Meta-Bayesian Nash Equilibrium: Existence via Kakutani's Fixed Point Theorem

TL;DR

This paper introduces meta-Bayesian Nash equilibrium, extending the concept to incomplete information environments and proving its existence using Kakutani's fixed point theorem under certain assumptions.

Contribution

It generalizes meta-Nash equilibrium to incomplete information settings and establishes existence results using fixed point theory.

Findings

Meta-Bayesian Nash equilibrium exists under specified conditions.

Private information at the meta-level influences game transformation.

Framework encompasses classical Bayesian and complete-information meta-games.

Abstract

We extend the concept of meta-Nash equilibrium, introduced by Eshaghi Gordji and Bagha [2026] for complete-information games, to environments with incomplete information. We define a meta-Bayesian Nash equilibrium as a profile of type-dependent mixed meta-strategies together with an environmental move such that no player type can profitably deviate and the environment cannot improve its expected payoff. For each transformed game, meta-payoffs are determined by the unique Bayesian Nash equilibrium of that game. Using Kakutani's fixed point theorem, we establish existence under finiteness assumptions on type spaces, meta-actions, and transformations, together with the assumption that each transformed game admits a unique Bayesian Nash equilibrium. Several illustrative examples, including adaptive subsidy competition, cybersecurity protocol selection, and platform rule formation, demonstrate that private information at the meta-level plays an essential role in endogenous game transformation. The framework contains both classical Bayesian games and complete-information meta-games as special cases.