Additive Security Games: Structure and Optimization

TL;DR

This paper characterizes Nash equilibria in additive security games, classifies equilibrium types, provides conditions and algorithms for their computation, and explores optimization and extensions to non-additive games.

Contribution

It offers a comprehensive structural analysis of security games with additive utility, including equilibrium classification, closed-form solutions, and algorithms for equilibrium computation.

Findings

Seven types of equilibria classified with feasibility conditions.

Closed-form expressions for expected outcomes at equilibrium.

NP-hardness results and polynomial algorithms for specific cases.

Abstract

In this work, we provide a structural characterization of the possible Nash equilibria in the well-studied class of security games with additive utility. Our analysis yields a classification of possible equilibria into seven types and we provide closed-form feasibility conditions for each type as well as closed-form expressions for the expected outcomes to the players at equilibrium. We provide uniqueness and multiplicity results for each type and utilize our structural approach to propose a novel algorithm to compute equilibria of each type when they exist. We then consider the special cases of security games with fully protective resources and zero-sum games. Under the assumption that the defender can perturb the payoffs to the attacker, we study the problem of optimizing the defender expected outcome at equilibrium. We show that this problem is weakly NP- hard in the case of Stackelberg equilibria and multiple attacker resources and present a pseudopolynomial time procedure to solve this problem for the case of Nash equilibria under mild assumptions. Finally, to address non-additive security games, we propose a notion of nearest additive game and demonstrate the existence and uniqueness of a such a nearest additive game for any non-additive game.