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118
Reducibility Among Equilibrium Problems
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY
, 2005
"... We address the fundamental question of whether the Nash equilibria of a game can be computed in polynomial time. We describe certain efficient reductions between this problem for normal form games with a fixed number of players and graphical games with fixed degree. Our main result is that the probl ..."
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Cited by 39 (1 self)
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We address the fundamental question of whether the Nash equilibria of a game can be computed in polynomial time. We describe certain efficient reductions between this problem for normal form games with a fixed number of players and graphical games with fixed degree. Our main result is that the problem of solving a game for any constant number of players, is reducible to solving a 4player game.
Noncooperative multicast and facility location games (Extended Abstract)
 IN PROCEEDINGS OF THE 7TH ACM CONFERENCE ON ELECTRONIC COMMERCE
, 2006
"... We consider a multicast game with selfish noncooperative players. There is a special source node and each player is interested in connecting to the source by making a routing decision that minimizes its payment. The mutual influence of the players is determined by a cost sharing mechanism, which in ..."
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Cited by 36 (2 self)
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We consider a multicast game with selfish noncooperative players. There is a special source node and each player is interested in connecting to the source by making a routing decision that minimizes its payment. The mutual influence of the players is determined by a cost sharing mechanism, which in our case evenly splits the cost of an edge among the players using it. We consider two different models: an integral model, where each player connects to the source by choosing a single path, and a fractional model, where a player is allowed to split the flow it receives from the source between several paths. In both models we explore the overhead incurred in network cost due to the selfish behavior of the users, as well as the computational complexity of finding a Nash equilibrium. The existence of a Nash equilibrium for the integral model was previously established by the means of a potential function. We prove that finding a Nash equilibrium that minimizes the potential function is NPhard. We focus on the price of anarchy of a Nash equilibrium resulting from the bestresponse dynamics of a game course, where the players join the game sequentially. For a game with n players, we establish an upper bound of O ( √ n log 2 n) on the price of anarchy, and a lower bound of Ω(log n / log log n). For the fractional model, we prove the existence of a Nash equilibrium via a potential function and give a polynomial time algorithm for computing an equilibrium that minimizes the potential function. Finally, we consider a weighted extension of the multicast game, and prove that in the fractional model, the game always has a Nash equilibrium.
Distributed selfish load balancing
 In Proc. 17th Ann. ACM–SIAM Symp. on Discrete Algorithms (SODA
, 2006
"... Abstract. Suppose that a set of m tasks are to be shared as equally as possible amongst a set of n resources. A gametheoretic mechanism to find a suitable allocation is to associate each task with a “selfish agent”, and require each agent to select a resource, with the cost of a resource being the n ..."
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Cited by 31 (3 self)
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Abstract. Suppose that a set of m tasks are to be shared as equally as possible amongst a set of n resources. A gametheoretic mechanism to find a suitable allocation is to associate each task with a “selfish agent”, and require each agent to select a resource, with the cost of a resource being the number of agents to select it. Agents would then be expected to migrate from overloaded to underloaded resources, until the allocation becomes balanced. Recent work has studied the question of how this can take place within a distributed setting in which agents migrate selfishly without any centralized control. In this paper we discuss a natural protocol for the agents which combines the following desirable features: It can be implemented in a strongly distributed setting, uses no central control, and has good convergence properties. For m ≫ n, the system becomes approximately balanced (an ǫNash equilibrium) in expected time O(log log m). We show using a martingale technique that the process converges to a perfectly balanced allocation in expected time O(log log m + n 4). We also give a lower bound of Ω(max{log log m, n}) for the convergence time. 1. Introduction. Suppose
Fast and Compact: A Simple Class of Congestion Games
 In Proc. of the 20th Nat. Conference on Artificial Intelligence (AAAI
, 2005
"... We study a simple, yet rich subclass of congestion games that we call singleton games. These games are exponentially more compact than general congestion games. In contrast with some other compact subclasses, we show tractability of many natural gametheoretic questions, such as finding a sample or ..."
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Cited by 30 (0 self)
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We study a simple, yet rich subclass of congestion games that we call singleton games. These games are exponentially more compact than general congestion games. In contrast with some other compact subclasses, we show tractability of many natural gametheoretic questions, such as finding a sample or optimal Nash equilibrium. For best and betterresponse dynamics, we establish polynomial upper and lower bounds on the rate of convergence and present experimental results. We also consider a natural generalization of singleton games and show that many tractability results carry over.
Selfish Routing with Incomplete Information
 Theory of Computing Systems
"... In his seminal work, Harsanyi [19] introduced an elegant approach to study noncooperative games with incomplete information. In our work, we use this approach to define a new selfish routing game with incomplete information that we call Bayesian routing game. Here, each of n selfish users wishes to ..."
