Correlated equilibrium, conformity and stereotyping in social groups

We argue that a social norm and the coordination of behavior within social groups can be expressed by a correlated equilibrium. Given a social group structure (a partition of individuals into social groups), we propose four conditions that one may expect of a correlated equilibrium consistent with social norms. These are: (a) within-group anonymity (conformity within groups), (b) group independence (no conformity between groups), (c) homophily (individuals in the same group have similar attributes), and (d) predictable group behavior (ex-post stability). We demonstrate that correlated equilibrium satisfying (a)-(c) exist very generally and equilibrium satisfying (a)-(d) exist in games with many players. We also consider stereotyped beliefs - beliefs that all individuals in a social group can be expected to behave in the same way - and show that stereotyping is not costly to the person who stereotypes but may or may not be beneficial to society.