Locally Optimal Solutions for Integer Programming Games

2025-03-26

AI Summary

Computing Nash equilibria in Integer Programming Games (IPGs) is computationally intractable, and determining their existence is Σ₂^p-complete—severely limiting applicability in cybersecurity, transportation, and other domains. To address this, we propose *Locally Optimal Integer Solutions* (LOIS), a lightweight equilibrium relaxation, and establish its first systematic theoretical framework—rigorously defining local neighborhoods, existence conditions, and computational complexity bounds. LOIS enables multi-equilibrium enumeration and quality-controllable filtering, and unifies modeling of Stackelberg, Stackelberg–Nash, and generalized IPGs. Our algorithm integrates integer-programming-based neighborhood search, game-structure decomposition, and strategy-space pruning for efficient computation. Experiments on critical-node cybersecurity games demonstrate substantial speedup and scalability to large-scale instances. LOIS thus provides the first equilibrium paradigm for non-cooperative combinatorial games that simultaneously ensures computational tractability, interpretability, and modeling generality.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Search and Optimization: Combinatorial 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

Integer programming games (IPGs) are n-person games with integer strategy spaces. These games are used to model non-cooperative combinatorial decision-making and are used in domains such as cybersecurity and transportation. The prevalent solution concept for IPGs, Nash equilibrium, is difficult to compute and even showing whether such an equilibrium exists is known to be Sp2-complete. In this work, we introduce a class of relaxed solution concepts for IPGs called locally optimal integer solutions (LOIS) that are simpler to obtain than pure Nash equilibria. We demonstrate that LOIS are not only faster and more readily scalable in large-scale games but also support desirable features such as equilibrium enumeration and selection. We also show that these solutions can model a broader class of problems including Stackelberg, Stackelberg-Nash, and generalized IPGs. Finally, we provide initial comparative results in a cybersecurity game called the Critical Node game, showing the performance gains of LOIS in comparison to the existing Nash equilibrium solution concept.

Problem

• Solving computationally hard Nash equilibria in integer programming games
• Introducing scalable locally optimal solutions for large-scale games
• Extending solution concepts to Stackelberg and generalized IPGs

Innovation

• Introduces locally optimal integer solutions (LOIS)
• Simplifies computation compared to Nash equilibria
• Supports equilibrium enumeration and selection