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Cited by 24 (2 self)
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In his seminal work, Harsanyi [19] introduced an elegant approach to study noncooperative games with incomplete information. In our work, we use this approach to define a new selfish routing game with incomplete information that we call Bayesian routing game. Here, each of n selfish users wishes to assign its traffic to one of m parallel links. However, users do not know each other’s traffic. Following Harsanyi’s approach, we introduce, for each user, a set of possible types. In our model, each type of a user corresponds to some traffic and the players ’ uncertainty about each other’s traffic is described by a probability distribution over all possible type profiles. We present a comprehensive collection of results about our Bayesian routing game. Our main findings are as follows: • Using a potential function, we prove that every Bayesian routing game has a pure Bayesian Nash equilibrium. More precisely, we show this existence for a more general class of games that we call weighted Bayesian congestion games. For Bayesian routing games with identical links and independent type
The complexity of game dynamics: Bgp oscillations, sink equilibria, and beyond
 In SODA ’08: Proceedings of the nineteenth annual ACMSIAM symposium on Discrete algorithms
, 2008
"... We settle the complexity of a wellknown problem in networking by establishing that it is PSPACEcomplete to tell whether a system of path preferences in the BGP protocol [25] can lead to oscillatory behavior; one key insight is that the BGP oscillation question is in fact one about Nash dynamics. W ..."
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Cited by 21 (4 self)
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We settle the complexity of a wellknown problem in networking by establishing that it is PSPACEcomplete to tell whether a system of path preferences in the BGP protocol [25] can lead to oscillatory behavior; one key insight is that the BGP oscillation question is in fact one about Nash dynamics. We show that the concept of sink equilibria proposed recently in [11] is also PSPACEcomplete to analyze and approximate for graphical games. Finally, we propose a new equilibrium concept inspired by game dynamics, unit recall equilibria, which we show to be close to universal (exists with high probability in a random game) and algorithmically promising. We also give a relaxation thereof, called componentwise unit recall equilibria, which we show to be both tractable and universal (guaranteed to exist in every game).
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.
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 ..."
Symmetry in Network Congestion Games: Pure Equilibria and Anarchy Cost
 In Proc. of the 3rd Workshop on Approximation and Online Algorithms (WAOA ’05
, 2005
"... Abstract. We study computational and coordination efficiency issues of Nash equilibria in symmetric network congestion games. We first propose a simple and natural greedy method that computes a pure Nash equilibrium with respect to traffic congestion in a network. In this algorithm each user plays o ..."
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Cited by 17 (6 self)
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Abstract. We study computational and coordination efficiency issues of Nash equilibria in symmetric network congestion games. We first propose a simple and natural greedy method that computes a pure Nash equilibrium with respect to traffic congestion in a network. In this algorithm each user plays only once and allocates her traffic to a path selected via a shortest path computation. We then show that this algorithm works for seriesparallel networks when users are identical or when users are of varying demands but have the same best response strategy for any initial network traffic. We also give constructions where the algorithm fails if either the above condition is violated (even for seriesparallel networks) or the network is not seriesparallel (even for identical users). Thus, we essentially indicate the limits of the applicability of this greedy approach. We also study the price of anarchy for the objective of maximum latency. We prove that for any network of m uniformly related links and log m for identical users, the price of anarchy is Θ( 1
The equilibrium existence problem in finite network congestion games
 In Proc. of the 2nd Int. Workshop on Internet and Network Economics (WINE
, 2006
"... Abstract. An open problem is presented regarding the existence of pure strategy Nash equilibrium (PNE) in network congestion games with a finite number of nonidentical players, in which the strategy set of each player is the collection of all paths in a given network that link the player’s origin a ..."
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Cited by 13 (1 self)
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Abstract. An open problem is presented regarding the existence of pure strategy Nash equilibrium (PNE) in network congestion games with a finite number of nonidentical players, in which the strategy set of each player is the collection of all paths in a given network that link the player’s origin and destination vertices, and congestion increases the costs of edges. A network congestion game in which the players differ only in their origin–destination pairs is a potential game, which implies that, regardless of the exact functional form of the cost functions, it has a PNE. A PNE does not necessarily exist if (i) the dependence of the cost of each edge on the number of users is player as well as edgespecific or (ii) the (possibly, edgespecific) cost is the same for all players but it is a function (not of the number but) of the total weight of the players using the edge, with each player i having a different weight wi. In a parallel twoterminal network, in which the origin and the destination are the only vertices different edges have in common, a PNE always exists even if the players differ in either their cost functions or weights, but not in both. However, for general twoterminal networks this is not so. The problem is to characterize the class of all twoterminal network topologies for which the existence of a PNE is guaranteed even with playerspecific costs, and the corresponding class for playerspecific weights. Some progress in solving this problem is reported.