Results 1  10
of
200
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 ..."
Abstract

Cited by 324 (23 self)
 Add to MetaCart
(Show Context)
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
2006), “Settling the complexity of twoplayer Nash equilibrium
 Proc. 47th Symp. Foundations of Computer Science (FOCS
"... ..."
(Show Context)
Market Equilibrium via a PrimalDual Algorithm for a Convex Program
"... We give the first polynomial time algorithm for exactly computing an equilibrium for the linear utilities case of the market model defined by Fisher. Our algorithm uses the primaldual paradigm in the enhanced setting of KKT conditions and convex programs. We pinpoint the added difficulty raised by ..."
Abstract

Cited by 129 (27 self)
 Add to MetaCart
(Show Context)
We give the first polynomial time algorithm for exactly computing an equilibrium for the linear utilities case of the market model defined by Fisher. Our algorithm uses the primaldual paradigm in the enhanced setting of KKT conditions and convex programs. We pinpoint the added difficulty raised by this setting and the manner in which our algorithm circumvents it.
Playing large games using simple strategies
 IN: PROC. OF THE 4TH ACM CONF. ON EL. COMMERCE (EC ’03). ASSOC. OF COMP. MACH
, 2003
"... We prove the existence of Nash equilibrium strategies with support logarithmic in the number of pure strategies. We also show that the payoffs to all players in any (exact) Nash equilibrium can be approximated by the payoffs to the players in some such logarithmic support Nash equilibrium. These ..."
Abstract

Cited by 123 (4 self)
 Add to MetaCart
(Show Context)
We prove the existence of Nash equilibrium strategies with support logarithmic in the number of pure strategies. We also show that the payoffs to all players in any (exact) Nash equilibrium can be approximated by the payoffs to the players in some such logarithmic support Nash equilibrium. These strategies are also uniform on a multiset of logarithmic size and therefore this leads to a quasipolynomial algorithm for computing an Nash equilibrium. To our knowledge this is the rst subexponential algorithm for finding an Nash equilibrium. Our results hold for any multipleplayer game as long as the number of players is a constant (i.e., it is independent of the number of pure strategies). A similar argument also proves that for a xed number of players m, the payos to all players in any mtuple of mixed strategies can be approximated by the payos in some mtuple of constant support strategies. We also prove that if the payoff matrices of a two person game have low rank then the game has an exact Nash equilibrium with small support. This implies that if the payoff matrices can be well approximated by low rank matrices, the game has an equilibrium with small support. It also implies that if the payo matrices have constant rank we can compute an exact Nash equilibrium in polynomial time.
The Structure and Complexity of Nash Equilibria for a Selfish Routing Game
, 2002
"... In this work, we study the combinatorial structure and the computational complexity of Nash equilibria for a certain game that models sel sh routing over a network consisting of m parallel links. We assume a collection of n users, each employing a mixed strategy, which is a probability distribu ..."
Abstract

Cited by 122 (28 self)
 Add to MetaCart
(Show Context)
In this work, we study the combinatorial structure and the computational complexity of Nash equilibria for a certain game that models sel sh routing over a network consisting of m parallel links. We assume a collection of n users, each employing a mixed strategy, which is a probability distribution over links, to control the routing of its own assigned trac. In a Nash equilibrium, each user sel shly routes its trac on those links that minimize its expected latency cost, given the network congestion caused by the other users. The social cost of a Nash equilibrium is the expectation, over all random choices of the users, of the maximum, over all links, latency through a link.
Efficient Computation of Equilibria for Extensive Twoperson Games
, 1996
"... The Nash equilibria of a twoperson, nonzerosum game are the solutions of a certain linear complementarity problem (LCP). In order to use this for solving a game in extensive form, the game must first be converted to a strategic description such as the normal form. The classical normal form, howev ..."
Abstract

Cited by 120 (8 self)
 Add to MetaCart
The Nash equilibria of a twoperson, nonzerosum game are the solutions of a certain linear complementarity problem (LCP). In order to use this for solving a game in extensive form, the game must first be converted to a strategic description such as the normal form. The classical normal form, however, is often exponentially large in the size of the game tree. If the game has perfect recall, a linearsized strategic description is the sequence form. For the resulting small LCP, we show that an equilibrium is found efficiently by Lemke’s algorithm, a generalization of the Lemke–Howson method.
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 ..."
Abstract

Cited by 88 (5 self)
 Add to MetaCart
(Show Context)
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.
Computing Nash equilibria: Approximation and smoothed complexity
 In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS
, 2006
"... By proving that the problem of computing a 1/n Θ(1)approximate Nash equilibrium remains PPADcomplete, we show that the BIMATRIX game is not likely to have a fully polynomialtime approximation scheme. In other words, no algorithm with time polynomial in n and 1/ǫ can compute an ǫapproximate Nash ..."
Abstract

Cited by 86 (11 self)
 Add to MetaCart
(Show Context)
By proving that the problem of computing a 1/n Θ(1)approximate Nash equilibrium remains PPADcomplete, we show that the BIMATRIX game is not likely to have a fully polynomialtime approximation scheme. In other words, no algorithm with time polynomial in n and 1/ǫ can compute an ǫapproximate Nash equilibrium of an n×n bimatrix game, unless PPAD ⊆ P. Instrumental to our proof, we introduce a new discrete fixedpoint problem on a highdimensional cube with a constant sidelength, such as on an ndimensional cube with sidelength 7, and show that they are PPADcomplete. Furthermore, we prove that it is unlikely, unless PPAD ⊆ RP, that the smoothed complexity of the LemkeHowson algorithm or any algorithm for computing a Nash equilibrium of a bimatrix game is polynomial in n and 1/σ under perturbations with magnitude σ. Our result answers a major open question in the smoothed analysis of algorithms and the approximation of Nash equilibria.
A deterministic subexponential algorithm for solving parity games
 SODA
, 2006
"... The existence of polynomial time algorithms for the solution of parity games is a major open problem. The fastest known algorithms for the problem are randomized algorithms that run in subexponential time. These algorithms are all ultimately based on the randomized subexponential simplex algorithms ..."
Abstract

Cited by 82 (3 self)
 Add to MetaCart
The existence of polynomial time algorithms for the solution of parity games is a major open problem. The fastest known algorithms for the problem are randomized algorithms that run in subexponential time. These algorithms are all ultimately based on the randomized subexponential simplex algorithms of Kalai and of Matousek, Sharir and Welzl. Randomness seems to play an essential role in these algorithms. We use a completely different, and elementary, approach to obtain a deterministic subexponential algorithm for the solution of parity games. The new algorithm, like the existing randomized subexponential algorithms, uses only polynomial space, and it is almost as fast as the randomized subexponential algorithms mentioned above.