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21
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
Computing correlated equilibria in MultiPlayer Games
 STOC'05
, 2005
"... We develop a polynomialtime algorithm for finding correlated equilibria (a wellstudied notion of rationality due to Aumann that generalizes the Nash equilibrium) in a broad class of succinctly representable multiplayer games, encompassing essentially all known kinds, including all graphical games, ..."
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Cited by 96 (6 self)
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We develop a polynomialtime algorithm for finding correlated equilibria (a wellstudied notion of rationality due to Aumann that generalizes the Nash equilibrium) in a broad class of succinctly representable multiplayer games, encompassing essentially all known kinds, including all graphical games, polymatrix games, congestion games, scheduling games, local effect games, as well as several generalizations. Our algorithm is based on a variant of the existence proof due to Hart and Schmeidler [11], and employs linear programming duality, the ellipsoid algorithm, Markov chain steady state computations, as well as applicationspecific methods for computing multivariate expectations.
On the complexity of twoplayer winlose games
 In Proceedings of FOCS’05
, 2005
"... The efficient computation of Nash equilibria is one of the most formidable challenges in computational complexity today. The problem remains open for twoplayer games. We show that the complexity of twoplayer Nash equilibria is unchanged when all outcomes are restricted to be 0 or 1. That is, wino ..."
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Cited by 37 (1 self)
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The efficient computation of Nash equilibria is one of the most formidable challenges in computational complexity today. The problem remains open for twoplayer games. We show that the complexity of twoplayer Nash equilibria is unchanged when all outcomes are restricted to be 0 or 1. That is, winorlose games are as complex as the general case for twoplayer games. 1 Game Theory Game theory asks the question: given a set of players playing a certain game, what happens? Computational game theory asks the question: given a representation of a game and some fixed criteria for reasonable play, how may we efficiently compute properties of the possible outcomes? Needless to say, there are many possible ways to define a game, and many more ways to efficiently represent these games. Since the computational complexity of an algorithm is defined as a function of the length of its input representation, different game representations may have significantly different algorithmic consequences. Much work is being done to investigate how to take advantage of some of the more exotic representations of games (see [4, 7, 8, 10] and the references therein). Nevertheless, for two player games, computational game theorists almost exclusively work with the representation known as a rational bimatrix game, which we define as follows. Definition 1 A rational bimatrix game is a game representation that consists of a matrix of pairs of rational numbers
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 23 (4 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.
Computing equilibria: A computational complexity perspective
, 2009
"... Computational complexity is the subfield of computer science that rigorously studies the intrinsic difficulty of computational problems. This survey explains how complexity theory defines “hard problems”; applies these concepts to several equilibrium computation problems; and discusses implications ..."
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Cited by 12 (2 self)
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Computational complexity is the subfield of computer science that rigorously studies the intrinsic difficulty of computational problems. This survey explains how complexity theory defines “hard problems”; applies these concepts to several equilibrium computation problems; and discusses implications for computation, games, and behavior. We assume
On strictly competitive multiplayer games
 In Proc. 21st Conf. Artificial Intelligence (AAAI
, 2006
"... We embark on an initial study of a new class of strategic (normalform) games, socalled ranking games, in which the payoff to each agent solely depends on his position in a ranking of the agents induced by their actions. This definition is motivated by the observation that in many strategic situati ..."
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Cited by 10 (2 self)
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We embark on an initial study of a new class of strategic (normalform) games, socalled ranking games, in which the payoff to each agent solely depends on his position in a ranking of the agents induced by their actions. This definition is motivated by the observation that in many strategic situations such as parlor games, competitive economic scenarios, and some social choice settings, players are merely interested in performing optimal relative to their opponents rather than in absolute measures. A simple but important subclass of ranking games are singlewinner games where in any outcome one agent wins and all others lose. We investigate the computational complexity of a variety of common gametheoretic solution concepts in ranking games and deliver hardness results for iterated weak dominance and mixed Nash equilibria when there are more than two players and pure Nash equilibria when the number of players is unbounded. This dashes hope that multiplayer ranking games can be solved efficiently, despite the structural restrictions of these games.
