[Paper Review] Modeling Other Players with Bayesian Beliefs for Games with Incomplete Information

Zuyuan Zhang, Mahdi Imani+1 more|arXiv (Cornell University)|May 23, 2024

Game Theory and ApplicationsDecision Sciences3 citations

TL;DR

This paper introduces Bayesian-CFR, a novel Counterfactual Regret Minimization algorithm for computing Bayesian Nash Equilibria in games with incomplete information. By using kernel-density estimation to update posterior beliefs about other players' types and strategies, the method minimizes Bayesian regret and significantly outperforms existing CFR-based baselines in Texas Hold’em poker, achieving exploitability as low as 0.02 in mixed-type player scenarios.

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.

Motivation & Objective

• Address the gap in applying Counterfactual Regret Minimization (CFR) to Bayesian games, where players have incomplete information about payoffs and other players’ types.
• Develop a method to dynamically update players’ beliefs about the game and other players’ private information using interaction history.
• Formulate Bayesian regret and design a minimization algorithm that converges to Bayesian Nash Equilibrium in extensive-form Bayesian games.
• Extend the framework to scalable settings via Bayesian-CFR+ and Deep Bayesian CFR, leveraging hierarchical and neural network-based representations.
• Demonstrate superior performance in real-world settings like Texas Hold’em, where players exhibit diverse and unknown behavioral types.

Proposed method

• Use kernel-density estimation to recursively update the posterior distribution of beliefs about other players’ types and game parameters based on local observation history.
• Define immediate Bayesian counterfactual regret and prove that minimizing it leads to minimization of overall Bayesian regret.
• Introduce Bayesian regret as a new regret metric that accounts for uncertainty in players’ types and beliefs.
• Design Bayesian-CFR as a regret minimization algorithm that maintains and updates belief models throughout gameplay using observed interactions.
• Extend Bayesian-CFR to Bayesian-CFR+ by incorporating cumulative counterfactual regret and to Deep Bayesian CFR using deep neural networks for belief representation.
• Ensure theoretical convergence by proving a regret bound that includes an additional term ΔΘ^T, dependent on time T and belief model dimension Θ.

(a) Pure-N

Experimental results

Research questions

• RQ1 Can Counterfactual Regret Minimization be effectively adapted to Bayesian games with incomplete information, where players must reason about others’ private types?
• RQ2 How can belief models about other players’ types be efficiently and consistently updated using interaction history in extensive-form games?
• RQ3 What is the theoretical regret bound of the proposed Bayesian-CFR algorithm with respect to Bayesian Nash Equilibrium?
• RQ4 How do belief-based regret minimization and kernel-density estimation improve performance in large-scale Bayesian games like Texas Hold’em?
• RQ5 To what extent do Bayesian-CFR and its extensions outperform standard CFR, CFR+, and deep reinforcement learning baselines in terms of exploitability and convergence?

Key findings

• Bayesian-CFR achieves an exploitability of 0.02 in mixed-type player scenarios, significantly outperforming CFR (0.31), CFR+ (0.27), and DQN (0.33) baselines.
• The ablation study shows that removing belief model updates increases exploitability from 0.19 (with full Bayesian-CFR) to 0.27 (without posterior updates), proving the critical role of belief learning.
• Bayesian-CFR achieves exploitability of 0.19 on average, which is very close to the ideal complete-information lower bound of 0.16, demonstrating strong performance under uncertainty.
• Bayesian-CFR+ achieves stable exploitability of 0.02 across all tested player types, indicating robustness and scalability in diverse and unknown behavioral settings.
• The proposed framework significantly reduces exploitability compared to non-belief-based methods, especially in mixed-type environments where player behavior is heterogeneous.
• Deep Bayesian CFR and Bayesian-CFR+ show improved convergence and stability over standard CFR and DQN, particularly in complex, large-scale Bayesian games with evolving player types.

(b) Pure-C