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40
Complexity of computing optimal Stackelberg strategies in security resource allocation games
 In Proceedings of the National Conference on Artificial Intelligence (AAAI
, 2010
"... Recently, algorithms for computing gametheoretic solutions have been deployed in realworld security applications, such as the placement of checkpoints and canine units at Los Angeles International Airport. These algorithms assume that the defender (security personnel) can commit to a mixed strateg ..."
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Recently, algorithms for computing gametheoretic solutions have been deployed in realworld security applications, such as the placement of checkpoints and canine units at Los Angeles International Airport. These algorithms assume that the defender (security personnel) can commit to a mixed strategy, a socalled Stackelberg model. As pointed out by Kiekintveld et al. (Kiekintveld et al. 2009), in these applications, generally, multiple resources need to be assigned to multiple targets, resulting in an exponential number of pure strategies for the defender. In this paper, we study how to compute optimal Stackelberg strategies in such games, showing that this can be done in polynomial time in some cases, and is NPhard in others.
The Status of the P versus NP Problem
"... When Moshe Vardi asked me to write this piece for CACM, my first reaction was the article could be written in two words Still open. ..."
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Cited by 25 (0 self)
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When Moshe Vardi asked me to write this piece for CACM, my first reaction was the article could be written in two words Still open.
Solving Stackelberg games with uncertain observability
 In AAMAS
, 2011
"... Recent applications of game theory in security domains use algorithms to solve a Stackelberg model, in which one player (the leader) first commits to a mixed strategy and then the other player (the follower) observes that strategy and bestresponds to it. However, in realworld applications, it is ha ..."
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Cited by 21 (5 self)
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Recent applications of game theory in security domains use algorithms to solve a Stackelberg model, in which one player (the leader) first commits to a mixed strategy and then the other player (the follower) observes that strategy and bestresponds to it. However, in realworld applications, it is hard to determine whether the follower is actually able to observe the leader’s mixed strategy before acting. In this paper, we model the uncertainty about whether the follower is able to observe the leader’s strategy as part of the game (as proposed in the extended version of Yin et al. [17]). We describe an iterative algorithm for solving these games. This algorithm alternates between calling a Nash equilibrium solver and a Stackelberg solver as subroutines. We prove that the algorithm finds a solution in a finite number of steps and show empirically that it runs fast on games of reasonable size. We also discuss other properties of this methodology based on the experiments.
The Computational Complexity of Nash Equilibria in Concisely Represented Games
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 52
, 2005
"... Games may be represented in many different ways, and different representations of games affect the complexity of problems associated with games, such as finding a Nash equilibrium. The traditional method of representing a game is to explicitly list all the payoffs, but this incurs an exponential blo ..."
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Cited by 20 (1 self)
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Games may be represented in many different ways, and different representations of games affect the complexity of problems associated with games, such as finding a Nash equilibrium. The traditional method of representing a game is to explicitly list all the payoffs, but this incurs an exponential blowup as the number of agents grows. We study two models of concisely represented games: circuit games, where the payoffs are computed by a given boolean circuit, and graph games, where each agent’s payoff is a function of only the strategies played by its neighbors in a given graph. For these two models, we study the complexity of four questions: determining if a given strategy is a Nash equilibrium, finding a Nash equilibrium, determining if there exists a pure Nash equilibrium, and determining if there exists a Nash equilibrium in which the payoffs to a player meet some given guarantees. In many cases, we obtain tight results, showing that the problems are complete for various complexity classes.
Computing Optimal Strategies to Commit to in ExtensiveForm Games
 Association for Computing Machinery
"... Computing optimal strategies to commit to in general normalform or Bayesian games is a topic that has recently been gaining attention, in part due to the application of such algorithms in various security and law enforcement scenarios. In this paper, we extend this line of work to the more general ..."
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Cited by 17 (8 self)
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Computing optimal strategies to commit to in general normalform or Bayesian games is a topic that has recently been gaining attention, in part due to the application of such algorithms in various security and law enforcement scenarios. In this paper, we extend this line of work to the more general case of commitment in extensiveform games. We show that in some cases, the optimal strategy can be computed in polynomial time; in others, computing it is NPhard.
Malicious Bayesian Congestion Games
 6th Workshop on Approximation and Online Algorithms (WAOA
, 2008
"... Abstract. In this paper, we introduce malicious Bayesian congestion games as an extension to congestion games where players might act in a malicious way. In such a game each player has two types. Either the player is a rational player seeking to minimize her own delay, or – with a certain probabilit ..."
