a society, collective representations may differ across groups. Moreover, organic solidarity is obviously an equilibrium conditions that fails in periods of social change. 3Geanakoplos (1992) provides an overview of the literature inspired by Aumann’s original Agree- ment Theorem. 2 model strategic interaction must include more than Bayesian rationality. Game theory would provide a mechanism for the formation of common priors if the Harsanyi doctrine (Harsanyi 1967–68) were correct. The Harsanyi doctrine holds that rational individuals can have divergent beliefs only if they have different information. Asymmetric information, according to Harsanyi, can be modeled by assuming common prior beliefs and formally incorporating the mechanisms of informational asymmetry in the model itself. This argument may be plausible under favorable observational conditions when the events in question are natural occurrences for which causal covering laws and probabilistic frequencies exist. However, it is not plausible when the events involve the subjective priors of other agents (Morris 1995, Gul 1998). Aumann’s (1998) classic response to Gul’s (1998) critique of the common prior assumption is that the common prior assumption, “embodies a reasonable and use- ful approach to interactive decision problems, though by no means the only such approach.” (p. 929) Aumann’s position, then, is that the concordance of beliefs across individuals (common priors) is sufficiently widespread in social groups that it may be taken as a starting-point for analytical purposes, although it does not derive from principles of Bayesian rationality. While this position is pragmatically defensible as a provisional step in the development of social theory, it is clearly not the final word on the topic. Indeed, game theory is rejected by many behavioral scientists who, like Durkheim and Parsons, consider the commonality of beliefs a highly problematic yet key aspect of human sociality, and study the mechanisms that foster such commonality. We say an event E is common knowledge for agents i = 1,...,n if the each agent i knows E, each i knows that each agent j knows E, each agent k knows that each j knows that each i knows E, and so on. So, for instance, we say that common priors are common knowledge if each agent knows that each agents has the same priors, each agent knows that each agent knows that each agent has the same priors, and so on. The importance of common knowledge for interactive epistemology was first stressed by Lewis (1969) and Aumann (1976), its critical role among the conditions for correlated and Nash equilibrium being established in Aumann (1987) and Aumann and Brandenburger (1995). Aumann’s (1976) epistemological analysis of common knowledge derives col- lective representations from axioms of epistemic logic, in the form of a theorem asserting that if E is a public event, meaning that it is self-evident for each of a set of agents, then at every state ω ∈ E, E is necessarily common knowledge (precise definitions are given later). However, as we shall see, if the knowledge system is constructed without specific inter-agent epistemological assumptions, the only public event that can be shown to follow from the tautologies of the system and as- sumptions concerning the distribution of first-order knowledge, is the trivial event 3 , the whole event space. We conclude that social norms and conventions, which do assert the common- ality of mental representations, are emergent properties of human social systems that must be posited along side of Bayesian rationality to explain human strate- gic interaction. The capacity to share beliefs is also an evolved characteristic of the human brain (Premack and Woodruff 1978, Heyes 1998, Tomasello, Carpen- ter, Call, Behne and Moll 2005). This capacity doubtless arose during the gene- culture coevolutionary process that gave rise to our species (Cavalli-Sforza and Feldman 1981, Boyd and Richerson 1985, Durham 1991). Indeed, human culture is only the latest in a series of emergent transitions characteristic of the biological evolution of complexity from the most primitive microbes to the present (Maynard Smith and Szathmary 1997, Morowitz 2002). Affirming the emergent character of social norms means they have the status of fundamental principles that cannot be derived analytically from other funda- mental principles. Affording this status to social norms requires that we rethink certain basic theoretical presuppositions, because it entails denying the principle of methodological individualism,