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108
The complexity of computing a Nash equilibrium
, 2006
"... We resolve the question of the complexity of Nash equilibrium by showing that the problem of computing a Nash equilibrium in a game with 4 or more players is complete for the complexity class PPAD. Our proof uses ideas from the recentlyestablished equivalence between polynomialtime solvability of n ..."
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Cited by 226 (14 self)
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We resolve the question of the complexity of Nash equilibrium by showing that the problem of computing a Nash equilibrium in a game with 4 or more players is complete for the complexity class PPAD. Our proof uses ideas from the recentlyestablished equivalence between polynomialtime solvability of normalform games and graphical games, and shows that these kinds of games can implement arbitrary members of a PPADcomplete class of Brouwer functions. 1
Computing Nash equilibria: Approximation and smoothed complexity
 In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS
, 2006
"... By proving that the problem of computing a 1/n Θ(1)approximate Nash equilibrium remains PPADcomplete, we show that the BIMATRIX game is not likely to have a fully polynomialtime approximation scheme. In other words, no algorithm with time polynomial in n and 1/ǫ can compute an ǫapproximate Nash ..."
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Cited by 69 (10 self)
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By proving that the problem of computing a 1/n Θ(1)approximate Nash equilibrium remains PPADcomplete, we show that the BIMATRIX game is not likely to have a fully polynomialtime approximation scheme. In other words, no algorithm with time polynomial in n and 1/ǫ can compute an ǫapproximate Nash equilibrium of an n×n bimatrix game, unless PPAD ⊆ P. Instrumental to our proof, we introduce a new discrete fixedpoint problem on a highdimensional cube with a constant sidelength, such as on an ndimensional cube with sidelength 7, and show that they are PPADcomplete. Furthermore, we prove that it is unlikely, unless PPAD ⊆ RP, that the smoothed complexity of the LemkeHowson algorithm or any algorithm for computing a Nash equilibrium of a bimatrix game is polynomial in n and 1/σ under perturbations with magnitude σ. Our result answers a major open question in the smoothed analysis of algorithms and the approximation of Nash equilibria.
Correlated Qlearning
 In Proceedings of the Twentieth International Conference on Machine Learning
, 2003
"... There have been several attempts to design multiagent Qlearning algorithms capable of learning equilibrium policies in generalsum Markov games, just as Qlearning learns optimal policies in Markov decision processes. We introduce correlated Qlearning, one such algorithm based on the correlated eq ..."
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Cited by 56 (2 self)
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There have been several attempts to design multiagent Qlearning algorithms capable of learning equilibrium policies in generalsum Markov games, just as Qlearning learns optimal policies in Markov decision processes. We introduce correlated Qlearning, one such algorithm based on the correlated equilibrium solution concept. Motivated by a fixed point proof of the existence of stationary correlated equilibrium policies in Markov games, we present a generic multiagent Qlearning algorithm of which many popular algorithms are immediate special cases. We also prove that certain variants of correlated (and Nash) Qlearning are guaranteed to converge to stationary correlated (and Nash) equilibrium policies in two special classes of Markov games, namely zerosum and commoninterest. Finally, we show empirically that correlated Qlearning outperforms Nash Qlearning, further justifying the former beyond noting that it is less computationally expensive than the latter.
Regret minimization and the price of total anarchy
, 2008
"... We propose weakening the assumption made when studying the price of anarchy: Rather than assume that selfinterested players will play according to a Nash equilibrium (which may even be computationally hard to find), we assume only that selfish players play so as to minimize their own regret. Regret ..."
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Cited by 40 (7 self)
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We propose weakening the assumption made when studying the price of anarchy: Rather than assume that selfinterested players will play according to a Nash equilibrium (which may even be computationally hard to find), we assume only that selfish players play so as to minimize their own regret. Regret minimization can be done via simple, efficient algorithms even in many settings where the number of action choices for each player is exponential in the natural parameters of the problem. We prove that despite our weakened assumptions, in several broad classes of games, this “price of total anarchy” matches the Nash price of anarchy, even though play may never converge to Nash equilibrium. In contrast to the price of anarchy and the recently introduced price of sinking [15], which require all players to behave in a prescribed manner, we show that the price of total anarchy is in many cases resilient to the presence of Byzantine players, about whom we make no assumptions. Finally, because the price of total anarchy is an upper bound on the price of anarchy even in mixed strategies, for some games our results yield as corollaries previously unknown bounds on the price of anarchy in mixed strategies.
