Branch-and-Cut for Mixed-Integer Generalized Nash Equilibrium Problems

2025-06-03

AI Summary

This paper addresses the computational challenge of computing exact pure Nash equilibria in Mixed-Integer Generalized Nash Equilibrium Problems (MIGNEPs). We propose the first provably convergent branch-and-bound algorithm for MIGNEPs: the original problem is reformulated as a bilevel optimization model based on the Nikaido–Isoda function, and novel equilibrium cuts and intersection cuts—adapted to the mixed-integer game setting—are integrated into the framework. Under mild assumptions, the algorithm guarantees finite termination and correctness of the computed equilibrium. Unlike existing heuristic or approximation methods, our approach constitutes the first exact solution framework for MIGNEPs. Extensive experiments on two classes of problems—knapsack games and capacity-constrained flow games—demonstrate its effectiveness, successfully identifying rigorous pure Nash equilibria.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous 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: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Computational social science

Abstract

Generalized Nash equilibrium problems with mixed-integer variables form an important class of games in which each player solves a mixed-integer optimization problem with respect to her own variables and the strategy space of each player depends on the strategies chosen by the rival players. In this work, we introduce a branch-and-cut algorithm to compute exact pure Nash equilibria for different classes of such mixed-integer games. The main idea is to reformulate the equilibrium problem as a suitable bilevel problem based on the Nikaido--Isoda function of the game. The proposed branch-and-cut method is applicable to generalized Nash equilibrium problems under quite mild assumptions. Depending on the specific setting, we use tailored equilibrium or intersection cuts. The latter are well-known in mixed-integer linear optimization and we adapt them to the game setting. We prove finite termination and correctness of the algorithm and present some first numerical results for two different types of knapsack games and another game based on capacitated flow problems.

Problem

• Solving mixed-integer Generalized Nash Equilibrium Problems
• Computing exact pure Nash equilibria via branch-and-cut
• Adapting intersection cuts for game-theoretic settings

Innovation

• Branch-and-cut for mixed-integer Nash equilibria
• Reformulation using Nikaido-Isoda bilevel problem
• Tailored equilibrium or intersection cuts