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 recently-established equivalence between polynomialtime solvability of normal-form games and graphical games, and shows that these kinds of games can implement arbitrary members of a PPAD-complete class of Brouwer functions. 1

We advance significantly beyond the recent progress on the algorithmic complexity of Nash equilibria by solving two major open problems in the approximation of Nash equilibria and in the smoothed analysis of algorithms. • We show that no algorithm with complexity poly(n, 1 ɛ) can compute an ɛ-approximate Nash equilibrium in a two-player game, in which each player has n pure strategies, unless PPAD ⊆ P. In other words, the problem of computing a Nash equilibrium in a twoplayer game does not have a fully polynomial-time approximation scheme unless PPAD ⊆ P. • We prove that no algorithm for computing a Nash equilibrium in a two-player game can have smoothed complexity poly(n, 1 σ) under input perturbation of magnitude σ, unless PPAD ⊆ RP. In particular, the smoothed complexity of the classic Lemke-Howson algorithm is not polynomial unless PPAD ⊆ RP. Instrumental to our proof, we introduce a new discrete fixed-point problem on a high-dimensional hypergrid with constant side-length, and show that it can host the embedding of the proof structure of any PPAD problem. We prove a key geometric lemma for finding a discrete fixed-point, a new concept defined on n +1vertices of a unit hypercube. This lemma enables us to overcome the curse of dimensionality in reasoning about fixed-points in high dimensions. 1

We prove that Bimatrix, the problem of finding a Nash equilibrium in a two-player 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 four-player 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 two-player Nash equilibria. In particular, we prove the following theorems: • Bimatrix does not have a fully polynomial-time approximation scheme unless every problem in PPAD is solvable in polynomial time. • The smoothed complexity of the classic Lemke-Howson 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: • Arrow-Debreu market equilibria are PPAD-hard to compute.

The efficient computation of Nash equilibria is one of the most formidable challenges in computational complexity today. The problem remains open for two-player games. We show that the complexity of two-player Nash equilibria is unchanged when all outcomes are restricted to be 0 or 1. That is, win-or-lose games are as complex as the general case for two-player 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

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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 multi-player 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 close-to-optimal strategies.

We define a generalized strategy eliminability criterion for bimatrix games that considers whether a given strategy is eliminable relative to given dominator & eliminee subsets of the players ’ strategies. We show that this definition spans a spectrum of eliminability criteria from strict dominance (when the sets are as small as possible) to Nash equilibrium (when the sets are as large as possible). We show that checking whether a strategy is eliminable according to this criterion is coNP-complete (both when all the sets are as large as possible and when the dominator sets each have size 1). We then give an alternative definition of the eliminability criterion and show that it is equivalent using the Minimax Theorem. We show how this alternative definition can be translated into a mixed integer program of polynomial size with a number of (binary) integer variables equal to the sum of the sizes of the eliminee sets, implying that checking whether a strategy is eliminable according to the criterion can be done in polynomial time, given that the eliminee sets are small. Finally, we study using the criterion for iterated elimination of strategies. Categories and Subject Descriptors

Abstract. A recent sequence of results established that computing Nash equilibria in normal form games is a PPAD-complete problem even in the case of two players [11,6,4]. By extending these techniques we prove a general theorem, showing that, for a far more general class of families of succinctly representable multiplayer games, the Nash equilibrium problem can also be reduced to the twoplayer case. In view of empirically successful algorithms available for this problem, this is in essence a positive result — even though, due to the complexity of the reductions, it is of no immediate practical significance. We further extend this conclusion to extensive form games and network congestion games, two classes which do not fall into the same succinct representation framework, and for which no positive algorithmic result had been known. 1

Significant work has been done on computational aspects of solving games under various solution concepts, such as Nash equilibrium, subgame perfect Nash equilibrium, correlated equilibrium, and (iterated) dominance. However, the fundamental concepts of rationalizability and CURB (Closed Under Rational Behavior) sets have not, to our knowledge, been studied from a computational perspective. First, for rationalizability we describe an LP-based polynomial algorithm that finds all strategies that are rationalizable against a mixture over a given set of opponent strategies. Then, we describe a series of increasingly sophisticated polynomial algorithms for finding all minimal CURB sets, one minimal CURB set, and the smallest minimal CURB set. Finally, we give theoretical results regarding the relationships between CURB sets and Nash equilibria, showing that finding a Nash equilibrium can be exponential only in the size of the smallest CURB set. We show that this can lead to an arbitrarily large reduction in the complexity of finding a Nash equilibrium. On the downside, we also show that the smallest CURB set can be arbitrarily larger than the supports of the enclosed Nash equilibrium.

Koller, Megiddo and von Stengel showed how to efficiently compute minimax strategies for two-player extensive-form zero-sum games with imperfect information but perfect recall using linear programming and avoiding conversion to normal form. Their algorithm has been used by AI researchers for constructing prescriptive strategies for concrete, often fairly large games. 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 two-player zero-sum game with imperfect information but perfect recall can be found in polynomial time. In addition, the equilibrium we find is normal-form perfect. We also describe an extension of our technique to general-sum games which is likely to be prove practical, even though it is not polynomial-time.