An optimization approach for approximate Nash equilibria
 In 3rd international Workshop on Internet and Network Economics, Proceedings of
, 2007
"... Abstract. In this paper we propose a new methodology for determining approximate Nash equilibria of noncooperative bimatrix games and, based on that, we provide an efficient algorithm that computes 0.3393approximate equilibria, the best approximation till now. The methodology is based on the formul ..."
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Cited by 28 (3 self)
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Abstract. In this paper we propose a new methodology for determining approximate Nash equilibria of noncooperative bimatrix games and, based on that, we provide an efficient algorithm that computes 0.3393approximate equilibria, the best approximation till now. The methodology is based on the formulation of an appropriate function of pairs of mixed strategies reflecting the maximum deviation of the players ’ payoffs from the best payoff each player could achieve given the strategy chosen by the other. We then seek to minimize such a function using descent procedures. As it is unlikely to be able to find global minima in polynomial time, given the recently proven intractability of the problem, we concentrate on the computation of stationary points and prove that they can be approximated arbitrarily close in polynomial time and that they have the above mentioned approximation property. Our result provides the best ɛ till now for polynomially computable ɛapproximate Nash equilibria of bimatrix games. Furthermore, our methodology for computing approximate Nash equilibria has not been used by others. 1
Spread: Foiling smart jammers using multilayer agility
 in INFOCOM 2007. 26th IEEE International Conference on Computer Communications. IEEE
, 2007
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The complexity of game dynamics: Bgp oscillations, sink equilibria, and beyond
 In SODA ’08: Proceedings of the nineteenth annual ACMSIAM symposium on Discrete algorithms
, 2008
"... We settle the complexity of a wellknown problem in networking by establishing that it is PSPACEcomplete to tell whether a system of path preferences in the BGP protocol [25] can lead to oscillatory behavior; one key insight is that the BGP oscillation question is in fact one about Nash dynamics. W ..."
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Cited by 21 (4 self)
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We settle the complexity of a wellknown problem in networking by establishing that it is PSPACEcomplete to tell whether a system of path preferences in the BGP protocol [25] can lead to oscillatory behavior; one key insight is that the BGP oscillation question is in fact one about Nash dynamics. We show that the concept of sink equilibria proposed recently in [11] is also PSPACEcomplete to analyze and approximate for graphical games. Finally, we propose a new equilibrium concept inspired by game dynamics, unit recall equilibria, which we show to be close to universal (exists with high probability in a random game) and algorithmically promising. We also give a relaxation thereof, called componentwise unit recall equilibria, which we show to be both tractable and universal (guaranteed to exist in every game).
Lossless abstraction of imperfect information games
 Journal of the ACM
, 2007
"... Abstract. Finding an equilibrium of an extensive form game of imperfect information is a fundamental problem in computational game theory, but current techniques do not scale to large games. To address this, we introduce the ordered game isomorphism and the related ordered game isomorphic abstractio ..."
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Cited by 21 (9 self)
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Abstract. Finding an equilibrium of an extensive form game of imperfect information is a fundamental problem in computational game theory, but current techniques do not scale to large games. To address this, we introduce the ordered game isomorphism and the related ordered game isomorphic abstraction transformation. For a multiplayer sequential game of imperfect information with observable actions and an ordered signal space, we prove that any Nash equilibrium in an abstracted smaller game, obtained by one or more applications of the transformation, can be easily converted into a Nash equilibrium in the original game. We present an algorithm, GameShrink, for abstracting the game using our isomorphism exhaustively. Its complexity is Õ(n2), where n is the number of nodes in a structure we call the signal tree. It is no larger than the game tree, and on nontrivial games it is drastically smaller, so GameShrink has time and space complexity sublinear in the size of the game tree. Using GameShrink, we find an equilibrium to a poker game with 3.1 billion nodes—over four orders of magnitude more than in the largest poker game solved previously. To address even larger games, we introduce approximation methods that do not preserve equilibrium, but nevertheless yield (ex post) provably closetooptimal strategies.