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47
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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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
Settling the Complexity of Computing TwoPlayer Nash Equilibria
"... We prove that Bimatrix, the problem of finding a Nash equilibrium in a twoplayer game, is complete for the complexity class PPAD (Polynomial Parity Argument, Directed version) introduced by Papadimitriou in 1991. Our result, building upon the work of Daskalakis, Goldberg, and Papadimitriou on the c ..."
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We prove that Bimatrix, the problem of finding a Nash equilibrium in a twoplayer game, is complete for the complexity class PPAD (Polynomial Parity Argument, Directed version) introduced by Papadimitriou in 1991. Our result, building upon the work of Daskalakis, Goldberg, and Papadimitriou on the complexity of fourplayer Nash equilibria [21], settles a long standing open problem in algorithmic game theory. It also serves as a starting point for a series of results concerning the complexity of twoplayer Nash equilibria. In particular, we prove the following theorems: • Bimatrix does not have a fully polynomialtime approximation scheme unless every problem in PPAD is solvable in polynomial time. • The smoothed complexity of the classic LemkeHowson algorithm and, in fact, of any algorithm for Bimatrix is not polynomial unless every problem in PPAD is solvable in randomized polynomial time. Our results also have a complexity implication in mathematical economics: • ArrowDebreu market equilibria are PPADhard to compute.
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 85 (11 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.
Finding equilibria in large sequential games of imperfect information
 IN ACM CONFERENCE ON ELECTRONIC COMMERCE
, 2006
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On the complexity of twoplayer winlose games
 In Proceedings of FOCS’05
, 2005
"... The efficient computation of Nash equilibria is one of the most formidable challenges in computational complexity today. The problem remains open for twoplayer games. We show that the complexity of twoplayer Nash equilibria is unchanged when all outcomes are restricted to be 0 or 1. That is, wino ..."
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Cited by 36 (1 self)
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The efficient computation of Nash equilibria is one of the most formidable challenges in computational complexity today. The problem remains open for twoplayer games. We show that the complexity of twoplayer Nash equilibria is unchanged when all outcomes are restricted to be 0 or 1. That is, winorlose games are as complex as the general case for twoplayer games. 1 Game Theory Game theory asks the question: given a set of players playing a certain game, what happens? Computational game theory asks the question: given a representation of a game and some fixed criteria for reasonable play, how may we efficiently compute properties of the possible outcomes? Needless to say, there are many possible ways to define a game, and many more ways to efficiently represent these games. Since the computational complexity of an algorithm is defined as a function of the length of its input representation, different game representations may have significantly different algorithmic consequences. Much work is being done to investigate how to take advantage of some of the more exotic representations of games (see [4, 7, 8, 10] and the references therein). Nevertheless, for two player games, computational game theorists almost exclusively work with the representation known as a rational bimatrix game, which we define as follows. Definition 1 A rational bimatrix game is a game representation that consists of a matrix of pairs of rational numbers
Lossless abstraction of imperfect information games
 JOURNAL OF THE ACM
, 2007
"... 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 transfor ..."
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Cited by 31 (14 self)
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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.
Computing sequential equilibria for twoplayer games
 In SODA ’06
, 2006
"... Koller, Megiddo and von Stengel showed how to efficiently compute minimax strategies for twoplayer extensiveform zerosum games with imperfect information but perfect recall using linear programming and avoiding conversion to normal form. Koller and Pfeffer pointed out that the strategies obtai ..."
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Cited by 27 (1 self)
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Koller, Megiddo and von Stengel showed how to efficiently compute minimax strategies for twoplayer extensiveform zerosum games with imperfect information but perfect recall using linear programming and avoiding conversion to normal form. Koller and Pfeffer pointed out that the strategies obtained by the algorithm are not necessarily sequentially rational and that this deficiency is often problematic for the practical applications. We show how to remove this deficiency by modifying the linear programs constructed by Koller, Megiddo and von Stengel so that pairs of strategies forming a sequential equilibrium are computed. In particular, we show that a sequential equilibrium for a twoplayer zerosum game with imperfect information but perfect recall can be found in polynomial time. In addition, the equilibrium we find is normalform perfect. Our technique generalizes to generalsum games, yielding an algorithm for such games which is likely to be prove practical, even though it is not polynomialtime. 1
Algorithms for approximating Nash equilibria
 IN: PROCEEDINGS OF THE 2ND WORKSHOP ON INTERNET AND NETWORK ECONOMICS (WINE’06
, 2006
"... 1 PROBLEM DEFINITION Nash [13] introduced the concept of Nash equilibria in noncooperative games and proved that any game possesses at least one such equilibrium. A wellknown algorithm for computing a Nash equilibrium of a 2player game is the LemkeHowson algorithm [11], however it has exponentia ..."
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Cited by 26 (5 self)
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1 PROBLEM DEFINITION Nash [13] introduced the concept of Nash equilibria in noncooperative games and proved that any game possesses at least one such equilibrium. A wellknown algorithm for computing a Nash equilibrium of a 2player game is the LemkeHowson algorithm [11], however it has exponential worstcase running time in the number of available pure strategies [15]. Recently, Daskalakis et al [4] showed that the problem of computing a Nash equilibrium in a game with 4 or more players is PPADcomplete; this result was later extended to games with 3 players [7]. Eventually, Chen and Deng [2] proved that the problem is PPADcomplete for 2player games as well. This fact emerged the computation of approximate Nash equilibria. There are several versions of approximate Nash equilibria that have been defined in the literature; however the focus of this entry is on the notions of ɛNash equilibrium and ɛwellsupported Nash equilibrium. An ɛNash equilibrium is a strategy profile such that no deviating player could achieve a payoff higher than the one that the specific profile gives her, plus ɛ. A stronger notion of approximate Nash equilibria is the ɛwellsupported Nash equilibria; these are strategy profiles such that each player plays only
GAMES OF FIXED RANK: A HIERARCHY OF BIMATRIX GAMES
, 2007
"... We propose and investigate a hierarchy of bimatrix games (A, B), whose (entrywise) sum of the payoff matrices of the two players is of rank k, where k is a constant. We will say the rank of such a game is k. For every fixed k, the class of rank kgames strictly generalizes the class of zerosum ga ..."
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Cited by 26 (1 self)
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We propose and investigate a hierarchy of bimatrix games (A, B), whose (entrywise) sum of the payoff matrices of the two players is of rank k, where k is a constant. We will say the rank of such a game is k. For every fixed k, the class of rank kgames strictly generalizes the class of zerosum games, but is a very special case of general bimatrix games. We study both the expressive power and the algorithmic behavior of these games. Specifically, we show that even for k = 1 the set of Nash equilibria of these games can consist of an arbitrarily large number of connected components. While the question of exact polynomial time algorithms to find a Nash equilibrium remains open for games of fixed rank, we present polynomial time algorithms for finding an εapproximation.
Nash equilibria in random games
 IN PROCEEDINGS OF 46TH ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 2007
"... We consider Nash equilibria in 2player random games and analyze a simple Las Vegas algorithm for finding an equilibrium. The algorithm is combinatorial and always finds a Nash equilibrium; on m × n payoff matrices, it runs in time O(m2n log log n + n2m log lo gm) with high probability. Our result ..."
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We consider Nash equilibria in 2player random games and analyze a simple Las Vegas algorithm for finding an equilibrium. The algorithm is combinatorial and always finds a Nash equilibrium; on m × n payoff matrices, it runs in time O(m2n log log n + n2m log lo gm) with high probability. Our result follows from showing that a 2player random game has a Nash equilibrium with supports of size two with high probability, at least 1 − O(1 / log n). Our main tool is a polytope