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18
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 87 (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 PureStrategy Nash Equilibria in Congestion and LocalEffect Games
 In Proc. of the 2nd Int. Workshop on Internet and Network Economics (WINE
, 2006
"... doi 10.1287/moor.1080.0322 ..."
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Computing Pure Nash Equilibria in Symmetric Action Graph Games
, 2007
"... We analyze the problem of computing pure Nash equilibria in action graph games (AGGs), which are a compact gametheoretic representation. While the problem is NPcomplete in general, for certain classes of AGGs there exist polynomial time algorithms. We propose a dynamicprogramming approach that con ..."
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Cited by 10 (1 self)
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We analyze the problem of computing pure Nash equilibria in action graph games (AGGs), which are a compact gametheoretic representation. While the problem is NPcomplete in general, for certain classes of AGGs there exist polynomial time algorithms. We propose a dynamicprogramming approach that constructs equilibria of the game from equilibria of restricted games played on subgraphs of the action graph. In particular, if the game is symmetric and the action graph has bounded treewidth, our algorithm determines the existence of pure Nash equilibrium in polynomial time.
On the Complexity of Paretooptimal Nash and Strong Equilibria
"... We consider the computational complexity of coalitional solution concepts in scenarios related to load balancing suchas anonymous and congestion games. Incongestion games, Paretooptimal Nash and strong equilibria, which are resilient to coalitional deviations, have recently been shown to yield sign ..."
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Cited by 5 (1 self)
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We consider the computational complexity of coalitional solution concepts in scenarios related to load balancing suchas anonymous and congestion games. Incongestion games, Paretooptimal Nash and strong equilibria, which are resilient to coalitional deviations, have recently been shown to yield significantly smaller inefficiency. Unfortunately, we show that several problems regarding existence, recognition, and computation of these concepts are hard, even in seemingly special classes of games. In anonymous games with constant number of strategies, we can efficiently recognize a state as Paretooptimal Nash or strong equilibrium, but deciding existence for a game remains hard. In the case of playerspecific singleton congestion games, we show that recognition and computation of both concepts can be done efficiently. In addition, in these games there are always short sequences of coalitional improvement moves to Paretooptimal Nash and strong equilibria that can be computed efficiently.
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 4 (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.
On the Hardness and Existence of QuasiStrict Equilibria
"... This paper investigates the computational properties of quasistrict equilibrium, an attractive equilibrium refinement proposed by Harsanyi, which was recently shown to always exist in bimatrix games. We prove that deciding the existence of a quasistrict equilibrium in games with more than two pla ..."
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Cited by 4 (3 self)
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This paper investigates the computational properties of quasistrict equilibrium, an attractive equilibrium refinement proposed by Harsanyi, which was recently shown to always exist in bimatrix games. We prove that deciding the existence of a quasistrict equilibrium in games with more than two players is NPcomplete. We further show that, in contrast to Nash equilibrium, the support of quasistrict equilibrium in zerosum games is unique and propose a linear program to compute quasistrict equilibria in these games. Finally, we prove that every symmetric multiplayer game where each player has two actions at his disposal contains an efficiently computable quasistrict equilibrium which may itself be asymmetric.
Pure Nash Equilibria: Complete Characterization of Hard and Easy Graphical Games
, 2010
"... We consider the computational complexity of pure Nash equilibria in graphical games. It is known that the problem is NPcomplete in general, but tractable (i.e., in P) for special classes of graphs such as those with bounded treewidth. It is then natural to ask: is it possible to characterize all tr ..."
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Cited by 3 (0 self)
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We consider the computational complexity of pure Nash equilibria in graphical games. It is known that the problem is NPcomplete in general, but tractable (i.e., in P) for special classes of graphs such as those with bounded treewidth. It is then natural to ask: is it possible to characterize all tractable classes of graphs for this problem? In this work, we provide such a characterization for the case of bounded indegree graphs, thereby resolving the gap between existing hardness and tractability results. In particular, we analyze the complexity of PUREGG(C, −), the problem of deciding the existence of pure Nash equilibria in graphical games whose underlying graphs are restricted to class C. We prove that, under reasonable complexity theoretic assumptions, for every recursively enumerable class C of directed graphs with bounded indegree, PUREGG(C, −) is in polynomial time if and only if the reduced graphs (the graphs resulting from iterated removal of sinks) of C have bounded treewidth. We also give a characterization for PURECHG(C, −), the problem of deciding the existence of pure Nash equilibria in colored hypergraphical games, a game representation that can express the additional structure that some of the players have identical local utility functions. We show that the tractable classes of boundedarity colored hypergraphical games are precisely those whose reduced graphs have bounded treewidth modulo homomorphic equivalence. Our proofs make novel use of Grohe’s characterization of the complexity of homomorphism problems.
