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29
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 227 (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
Settling the complexity of twoplayer Nash equilibrium
 In Proc. 47th FOCS
, 2006
"... We prove that the problem of finding a Nash equilibrium in a twoplayer game is PPADcomplete. 1 ..."
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Cited by 106 (5 self)
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We prove that the problem of finding a Nash equilibrium in a twoplayer game is PPADcomplete. 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 46 (3 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.
On the Complexity of Nash Equilibria and Other Fixed Points (Extended Abstract)
 IN PROC. FOCS
, 2007
"... We reexamine what it means to compute Nash equilibria and, more generally, what it means to compute a fixed point of a given Brouwer function, and we investigate the complexity of the associated problems. Specifically, we study the complexity of the following problem: given a finite game, Γ, with 3 ..."
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Cited by 39 (4 self)
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We reexamine what it means to compute Nash equilibria and, more generally, what it means to compute a fixed point of a given Brouwer function, and we investigate the complexity of the associated problems. Specifically, we study the complexity of the following problem: given a finite game, Γ, with 3 or more players, and given ɛ> 0, compute an approximation within ɛ of some (actual) Nash equilibrium. We show that approximation of an actual Nash Equilibrium, even to within any nontrivial constant additive factor ɛ < 1/2 in just one desired coordinate, is at least as hard as the long standing squareroot sum problem, as well as a more general arithmetic circuit decision problem that characterizes Ptime in a unitcost model of computation with arbitrary precision rational arithmetic; thus placing the approximation problem in P, or even NP, would resolve major open problems in the complexity of numerical computation. We show similar results for market equilibria: it is hard to estimate with any nontrivial accuracy the equilibrium prices in an exchange economy with a unique equilibrium, where the economy is given by explicit algebraic formulas for the excess demand functions. We define a class, FIXP, which captures search problems that can be cast as fixed point
S.H.: Settling the complexity of ArrowDebreu equilibria in markets with additively separable utilities
 In: Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science
, 2009
"... We prove that the problem of computing an ArrowDebreu market equilibrium is PPADcomplete even when all traders use additively separable, piecewiselinear and concave utility functions. In fact, our proof shows that this marketequilibrium problem does not have a fully polynomialtime approximation ..."
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Cited by 18 (3 self)
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We prove that the problem of computing an ArrowDebreu market equilibrium is PPADcomplete even when all traders use additively separable, piecewiselinear and concave utility functions. In fact, our proof shows that this marketequilibrium problem does not have a fully polynomialtime approximation scheme unless every problem in PPAD is solvable in polynomial time.
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
"... doi 10.1287/moor.1080.0322 ..."
The approximation complexity of winlose games
 In Proceedings of SODA
, 2007
"... We further our algorithmic and structural understanding of Nash equilibria. Specifically: We distill the hard core of the complexity of Nash equilibria, showing that even correctly computing a logarithmic number of bits of the equilibrium strategies of a twoplayer winlose game is as hard as the g ..."
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Cited by 8 (2 self)
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We further our algorithmic and structural understanding of Nash equilibria. Specifically: We distill the hard core of the complexity of Nash equilibria, showing that even correctly computing a logarithmic number of bits of the equilibrium strategies of a twoplayer winlose game is as hard as the general problem. We prove the following structural result about Nash equilibria: “the set of approximate equilibria of a zerosum game is convex. ” 1 1
Computing equilibria: A computational complexity perspective
, 2009
"... Computational complexity is the subfield of computer science that rigorously studies the intrinsic difficulty of computational problems. This survey explains how complexity theory defines “hard problems”; applies these concepts to several equilibrium computation problems; and discusses implications ..."
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Cited by 7 (2 self)
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Computational complexity is the subfield of computer science that rigorously studies the intrinsic difficulty of computational problems. This survey explains how complexity theory defines “hard problems”; applies these concepts to several equilibrium computation problems; and discusses implications for computation, games, and behavior. We assume
Welfare maximization in congestion games
 IEEE JSAC special issue on NonCooperative Behavior in Networking. Preliminary version in EC’06
"... Congestion games are noncooperative games where the utility of a player from using a certain resource depends on the total number of players that are using the same resource. While most work so far took a distributed gametheoretic approach to this problem, this paper studies centralized solutions ..."
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Cited by 7 (2 self)
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Congestion games are noncooperative games where the utility of a player from using a certain resource depends on the total number of players that are using the same resource. While most work so far took a distributed gametheoretic approach to this problem, this paper studies centralized solutions for congestion games. The first part of the paper analyzes the problem from a computational perspective. We analyze the computational complexity of the welfaremaximization problem, for which we provide both approximation algorithms and lower bounds. We study this optimization problem under different kinds of congestion effects (externalities) among the players: positive, negative, and unrestricted. Our main algorithmic result is a constant approximation algorithm for congestion games with unrestricted externalities. In the second part of the paper, we also take the strategic behavior of the players into account, and present centralized truthful mechanisms for congestiongame environments. Our main result in this part is an incentivecompatible mechanism for mresource nplayer congestion games that achieves an O ( √ m log n) approximation to the optimal welfare. We also describe an important and useful connection between congestion games and combinatorial auctions. This connection allows us to use insights and methods from the combinatorialauction literature for solving congestiongame problems.