A Bayesian nonatomic game and its applicability to finite-player situations

We investigate a nonatomic game (NG) involving a random state of the world. Every player receives only a signal of the state’s realization. Not knowing the true state, the player would strive for a higher average payoff conditioned on her received signal. We demonstrate that equilibria exist for the game under reasonable conditions. Not only are they in existence, but these equilibrium points are also useful. When the state space is finite, such an equilibrium would help induce asymptotically equilibrium behaviors as n tends to +∞ in n-player Bayesian games whose player profiles are randomly generated from the original NG’s player distribution. More important, pure ϵ-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.