Layered Graph Security Games

TL;DR

This paper introduces a layered graph approach to model large strategy spaces in security games, providing polynomial-time solutions for linear utilities and practical algorithms for complex cases, with extensive experimental validation.

Contribution

It presents a novel layered graph formulation for security games and develops efficient algorithms for computing equilibria, especially under linear utility models.

Findings

Polynomial-time computation of Nash equilibria under linear utilities.

Intractability of best-response computation with binary utilities.

Graph structure and target values significantly impact algorithm performance.

Abstract

Security games model strategic interactions in adversarial real-world applications. Such applications often involve extremely large but highly structured strategy sets (e.g., selecting a distribution over all patrol routes in a given graph). In this paper, we represent each player's strategy space using a layered graph whose paths represent an exponentially large strategy space. Our formulation entails not only classic pursuit-evasion games, but also other security games, such as those modeling anti-terrorism and logistical interdiction. We study two-player zero-sum games under two distinct utility models: linear and binary utilities. We show that under linear utilities, Nash equilibrium can be computed in polynomial time, while binary utilities may lead to situations where even computing a best-response is computationally intractable. To this end, we propose a practical algorithm based on incremental strategy generation and mixed integer linear programs. We show through extensive experiments that our algorithm efficiently computes $\epsilon$-equilibrium for many games of interest. We find that target values and graph structure often have a larger influence on running times as compared to the size of the graph per se.