Results 1  10
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14
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 ..."
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 9 (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 4 (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.
Weighted boolean formula games
 In Proceedings of the 3rd International Workshop on Internet and Network Economics (WINE
, 2007
"... We introduce a new class of succinct games, called weighted boolean formula games. Here, each player has a set of boolean formulas he wants to get satisfied. The boolean formulas of all players involve a ground set of boolean variables, and every player controls some of these variables. The payoff o ..."
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Cited by 3 (0 self)
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We introduce a new class of succinct games, called weighted boolean formula games. Here, each player has a set of boolean formulas he wants to get satisfied. The boolean formulas of all players involve a ground set of boolean variables, and every player controls some of these variables. The payoff of a player is the weighted sum of the values of his boolean formulas. For these games, we consider pure Nash equilibria [42] and their wellstudied refinement of payoffdominant equilibria [30], where every player is no worseoff than in any other pure Nash equilibrium. We study both structural and complexity properties for both decision and search problems with respect to the two concepts: • We consider a subclass of weighted boolean formula games, called mutual weighted boolean formula games, which make a natural mutuality assumption on the payoffs of distinct players. We present a very simple exact potential for mutual weighted boolean formula games. We also prove that each weighted, linearaffine (network) congestion game with playerspecific constants is polynomial, sound NashHarsanyiSelten homomorphic to a mutual weighted boolean formula game. In a general way, we prove that each weighted, linearaffine (network)
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 3 (2 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.
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 3 (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.
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.
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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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.
Ranking Games that have . . .
, 2010
"... This paper studies  from the perspective of efficient computation  a type of competition that is widespread throughout the plant and animal kingdoms, higher education, politics and artificial contests. In this setting, an agent gains utility from his relative performance (on some measurable cri ..."
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This paper studies  from the perspective of efficient computation  a type of competition that is widespread throughout the plant and animal kingdoms, higher education, politics and artificial contests. In this setting, an agent gains utility from his relative performance (on some measurable criterion) against other agents, as opposed to his absolute performance. We model this situation using ranking games in which each strategy corresponds to a level of competitiveness, and incurs an upfront cost that is higher for more competitive strategies. We study the Nash equilibria of these games, and polynomialtime algorithms for computing them. For games in which there is no tie between agents’ levels of competitiveness we give a polynomialtime algorithm for computing an exact equilibrium in the 2player case, and a characterization of Nash equilibria that shows an interesting parallel between these games and unrestricted 2player games in normal form. When ties are allowed, via a reduction from these games to a subclass of anonymous games, we give polynomialtime approximation schemes for two special cases: constantsized set of strategies, and constant number of players. The latter result is improved to a fully polynomialtime approximation scheme when the constant number of players only compete to win the game, i.e. to be ranked first.
ON ITERATED DOMINANCE, MATRIX ELIMINATION, AND MATCHED PATHS
 SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER
"... We study computational problems arising from the iterated removal of weakly dominated actions in anonymous games. Our main result shows that it is NPcomplete to decide whether an anonymous game with three actions can be solved via iterated weak dominance. The twoaction case can be reformulated as ..."
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We study computational problems arising from the iterated removal of weakly dominated actions in anonymous games. Our main result shows that it is NPcomplete to decide whether an anonymous game with three actions can be solved via iterated weak dominance. The twoaction case can be reformulated as a natural elimination problem on a matrix, the complexity of which turns out to be surprisingly difficult to characterize and ultimately remains open. We however establish connections to a matching problem along paths in a directed graph, which is computationally hard in general but can also be used to identify tractable cases of matrix elimination. We finally identify different classes of anonymous games where iterated dominance is in P and NPcomplete, respectively.