Evolution of conventions in games between behavioural rules

I examine when, how, and which conventions arise in N-player games. Each player draws a random sample of strategies used in the recent past, and then chooses a strategy in response to this sample. A player’s response is determined by a behavioural rule, which maps from the set of recently used strategy profiles to a subset of his own strategy set, and each element in the latter set is chosen with positive probability. A random sample of strategies is monomorphic if it contains only one distinct strategy for each of the other players. The behavioural rule of a player is responsive if, on drawing a monomorphic sample, there is a positive probability of playing a best-response to the other players’ strategy profile that is induced by their respective strategies in that sample; in addition, if the said induced strategy profile supports a strict Nash equilibrium, then a strategy played by him in the recent past is chosen with the complementary probability. A game is weakly acyclic if there exists a ‘best-response path’ from each outcome that is not a strict Nash equilibrium to a strict Nash equilibrium. I show that: (i) a convention forms whenever the players’ behavioural rules are responsive, and the game is weakly acyclic, (ii) in bi-matrix games, individuals described by the behavioural rule of extreme optimism—whereby, conditional on the random sample, they play a best-response to the most optimistic belief about the other player’s strategy choice—perform better than individuals described by any other responsive behavioural rule in the sense that the convention that is most preferred by the former is always in the stochastically stable set, and (iii) in bi-matrix pure coordination games, the said convention is the uniquely stochastically stable state if the other player’s behavioural rule is ‘mildly different’ from extreme optimism.