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31
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 226 (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
COMPUTATION OF EQUILIBRIA in Finite Games
, 1996
"... We review the current state of the art of methods for numerical computation of Nash equilibria for nitenperson games. Classical path following methods, such as the LemkeHowson algorithm for two person games, and Scarftype fixed point algorithms for nperson games provide globally convergent metho ..."
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Cited by 118 (1 self)
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We review the current state of the art of methods for numerical computation of Nash equilibria for nitenperson games. Classical path following methods, such as the LemkeHowson algorithm for two person games, and Scarftype fixed point algorithms for nperson games provide globally convergent methods for finding a sample equilibrium. For large problems, methods which are not globally convergent, such as sequential linear complementarity methods may be preferred on the grounds of speed. None of these methods are capable of characterizing the entire set of Nash equilibria. More computationally intensive methods, which derive from the theory of semialgebraic sets are required for finding all equilibria. These methods can also be applied to compute various equilibrium refinements.
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 ..."
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Cited by 101 (22 self)
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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.
Playing Large Games using Simple Strategies
, 2003
"... We prove the existence of #Nash equilibrium strategies with support logarithmic in the number of pure strategies. We also show that the payo#s to all players in any (exact) Nash equilibrium can be #approximated by the payo#s to the players in some such logarithmic support #Nash equilibrium. These ..."
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Cited by 92 (1 self)
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We prove the existence of #Nash equilibrium strategies with support logarithmic in the number of pure strategies. We also show that the payo#s to all players in any (exact) Nash equilibrium can be #approximated by the payo#s 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 first 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 fixed number of players m, the payo#s to all players in any mtuple of mixed strategies can be #approximated by the payo#s in some mtuple of constant support strategies.
Efficient Computation of Behavior Strategies
 GAMES AND ECONOMIC BEHAVIOR
, 1996
"... We propose the sequence form as a new strategic description for an extensive game with perfect recall. It is similar to the normal form but has linear instead of exponential complexity, and allows a direct representation and efficient computation of behavior strategies. Pure strategies and their mix ..."
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Cited by 46 (8 self)
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We propose the sequence form as a new strategic description for an extensive game with perfect recall. It is similar to the normal form but has linear instead of exponential complexity, and allows a direct representation and efficient computation of behavior strategies. Pure strategies and their mixed strategy probabilities are replaced by sequences of consecutive choices and their realization probabilities. A zerosum game is solved by a corresponding linear program that has linear size in the size of the game tree. General twoperson games are studied in the paper by Koller, Megiddo, and von Stengel in this journal issue.
Exponentially Many Steps for Finding a Nash Equilibrium in a Bimatrix Game
 IN PROCEEDINGS OF THE ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS
, 2004
"... The LemkeHowson algorithm is the classical algorithm for the problem NASH of finding one Nash equilibrium of a bimatrix game. It provides a constructive, elementary proof of existence of an equilibrium, by a typical "directed parity argument", which puts NASH into the complexity class PPA ..."
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Cited by 45 (1 self)
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The LemkeHowson algorithm is the classical algorithm for the problem NASH of finding one Nash equilibrium of a bimatrix game. It provides a constructive, elementary proof of existence of an equilibrium, by a typical "directed parity argument", which puts NASH into the complexity class PPAD. This paper presents a class of bimatrix games for which the LemkeHowson algorithm takes, even in the best case, exponential time in the dimension d of the game, requiring #((# 3=4 ) d ) many steps, where # is the Golden Ratio. The "parity argument" for NASH is thus explicitly shown to be inefficient. The games are constructed using pairs of dual cyclic polytopes with 2d suitably labeled facets in dspace.
Finding Mixed Strategies with Small Supports in Extensive Form Games
 International Journal of Game Theory
, 1995
"... The complexity of algorithms that compute strategies or operate on them typically depends on the representation length of the strategies involved. One measure for the size of a mixed strategy is the number of strategies in its supportthe set of pure strategies to which it gives positive probabili ..."
