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arXiv · 2602.15233

Computing Perfect Bayesian Equilibria, with Application to Empirical Game-Theoretic Analysis

Christine Konicki·Mithun Chakraborty·Michael P. Wellman
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Abstract

Perfect Bayesian Equilibrium (PBE) is a refinement of the Nash equilibrium for imperfect-information extensive-form games (EFGs) that enforces consistency between the two components of a solution: agents' strategy profile describing their decisions at information sets and the belief system quantifying their uncertainty over histories within an information set. We present a scalable approach for computing a PBE of an arbitrary two-player EFG. We adopt the definition of PBE enunciated by Bonanno in 2011 using a consistency concept based on the theory of belief revision due to Alchourr\'{o}n, G\"{a}rdenfors, and Makinson. Our algorithm for finding a PBE is an adaptation of Counterfactual Regret Minimization (CFR) that minimizes the expected regret at each information set given a belief system, while maintaining the necessary consistency criteria. We prove that our algorithm is correct for two-player zero-sum games and has a reasonable slowdown in time-complexity relative to classical CFR given the additional computation needed for refinement. We also experimentally demonstrate the competent performance of PBE-CFR in terms of equilibrium quality and running time on medium-to-large non-zero-sum EFGs. Finally, we investigate the effectiveness of using PBE for strategy exploration in empirical game-theoretic analysis. Specifically, we compute PBE as a meta-strategy solver (MSS) in a tree-exploiting variant of Policy Space Response Oracles (TE-PSRO). Our experiments show that PBE as an MSS leads to higher-quality empirical EFG models with complex imperfect information structures compared to MSSs based on an unrefined Nash equilibrium.

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Christine Konicki, Mithun Chakraborty, Michael P. Wellman. 2026-02-16. Computing Perfect Bayesian Equilibria, with Application to Empirical Game-Theoretic Analysis. https://arxiv.org/abs/2602.15233

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