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86
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 324 (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 twoplayer Nash equilibrium
 in: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
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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 68 (8 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
A note on approximate Nash equilibria
 IN WINE ’06
, 2006
"... In view of the intractability of finding a Nash equilibrium, it is important to understand the limits of approximation in this context. A subexponential approximation scheme is known [LMM03], and no approximation better than 1 is possible by any algorithm that examines 4 equilibria involving fewer ..."
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Cited by 54 (8 self)
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In view of the intractability of finding a Nash equilibrium, it is important to understand the limits of approximation in this context. A subexponential approximation scheme is known [LMM03], and no approximation better than 1 is possible by any algorithm that examines 4 equilibria involving fewer than log n strategies [Alt94]. We give a simple, lineartime algorithm examining just two strategies per player and resulting in a 1approximate Nash equilibrium in any 2player game. For the 2 more demanding notion of wellsupported approximate equilibrium due to [DGP06] no nontrivial bound is known; we show that the problem can be reduced to the case of winlose games (games with all utilities 0 − 1), and that an approximation of 5 is possible contingent upon a 6 graphtheoretic conjecture.
An optimization approach for approximate Nash equilibria
 In 3rd international Workshop on Internet and Network Economics, Proceedings of
, 2007
"... Abstract. In this paper we propose a new methodology for determining approximate Nash equilibria of noncooperative bimatrix games and, based on that, we provide an efficient algorithm that computes 0.3393approximate equilibria, the best approximation till now. The methodology is based on the formul ..."
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Cited by 42 (4 self)
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Abstract. In this paper we propose a new methodology for determining approximate Nash equilibria of noncooperative bimatrix games and, based on that, we provide an efficient algorithm that computes 0.3393approximate equilibria, the best approximation till now. The methodology is based on the formulation of an appropriate function of pairs of mixed strategies reflecting the maximum deviation of the players ’ payoffs from the best payoff each player could achieve given the strategy chosen by the other. We then seek to minimize such a function using descent procedures. As it is unlikely to be able to find global minima in polynomial time, given the recently proven intractability of the problem, we concentrate on the computation of stationary points and prove that they can be approximated arbitrarily close in polynomial time and that they have the above mentioned approximation property. Our result provides the best ɛ till now for polynomially computable ɛapproximate Nash equilibria of bimatrix games. Furthermore, our methodology for computing approximate Nash equilibria has not been used by others. 1
The complexity of game dynamics: Bgp oscillations, sink equilibria, and beyond
 In SODA ’08: Proceedings of the nineteenth annual ACMSIAM symposium on Discrete algorithms
, 2008
"... We settle the complexity of a wellknown problem in networking by establishing that it is PSPACEcomplete to tell whether a system of path preferences in the BGP protocol [25] can lead to oscillatory behavior; one key insight is that the BGP oscillation question is in fact one about Nash dynamics. W ..."
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Cited by 34 (4 self)
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We settle the complexity of a wellknown problem in networking by establishing that it is PSPACEcomplete to tell whether a system of path preferences in the BGP protocol [25] can lead to oscillatory behavior; one key insight is that the BGP oscillation question is in fact one about Nash dynamics. We show that the concept of sink equilibria proposed recently in [11] is also PSPACEcomplete to analyze and approximate for graphical games. Finally, we propose a new equilibrium concept inspired by game dynamics, unit recall equilibria, which we show to be close to universal (exists with high probability in a random game) and algorithmically promising. We also give a relaxation thereof, called componentwise unit recall equilibria, which we show to be both tractable and universal (guaranteed to exist in every game).
How hard is it to approximate the best Nash equilibrium?
, 2009
"... The quest for a PTAS for Nash equilibrium in a twoplayer game seeks to circumvent the PPADcompleteness of an (exact) Nash equilibrium by finding an approximate equilibrium, and has emerged as a major open question in Algorithmic Game Theory. A closely related problem is that of finding an equilibri ..."
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Cited by 32 (0 self)
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The quest for a PTAS for Nash equilibrium in a twoplayer game seeks to circumvent the PPADcompleteness of an (exact) Nash equilibrium by finding an approximate equilibrium, and has emerged as a major open question in Algorithmic Game Theory. A closely related problem is that of finding an equilibrium maximizing a certain objective, such as the social welfare. This optimization problem was shown to be NPhard by Gilboa and Zemel [Games and Economic Behavior 1989]. However, this NPhardness is unlikely to extend to finding an approximate equilibrium, since the latter admits a quasipolynomial time algorithm, as proved by Lipton, Markakis and Mehta [Proc. of 4th EC, 2003]. We show that this optimization problem, namely, finding in a twoplayer game an approximate equilibrium achieving large social welfare is unlikely to have a polynomial time algorithm. One interpretation of our results is that the quest for a PTAS for Nash equilibrium should not extend to a PTAS for finding the best Nash equilibrium, which stands in contrast to certain algorithmic techniques used so far (e.g. sampling and enumeration). Technically, our result is a reduction from a notoriously difficult problem in modern Combinatorics, of finding a planted (but hidden) clique in a random graph G(n, 1/2). Our reduction starts from an instance with planted clique size k = O(log n). For comparison, the currently known algorithms due to Alon, Krivelevich and Sudakov [Random Struct. & Algorithms, 1998], and Krauthgamer and Feige [Random Struct. & Algorithms, 2000], are effective for a much larger clique size k = Ω(√n).
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 31 (5 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.
Computing equilibria in anonymous games
 in 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS
, 2007
"... We present efficient approximation algorithms for finding Nash equilibria in anonymous games, that is, games in which the players utilities, though different, do not differentiate between other players. Our results pertain to such games with many players but few strategies. We show that any such gam ..."
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Cited by 31 (5 self)
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We present efficient approximation algorithms for finding Nash equilibria in anonymous games, that is, games in which the players utilities, though different, do not differentiate between other players. Our results pertain to such games with many players but few strategies. We show that any such game has an approximate pure Nash equilibrium, computable in polynomial time, with approximation O(s 2 λ), where s is the number of strategies and λ is the Lipschitz constant of the utilities. Finally, we show that there is a PTAS for finding an ɛapproximate Nash equilibrium when the number of strategies is two. 1