Ranking Games
, 2008
"... The outcomes of many strategic situations such as parlor games or competitive economic scenarios are rankings of the participants, with higher ranks generally at least as desirable as lower ranks. Here we define ranking games as a class of nplayer normalform games with a payoff structure reflectin ..."
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Cited by 8 (2 self)
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The outcomes of many strategic situations such as parlor games or competitive economic scenarios are rankings of the participants, with higher ranks generally at least as desirable as lower ranks. Here we define ranking games as a class of nplayer normalform games with a payoff structure reflecting the players’ von NeumannMorgenstern preferences over their individual ranks. We investigate the computational complexity of a variety of common gametheoretic solution concepts in ranking games and deliver hardness results for iterated weak dominance and mixed Nash equilibrium when there are more than two players, and for pure Nash equilibrium when the number of players is unbounded but the game is described succinctly. This dashes hope that multiplayer ranking games can be solved efficiently, despite their profound structural restrictions. Based on these findings, we provide matching upper and lower bounds for three comparative ratios, each of which relates two different solution concepts: the price of cautiousness, the mediation value, and the enforcement value.
On the complexity of Nash equilibria of ActionGraph Games
 In SODA: Proceedings of the ACMSIAM Symposium on Discrete Algorithms
, 2009
"... In light of much recent interest in finding a model of multiplayer multiaction games that allows for efficient computation of Nash equilibria yet remains as expressive as possible, we investigate the computational complexity of Nash equilibria in the recently proposed model of actiongraph games (A ..."
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Cited by 7 (1 self)
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In light of much recent interest in finding a model of multiplayer multiaction games that allows for efficient computation of Nash equilibria yet remains as expressive as possible, we investigate the computational complexity of Nash equilibria in the recently proposed model of actiongraph games (AGGs). AGGs, introduced by Bhat and LeytonBrown, are succinct representations of games that encapsulate both local dependencies as in graphical games, and partial indifference to other agents ’ identities as in anonymous games, which occur in many natural settings such as financial markets. This is achieved by specifying a graph on the set of actions, so that the payoff of an agent for selecting a strategy depends only on the number of agents playing each of the neighboring strategies in the action graph. We present a simple Fully Polynomial Time Approximation Scheme for computing mixed Nash equilibria of AGGs with constant degree, constant treewidth and a constant number of agent types (but an arbitrary number of strategies), and extend this algorithm to a broader set of instances. However, the main results of this paper are negative, showing that when either of the latter conditions are relaxed the problem becomes intractable. In particular, we show that even if the action graph is a tree but the number of agenttypes is unconstrained, it is NP– complete to decide the existence of a purestrategy Nash equilibrium and PPAD–complete to compute a mixed Nash equilibrium (even an approximate one). Similarly for AGGs with a constant number of agent types but unconstrained treewidth. These hardness results suggest that, in some sense, our FPTAS is as strong a positive result as one can expect. In the broader context of trying to pin down the boundary where the equilibria of multiplayer games can be computed efficiently, these results complement recent hardness results for graphical games and algorithmic results for anonymous games.
Equilibria of Graphical Games with Symmetries
, 2007
"... We study graphical games where the payoff function of each player satisfies one of four types of symmetries in the actions of his neighbors. We establish that deciding the existence of a pure Nash equilibrium is NPhard in graphical games with each of the four types of symmetry. Using a characteriza ..."
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Cited by 5 (1 self)
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We study graphical games where the payoff function of each player satisfies one of four types of symmetries in the actions of his neighbors. We establish that deciding the existence of a pure Nash equilibrium is NPhard in graphical games with each of the four types of symmetry. Using a characterization of games with pure equilibria in terms of even cycles in the neighborhood graph, as well as a connection to a generalized satisfiability problem, we identify tractable subclasses of the games satisfying the most restrictive type of symmetry. In the process, we characterize a satisfiability problem that remains NPhard in the presence of a matching, a result that may be of independent interest. Finally, games with symmetries of two of the four types are shown to possess a symmetric mixed equilibrium which can be computed in polynomial time. We have thus identified a class of games where the pure equilibrium problem is computationally harder than the mixed equilibrium problem, unless P=NP.