Risk-Averse Bayesian Games with an Unknown Type Distribution: Bayesian Learning, Equilibrium Analysis, and Finite-Sample Guarantees

RESUMO

Classical Bayesian games assume that the joint distribution of players' types is common knowledge, an assumption that rarely holds in practical applications. We address this issue by studying a group of myopic players who repeatedly interact under the Bayesian Nash conjecture and learn a parametric joint type distribution from type profiles observed over time within a Bayesian learning framework. Players may hold heterogeneous priors and may be risk averse toward both epistemic uncertainty about the distributional parameter and aleatoric uncertainty in their rivals' types. We propose two models: a Bayesian Nash equilibrium among risk-averse Bayesian learners (BNE-RABL), in which epistemic and aleatoric risks are evaluated separately, and a BNE based on Bayesian predictive distributions (BNE-BPD), in which the two sources of uncertainty are integrated into a Bayesian predictive distribution and evaluated through a single risk measure. Under suitable conditions, we establish the existence and uniqueness of both equilibria, and derive non-asymptotic convergence rates of the equilibrium sequences toward the corresponding oracle BNE under the true type distribution as the game is repeatedly played and more type profiles are observed. These results further show that the discrepancy between the BNE-RABL and BNE-BPD strategies vanishes as the sample size grows, which illustrates that the two models can be regarded as effective approximations to each other. We apply the proposed models to a price competition problem and numerically illustrate the theoretical results.