Grose & Paternotte’s Social Norms

Grose, Jonathan, and Cedric Paternotte. 2013. Social Norms: Repeated Interactions, Punishment, and Context DependencePublic Reason 5 (1): 19-30.


We argue that recent game theoretic approaches to social norms differ on some fundamental issues, our focus being on recent accounts by Ken Binmore and Cristina Bicchieri. After a brief introduction, we present the deepest cause for their disagreement, namely whether the action of norms should be modelled as a one-shot game, the option favoured by Bicchieri, or by a repeated game, as Binmore does. Although these choices appear to leave room for the two accounts to be complementary, we then argue that this is not possible. First, differing attitudes to modelling punishment, a central feature of all informal work on social norms, prevent any straightforward integration of the two theories. Second, the solution cannot consist in merely choosing between the two accounts, as they both fail to deal with the way in which triggered norms depend on context, in static as well as diachronic frameworks.


When it comes to different modelling approaches, the recent literature provides two presentations of social norms. Ken Binmore outlines a model that frames social norms in terms of repeated interactions (1994, 1998, 2005). In contrast, Cristina Bicchieri’s model is of one-shot interactions and involves utility transformations, triggered by social context that make norm conformity the rational behaviour (2006, 2008). Both of these authors repeatedly refer to each other’s work as belonging to a common programme but, we argue, their difference on the question of repeated interaction models makes integration of their approaches problematic.


When it comes to reasons for this preference transformation when a norm is triggered, Bicchieri makes a number of suggestions. Agents may be motivated by the desire to please others, recognition that others normative expectations are reasonable and, most importantly for our purposes, fear of punishment (Bicchieri 2006, 29).

The “folk theorem” of repeated game theory demonstrates that indefinitely repeated games
have multiple equilibria, that is, multiple different ways in which actions can be mutually appropriate, and thus players face an equilibrium selection problem.
(Myerson 1991, §7.5) For Binmore, social norms solve this problem by making a particular behaviour salient in a particular context. Our shared cultural heritage is what allows us to coordinate on an equilibrium. Notice that in the case of Bicchieri’s (and Gintis’s) norms, the coordination role of norms need not arise because the utility transformation makes norm-conformity the one rational outcome.

Integration would take the form of the Bicchieri’s games forming the base games for Binmore’s repeated interactions. However, we first argue, at least on the important issue of punishment, the different accounts cannot be straightforwardly integrated. We then show that they both fail to adequately describe how context-dependent social norms are, for related but different reasons.


[While] it is clear that, according to Bicchieri, sanctions play a role in transforming utilities, it is unclear precisely in what way this is cashed out in the model.
A second formalization of sanctioning is to expand the game to make the punishment action an explicit move. This has the advantage of prompting the modeller to pay attention to the possible consequences of punishing for the punisher. In particular, punishing is very often taken to involve paying costs oneself. We will not expand further on this option since it is not one taken by either Bicchieri, Gintis or Binmore, to whose repeated interaction model we now move.

A strength of interpreting punishment in terms of strategies in repeated games is that it naturally makes explicit that punishment can take many forms in terms of its duration and by what it is provoked. … However … there is nothing distinctive about punishing as represented by a repeated game strategy compared to any other behaviour that is conditional on ones partner’s actions in previous rounds of play.


[When] during a repeated interaction should we expect a social norm to be stable and when unstable?

For Bicchieri, a social norm is cued in certain situations by the agents’ beliefs about others’ behaviour and expectations. As seen in section two, this triggers a change in preference that makes it rational for agents to choose certain actions. For Binmore, whether a norm is triggered depends on the similarity of the situation with a known one. Agents’ actions are not explained by a preference change, but by their behaving as they are used to in a similar situation. In other words, when facing new situations in the laboratory, agents either conserve preferences or habits from their outside life (Woodward 2008).

… The term “context” refers to anything that cannot be expressed by games’ parameters. …

In both approaches, similarity between games may cue a social norm and resulting expectations influence the agents’ behaviour. [The] main difference lies in the link between the base game and the game that agents are actually playing. … In both cases, the characteristics of game A partly influence the game that players are really playing; only the way to determine the latter from the former varies.


When agents interact only once, context-dependence is not deeply problematic, as it all depends on which expectations agents have or which real-life situations they deem similar to the one at hand. Surely, this makes social norm following behaviour hard to predict, as a theorist would need to know all possibly related real-life situations and all expectations linked to a social norm in a given population. Still, the analyses provided by both one-shot and repeated interaction accounts are clear. What happens to a social norm when the same game gets repeated? What makes agents keep sticking to it or start following another one along the way? On this point, the two accounts described above start to differ significantly, even if none of them provide a satisfactory answer.

[Whenever] agents do not play according to a finitely repeated game’s equilibrium, such as when their contributions increase in the last period, one can always say that they are still behaving as if in real life, when it is hardly ever sure that an interaction will not be repeated some time in the future. Any behaviour in the last repetition could thus be seen as part of an equilibrium of the infinitely repeated game. Even if players perfectly understand that the numbers of repetitions in an experiment is finite, they might be behaving partly intuitively, based on the similarity between laboratory and real-life situations. Put differently, to be satisfying the explanation should tell us when agents act strategically (by considering the payoffs and structure of the actual interaction) and when habitually.

[In] Bicchieri’s one-shot interaction account of social norms, expectations have a triggering role, that is, they can lead to a change in preferences. So a change in expectations may well cause an agent swapping types, and in particular can lead to the appearance of new types. This cannot be made part of Bayesian Nash equilibria, in which a list of possible types is set from the beginning and cannot evolve.

Overall, both kinds of accounts seem able to fit any data, thanks to the liberal definitions of context. Expectations can be part of the context; as they routinely change during any repeated interaction, they may trigger a change in norms at any time and thus allow one to explain any behaviour. Learning processes determine when the context’s influence stops overcoming benefit-related considerations; but in the absence of a precise definition of such processes, the effect of context can also be used to explain any behaviour.


We have argued that despite surface-level similarities, game-theoretic accounts of social norms are not easily integrated. This is due to the existence of two main kinds of accounts, based on one-shot or on repeated interactions. This distinction gives rise to different treatments of the role played by punishment. Moreover, the problem is not merely to choose between them, as they both suffer similarly from difficulties to account for the context-dependence of social norms while conserving their explanatory power.


This paper is ostensibly aimed at ‘explaining’ actual norms rather than seeking to suggest better norms. Thus if we are concerned with such issues as social justice, conflict, climate change, financial crises, epidemics or the misuse of technologies, we might reasonably see the paper as providing insights into problematic norms rather than those yet to be created norms that might help us out of our various messes.

Yet …

Dave Marsay

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