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169
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 329 (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
The price of stability for network design with fair cost allocation
 In Proceedings of the 45th Annual Symposium on Foundations of Computer Science (FOCS
, 2004
"... Abstract. Network design is a fundamental problem for which it is important to understand the effects of strategic behavior. Given a collection of selfinterested agents who want to form a network connecting certain endpoints, the set of stable solutions — the Nash equilibria — may look quite differ ..."
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Cited by 281 (30 self)
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Abstract. Network design is a fundamental problem for which it is important to understand the effects of strategic behavior. Given a collection of selfinterested agents who want to form a network connecting certain endpoints, the set of stable solutions — the Nash equilibria — may look quite different from the centrally enforced optimum. We study the quality of the best Nash equilibrium, and refer to the ratio of its cost to the optimum network cost as the price of stability. The best Nash equilibrium solution has a natural meaning of stability in this context — it is the optimal solution that can be proposed from which no user will defect. We consider the price of stability for network design with respect to one of the most widelystudied protocols for network cost allocation, in which the cost of each edge is divided equally between users whose connections make use of it; this fairdivision scheme can be derived from the Shapley value, and has a number of basic economic motivations. We show that the price of stability for network design with respect to this fair cost allocation is O(log k), where k is the number of users, and that a good Nash equilibrium can be achieved via bestresponse dynamics in which users iteratively defect from a starting solution. This establishes that the fair cost allocation protocol is in fact a useful mechanism for inducing strategic behavior to form nearoptimal equilibria. We discuss connections to the class of potential games defined by Monderer and Shapley, and extend our results to cases in which users are seeking to balance network design costs with latencies in the constructed network, with stronger results when the network has only delays and no construction costs. We also present bounds on the convergence time of bestresponse dynamics, and discuss extensions to a weighted game.
Intrinsic Robustness of the Price of Anarchy
 STOC'09
, 2009
"... The price of anarchy (POA) is a worstcase measure of the inefficiency of selfish behavior, defined as the ratio of the objective function value of a worst Nash equilibrium of a game and that of an optimal outcome. This measure implicitly assumes that players successfully reach some Nash equilibrium ..."
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Cited by 101 (12 self)
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The price of anarchy (POA) is a worstcase measure of the inefficiency of selfish behavior, defined as the ratio of the objective function value of a worst Nash equilibrium of a game and that of an optimal outcome. This measure implicitly assumes that players successfully reach some Nash equilibrium. This drawback motivates the search for inefficiency bounds that apply more generally to weaker notions of equilibria, such as mixed Nash and correlated equilibria; or to sequences of outcomes generated by natural experimentation strategies, such as successive best responses or simultaneous regretminimization. We prove a general and fundamental connection between the price of anarchy and its seemingly stronger relatives in classes of games with a sum objective. First, we identify a “canonical sufficient condition ” for an upper bound of the POA for pure Nash equilibria, which we call a smoothness argument. Second, we show that every bound derived via a smoothness argument extends automatically, with no quantitative degradation in the bound, to mixed Nash equilibria, correlated equilibria, and the average objective function value of regretminimizing players (or “price of total anarchy”). Smoothness arguments also have automatic implications for the inefficiency of approximate and BayesianNash equilibria and, under mild additional assumptions, for bicriteria bounds and for polynomiallength bestresponse sequences. We also identify classes of games — most notably, congestion games with cost functions restricted to an arbitrary fixed set — that are tight, in the sense that smoothness arguments are guaranteed to produce an optimal worstcase upper bound on the POA, even for the smallest set of interest (pure Nash equilibria). Byproducts of our proof of this result include the first tight bounds on the POA in congestion games with nonpolynomial cost functions, and the first
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.
Selfish Unsplittable Flows
 Theoretical Computer Science
, 2004
"... What is the price of anarchy when unsplittable demands are routed selfishly in general networks with loaddependent edge delays? Motivated by this question we generalize the model of [14] to the case of weighted congestion games. We show that varying demands of users crucially affect the nature o ..."
