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54
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 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.
FlightPath: Obedience vs choice in cooperative services
 In OSDI 2008
, 2008
"... Abstract: We present FlightPath, a novel peertopeer streaming application that provides a highly reliable data stream to a dynamic set of peers. We demonstrate that FlightPath reduces jitter compared to previous works by several orders of magnitude. Furthermore, FlightPath uses a number of runtim ..."
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Cited by 49 (7 self)
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Abstract: We present FlightPath, a novel peertopeer streaming application that provides a highly reliable data stream to a dynamic set of peers. We demonstrate that FlightPath reduces jitter compared to previous works by several orders of magnitude. Furthermore, FlightPath uses a number of runtime adaptations to maintain low jitter despite 10 % of the population behaving maliciously and the remaining peers acting selfishly. At the core of FlightPath’s success are approximate equilibria. These equilibria allow us to design incentives to limit selfish behavior rigorously, yet they provide sufficient flexibility to build practical systems. We show how to use an εNash equilibrium, instead of a strict Nash, to engineer a live streaming system that uses bandwidth efficiently, absorbs flash crowds, adapts to sudden peer departures, handles churn, and tolerates malicious activity. 1
Gradientbased algorithms for finding nash equilibria in extensive form games
 In Proceedings of the Eighteenth International Conference on Game Theory
, 2007
"... We present a computational approach to the saddlepoint formulation for the Nash equilibria of twoperson, zerosum sequential games of imperfect information. The algorithm is a firstorder gradient method based on modern smoothing techniques for nonsmooth convex optimization. The algorithm requires ..."
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Cited by 44 (15 self)
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We present a computational approach to the saddlepoint formulation for the Nash equilibria of twoperson, zerosum sequential games of imperfect information. The algorithm is a firstorder gradient method based on modern smoothing techniques for nonsmooth convex optimization. The algorithm requires O(1/ɛ) iterations to compute an ɛequilibrium, and the work per iteration is extremely low. These features enable us to find approximate Nash equilibria for sequential games with a tree representation of about 10 10 nodes. This is three orders of magnitude larger than what previous algorithms can handle. We present two heuristic improvements to the basic algorithm and demonstrate their efficacy on a range of realworld games. Furthermore, we demonstrate how the algorithm can be customized to a specific class of problems with enormous memory savings. 1
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).
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
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 31 (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).
Polynomial algorithms for approximating Nash equilibria of bimatrix games
 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
Approximating Nash equilibria using smallsupport strategies
 ACM CONFERENCE ON ELECTRONIC COMMERCE
, 2007
"... We study the problem of finding approximate Nash equilibria of two player games. We show that for any 0 < ǫ < 1, there is no 1approximate equilibrium with strategies of 1+ǫ log n support O( ǫ2). ..."
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Cited by 25 (1 self)
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We study the problem of finding approximate Nash equilibria of two player games. We show that for any 0 < ǫ < 1, there is no 1approximate equilibrium with strategies of 1+ǫ log n support O( ǫ2).