Modeling Other Players with Bayesian Beliefs for Games with Incomplete Information

TL;DR

This paper introduces Bayesian-CFR, a new algorithm for Bayesian games with incomplete information, which effectively computes Bayesian Nash equilibria by updating beliefs and outperforming existing methods in poker games.

Contribution

It presents a novel CFR algorithm tailored for Bayesian games, including belief update methods and extensions like Bayesian-CFR+ and Deep Bayesian CFR.

Findings

Bayesian-CFR outperforms existing algorithms in Texas Hold'em.

The belief update method converges to the true distribution.

The approach effectively computes Bayesian Nash equilibria.

Abstract

Bayesian games model interactive decision-making where players have incomplete information -- e.g., regarding payoffs and private data on players' strategies and preferences -- and must actively reason and update their belief models (with regard to such information) using observation and interaction history. Existing work on counterfactual regret minimization have shown great success for games with complete or imperfect information, but not for Bayesian games. To this end, we introduced a new CFR algorithm: Bayesian-CFR and analyze its regret bound with respect to Bayesian Nash Equilibria in Bayesian games. First, we present a method for updating the posterior distribution of beliefs about the game and other players' types. The method uses a kernel-density estimate and is shown to converge to the true distribution. Second, we define Bayesian regret and present a Bayesian-CFR minimization algorithm for computing the Bayesian Nash equilibrium. Finally, we extend this new approach to other existing algorithms, such as Bayesian-CFR+ and Deep Bayesian CFR. Experimental results show that our proposed solutions significantly outperform existing methods in classical Texas Hold'em games.