Learning in Games Where Agents Sample

This paper proposes an equilibrium concept --the Sampling Bayesian Equilibrium-- for games in which players observe the actions of only a small random sample of other players. I show the existence of these equilibrium points for i) a class of coordination global games and  for ii) general static games in normal form. For the first, I further show the existence of a unique interior Sampling Bayesian Equilibrium, easing comparative statics over the set of equilibria. Using asymptotic Bayesian analysis, in particular Bernstein- von Mises theorem, I show that most equilibrium points in the complete information games (where agents have perfect foresight over the actions of all other players) can be obtained as limits of pure-strategy Sampling Bayesian Equilibria of the perturbed games, as agents learn and sample sizes tend to infinity. These purification results are robust to a wide class of prior distributions over strategy profiles and are consistent with Nash’s ‘mass-action’ interpretation of mixed strategies.