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38
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
On the price of anarchy and stability of correlated equilibria of linear congestion games
, 2005
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Finding equilibria in large sequential games of imperfect information
 In ACM Conference on Electronic Commerce
, 2006
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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 23 (4 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
Lossless abstraction of imperfect information games
 Journal of the ACM
, 2007
"... Abstract. Finding an equilibrium of an extensive form game of imperfect information is a fundamental problem in computational game theory, but current techniques do not scale to large games. To address this, we introduce the ordered game isomorphism and the related ordered game isomorphic abstractio ..."
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Cited by 21 (9 self)
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Abstract. Finding an equilibrium of an extensive form game of imperfect information is a fundamental problem in computational game theory, but current techniques do not scale to large games. To address this, we introduce the ordered game isomorphism and the related ordered game isomorphic abstraction transformation. For a multiplayer sequential game of imperfect information with observable actions and an ordered signal space, we prove that any Nash equilibrium in an abstracted smaller game, obtained by one or more applications of the transformation, can be easily converted into a Nash equilibrium in the original game. We present an algorithm, GameShrink, for abstracting the game using our isomorphism exhaustively. Its complexity is Õ(n2), where n is the number of nodes in a structure we call the signal tree. It is no larger than the game tree, and on nontrivial games it is drastically smaller, so GameShrink has time and space complexity sublinear in the size of the game tree. Using GameShrink, we find an equilibrium to a poker game with 3.1 billion nodes—over four orders of magnitude more than in the largest poker game solved previously. To address even larger games, we introduce approximation methods that do not preserve equilibrium, but nevertheless yield (ex post) provably closetooptimal strategies.
Symmetries and the Complexity of Pure Nash Equilibrium
, 2006
"... Strategic games may exhibit symmetries in a variety of ways. A common aspect of symmetry, enabling the compact representation of games even when the number of players is unbounded, is that players cannot (or need not) distinguish between the other players. We define four classes of symmetric games b ..."
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Cited by 20 (3 self)
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Strategic games may exhibit symmetries in a variety of ways. A common aspect of symmetry, enabling the compact representation of games even when the number of players is unbounded, is that players cannot (or need not) distinguish between the other players. We define four classes of symmetric games by considering two additional properties: identical payoff functions for all players and the ability to distinguish oneself from the other players. Based on these varying notions of symmetry, we investigate the computational complexity of pure Nash equilibria. It turns out that in all four classes of games equilibria can be found efficiently when only a constant number of actions is available to each player, a problem that has been shown intractable for other succinct representations of multiplayer games. We further show that identical payoff functions simplify the search for equilibria, while a growing number of actions renders it intractable. Finally, we show that our results extend to wider classes of threshold symmetric games where players are unable to determine the exact number of players playing a certain action.
GAMES OF FIXED RANK: A HIERARCHY OF BIMATRIX GAMES
, 2007
"... We propose and investigate a hierarchy of bimatrix games (A, B), whose (entrywise) sum of the payoff matrices of the two players is of rank k, where k is a constant. We will say the rank of such a game is k. For every fixed k, the class of rank kgames strictly generalizes the class of zerosum ga ..."
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Cited by 19 (1 self)
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We propose and investigate a hierarchy of bimatrix games (A, B), whose (entrywise) sum of the payoff matrices of the two players is of rank k, where k is a constant. We will say the rank of such a game is k. For every fixed k, the class of rank kgames strictly generalizes the class of zerosum games, but is a very special case of general bimatrix games. We study both the expressive power and the algorithmic behavior of these games. Specifically, we show that even for k = 1 the set of Nash equilibria of these games can consist of an arbitrarily large number of connected components. While the question of exact polynomial time algorithms to find a Nash equilibrium remains open for games of fixed rank, we present polynomial time algorithms for finding an εapproximation.
Market equilibria in polynomial time for fixed number of goods or agents
 In FOCS
, 2008
"... We consider markets in the classical ArrowDebreu model. There are n agents and m goods. Each buyer has a concave utility function (of the bundle of goods he/she buys) and an initial bundle. At an “equilibrium ” set of prices for goods, if each individual buyer separately exchanges the initial bundl ..."
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Cited by 16 (3 self)
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We consider markets in the classical ArrowDebreu model. There are n agents and m goods. Each buyer has a concave utility function (of the bundle of goods he/she buys) and an initial bundle. At an “equilibrium ” set of prices for goods, if each individual buyer separately exchanges the initial bundle for an optimal bundle at the set prices, the market clears, i.e., all goods are exactly consumed. Classical theorems guarantee the existence of equilibria, but computing them has been the subject of much recent research. In the related area of MultiAgent Games, much attention has been paid to the complexity as well as algorithms. While most general problems are hard, polynomial time algorithms have been developed for restricted classes of games, when one assumes the number of strategies is constant [20, 11]. For the Market Equilibrium problem, several important special cases of utility functions have been tackled. Here we begin a program for this problem similar to that for multiagent games, where general utilities are considered. We begin by showing that if the utilities are separable piecewise linear concave (PLC) functions, and the number of goods (or alternatively the number of buyers) is constant, then we can compute an exact equilibrium in polynomial time. Our technique for the constant number of goods is to decompose the space of price vectors into cells using certain hyperplanes, so that in each cell, each buyer’s threshold marginal utility is known. Still, one needs to solve a linear optimization problem in each cell. We then show the main result that for general (nonseparable) PLC utilities, an exact equilibrium can be found in polynomial time provided the number of goods is constant. The starting point of the algorithm is a “celldecomposition ” of the space of price vectors using polynomial surfaces (instead of hyperplanes). We use results from computational algebraic geometry to bound the number of such cells. For solving the problem inside each cell, we introduce and use a novel LPduality
Algorithmic Rationality: Game Theory with Costly Computation
, 2007
"... We develop a general gametheoretic framework for reasoning about strategic agents performing possibly costly computation. In this framework, many traditional gametheoretic results (such as the existence of a Nash equilibrium) no longer hold. Nevertheless, we can use the framework to provide psycho ..."
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Cited by 9 (7 self)
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We develop a general gametheoretic framework for reasoning about strategic agents performing possibly costly computation. In this framework, many traditional gametheoretic results (such as the existence of a Nash equilibrium) no longer hold. Nevertheless, we can use the framework to provide psychologically appealing explanations to observed behavior in wellstudied games (such as finitely repeated prisoner’s dilemma and rockpaperscissors). Furthermore, we Consider the following game. You are given a random odd nbit number x and you are supposed to decide whether x is prime or composite. If you guess correctly you receive $2, if you guess incorrectly you instead have to pay a penalty of $1000. Additionally you have the choice of “playing safe” by giving up, in which case you receive $1. In traditional game theory, computation is considered