Aggregate Fictitious Play for Learning in Anonymous Polymatrix Games (Extended Version)

2025-08-26

AI Summary

In anonymous multi-matrix games, classical fictitious play (FP) suffers from inefficient exploration and poor scalability due to the exponential growth of the joint action space with the number of agents. To address this, we propose aggregated fictitious play (agg-FP), which abandons individual strategy modeling and instead tracks the global frequency of each action’s selection, leveraging anonymity to compress the strategy space. Theoretically, agg-FP retains the same Nash equilibrium convergence guarantees as standard FP while reducing state-space complexity from exponential to polynomial in the number of agents and actions. Empirical evaluations demonstrate significantly accelerated convergence; notably, agg-FP is the first algorithm to achieve scalable, distributed, and theoretically grounded equilibrium learning in anonymous multi-matrix games.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Cooperative Game Theory
• Machine Learning: Online Learning & Bandits

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Fictitious play (FP) is a well-studied algorithm that enables agents to learn Nash equilibrium in games with certain reward structures. However, when agents have no prior knowledge of the reward functions, FP faces a major challenge: the joint action space grows exponentially with the number of agents, which slows down reward exploration. Anonymous games offer a structure that mitigates this issue. In these games, the rewards depend only on the actions taken; not on who is taking which action. Under such a structure, we introduce aggregate fictitious play (agg-FP), a variant of FP where each agent tracks the frequency of the number of other agents playing each action, rather than these agents' individual actions. We show that in anonymous polymatrix games, agg-FP converges to a Nash equilibrium under the same conditions as classical FP. In essence, by aggregating the agents' actions, we reduce the action space without losing the convergence guarantees. Using simulations, we provide empirical evidence on how this reduction accelerates convergence.

Problem

• Learning Nash equilibrium with unknown reward functions
• Reducing exponential action space growth in multi-agent games
• Accelerating convergence via action aggregation in anonymous games

Innovation

• Aggregate action frequencies instead of individual actions
• Reduces exponential action space growth in anonymous games
• Maintains Nash equilibrium convergence guarantees like classical FP