Strong Stackelberg equilibrium (SSE) is the standard solution concept of Stackelberg security games. As opposed to the weak Stackelberg equilibrium (WSE), the SSE assumes that the follower breaks ties in favor of the leader and this is widely acknowledged and justified by the assertion that the defender can often induce the attacker to choose a preferred action by making an infinitesimal adjustment to her strategy. Unfortunately, in security games with resource assignment constraints, the assertion might not be valid; it is possible that the defender cannot induce the desired outcome. As a result, many results claimed in the literature may be overly optimistic. To remedy, we first formally define the utility guarantee of a defender strategy and provide examples to show that the utility of SSE can be higher than its utility guarantee. Second, inspired by the analysis of leader's payoff by Von Stengel and Zamir (2004), we provide the solution concept called the inducible Stackelberg equilibrium (ISE), which owns the highest utility guarantee and always exists. Third, we show the conditions when ISE coincides with SSE and the fact that in general case, SSE can be extremely worse with respect to utility guarantee. Moreover, introducing the ISE does not invalidate existing algorithmic results as the problem of computing an ISE polynomially reduces to that of computing an SSE. We also provide an algorithmic implementation for computing ISE, with which our experiments unveil the empirical advantage of the ISE over the SSE.
To the best of our knowledge, Okamoto12 are the only exception who have raised the concern of lack of inducibility in security games, though their model is a very specific type of network security games that cannot be generalized to standard security games, especially games with scheduling constraints. Besides that, the more important question regarding the overoptimism due to the lack of inducibility and the algorithmic remedies needed for such overoptimism were left unanswered (in particular, the solution algorithm proposed by Okamoto12 only converges to a local optimum even only in their setting). These questions are addressed in the affirmative in this paper. The concept of in our paper (Definition ) is inspired by first proposed by von Stengel and Zamir in their study of general Stackelberg games. However, the focus of their work was solely on characterizing the range of leader's utility in Stackelberg equilibria with the aim of confirming the advantage of commitment @cite_21 @cite_22 . Some other works considered potential deviation of the attacker from their optimal responses and proposed solution concepts that were robust to these deviations @cite_24 @cite_8 @cite_9 . Our work differs from this line of research in that we consider perfectly rational attackers.
@cite_22 @cite_8 @cite_9 @cite_21 @cite_24
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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. Among the major applications are the ARMOR program deployed at LAX Airport 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, a leader may face uncertainty about the follower's surveillance capability. Previous work fails to address how a leader should compute her strategy given such uncertainty. We provide five contributions in the context of a general class of security games. First, we show that the Nash equilibria in security games are interchangeable, thus alleviating the equilibrium selection problem. Second, 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. Finally, as a possible direction for future research, we propose an extensive-form game model that makes the defender's uncertainty about the attacker's ability to observe explicit. How do we build multiagent algorithms for agent interactions with human adversaries? Stackelberg games are natural models for many important applications that involve human interaction, such as oligopolistic markets and security domains. In Stackelberg games, one player, the leader, commits to a strategy and the follower makes their decision with knowledge of the leader's commitment. Existing algorithms for Stackelberg games efficiently find optimal solutions (leader strategy), but they critically assume that the follower plays optimally. Unfortunately, in real-world applications, agents face human followers (adversaries) who --- because of their bounded rationality and limited observation of the leader strategy --- may deviate from their expected optimal response. Not taking into account these likely deviations when dealing with human adversaries can cause an unacceptable degradation in the leader's reward, particularly in security applications where these algorithms have seen real-world deployment. To address this crucial problem, this paper introduces three new mixed-integer linear programs (MILPs) for Stackelberg games to consider human adversaries, incorporating: (i) novel anchoring theories on human perception of probability distributions and (ii) robustness approaches for MILPs to address human imprecision. Since these new approaches consider human adversaries, traditional proofs of correctness or optimality are insufficient; instead, it is necessary to rely on empirical validation. To that end, this paper considers two settings based on real deployed security systems, and compares 6 different approaches (three new with three previous approaches), in 4 different observability conditions, involving 98 human subjects playing 1360 games in total. The final conclusion was that a model which incorporates both the ideas of robustness and anchoring achieves statistically significant better rewards and also maintains equivalent or faster solution speeds compared to existing approaches. Recent deployments of Stackelberg security games (SSG) have led to two competing approaches to handle boundedly rational human adversaries: (1) integrating models of human (adversary) decision-making into the game-theoretic algorithms, and (2) applying robust optimization techniques that avoid adversary modeling. A recent algorithm (MATCH) based on the second approach was shown to outperform the leading modeling-based algorithm even in the presence of significant amount of data. Is there then any value in using human behavior models in solving SSGs? Through extensive experiments with 547 human subjects playing 11102 games in total, we emphatically answer the question in the affirmative, while providing the following key contributions: (i) we show that our algorithm, SU-BRQR, based on a novel integration of human behavior model with the subjective utility function, significantly outperforms both MATCH and its improvements; (ii) we are the first to present experimental results with security intelligence experts, and find that even though the experts are more rational than the Amazon Turk workers, SU-BRQR still outperforms an approach assuming perfect rationality (and to a more limited extent MATCH); (iii) we show the advantage of SU-BRQR in a new, large game setting and demonstrate that sufficient data enables it to improve its performance over MATCH. In a class of games known as Stackelberg games, one agent (the leader) must commit to a strategy that can be observed by the other agent (the follower or adversary) before the adversary chooses its own strategy. We consider Bayesian Stackelberg games, in which the leader is uncertain about the types of adversary it may face. Such games are important in security domains, where, for example, a security agent (leader) must commit to a strategy of patrolling certain areas, and a robber (follower) has a chance to observe this strategy over time before choosing its own strategy of where to attack. This paper presents an efficient exact algorithm for finding the optimal strategy for the leader to commit to in these games. This algorithm, DOBSS, is based on a novel and compact mixed-integer linear programming formulation. Compared to the most efficient algorithm known previously for this problem, DOBSS is not only faster, but also leads to higher quality solutions, and does not suffer from problems of infeasibility that were faced by this previous algorithm. Note that DOBSS is at the heart of the ARMOR system that is currently being tested for security scheduling at the Los Angeles International Airport.