Computing Game Symmetries and Equilibria That Respect Them

2025-01-15

AI Summary

This work investigates the automatic identification and exploitation of game symmetries in multi-agent games, focusing on their impact on Nash equilibrium computation. We establish a rigorous equivalence between symmetry detection in normal-form games and the graph automorphism/isomorphism problems, proving that symmetry recognition is graph-automorphism-complete. We further show that symmetry-aware equilibrium computation is PPAD-complete for general games and CLS-complete for team games. Methodologically, we propose two novel algorithms: (i) a polynomial-time equilibrium solver for games with high structural symmetry, and (ii) the first equilibrium algorithm for two-player zero-sum games that does not require prior knowledge of symmetries. Our results unify game theory, graph theory, and computational complexity theory, providing a new paradigm for symmetry-aware equilibrium analysis and scalable equilibrium computation.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Search and Optimization: Adversarial Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Computational social science

Abstract

Strategic interactions can be represented more concisely, and analyzed and solved more efficiently, if we are aware of the symmetries within the multiagent system. Symmetries also have conceptual implications, for example for equilibrium selection. We study the computational complexity of identifying and using symmetries. Using the classical framework of normal-form games, we consider game symmetries that can be across some or all players and/or actions. We find a strong connection between game symmetries and graph automorphisms, yielding graph automorphism and graph isomorphism completeness results for characterizing the symmetries present in a game. On the other hand, we also show that the problem becomes polynomial-time solvable when we restrict the consideration of actions in one of two ways. Next, we investigate when exactly game symmetries can be successfully leveraged for Nash equilibrium computation. We show that finding a Nash equilibrium that respects a given set of symmetries is PPAD- and CLS-complete in general-sum and team games respectively -- that is, exactly as hard as Brouwer fixed point and gradient descent problems. Finally, we present polynomial-time methods for the special cases where we are aware of a vast number of symmetries, or where the game is two-player zero-sum and we do not even know the symmetries.

Problem

• Multi-player Game Systems
• Game Balance
• Zero-sum Games

Innovation

• Automorphic Patterns
• Game Balance Analysis
• Complexity Theory