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24
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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Cited by 31 (9 self)
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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.
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 11 (4 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.
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 9 (5 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 9 (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
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 5 (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.
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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Cited by 5 (3 self)
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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 4 (2 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
Nash equilibria: Complexity, symmetries, and approximation
 Computer Science Review
"... Dedicated to Christos Papadimitriou, the eternal adolescent We survey recent joint work with Christos Papadimitriou and Paul Goldberg on the computational complexity of Nash equilibria. We show that finding a Nash equilibrium in normal form games is computationally intractable, but in an unusual way ..."
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Cited by 4 (0 self)
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Dedicated to Christos Papadimitriou, the eternal adolescent We survey recent joint work with Christos Papadimitriou and Paul Goldberg on the computational complexity of Nash equilibria. We show that finding a Nash equilibrium in normal form games is computationally intractable, but in an unusual way. It does belong to the class NP; but Nash’s theorem, showing that a Nash equilibrium always exists, makes the possibility that it is also NPcomplete rather unlikely. We show instead that the problem is as hard computationally as finding Brouwer fixed points, in a precise technical sense, giving rise to a new complexity class called PPAD. The existence of the Nash equilibrium was established via Brouwer’s fixedpoint theorem; hence, we provide a computational converse to Nash’s theorem. To alleviate the negative implications of this result for the predictive power of the Nash equilibrium, it seems natural to study the complexity of approximate equilibria: an efficient approximation scheme would imply that players could in principle come arbitrarily close to a Nash equilibrium given enough time. We review recent work on computing approximate equilibria and conclude by studying how symmetries may affect the structure and approximation of Nash equilibria. Nash showed that every symmetric game has a symmetric equilibrium. We complement this theorem with a rich set of structural results for a broader, and more interesting class of games with symmetries, called anonymous games. 1
Approximation guarantees for fictitious play
 In Proceedings of the 47th Annual Allerton Conference on Communication, Control, and Computing
, 2009
"... Abstract—Fictitious play is a simple, wellknown, and oftenused algorithm for playing (and, especially, learning to play) games. However, in general it does not converge to equilibrium; even when it does, we may not be able to run it to convergence. Still, we may obtain an approximate equilibrium. I ..."
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Cited by 3 (0 self)
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Abstract—Fictitious play is a simple, wellknown, and oftenused algorithm for playing (and, especially, learning to play) games. However, in general it does not converge to equilibrium; even when it does, we may not be able to run it to convergence. Still, we may obtain an approximate equilibrium. In this paper, we study the approximation properties that fictitious play obtains when it is run for a limited number of rounds. We show that if both players randomize uniformly over their actions in the first r rounds of fictitious play, then the result is an ǫequilibrium, where ǫ = (r + 1)/(2r). (Since we are examining only a constant number of pure strategies, we know that ǫ < 1/2 is impossible, due to a result of Feder et al.) We show that this bound is tight in the worst case; however, with an experiment on random games, we illustrate that fictitious play usually obtains a much better approximation. We then consider the possibility that the players fail to choose the same r. We show how to obtain the optimal approximation guarantee when both the opponent’s r and the game are adversarially chosen (but there is an upper bound R on the opponent’s r), using a linear program formulation. We show that if the action played in the ith round of fictitious play is chosen with probability proportional to: 1 for i = 1 and 1/(i −1) for all 2 ≤ i ≤ R+1, this gives an approximation guarantee of 1 − 1/(2 + ln R). We also obtain a lower bound of 1 − 4/ln R. This provides an actionable prescription for how long to run fictitious play. I.
Algorithms for abstracting and solving imperfect information games
, 2007
"... Game theory is the mathematical study of rational behavior in strategic environments. In many settings, most notably twoperson zerosum games, game theory provides particularly strong and appealing solution concepts. Furthermore, these solutions are efficiently computable in the complexitytheory s ..."
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Cited by 2 (1 self)
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Game theory is the mathematical study of rational behavior in strategic environments. In many settings, most notably twoperson zerosum games, game theory provides particularly strong and appealing solution concepts. Furthermore, these solutions are efficiently computable in the complexitytheory sense. However, in most interesting potential applications in artificial intelligence, the solutions are difficult to compute using current techniques due primarily to the extremely large statespaces of the environments. In this thesis, we propose new algorithms for tackling these computational difficulties. In one stream of research, we introduce automated abstraction algorithms for sequential games of imperfect information. These algorithms take as input a description of a game and produce a description of a strategically similar, but smaller, game as output. We present algorithms that are lossless (i.e., equilibriumpreserving), as well as algorithms that are lossy, but which can yield much smaller games while still retaining the most important features of the original game. In a second stream of research, we develop specialized optimization algorithms for finding ɛequilibria in sequential games of imperfect information. The algorithms are based on recent advances in nonsmooth convex optimization (namely the excessive gap technique) and provide significant improvements