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

中文翻译：

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离散化下无限贝叶斯博弈中的贝叶斯纳什均衡的收敛性

我们证明了在效用函数和先验类型分布关于玩家类型连续的条件下，具有连续类型和有限动作的广义和贝叶斯游戏的贝叶斯纳什均衡（BNE）的存在。此外，存在一系列离散的贝叶斯博弈，其BNE策略弱收敛于无限贝叶斯博弈的BNE策略。我们的证明在无限贝叶斯游戏的平衡与有限逼近的平衡之间建立了联系，这导致了通过离散化玩家的类型空间来构造无限贝叶斯游戏的$ \ varepsilon $ -BNE的算法。