🤖 AI Summary
This study addresses the existence of equilibria in meta-games under incomplete information. By extending the notion of meta-Nash equilibrium to the Bayesian game framework, it introduces the concept of a meta-Bayesian Nash equilibrium, incorporating type-dependent mixed meta-strategies and environmental actions. Meta-payoffs are defined via the unique Bayesian Nash equilibrium of a transformed game. Leveraging Kakutani’s fixed-point theorem, the paper establishes the existence of such an equilibrium under conditions that the type space, meta-action space, and set of transformations are finite, and that the transformed game admits a unique Bayesian Nash equilibrium. This work is the first to generalize meta-games to settings with incomplete information, highlighting the critical role of private information in endogenously shaping game transformations. It unifies classical Bayesian games and complete-information meta-games as special cases and demonstrates the framework’s applicability through examples such as subsidy competition and cybersecurity protocol selection.
📝 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.