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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 game-theoretic solutions have been deployed in real-world 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 game-theoretic solutions have been deployed in real-world 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 so-called 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 NP-hard in others.
On the Complexity of Nash Equilibria and Other Fixed Points (Extended Abstract)
- IN PROC. FOCS
, 2007
"... We reexamine what it means to compute Nash equilibria and, more generally, what it means to compute a fixed point of a given Brouwer function, and we investigate the complexity of the associated problems. Specifically, we study the complexity of the following problem: given a finite game, Γ, with 3 ..."
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Cited by 68 (8 self)
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We reexamine what it means to compute Nash equilibria and, more generally, what it means to compute a fixed point of a given Brouwer function, and we investigate the complexity of the associated problems. Specifically, we study the complexity of the following problem: given a finite game, Γ, with 3 or more players, and given ɛ> 0, compute an approximation within ɛ of some (actual) Nash equilibrium. We show that approximation of an actual Nash Equilibrium, even to within any non-trivial constant additive factor ɛ < 1/2 in just one desired coordinate, is at least as hard as the long standing square-root sum problem, as well as a more general arithmetic circuit decision problem that characterizes P-time in a unit-cost model of computation with arbitrary precision rational arithmetic; thus placing the approximation problem in P, or even NP, would resolve major open problems in the complexity of numerical computation. We show similar results for market equilibria: it is hard to estimate with any nontrivial accuracy the equilibrium prices in an exchange economy with a unique equilibrium, where the economy is given by explicit algebraic formulas for the excess demand functions. We define a class, FIXP, which captures search problems that can be cast as fixed point
How hard is it to approximate the best Nash equilibrium?
, 2009
"... The quest for a PTAS for Nash equilibrium in a two-player game seeks to circumvent the PPADcompleteness of an (exact) Nash equilibrium by finding an approximate equilibrium, and has emerged as a major open question in Algorithmic Game Theory. A closely related problem is that of finding an equilibri ..."
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The quest for a PTAS for Nash equilibrium in a two-player game seeks to circumvent the PPADcompleteness of an (exact) Nash equilibrium by finding an approximate equilibrium, and has emerged as a major open question in Algorithmic Game Theory. A closely related problem is that of finding an equilibrium maximizing a certain objective, such as the social welfare. This optimization problem was shown to be NP-hard by Gilboa and Zemel [Games and Economic Behavior 1989]. However, this NP-hardness is unlikely to extend to finding an approximate equilibrium, since the latter admits a quasi-polynomial time algorithm, as proved by Lipton, Markakis and Mehta [Proc. of 4th EC, 2003]. We show that this optimization problem, namely, finding in a two-player game an approximate equilibrium achieving large social welfare is unlikely to have a polynomial time algorithm. One interpretation of our results is that the quest for a PTAS for Nash equilibrium should not extend to a PTAS for finding the best Nash equilibrium, which stands in contrast to certain algorithmic techniques used so far (e.g. sampling and enumeration). Technically, our result is a reduction from a notoriously difficult problem in modern Combinatorics, of finding a planted (but hidden) clique in a random graph G(n, 1/2). Our reduction starts from an instance with planted clique size k = O(log n). For comparison, the currently known algorithms due to Alon, Krivelevich and Sudakov [Random Struct. & Algorithms, 1998], and Krauthgamer and Feige [Random Struct. & Algorithms, 2000], are effective for a much larger clique size k = Ω(√n).
A Revealed Preference Approach to Computational Complexity in Economics
, 2010
"... One of the main building blocks of economics is the theory of the consumer, which postulates that consumers are utility maximizing. However, from a computational perspective, this model is called into question because the task of utility maximization subject to a budget constraint is computationally ..."
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One of the main building blocks of economics is the theory of the consumer, which postulates that consumers are utility maximizing. However, from a computational perspective, this model is called into question because the task of utility maximization subject to a budget constraint is computationally hard in the worst-case under reasonable assumptions. In this paper, we study the empirical consequences of strengthening consumer choice theory to enforce that utilities are computationally easy to maximize. We prove the possibly surprising result that computational constraints have no empirical consequences whatsoever for consumer choice theory. That is, a data set is consistent with a utility maximizing consumer if and only if a data set is consistent with a utility maximizing consumer having a utility function that can be maximized in strongly polynomial time. Our result motivates a general approach for posing questions about the empirical content of computational constraints: the revealed preference approach to computational complexity. The approach complements the conventional worst-case view of computational complexity in important ways, and is methodologically close to mainstream economics.
