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Computing Game Symmetries and Equilibria That Respect Them

Emanuel Tewolde, Brian Hu Zhang, Caspar Oesterheld, Tuomas Sandholm, Vincent Conitzer

TL;DR

This work formalizes game symmetries as permutations over players and actions that preserve payoffs, establishing a deep link to graph automorphism and isomorphism problems. It provides a complete map of when symmetries can be exploited to compute Nash equilibria, showing PPAD‑complete hardness for symmetry respecting equilibria in general and CLS‑hardness for team games, while delivering polynomial‑time solutions for two‑player zero‑sum games and for regimes with many symmetries or few action orbits. The authors also present practical algorithms for identifying symmetries via hypergraph automorphisms and for solving symmetry respecting equilibria through orbit reductions and KKT/convex optimization techniques. The results clarify when symmetry helps or hinders solution methods and offer a path toward scalable analysis in large yet highly symmetric multiagent systems, with clear directions for extending to other representations and approximate symmetries.

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.

Computing Game Symmetries and Equilibria That Respect Them

TL;DR

This work formalizes game symmetries as permutations over players and actions that preserve payoffs, establishing a deep link to graph automorphism and isomorphism problems. It provides a complete map of when symmetries can be exploited to compute Nash equilibria, showing PPAD‑complete hardness for symmetry respecting equilibria in general and CLS‑hardness for team games, while delivering polynomial‑time solutions for two‑player zero‑sum games and for regimes with many symmetries or few action orbits. The authors also present practical algorithms for identifying symmetries via hypergraph automorphisms and for solving symmetry respecting equilibria through orbit reductions and KKT/convex optimization techniques. The results clarify when symmetry helps or hinders solution methods and offer a path toward scalable analysis in large yet highly symmetric multiagent systems, with clear directions for extending to other representations and approximate symmetries.

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.
Paper Structure (62 sections, 46 theorems, 46 equations, 2 figures, 1 table)

This paper contains 62 sections, 46 theorems, 46 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation
  3. Structure and a First Overview
  4. Are Symmetries Actually Helpful for Solving Games?
  5. Potential Harm
  6. Potential Slow-down
  7. Results and General Conclusions
  8. A Positive Result When There Are Many Symmetries
  9. A Remark On Approximate Symmetries
  10. Preliminaries on Game Symmetries
  11. Computation of Game Symmetries
  12. Complexity Results
  13. Efficient Computation
  14. Preliminaries on Nash Equilibria That Respect Game Symmetries
  15. Strategies, Nash Equilibria, Respecting Symmetries
  16. ...and 47 more sections

Key Result

Proposition 7

We can determine whether a game is totally symmetric, for $A^* = A^1 = \dots = A^N$ with the current action numbering, in polytime $\mathcal{O}(N^2 \cdot |A|)$.

Figures (2)

  • Figure 1: A two-player coordination game. If both players pick the same color, they each receive the associated utility points. If they miscoordinate, both receive $0$ points. Without knowing who you are playing with, what color would you choose?
  • Figure 2: Two extensions to Rock-Paper-Scissors. In both, there is only one symmetries-respecting strategy profile. The right game is known as Rock-Paper-Scissors-Lizard-Spock.

Theorems & Definitions (89)

  • Definition 1
  • Remark 2
  • Definition 3: Nash51
  • Definition 4: NeumannM44
  • Remark 5
  • Definition 6
  • Definition 7
  • Proposition 7
  • Proposition 7
  • Theorem 1
  • ...and 79 more