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Cited by 82 (10 self)
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What is the price of anarchy when unsplittable demands are routed selfishly in general networks with loaddependent edge delays? Motivated by this question we generalize the model of [14] to the case of weighted congestion games. We show that varying demands of users crucially affect the nature of these games, which are no longer isomorphic to exact potential games, even for very simple instances. Indeed we construct examples where even a singlecommodity (weighted) network congestion game may have no pure Nash equilibrium.
Pure Nash Equilibria: Hard and Easy Games
"... In this paper we investigate complexity issues related to pure Nash equilibria of strategic games. We show that, even in very restrictive settings, determining whether a game has a pure Nash Equilibrium is NPhard, while deciding whether a game has a strong Nash equilibrium is Stcomplete. We then s ..."
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Cited by 81 (4 self)
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In this paper we investigate complexity issues related to pure Nash equilibria of strategic games. We show that, even in very restrictive settings, determining whether a game has a pure Nash Equilibrium is NPhard, while deciding whether a game has a strong Nash equilibrium is Stcomplete. We then study practically relevant restrictions that lower the complexity. In particular, we are interested in quantitative and qualitative restrictions of the way each player's move depends on moves of other players. We say that a game has small neighborhood if the &quot; utility function for each player depends only on (the actions of) a logarithmically small number of other players, The dependency structure of a game G can he expressed by a graph G(G) or by a hypergraph II(G). Among other results, we show that if jC has small neighborhood and if II(G) has botmdecl hypertree width (or if G(G) has bounded treewidth), then finding pure Nash and Pareto equilibria is feasible in polynomial time. If the game is graphical, then these problems are LOGCFLcomplete and thus in the class _NC ~ of highly parallelizable problems. 1 Introduction and Overview of Results The theory of strategic games and Nash equilibria has important applications in economics and decision making [31, 2]. Determining whether Nash equilibria exist, and effectively computing
Convergence to Approximate Nash Equilibria in Congestion Games
 In SODA ’07
, 2007
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Selfish Caching in Distributed Systems: A GameTheoretic Analysis
 in Proc. ACM Symposium on Principles of Distributed Computing (ACM PODC
, 2004
"... We analyze replication of resources by server nodes that act selfishly, using a gametheoretic approach. We refer to this as the selfish caching problem. In our model, nodes incur either cost for replicating resources or cost for access to a remote replica. We show the existence of pure strategy Nas ..."
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Cited by 62 (2 self)
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We analyze replication of resources by server nodes that act selfishly, using a gametheoretic approach. We refer to this as the selfish caching problem. In our model, nodes incur either cost for replicating resources or cost for access to a remote replica. We show the existence of pure strategy Nash equilibria and investigate the price of anarchy, which is the relative cost of the lack of coordination. The price of anarchy can be high due to undersupply problems, but with certain network topologies it has better bounds. With a payment scheme the game can always implement the social optimum in the best case by giving servers incentive to replicate.
On the impact of combinatorial structure on congestion games
 in Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS
"... We study the impact of combinatorial structure in congestion games on the complexity of computing pure Nash equilibria and the convergence time of best response sequences. In particular, we investigate which properties of the strategy spaces of individual players ensure a polynomial convergence time ..."
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Cited by 61 (12 self)
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We study the impact of combinatorial structure in congestion games on the complexity of computing pure Nash equilibria and the convergence time of best response sequences. In particular, we investigate which properties of the strategy spaces of individual players ensure a polynomial convergence time. We show, if the strategy space of each player consists of the bases of a matroid over the set of resources, then the lengths of all best response sequences are polynomially bounded in the number of players and resources. We can also prove that this result is tight, that is, the matroid property is a necessary and sufficient condition on the players ’ strategy spaces for guaranteeing polynomial time convergence to a Nash equilibrium. In addition, we present an approach that enables us to devise hardness proofs for various kinds of combinatorial games, including first results about the hardness of market sharing games and congestion games for overlay network design. Our approach also yields a short proof for the PLScompleteness of network congestion games. 1