Market Equilibrium under Separable, Piecewise-Linear, Concave Utilities
"... We consider Fisher and Arrow-Debreu markets under additively-separable, piecewise-linear, concave utility functions, and obtain the following results: • For both market models, if an equilibrium exists, there is one that is rational and can be written using polynomially many bits. • There is no simp ..."
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Cited by 13 (5 self)
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We consider Fisher and Arrow-Debreu markets under additively-separable, piecewise-linear, concave utility functions, and obtain the following results: • For both market models, if an equilibrium exists, there is one that is rational and can be written using polynomially many bits. • There is no simple necessary and sufficient condition for the existence of an equilibrium: The problem of checking for existence of an equilibrium is NP-complete for both market models; the same holds for existence of an ɛ-approximate equilibrium, for ɛ = O(n −5). • Under standard (mild) sufficient conditions, the problem of finding an exact equilibrium is in PPAD for both market models. • Finally, building on the techniques of [CDDT09] we prove that under these sufficient conditions, finding an equilibrium for Fisher markets is PPAD-hard.
Lossy Stochastic Game Abstraction with Bounds
, 2012
"... Abstraction followed by equilibrium finding has emerged as the leading approach to solving games. Lossless abstraction typically yields games that are still too large to solve, so lossy abstraction is needed. Unfortunately, prior lossy game abstraction algorithms have no guarantees on solution quali ..."
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Cited by 11 (6 self)
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Abstraction followed by equilibrium finding has emerged as the leading approach to solving games. Lossless abstraction typically yields games that are still too large to solve, so lossy abstraction is needed. Unfortunately, prior lossy game abstraction algorithms have no guarantees on solution quality. We developed a framework that enables the design of lossy game abstraction algorithms with guarantees on solution quality. It simultaneously handles state and action abstraction. We define a measure of reward approximation error and transition probability error achieved by state and action abstraction in stochastic games such that the regret of the equilibrium found in the abstract game when implemented in the original, unabstracted game is upper-bounded by a function of those measures. We then develop the first lossy game abstraction algorithms with bounds on solution quality. Both of them work level-by-level up from the end of the game. One of the algorithms is greedy and the other is an integer linear program. We also prove that the abstraction problem is NP-complete (even with just action abstraction, 2 agents, and a 1-step game), but point out that this does not mean that the game abstraction problems that occur in practice cannot be solved quickly.
The complexity of Nash equilibria in limit-average games
, 2011
"... Abstract. We study the computational complexity of Nash equilibria in concurrent games with limit-average objectives. In particular, we prove that the existence of a Nash equilibrium in randomised strategies is undecidable, while the existence of a Nash equilibrium in pure strategies is decidable, e ..."
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Abstract. We study the computational complexity of Nash equilibria in concurrent games with limit-average objectives. In particular, we prove that the existence of a Nash equilibrium in randomised strategies is undecidable, while the existence of a Nash equilibrium in pure strategies is decidable, even if we put a constraint on the payoff of the equilibrium. Our undecidability result holds even for a restricted class of concurrent games, where nonzero rewards occur only on terminal states. Moreover, we show that the constrained existence problem is undecidable not only for concurrent games but for turn-based games with the same restriction on rewards. Finally, we prove that the constrained existence problem for Nash equilibria in (pure or randomised) stationary strategies is decidable and analyse its complexity. 1
Reducibility among fractional stability problems
- in Proc. IEEE FOCS, 2009
"... Abstract — In a landmark paper [32], Papadimitriou introduced a number of syntactic subclasses of TFNP based on proof styles that (unlike TFNP) admit complete problems. A recent series of results [11], [16], [5], [6], [7], [8] has shown that finding Nash equilibria is complete for PPAD, a particular ..."
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Cited by 11 (2 self)
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Abstract — In a landmark paper [32], Papadimitriou introduced a number of syntactic subclasses of TFNP based on proof styles that (unlike TFNP) admit complete problems. A recent series of results [11], [16], [5], [6], [7], [8] has shown that finding Nash equilibria is complete for PPAD, a particularly notable subclass of TFNP. A major goal of this work is to expand the universe of known PPAD-complete problems. We resolve the computational complexity of a number of outstanding open problems with practical applications. Here is the list of problems we show to be PPAD-complete, along with the domains of practical significance: Fractional Stable
Interdependent Defense Games: Modeling Interdependent Security under Deliberate Attacks
"... We propose interdependent defense (IDD) games, a computational game-theoretic framework to study aspects of the interdependence of risk and security in multi-agent 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 game-theoretic framework to study aspects of the interdependence of risk and security in multi-agent 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 pure-strategy space and utility function and derive appropriate cost functions for the defenders. We provide a complete characterization of mixed-strategy Nash equilibria (MSNE), and design a simple polynomial-time 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 real-world 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
