Spatial Branch-and-Bound for Computing Multiplayer Nash Equilibrium

2025-08-13

AI Summary

Computing Nash equilibria in general-sum and normal-form multiplayer non-cooperative games suffers from high computational complexity, and existing non-convex optimization methods often lack convergence guarantees and scalability. This paper reformulates the Nash equilibrium problem as a Polynomial Complementarity Problem (PCP) — the first such formulation — and designs a sound and complete spatial branch-and-bound algorithm. By integrating nonlinear constraint propagation with interval analysis, we establish a rigorous theoretical link between approximate solutions and ε-Nash equilibria. Experiments demonstrate that our method significantly outperforms existing complete algorithms on multiplayer games, offering global convergence guarantees, superior scalability, and high-quality approximate solutions. It thus overcomes fundamental theoretical and practical limitations of conventional non-convex optimization frameworks for equilibrium computation.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Search and Optimization: Non-convex Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

Equilibria of realistic multiplayer games constitute a key solution concept both in practical applications, such as online advertising auctions and electricity markets, and in analytical frameworks used to study strategic voting in elections or assess policy impacts in integrated assessment models. However, efficiently computing these equilibria requires games to have a carefully designed structure and satisfy numerous restrictions; otherwise, the computational complexity becomes prohibitive. In particular, finding even approximate Nash equilibria in general-sum normal-form games with two or more players is known to be PPAD-complete. Current state-of-the-art algorithms for computing Nash equilibria in multiplayer normal-form games either suffer from poor scalability due to their reliance on non-convex optimization solvers, or lack guarantees of convergence to a true equilibrium. In this paper, we propose a formulation of the Nash equilibrium computation problem as a polynomial complementarity problem and develop a complete and sound spatial branch-and-bound algorithm based on this formulation. We provide a qualitative analysis arguing why one should expect our approach to perform well, and show the relationship between approximate solutions to our formulation and that of computing an approximate Nash equilibrium. Empirical evaluations demonstrate that our algorithm substantially outperforms existing complete methods.

Problem

• Computing multiplayer Nash equilibria efficiently
• Overcoming PPAD-complete complexity in general-sum games
• Improving scalability and convergence guarantees in equilibrium computation

Innovation

• Formulate Nash equilibrium as polynomial complementarity problem
• Develop spatial branch-and-bound algorithm
• Outperform existing complete methods empirically