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Cited by 24 (1 self)
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The complexity of algorithms that compute strategies or operate on them typically depends on the representation length of the strategies involved. One measure for the size of a mixed strategy is the number of strategies in its supportthe set of pure strategies to which it gives positive probability. This paper investigates the existence of "small" mixed strategies in extensive form games, and how such strategies can be used to create more efficient algorithms. The basic idea is that, in an extensive form game, a mixed strategy induces a small set of realization weights that completely describe its observable behavior. This fact can be used to show that for any mixed strategy ¯, there exists a realizationequivalent mixed strategy ¯ 0 whose size is at most the size of the game tree. For a player with imperfect recall, the problem of finding such a strategy ¯ 0 (given the realization weights) is NPhard. On the other hand, if ¯ is a behavior strategy, ¯ 0 can be constructed from...
A fictitious play approach to largescale optimization
, 2003
"... In this paper we investigate the properties of the sampled version of the fictitious play algorithm, familiar from game theory, for games with identical payoffs, and propose a heuristic based on fictitious play as a solution procedure for discrete optimization problems of the form max{u(y) : y = (y ..."
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Cited by 21 (4 self)
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In this paper we investigate the properties of the sampled version of the fictitious play algorithm, familiar from game theory, for games with identical payoffs, and propose a heuristic based on fictitious play as a solution procedure for discrete optimization problems of the form max{u(y) : y = (y 1,..., y n) ∈ Y 1 × · · · × Y n}, i.e., in which the feasible region is a Cartesian product of finite sets Y i, i ∈ N = {1,..., n}. The contributions of this paper are twofold. In the first part of the paper we broaden the existing results on convergence properties of the fictitious play algorithm on games with identical payoffs to include an approximate fictitious play algorithm which allows for errors in players ’ best replies. Moreover, we introduce samplingbased approximate fictitious play which possesses the above convergence properties, and at the same time provides a computationally efficient method for implementing fictitious play. In the second part of the paper we motivate the use of algorithms based on sampled fictitious play to solve optimization problems in the above form with particular focus on the problems in which the objective function u(·) comes from a “black box,” such as a simulation model, where significant computational effort is required for each function evaluation.
Stationary Equilibria in Stochastic Games: Structure, Selection, and Computation
, 2000
"... This paper is the first to introduce an algorithm to compute stationary equilibria in stochastic games, and shows convergence of the algorithm for almost all such games. Moreover, since in general the number of stationary equilibria is overwhelming, we payattention to the issue of equilibrium select ..."
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Cited by 13 (1 self)
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This paper is the first to introduce an algorithm to compute stationary equilibria in stochastic games, and shows convergence of the algorithm for almost all such games. Moreover, since in general the number of stationary equilibria is overwhelming, we payattention to the issue of equilibrium selection. We do this by extending the linear tracing procedure to the class of stochastic games, called the stochastic tracing procedure. From a computational point of view, the class of stochastic games possesses substantial difficulties compared to normal form games. Apart from technical difficulties, there are also conceptual difficulties, for instance the question how to extend the linear tracing procedure to the environment of stochastic games. We prove that there is a generic subclass of the class of stochastic games for which the stochastic tracing procedure is a compact onedimensional piecewise differentiable manifold with boundary. Furthermore, we prove that the stochastic tracing procedure generates a unique path leading from any exogenously specified prior belief, to a stationary equilibrium. A wellchosen transformation of variables is used to formulate an everywhere differentiable homotopy function, whose zeros describe the (unique) path generated by the stochastic tracing procedure. Because of differentiability we are able to follow this path using standard pathfollowing techniques. This yields a globally convergent algorithm that is easily and robustly implemented on a computer using existing software routines. As a byproduct of our results, we extend a recent result on the generic finiteness of stationary equilibria in stochastic games to oddness of equilibria.
Universality of Nash equilibria
 Mathematics of Operations Research
, 2003
"... ABSTRACT. Every real algebraic variety is isomorphic to the set of totally mixed Nash equilibria of some threeperson game, and also to the set of totally mixed Nash equilibria of an Nperson game in which each player has two pure strategies. From the NashTognoli Theorem it follows that every compa ..."
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Cited by 10 (2 self)
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ABSTRACT. Every real algebraic variety is isomorphic to the set of totally mixed Nash equilibria of some threeperson game, and also to the set of totally mixed Nash equilibria of an Nperson game in which each player has two pure strategies. From the NashTognoli Theorem it follows that every compact differentiable manifold can be encoded as the set of totally mixed Nash equilibria of some game. Moreover, there exist isolated Nash equilibria of arbitrary topological degree. 1.