Computing Equilibria in Integer Programming Games with Shared Constraints

2026-10-03

AI Summary

This study addresses the computation of pure Nash equilibria in integer games with shared constraints, focusing on feasibility dependencies arising from opponents' strategy changes. Methodologically, it proposes a cutting-plane algorithm based on conditional equilibrium inequalities to precisely characterize the equilibrium set, embedded within a branch-and-cut framework to optimize linear objectives. Additionally, a generalized no-regret dynamics algorithm combined with bisection is designed to certify approximation factors, establishing the enforceability of vertex strategies under concave costs. Evaluated on 3,300 instances, the approach efficiently computes socially optimal equilibria for the vast majority of cases, effectively validating both equilibrium non-existence and approximation bounds.

Abstract

We develop a cutting-plane algorithm for computing pure Nash equilibria in finite integer games with shared constraints, where a deviation may be feasible against one opponent profile and infeasible against another. Conditional equilibrium inequalities capture this dependence through an explicit activation term. We give affine encodings of costs and activation conditions and prove that the resulting inequalities characterize the equilibrium set exactly. Embedded as lazy constraints in branch-and-cut, they yield the Generalized Zero-Regret algorithm for optimizing a linear objective over exact or approximate equilibria; a bisection procedure maintains certified bounds on the smallest achievable approximation factor. We also prove that, when the conditional polytopes have integral vertices, a concave cost is constant on the minimal face containing an equilibrium strategy, so strict concavity forces vertex strategies; for uniform integer-splittable bin packing, a cost-preserving transformation yields vertex equilibria even without strict concavity. We give formulations for bin packing, network formation, and knapsack games with shared capacities and evaluate the algorithm on 3,300 instances, computing a socially optimal equilibrium on all but 43, certifying nonexistence on 16, and certifying approximation factors within a few percent where no exact equilibrium exists.

Problem

• Integer Programming Games
• Shared Constraints
• Pure Nash Equilibria
• Equilibrium Computation

Innovation

• Cutting-plane algorithm
• Pure Nash equilibria
• Shared constraints
• Branch-and-cut
• Conditional equilibrium inequalities