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Cited by 12 (0 self)
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Abstract. In this paper, we introduce malicious Bayesian congestion games as an extension to congestion games where players might act in a malicious way. In such a game each player has two types. Either the player is a rational player seeking to minimize her own delay, or – with a certain probability – the player is malicious in which case her only goal is to disturb the other players as much as possible. We show that such games do in general not possess a Bayesian Nash equilibrium in pure strategies (i.e. a pure Bayesian Nash equilibrium). Moreover, given a game, we show that it is NPcomplete to decide whether it admits a pure Bayesian Nash equilibrium. This result even holds when resource latency functions are linear, each player is malicious with the same probability, and all strategy sets consist of singleton sets of resources. For a slightly more restricted class of malicious Bayesian congestion games, we provide easy checkable properties that are necessary and sufficient for the existence of a pure Bayesian Nash equilibrium. In the second part of the paper we study the impact of the malicious types on the overall performance of the system (i.e. the social cost). To measure this impact, we use the Price of Malice. We provide (tight) bounds on the Price of Malice for an interesting class of malicious Bayesian congestion games. Moreover, we show that for certain congestion games the advent of malicious types can also be beneficial to the system in the sense that the social cost of the worst case equilibrium decreases. We provide a tight bound on the maximum factor by which this happens. 1
Commitment to Correlated Strategies
"... The standard approach to computing an optimal mixed strategy to commit to is based on solving a set of linear programs, one for each of the follower’s pure strategies. We show that these linear programs can be naturally merged into a single linear program; that this linear program can be interpreted ..."
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The standard approach to computing an optimal mixed strategy to commit to is based on solving a set of linear programs, one for each of the follower’s pure strategies. We show that these linear programs can be naturally merged into a single linear program; that this linear program can be interpreted as a formulation for the optimal correlated strategy to commit to, giving an easy proof of a result by von Stengel and Zamir that the leader’s utility is at least the utility she gets in any correlated equilibrium of the simultaneousmove game; and that this linear program can be extended to compute optimal correlated strategies to commit to in games of three or more players. (Unlike in twoplayer games, in games of three or more players, the notions of optimal mixed and correlated strategies to commit to are truly distinct.) We give examples, and provide experimental results that indicate that for 50 × 50 games, this approach is usually significantly faster than the multipleLPs approach.
Computing optimal strategies to commit to in stochastic games
 In AAAI
, 2012
"... Significant progress has been made recently in the following two lines of research in the intersection of AI and game theory: (1) the computation of optimal strategies to commit to (Stackelberg strategies), and (2) the computation of correlated equilibria of stochastic games. In this paper, we unite ..."
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Cited by 12 (4 self)
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Significant progress has been made recently in the following two lines of research in the intersection of AI and game theory: (1) the computation of optimal strategies to commit to (Stackelberg strategies), and (2) the computation of correlated equilibria of stochastic games. In this paper, we unite these two lines of research by studying the computation of Stackelberg strategies in stochastic games. We provide theoretical results on the value of being able to commit and the value of being able to correlate, as well as complexity results about computing Stackelberg strategies in stochastic games. We then modify the QPACE algorithm (MacDermed et al. 2011) to compute Stackelberg strategies, and provide experimental results. 1
Interdependent Defense Games: Modeling Interdependent Security under Deliberate Attacks
"... We propose interdependent defense (IDD) games, a computational gametheoretic framework to study aspects of the interdependence of risk and security in multiagent systems under deliberate external attacks. Our model builds upon interdependent security (IDS) games, a model due to Heal and Kunreuther ..."
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Cited by 9 (1 self)
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We propose interdependent defense (IDD) games, a computational gametheoretic framework to study aspects of the interdependence of risk and security in multiagent systems under deliberate external attacks. Our model builds upon interdependent security (IDS) games, a model due to Heal and Kunreuther that considers the source of the risk to be the result of a fixed randomizedstrategy. We adapt IDS games to model the attacker’s deliberate behavior. We define the attacker’s purestrategy space and utility function and derive appropriate cost functions for the defenders. We provide a complete characterization of mixedstrategy Nash equilibria (MSNE), and design a simple polynomialtime algorithm for computing all of them, for an important subclass of IDD games. In addition, we propose a randominstance generator of (general) IDD games based on a version of the realworld Internetderived Autonomous Systems (AS) graph (with around 27K nodes and 100K edges), and present promising empirical results using a simple learning heuristics to compute (approximate) MSNE in such games. 1
Computing GameTheoretic Solutions and Applications to Security
, 2012
"... The multiagent systems community has adopted game theory as a framework for the design of systems of multiple selfinterested agents. For this to be effective, efficient algorithms must be designed to compute the solutions that game theory prescribes. In this paper, I summarize some of the state of ..."
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The multiagent systems community has adopted game theory as a framework for the design of systems of multiple selfinterested agents. For this to be effective, efficient algorithms must be designed to compute the solutions that game theory prescribes. In this paper, I summarize some of the state of the art on this topic, focusing particularly on how this line of work has contributed to several highly visible deployed security applications, developed at the University of Southern California.