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Locally Optimal Solutions for Integer Programming Games

Pravesh Koirala, Mel Krusniak, Forrest Laine

TL;DR

The paper addresses the computational difficulty of pure Nash equilibria in integer programming games (IPGs) by introducing Locally Optimal Integer Solutions (LOIS) as a tractable relaxation. LOIS-m yields local, neighborhood-optimal solutions that can be encoded as implication constraints (ICs) and, for quadratic payoffs with linear constraints, as linear integer constraints (LICs), enabling efficient solver-based computation. The framework supports equilibrium enumeration and welfare-based selection and extends to generalized IPGs and Stackelberg variants, with a cybersecurity case study (Critical Node Game) showing substantial speedups over Nash-based approaches. This approach offers a practical, scalable alternative for large-scale IPGs, albeit with trade-offs in solution quality relative to exact Nash equilibria, and it opens avenues for broader applications and richer local-solution variants.

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.

Locally Optimal Solutions for Integer Programming Games

TL;DR

The paper addresses the computational difficulty of pure Nash equilibria in integer programming games (IPGs) by introducing Locally Optimal Integer Solutions (LOIS) as a tractable relaxation. LOIS-m yields local, neighborhood-optimal solutions that can be encoded as implication constraints (ICs) and, for quadratic payoffs with linear constraints, as linear integer constraints (LICs), enabling efficient solver-based computation. The framework supports equilibrium enumeration and welfare-based selection and extends to generalized IPGs and Stackelberg variants, with a cybersecurity case study (Critical Node Game) showing substantial speedups over Nash-based approaches. This approach offers a practical, scalable alternative for large-scale IPGs, albeit with trade-offs in solution quality relative to exact Nash equilibria, and it opens avenues for broader applications and richer local-solution variants.

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.

Paper Structure

This paper contains 21 sections, 1 theorem, 34 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Existing algorithms for IPGs
  4. Local solutions
  5. Definitions
  6. Optimality conditions for LOIS-m
  7. LOIS-m solution for IPGs
  8. Quadratic integer programs with linear constraints
  9. Quadratic IPGs with linear constraints (QPIGs)
  10. Encoding LICs as linear integer constraints
  11. Complexity of LOIS
  12. Equilibrium enumeration and selection
  13. LOIS for generalized IPGs
  14. Stackelberg and Stackelberg-Nash IPGs
  15. Interpretation of LOIS
  16. ...and 6 more sections

Key Result

lemma 1

For a constant $\delta \in \mathbb{Z}^n$, the inequality $f(x + \delta) - f(x) < 0$ is linear in $x$.

Figures (1)

  • Figure 1: Algorithmic building blocks of major algorithms for solving IPGs. Each algorithm must repeatedly solve a mixed-integer program in the play phase which is the major source of complexity in finding solutions for IPGs. Figure is taken from carvalho2023integer

Theorems & Definitions (10)

  • definition 1: IPG
  • definition 2: Pure Nash equilibrium for IPGs
  • definition 3: m-order integer neighborhood
  • definition 4: Locally optimal integer solution of $m$-th order
  • definition 5: Implication constraints (ICs)
  • definition 6: Mathematical Program with Implication Constraints (MPIC)
  • definition 7: Linear implication constraints (LICs)
  • lemma 1
  • definition 8: Price of aggression (PoA)
  • definition 9: Price of security (PoS)