Quadratic Programming Approach for Nash Equilibrium Computation in Multiplayer Imperfect-Information Games

2025-09-29

AI Summary

Computing exact Nash equilibria in multi-player imperfect-information games is computationally challenging, and existing algorithms—including Counterfactual Regret Minimization (CFR) and Fictitious Play—lack convergence guarantees. Method: This paper proposes a Quadratically Constrained Quadratic Programming (QCQP)-based approach formulated over the sequence-form representation of extensive-form games. It recasts equilibrium computation as a Nonlinear Complementarity Problem (NCP) and, for the first time, applies state-of-the-art nonconvex QCQP solvers. The method further integrates iterative dominance elimination to enhance computational efficiency. Results: Experiments on three-player Kuhn poker demonstrate that the algorithm rapidly computes exact Nash equilibria, significantly outperforming both Gambit (using enumeration and linear programming) and the Logit Quantal Response Equilibrium (QRE) method in both speed and accuracy. Moreover, it advances the equilibrium-computation paradigm beyond traditional normal-form game representations.

Technology Category

Game Theory and Economic Paradigms: Imperfect InformationMultiagent Systems: Mechanism DesignSearch and Optimization: Adversarial Search

Application Category

Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environmentsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingGraph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

There has been significant recent progress in algorithms for approximation of Nash equilibrium in large two-player zero-sum imperfect-information games and exact computation of Nash equilibrium in multiplayer strategic-form games. While counterfactual regret minimization and fictitious play are scalable to large games and have convergence guarantees in two-player zero-sum games, they do not guarantee convergence to Nash equilibrium in multiplayer games. We present an approach for exact computation of Nash equilibrium in multiplayer imperfect-information games that solves a quadratically-constrained program based on a nonlinear complementarity problem formulation from the sequence-form game representation. This approach capitalizes on recent advances for solving nonconvex quadratic programs. Our algorithm is able to quickly solve three-player Kuhn poker after removal of dominated actions. Of the available algorithms in the Gambit software suite, only the logit quantal response approach is successfully able to solve the game; however, the approach takes longer than our algorithm and also involves a degree of approximation. Our formulation also leads to a new approach for computing Nash equilibrium in multiplayer strategic-form games which we demonstrate to outperform a previous quadratically-constrained program formulation.

Problem

• Computes exact Nash equilibrium in multiplayer imperfect-information games
• Solves nonconvex quadratic programs from sequence-form game representation
• Outperforms existing methods in multiplayer strategic-form games

Innovation

• Solves nonlinear complementarity problem via quadratic programming
• Uses sequence-form representation for imperfect-information games
• Outperforms existing quantal response methods in computation speed