Continuous Behavioural Function Equilibria and Approximation Schemes in Bayesian Games with Non-Finite Type and Action Spaces

TL;DR

This paper establishes verifiable growth conditions for continuous behavioral function equilibria in complex Bayesian games and introduces approximation schemes using polynomial decision rules and optimal discretization, with proven error bounds.

Contribution

It provides new verifiable conditions for equi-continuity in Bayesian games and develops approximation methods with error analysis for equilibrium computation.

Findings

Growth conditions ensure equi-continuity of behavioral functions.

Polynomial decision rules reduce the problem to finite-dimensional stochastic equilibrium.

Optimal quantization offers effective discretization with error bounds.

Abstract

Meirowitz [17] showed existence of continuous behavioural function equilibria for Bayesian games with non-finite type and action spaces. A key condition for the proof of the existence result is equi-continuity of behavioural functions which, according to Meirowitz [17, page 215], is likely to fail or difficult to verify. In this paper, we advance the research by presenting some verifiable conditions for the required equi-continuity, namely some growth conditions of the expected utility functions of each player at equilibria. In the case when the growth is of second order, we demonstrate that the condition is guaranteed by strong concavity of the utility function. Moreover, by using recent research on polynomial decision rules and optimal discretization approaches in stochastic and robust optimization, we propose some approximation schemes for the Bayesian equilibrium problem: first, by restricting the behavioral functions to polynomial functions of certain order over the space of types, we demonstrate that solving a Bayesian polynomial behavioural function equilibrium is down to solving a finite dimensional stochastic equilibrium problem; second, we apply the optimal quantization method due to Pflug and Pichler [18] to develop an effective discretization scheme for solving the latter. Error bounds are derived for the respective approximation schemes under moderate conditions and both aca- demic examples and numerical results are presented to explain the Bayesian equilibrium problem and their approximation schemes.