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

TL;DR

This paper establishes the existence, uniqueness, and stability of continuous and monotone Bayesian Nash equilibria using the Banach fixed-point theorem, advancing theoretical understanding in game theory.

Contribution

It introduces a new approach to prove both existence and uniqueness of Bayesian Nash equilibria simultaneously under moderate conditions.

Findings

Proves existence and uniqueness using Banach fixed-point theorem.

Analyzes stability of equilibria under distribution perturbations.

Provides theoretical support for data-driven Bayesian equilibrium models.

Abstract

Since the seminal work by Meirowitz, there has been growing attention on 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.