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

TL;DR

This work tackles the challenge of exactly computing Nash equilibria in multiplayer imperfect-information games, where traditional iterative methods lack convergence guarantees. It introduces a nonlinear complementarity problem (NCP) formulation derived from an extended sequence-form, recast as a quadratically-constrained feasibility program and solved via nonconvex quadratic programming. The approach achieves exact Nash equilibria for a nontrivial 3-player Kuhn poker after removing dominated actions in under 3 seconds, outperforming Gambit’s exact methods on this reduced game, while the full game remains intractable within 24 hours. Comparisons with Gambit’s logit QRE show scalable approximation capabilities for larger games, highlighting a practical trade-off between exactness and scalability and underscoring preprocessing (dominance removal) as a valuable preprocessing step. The results suggest potential for specialized NCP solvers and hybrid strategies that combine exact methods for small instances with approximate methods for larger ones.

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.