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We prove the existence of Bayesian Nash Equilibrium (BNE) of general-sumBayesian games with continuous types and finite actions under the conditionsthat the utility functions and the prior type distributions are continuousconcerning the players' types. Moreover, there exists a sequence of discretizedBayesian games whose BNE strategies converge weakly to a BNE strategy of theinfinite Bayesian game. Our proof establishes a connection between theequilibria of the infinite Bayesian game and those of finite approximations,which leads to an algorithm to construct $\varepsilon$-BNE of infinite Bayesiangames by discretizing players' type spaces.

Convergence of Bayesian Nash Equilibrium in Infinite Bayesian Games under Discretization