Layered Graph Security Games

TL;DR

This work introduces Layered Graph Security Games (LGSGs), a framework in which each player selects a path in a layered DAG, enabling compact representation of exponentially large strategy spaces. It analyzes two utility models, linear and binary, showing that linear utilities admit polynomial-time computation via flow methods, while binary utilities pose NP-hard challenges that necessitate MILP-based best-response oracles and a double oracle strategy-generation approach. The authors propose DO-based solvers with approximate best-responses and various speedups, demonstrating scalability on grid-world and real-city maps and revealing that equilibria tend to have small supports, highlighting the role of structure over size. The results provide practical methods for solving complex security games with realistic interdictive interactions across pursuit-evasion, anti-terrorism, and logistical interdiction domains, with significant implications for security planning and resource allocation.

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 $ε$-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.

Figures (19)

• Figure 1: Layered graphs of Example \ref{['eg:hexa-eight']}. $\mathcal{V}$ has 5 layers with a single source and sink. Disconnected vertices in $\mathcal{G}_a$ and $\mathcal{G}_d$ are in white.
• Figure 2: Physical graph and the layered graphs $\mathcal{G}_d, \mathcal{G}_a$ obtained by unrolling over 3 steps. The defender and attacker starts at A and B. Note $\mathsf{G}_d$ has an extra loop at A. Unreachable vertices are in white.
• Figure 3: Real-world physical graphs used to generate LGSGs for our application domains, together with examples of defender's (red) and attacker's (blue) equilibrial paths in (a) PE, (b) AT, and (c) LI.
• Figure 4: Computation times (black lines) and sparsity metrics (orange lines, for vanilla DO with exact best-responses) for PE and AT domains on grid world and Lower Manhattan.
• Figure 5: Computation times and sparsity metrics for AP and LI domains demonstrating the effects of additional factors on the performance of our DO algorithm with approximate best-responses.

Theorems & Definitions (10)

• Example 1
• Proposition 1
• Proposition 2: Kuhn's theorem for $u_{\textsc{lin}}$
• Proposition 3
• Proposition 4
• Conjecture 1
• Proposition 5
• proof
• proof
• proof