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Computing ESS in Multiplayer Games

Sam Ganzfried

TL;DR

Problem addressed: computing evolutionarily stable strategies (ESS) in nondegenerate symmetric multiplayer normal-form games is challenging due to nonconvexity and combinatorial search. The authors propose a support-enumeration algorithm that, for each potential support, computes a symmetric Nash equilibrium via a nonconvex quadratically-constrained program and then tests ESS with a QCQP, with a degeneracy-detection step. The method extends to n players by quadraticizing higher-order payoff terms and relies on preprocessing shortcuts to prune candidates, enabling practical computation even for eight strategies. Experiments on benchmark and random games demonstrate the approach finds all ESS in nondegenerate cases with reasonable runtimes and reveals the impact of preprocessing. This work provides a first scalable computational tool for evolutionary stability in multiplayer contexts, with potential applications in tumor ecology and behavioral ecology.

Abstract

We present an algorithm for computing all evolutionarily stable strategies in nondegenerate normal-form games with three or more players.

Computing ESS in Multiplayer Games

TL;DR

Problem addressed: computing evolutionarily stable strategies (ESS) in nondegenerate symmetric multiplayer normal-form games is challenging due to nonconvexity and combinatorial search. The authors propose a support-enumeration algorithm that, for each potential support, computes a symmetric Nash equilibrium via a nonconvex quadratically-constrained program and then tests ESS with a QCQP, with a degeneracy-detection step. The method extends to n players by quadraticizing higher-order payoff terms and relies on preprocessing shortcuts to prune candidates, enabling practical computation even for eight strategies. Experiments on benchmark and random games demonstrate the approach finds all ESS in nondegenerate cases with reasonable runtimes and reveals the impact of preprocessing. This work provides a first scalable computational tool for evolutionary stability in multiplayer contexts, with potential applications in tumor ecology and behavioral ecology.

Abstract

We present an algorithm for computing all evolutionarily stable strategies in nondegenerate normal-form games with three or more players.

Paper Structure

This paper contains 13 sections, 17 equations, 6 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Algorithm
  3. Experiments
  4. Conclusion
  5. Payoff Tensors for the Example Games
  6. Game 1 (two–strategy game with pure and mixed ESS)
  7. Game 2 (neutral two–strategy game)
  8. Game 3 (strict dominance of strategy 0)
  9. Game 4 (three–strategy rock–paper–scissors)
  10. Game 5 (strict pure ESS at $(1,0,0)$)
  11. Game 6 (Proliferator–Producer–Invasive tumor game)
  12. Game 7 (three–strategy game with six SNE and two ESS)
  13. Game 8 (random three–strategy game with mixed ESS)

Theorems & Definitions (3)

  • Definition 1
  • Definition 2
  • Definition 3