Regret Minimization in Population Network Games: Vanishing Heterogeneity and Convergence to Equilibria

2025-07-23

AI Summary

This paper addresses the unclear equilibrium formation mechanism in large-scale heterogeneous multi-agent games. We propose an analytical framework based on regret distribution dynamics: modeling agents’ strategy states as probability distributions and characterizing their evolution under Smooth Regret Matching via a continuity equation. We discover that heterogeneity spontaneously decays over time—specifically, the variance of the regret distribution monotonically decreases, driving the population toward uniform behavior. We theoretically prove that this mechanism guarantees convergence to the Quantal Response Equilibrium (QRE) in both competitive and cooperative games. Our core contribution is the first identification of distributional variance decay as the intrinsic dynamical mechanism underlying heterogeneity dissipation, thereby establishing a novel, interpretable, and predictive paradigm for equilibrium selection in multi-agent systems.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Sequential Decision Making

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Web Mining and Content Analysis: Models for Web evolution
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Understanding and predicting the behavior of large-scale multi-agents in games remains a fundamental challenge in multi-agent systems. This paper examines the role of heterogeneity in equilibrium formation by analyzing how smooth regret-matching drives a large number of heterogeneous agents with diverse initial policies toward unified behavior. By modeling the system state as a probability distribution of regrets and analyzing its evolution through the continuity equation, we uncover a key phenomenon in diverse multi-agent settings: the variance of the regret distribution diminishes over time, leading to the disappearance of heterogeneity and the emergence of consensus among agents. This universal result enables us to prove convergence to quantal response equilibria in both competitive and cooperative multi-agent settings. Our work advances the theoretical understanding of multi-agent learning and offers a novel perspective on equilibrium selection in diverse game-theoretic scenarios.

Problem

• Analyzes how regret-matching unifies heterogeneous agents' behavior
• Studies vanishing heterogeneity in regret distribution over time
• Proves convergence to quantal response equilibria in multi-agent systems

Innovation

• Smooth regret-matching unifies heterogeneous agents
• Regret distribution variance diminishes over time
• Convergence to quantal response equilibria proven