A Bayesian Nonatomic Game and Its Applicability to Finite-player Situations

Abstract

We investigate a nonatomic game (NG) on top of whose more or less conventional setups there is also a random state of the world. Every player receives only a signal of the state's realization upon which her decision is based. The state of the world, her own action, and the external environment formed by other players taking their actions may all influence a player's payoff. Not knowing the true state, the player would strive for a higher average payoff conditioned on her received signal. We demonstrate that equilibria can be reached for the game under reasonable conditions. Not only are they in existence, these equilibrium points are also useful. When the state space is finite, each of them would become ever more likely to help an $n$-player Bayesian game whose player profile is randomly generated from the original NG's player distribution to reach ever nearer equilibrium as n tends to +∞. More important, pure E-equilibria would likely emerge even though the NG equilibrium might well be mixed to start with. Pure versions of the latter would exist when the NG is anonymous.