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44
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 329 (23 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
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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Cited by 88 (5 self)
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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.
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
Market equilibria in polynomial time for fixed number of goods or agents
 In FOCS
, 2008
"... We consider markets in the classical ArrowDebreu 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 bundl ..."
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Cited by 25 (3 self)
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We consider markets in the classical ArrowDebreu 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 MultiAgent 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 piecewise 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 (nonseparable) 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 “celldecomposition ” 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 LPduality
On the Complexity of PureStrategy Nash Equilibria in Congestion and LocalEffect Games
 In Proc. of the 2nd Int. Workshop on Internet and Network Economics (WINE
, 2006
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The Computational Complexity of Nash Equilibria in Concisely Represented Games
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 52
, 2005
"... Games may be represented in many different ways, and different representations of games affect the complexity of problems associated with games, such as finding a Nash equilibrium. The traditional method of representing a game is to explicitly list all the payoffs, but this incurs an exponential blo ..."
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Cited by 20 (1 self)
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Games may be represented in many different ways, and different representations of games affect the complexity of problems associated with games, such as finding a Nash equilibrium. The traditional method of representing a game is to explicitly list all the payoffs, but this incurs an exponential blowup as the number of agents grows. We study two models of concisely represented games: circuit games, where the payoffs are computed by a given boolean circuit, and graph games, where each agent’s payoff is a function of only the strategies played by its neighbors in a given graph. For these two models, we study the complexity of four questions: determining if a given strategy is a Nash equilibrium, finding a Nash equilibrium, determining if there exists a pure Nash equilibrium, and determining if there exists a Nash equilibrium in which the payoffs to a player meet some given guarantees. In many cases, we obtain tight results, showing that the problems are complete for various complexity classes.
A generalized strategy eliminability criterion and computational methods for applying it
 In Proceedings of the National Conference on Artificial Intelligence (AAAI
, 2005
"... 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 domina ..."
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Cited by 15 (6 self)
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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 coNPcomplete (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
Separable and lowrank continuous games
 Int. J. Game Theory,37,475–504
, 2008
"... In this paper, we study nonzerosum separable games, which are continuous games whose payoffs take a sumofproducts 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 suppo ..."
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Cited by 14 (10 self)
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In this paper, we study nonzerosum separable games, which are continuous games whose payoffs take a sumofproducts 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 twoplayer finite games in terms of the ranks of the payoff matrices, we define the notion of the rank of an nplayer 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 twoplayer separable games with fixed strategy spaces in time polynomial in the rank of the game. 1
On the Approximation and Smoothed Complexity of Leontief Market Equilibria
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 31 (2006)
, 2006
"... We show that the problem of finding an ɛapproximate Nash equilibrium of an n × n twoperson games can be reduced to the computation of an (ɛ/n) 2approximate market equilibrium of a Leontief economy. Together with a recent result of Chen, Deng and Teng, this polynomial reduction implies that the Le ..."
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Cited by 10 (4 self)
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We show that the problem of finding an ɛapproximate Nash equilibrium of an n × n twoperson games can be reduced to the computation of an (ɛ/n) 2approximate market equilibrium of a Leontief economy. Together with a recent result of Chen, Deng and Teng, this polynomial reduction implies that the Leontief market exchange problem does not have a fully polynomialtime approximation scheme, that is, there is no algorithm that can compute an ɛapproximate market equilibrium in time polynomial in m, n, and 1/ɛ, unless PPAD ⊆ P, We also extend the analysis of our reduction to show, unless PPAD ⊆ RP, that the smoothed complexity of the Scarf’s general fixedpoint approximation algorithm (when applying to solve the approximate Leontief market exchange problem) or of any algorithm for computing an approximate market equilibrium of Leontief economies is not polynomial in n and 1/σ, under Gaussian or uniform perturbations with magnitude σ.