Stackelberg vs. Nash in Security Games: An Extended Investigation of Interchangeability, Equivalence, and Uniqueness

There has been significant recent interest in game theoretic approaches to security, with much of the recent research focused on utilizing the leader-follower Stackelberg game model; for example, these games are at the heart of major applications such as the ARMOR program deployed for security at the LAX airport since 2007 and the IRIS program in use by the US Federal Air Marshals (FAMS). The foundational assumption for using Stackelberg games is that security forces (leaders), acting first, commit to a randomized strategy; while their adversaries (followers) choose their best response after surveillance of this randomized strategy. Yet, in many situations, the followers may act without observation of the leader’s strategy, essentially converting the game into a simultaneous-move game model. Previous work fails to address how a leader should compute her strategy given this fundamental uncertainty about the type of game faced. Focusing on the complex games that are directly inspired by real-world security applications, the paper provides four contributions in the context of a general class of security games. First, exploiting the structure of these security games, the paper shows that the Nash equilibria in security games are interchangeable, thus alleviating the equilibrium selection problem. Second, resolving the leader’s dilemma, it shows that under a natural restriction on security games, any Stackelberg strategy is also a Nash equilibrium strategy; and furthermore, the solution is unique in a class of security games of which ARMOR is a key exemplar. Third, when faced with a follower that can attack multiple targets, many of these properties no longer hold. Fourth, we show experimentally that in most (but not all) games where the restriction does not hold, the Stackelberg strategy is still a Nash equilibrium strategy, but this is no longer true when the attacker can attack multiple targets. These contributions have major implications for the real-world applications. As a possible direction for future research on cases where the Stackelberg strategy is not a Nash equilibrium strategy, we propose an extensive-form game model that makes the defender’s uncertainty about the attacker’s ability to observe explicit.