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

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