Existence and Uniqueness Theorem of Continuous and Monotone Bayesian Nash Equilibrium and Stability Analysis

Since the seminal work by Meirowitz [Meirowitz A (2003) On the existence of equilibria to Bayesian games with non-finite type and action spaces. Econom. Lett. 78(2):213–218], there has been growing attention to the existence and uniqueness of continuous Bayesian Nash equilibria. In the existing literature, existence is typically established using Schauder’s fixed-point theorem, relying on the equicontinuity of players’ best response functions. Uniqueness, on the other hand, is usually derived under additional monotonicity conditions. In this paper, we revisit the issues of existence and uniqueness, and advance the literature by establishing both simultaneously using the Banach fixed-point theorem under a set of moderate conditions. Furthermore, we analyze the stability of such equilibria with respect to perturbations in the joint probability distribution of type parameters, offering theoretical support for the application of Bayesian Nash equilibrium models in data-driven contexts.

Funding: This research is supported by a CUHK start-up grant.