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 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.

doi 10.1287/moor.1080.0322

Abstract not found

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

We present a technique for reducing a normal-form (aka. (bi)matrix) game, O, to a smaller normal-form game, R, for the purpose of computing a Nash equilibrium. This is done by computing a Nash equilibrium for a subcomponent, G, of O for which a certain condition holds. We also show that such a subcomponent G on which to apply the technique can be found in polynomial time (if it exists), by showing that the condition that G needs to satisfy can be modeled as a Horn satisfiability problem. We show that the technique does not extend to computing Pareto-optimal or welfaremaximizing equilibria. We present a class of games, which we call ALAGIU (Any Lower Action Gives Identical Utility) games, for which recursive application of (special cases of) the technique is su#cient for finding a Nash equilibrium in linear time. Finally, we discuss using the technique to compute approximate Nash equilibria.

We consider markets in the classical Arrow-Debreu model. There are n agents and m goods. Each buyer has a concave utility function (of the bundle of goods he/she buys) and an initial bundle. At an “equilibrium ” set of prices for goods, if each individual buyer separately exchanges the initial bundle for an optimal bundle at the set prices, the market clears, i.e., all goods are exactly consumed. Classical theorems guarantee the existence of equilibria, but computing them has been the subject of much recent research. In the related area of Multi-Agent Games, much attention has been paid to the complexity as well as algorithms. While most general problems are hard, polynomial time algorithms have been developed for restricted classes of games, when one assumes the number of strategies is constant [20, 11]. For the Market Equilibrium problem, several important special cases of utility functions have been tackled. Here we begin a program for this problem similar to that for multiagent games, where general utilities are considered. We begin by showing that if the utilities are separable piece-wise linear concave (PLC) functions, and the number of goods (or alternatively the number of buyers) is constant, then we can compute an exact equilibrium in polynomial time. Our technique for the constant number of goods is to decompose the space of price vectors into cells using certain hyperplanes, so that in each cell, each buyer’s threshold marginal utility is known. Still, one needs to solve a linear optimization problem in each cell. We then show the main result- that for general (non-separable) PLC utilities, an exact equilibrium can be found in polynomial time provided the number of goods is constant. The starting point of the algorithm is a “cell-decomposition ” of the space of price vectors using polynomial surfaces (instead of hyperplanes). We use results from computational algebraic geometry to bound the number of such cells. For solving the problem inside each cell, we introduce and use a novel LP-duality

Abstract. This is the 26th edition of a column that covers new developments in the theory of NP-completeness. The presentation is modeled on that which M. R. Garey and I used in our book “Computers and Intractability: A Guide to the Theory of NP-Completeness, ” W. H. Freeman & Co., New York, 1979, hereinafter referred to as “[G&J]. ” Previous columns, the first 23 of which appeared in J. Algorithms, will be referred to by a combination of their sequence number and year of appearance, e.g., “Column 1 [1981]. ” Full bibliographic details on the previous columns, as well as downloadable unofficial versions of them, can be found at

In this paper we make a comprehensive study of the complexity of the problem of deciding the existence of equilibria in strategic games with incomplete information, in case of pure strategies. In particular, we show that this is NP-complete in general Bayesian Games in Standard Normal Form, and that it becomes PP-hard (and, in fixed-precision scenarios, PP-complete), when the game is represented succinctly in General Normal Form. Suitable restrictions in case of graphical games that make the problem tractable are also discussed.

In this paper, we study nonzero-sum separable games, which are continuous games whose payoffs take a sum-of-products form. Included in this subclass are all finite games and polynomial games. We investigate the structure of equilibria in separable games. We show that these games admit finitely supported Nash equilibria. Motivated by the bounds on the supports of mixed equilibria in two-player finite games in terms of the ranks of the payoff matrices, we define the notion of the rank of an n-player continuous game and use this to provide bounds on the cardinality of the support of equilibrium strategies. We present a general characterization theorem that states that a continuous game has finite rank if and only if it is separable. Using our rank results, we present an efficient algorithm for computing approximate equilibria of two-player separable games with fixed strategy spaces in time polynomial in the rank of the game. 1