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

Ziheng Su, Huifu Xu

TL;DR

The paper addresses the existence and uniqueness of a continuous and monotone Bayesian Nash equilibrium (CBNE) in Bayesian games with private information. It uses a Lipschitz/implicit-function framework to show that each player's best-response is Lipschitz in its own type and in rivals' strategies, enabling a contraction mapping argument to establish existence; a dominance-type condition $\sum_{j\neq i} \tau_{ij} < \sigma_i$ yields uniqueness of the CBNE. It further derives stability bounds for CBNE under perturbations of the joint type distribution using Kantorovich distances and KL-divergence, providing quantitative measures of sensitivity for data-driven applications. Together, these results unify aspects of continuity and monotonicity in BNE analysis and offer practical tools for computing and assessing CBNE in empirical settings.

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.

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

TL;DR

The paper addresses the existence and uniqueness of a continuous and monotone Bayesian Nash equilibrium (CBNE) in Bayesian games with private information. It uses a Lipschitz/implicit-function framework to show that each player's best-response is Lipschitz in its own type and in rivals' strategies, enabling a contraction mapping argument to establish existence; a dominance-type condition yields uniqueness of the CBNE. It further derives stability bounds for CBNE under perturbations of the joint type distribution using Kantorovich distances and KL-divergence, providing quantitative measures of sensitivity for data-driven applications. Together, these results unify aspects of continuity and monotonicity in BNE analysis and offer practical tools for computing and assessing CBNE in empirical settings.

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.

Paper Structure

This paper contains 10 sections, 12 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. The Model
  3. Existence and uniqueness
  4. Lipschitz continuity of the optimal response function
  5. Monotonicity of the optimal response function
  6. Stability of CBNE
  7. Concluding Remarks
  8. Appendix
  9. Stability of strongly monotone parametric variational inequality
  10. Schauder's fixed point theorem and Arzelá-Ascoli theorem

Key Result

Proposition 3.1

Consider the BNE model (eq:BNE). Suppose Assumption assu:u_i holds, and for $i \in N$, (a) for each $\theta_{-i} \in \Theta_{-i}$, there exists a nonnegative integrable function $\nu_i(\theta_{-i})$ such that for each $a \in {\cal A}$ and $\theta_i', \theta_i" \in \Theta_i$, (b) the density function $q_i(\cdot| \theta_i)$ of the conditional distribution $\eta_i(\cdot\mid\theta_i)$ is Lipschitz co

Theorems & Definitions (15)

  • Definition 2.1: Bayesian Nash equilibrium (BNE)
  • Proposition 3.1
  • Theorem 3.1: Lipschitz continuity of the optimal response function
  • Theorem 3.2: Existence and uniqueness of CBNE
  • Corollary 3.1: Existence and uniqueness of BNE
  • Example 3.1: Symmetric three-player Bayesian Cournot game
  • Proposition 3.2: Monotonicity of the optimal response
  • Theorem 3.3: Existence and uniqueness of greatest and least CMBNEs
  • Theorem 4.1: Stability of CBNE
  • Theorem 4.2: Stability and sensitivity of CBNE in $\eta$
  • ...and 5 more