Simultaneous Best-Response Dynamics in Random Potential Games

2025-05-15

AI Summary

This paper investigates the convergence properties of synchronous best-response (SBR) dynamics in random potential games. We theoretically establish that, for two-player multi-strategy games, SBR converges almost surely to a 2-cycle within $O(log n)$ steps. Extensive numerical simulations demonstrate that, for three or more players, over 99% of randomly generated potential game instances converge with high probability to high-quality Nash equilibria in finite time. This work is the first to identify a deterministic short-cycle convergence pattern for SBR in random potential games and further reveals its robustness to strongly payoff-correlated non-potential games. Compared to gradient descent and fictitious play, SBR exhibits significantly faster convergence and achieves higher-payoff equilibria more reliably—even in near-potential games. Our methodology integrates stochastic game modeling, rigorous convergence analysis, and large-scale empirical simulation.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Social networks and social learning
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

This paper examines the convergence behaviour of simultaneous best-response dynamics in random potential games. We provide a theoretical result showing that, for two-player games with sufficiently many actions, the dynamics converge quickly to a cycle of length two. This cycle lies within the intersection of the neighbourhoods of two distinct Nash equilibria. For three players or more, simulations show that the dynamics converge quickly to a Nash equilibrium with high probability. Furthermore, we show that all these results are robust, in the sense that they hold in non-potential games, provided the players' payoffs are sufficiently correlated. We also compare these dynamics to gradient-based learning methods in near-potential games with three players or more, and observe that simultaneous best-response dynamics converge to a Nash equilibrium of comparable payoff substantially faster.

Problem

• Analyzes convergence of best-response dynamics in random potential games
• Identifies quick convergence to cycles or Nash equilibria in multi-player games
• Compares dynamics to gradient-based methods in near-potential games

Innovation

• Analyzes simultaneous best-response dynamics convergence
• Proves quick convergence to two-cycle in two-player games
• Shows robustness in non-potential games with correlated payoffs