COMPUTING PURE STRATEGY NASH EQUILIBRIA IN COMPACT, SYMMETRIC GAMES WITH A FIXED NUMBER OF ACTIONS
"... Abstract. We analyze the complexity of computing pure strategy Nash equilibria (PSNE) in symmetric games with a fixed number of actions. We restrict ourselves to “compact ” representations, meaning that the number of players can be exponential in the representation size. We show that in the general ..."
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Abstract. We analyze the complexity of computing pure strategy Nash equilibria (PSNE) in symmetric games with a fixed number of actions. We restrict ourselves to “compact ” representations, meaning that the number of players can be exponential in the representation size. We show that in the general case, where utility functions are represented as arbitrary circuits, the problem of deciding the existence of PSNE is NPcomplete. For the special case of games with two actions, we show that there always exist a PSNE and give a polynomialtime algorithm for finding one. We then focus on a specific compact representation: piecewiselinear functions. We give polynomialtime algorithms for finding a sample PSNE and for counting the number of PSNE. Our approach makes use of Barvinok and Wood’s rational generating function method [3], which enables us to encode the set of PSNE as a generating function of polynomial size. 1.
Nash Equilibria in Symmetric Games with Partial Observation
"... We investigate a model for representing large multiplayer games, which satisfy strong symmetry properties. This model is made of multiple copies of an arena; each player plays in his own arena, and can partially observe what the other players do. Therefore, this game has partial information and symm ..."
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Cited by 2 (1 self)
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We investigate a model for representing large multiplayer games, which satisfy strong symmetry properties. This model is made of multiple copies of an arena; each player plays in his own arena, and can partially observe what the other players do. Therefore, this game has partial information and symmetry constraints, which make the computation of Nash equilibria difficult. We show several undecidability results, and for boundedmemory strategies, we precisely characterize the complexity of computing pure Nash equilibria (for qualitative objectives) in this game model. 1
On the Complexity of Nash Dynamics and Sink equilibria
, 2009
"... Studying Nash dynamics is an important approach for analyzing the outcome of games with repeated selfish behavior of selfinterested agents. Sink equilibria has been introduced by Goemans, Mirrokni, and Vetta for studying social cost on Nash dynamics over pure strategies in games. However, they do n ..."
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Cited by 2 (1 self)
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Studying Nash dynamics is an important approach for analyzing the outcome of games with repeated selfish behavior of selfinterested agents. Sink equilibria has been introduced by Goemans, Mirrokni, and Vetta for studying social cost on Nash dynamics over pure strategies in games. However, they do not address the complexity of sink equilibria in these games. Recently, Fabrikant and Papadimitriou initiated the study of the complexity of Nash dynamics in two classes of games. In order to completely understand the complexity of Nash dynamics in a variety of games, we study the following three questions for various games: (i) given a state in game, can we verify if this state is in a sink equilibrium or not? (ii) given an instance of a game, can we verify if there exists any sink equilibrium other than pure Nash equilibria? and (iii) given an instance of a game, can we verify if there exists a pure Nash equilibrium (i.e, a sink equilibrium with one state)? In this paper, we almost answer all of the above questions for a variety of classes of games with succinct representation, including anonymous games, playerspecific and weighted congestion games, validutility games, and twosided market games. In particular, for most of these problems, we show that (i) it is PSPACEcomplete to verify if a given state is in a sink equilibrium, (ii) it is NPhard to verify if there exists a pure Nash equilibrium in the game or not, (iii) it is PSPACEcomplete to verify if there exists any sink equilibrium other than pure Nash equilibria. To solve these problems, we illustrate general techniques that could be used to answer similar questions in other classes of games.