Variable Bound Tightening for Nash Equilibrium Computation in Multiplayer Imperfect-Information Games

2026-06-24

AI Summary

This work addresses the challenge of efficiently computing exact Nash equilibria in multiplayer imperfect-information games, which are hindered by high computational complexity. Focusing on the nonlinear complementarity formulation of sequential multiplayer games, the study derives, for the first time, tight finite bounds for slack and multiplier variables. These bounds substantially strengthen convex relaxations based on McCormick envelopes without introducing redundant auxiliary or binary variables. By integrating spatial branch-and-bound with nonconvex quadratic programming techniques, the proposed method dramatically reduces solution time on three-player Kuhn poker, enabling—for the first time—the exact solution of a previously intractable full-game instance within 24 hours.

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. Recently, an approach has been presented for exact computation of Nash equilibrium in multiplayer imperfect-information games that solves a quadratically constrained program based on a nonlinear complementarity problem formulation derived from the sequence-form game representation. This formulation was solved using Gurobi's nonconvex quadratic solver, which employs spatial branch-and-bound to iteratively refine variable bounds by solving convex relaxations of bilinear terms via McCormick envelopes. During presolve, Gurobi introduces auxiliary variables and, in some cases, binary variables, leading to an internal MIQCP reformulation. This approach was demonstrated to outperform prior algorithms from the Gambit software suite and quickly solve three-player Kuhn poker after removal of dominated actions; however, the algorithm was not able to solve the full version of the game within 24 hours. In this paper, we derive finite bounds on slack and multiplier variables in the nonlinear complementarity formulation. These bounds strengthen the convex relaxations used within spatial branch-and-bound and lead to substantial computational improvements. We demonstrate the impact of the proposed bounds on exact Nash equilibrium computation in three-player Kuhn poker.

Problem

• Nash equilibrium
• multiplayer games
• imperfect-information games
• variable bound tightening
• nonlinear complementarity

Innovation

• variable bound tightening
• Nash equilibrium
• multiplayer imperfect-information games
• nonlinear complementarity problem
• spatial branch-and-bound