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199
Computing Equilibria in MultiPlayer Games
 In Proceedings of the Annual ACMSIAM Symposium on Discrete Algorithms (SODA
, 2004
"... We initiate the systematic study of algorithmic issues involved in finding equilibria (Nash and correlated) in games with a large number of players; such games, in order to be computationally meaningful, must be presented in some succinct, gamespecific way. We develop a general framework for obta ..."
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Cited by 55 (4 self)
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We initiate the systematic study of algorithmic issues involved in finding equilibria (Nash and correlated) in games with a large number of players; such games, in order to be computationally meaningful, must be presented in some succinct, gamespecific way. We develop a general framework for obtaining polynomialtime algorithms for optimizing over correlated equilibria in such settings, and show how it can be applied successfully to symmetric games (for which we actually find an exact polytopal characterization), graphical games, and congestion games, among others. We also present complexity results implying that such algorithms are not possible in certain other such games. Finally, we present a polynomialtime algorithm, based on quantifier elimination, for finding a Nash equilibrium in symmetric games when the number of strategies is relatively small.
A New Model for Selfish Routing
 Proceedings of the 21st International Symposium on Theoretical Aspects of Computer Science (STACS’04), LNCS 2996
, 2004
"... Abstract. In this work, we introduce and study a new model for selfish routing over noncooperative networks that combines features from the two such best studied models, namely the KP model and the Wardrop model in an interesting way. We consider a set of n users, each using a mixed strategy to shi ..."
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Cited by 53 (10 self)
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Abstract. In this work, we introduce and study a new model for selfish routing over noncooperative networks that combines features from the two such best studied models, namely the KP model and the Wardrop model in an interesting way. We consider a set of n users, each using a mixed strategy to ship its unsplittable traffic over a network consisting of m parallel links. In a Nash equilibrium, no user can increase its Individual Cost by unilaterally deviating from its strategy. To evaluate the performance of such Nash equilibria, we introduce Quadratic Social Cost as a certain sum of Individual Costs – namely, the sum of the expectations of the squares of the incurred link latencies. This definition is unlike the KP model, where Maximum Social Cost has been defined as the maximum of Individual Costs. We analyse the impact of our modeling assumptions on the computation of Quadratic Social Cost, on the structure of worstcase Nash equilibria, and on bounds on the Quadratic Coordination Ratio.
Coordination mechanisms
 PROCEEDINGS OF THE 31ST INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES AND PROGRAMMING, IN: LECTURE NOTES IN COMPUTER SCIENCE
, 2004
"... We introduce the notion of coordination mechanisms to improve the performance in systems with independent selfish and noncolluding agents. The quality of a coordination mechanism is measured by its price of anarchy—the worstcase performance of a Nash equilibrium over the (centrally controlled) soc ..."
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Cited by 51 (5 self)
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We introduce the notion of coordination mechanisms to improve the performance in systems with independent selfish and noncolluding agents. The quality of a coordination mechanism is measured by its price of anarchy—the worstcase performance of a Nash equilibrium over the (centrally controlled) social optimum. We give upper and lower bounds for the price of anarchy for selfish task allocation and congestion games.
Exact Price of Anarchy for Polynomial Congestion Games
, 2006
"... We show exact values for the price of anarchy of weighted and unweighted congestion games with polynomial latency functions. The given values also hold for weighted and unweighted network congestion games. ..."
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Cited by 45 (8 self)
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We show exact values for the price of anarchy of weighted and unweighted congestion games with polynomial latency functions. The given values also hold for weighted and unweighted network congestion games.
A PriceAnticipating Resource Allocation Mechanism for Distributed Shared
 Clusters”, 6th ACM Conference on Electronic Commerce
, 2005
"... In this paper we formulate the fixed budget resource allocation game to understand the performance of a distributed marketbased resource allocation system. Multiple users decide how to distribute their budget (bids) among multiple machines according to their individual preferences to maximize their ..."
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Cited by 42 (8 self)
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In this paper we formulate the fixed budget resource allocation game to understand the performance of a distributed marketbased resource allocation system. Multiple users decide how to distribute their budget (bids) among multiple machines according to their individual preferences to maximize their individual utility. We look at both the efficiency and the fairness of the allocation at the equilibrium, where fairness is evaluated through the measures of utility uniformity and envyfreeness. We show analytically and through simulations that despite being highly decentralized, such a system converges quickly to an equilibrium and unlike the social optimum that achieves high efficiency but poor fairness, the proposed allocation scheme achieves a nice balance of high degrees of efficiency and fairness at the equilibrium. 1.
NonCooperative Multicast and Facility Location Games
"... 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 40 (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
, 2006
"... 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 ..."
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Cited by 39 (2 self)
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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{loglog m, n}) for the convergence time.
Nash Equilibria in Discrete Routing Games with Convex Latency Functions
, 2004
"... In a discrete routing game, each of n selfish users employs a mixed strategy to ship her (unsplittable) traffic over m parallel links. The (expected) latency on a link is determined by an arbitrary nondecreasing, nonconstant and convex latency function φ. In a Nash equilibrium, each user alone is ..."
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Cited by 39 (12 self)
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In a discrete routing game, each of n selfish users employs a mixed strategy to ship her (unsplittable) traffic over m parallel links. The (expected) latency on a link is determined by an arbitrary nondecreasing, nonconstant and convex latency function φ. In a Nash equilibrium, each user alone is minimizing her (Expected) Individual Cost, which is the (expected) latency on the link she chooses. To evaluate Nash equilibria, we formulate Social Cost as the sum of the users ’ (Expected) Individual Costs. The Price of Anarchy is the worstcase ratio of Social Cost for a Nash equilibrium over the least possible Social Cost. A Nash equilibrium is pure if each user deterministically chooses a single link; a Nash equilibrium is fully mixed if each user chooses each link with nonzero probability. We obtain: For the case of identical users, the Social Cost of any Nash equilibrium is no more than the Social Cost of the fully mixed Nash equilibrium, which may exist only uniquely. Moreover, instances admitting a fully mixed Nash equilibrium enjoy an efficient characterization. For the case of identical users, we derive two upper bounds on the Price of Anarchy: For the case of identical links with a monomial latency function φ(x) = x d, the Price of Anarchy is the Bell number of order d + 1. For pure Nash equilibria, a generic upper bound from the Wardrop model can be transfered to discrete routing games. For polynomial latency functions with nonnegative coefficients and degree d, this yields an upper bound of d + 1. For the
Resource selection games with unknown number of players
, 2006
"... In the context of preBayesian games we analyze resource selection games with unknown number of players. We prove the existence and uniqueness of a symmetric safetylevel equilibrium in such games and show that in a game with strictly increasing linear cost functions every player benefits from the c ..."
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Cited by 38 (9 self)
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In the context of preBayesian games we analyze resource selection games with unknown number of players. We prove the existence and uniqueness of a symmetric safetylevel equilibrium in such games and show that in a game with strictly increasing linear cost functions every player benefits from the common ignorance about the number of players. In order to perform the analysis we define safetylevel equilibrium for preBayesian games, and prove that it exists in a compactcontinuousconcave setup; in particular it exists in a finite setup. 1
Fast and compact: A simple class of congestion games
 In 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 33 (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.