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The Price of Anarchy of Finite Congestion Games
 In Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC
, 2005
"... Abstract We consider the price of anarchy of pure Nash equilibria in congestion games with linearlatency functions. For asymmetric games, the price of anarchy of maximum social cost is \Theta (p N),where N is the number of players. For all other cases of symmetric or asymmetric games andfor both max ..."
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Cited by 166 (6 self)
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Abstract We consider the price of anarchy of pure Nash equilibria in congestion games with linearlatency functions. For asymmetric games, the price of anarchy of maximum social cost is \Theta (p N),where N is the number of players. For all other cases of symmetric or asymmetric games andfor both maximum and average social cost, the price of anarchy is 5 /2. We extend the results tolatency functions that are polynomials of bounded degree. We also extend some of the results to mixed Nash equilibria.
Selfish load balancing and atomic congestion games
 Algorithmica
, 2004
"... Abstract We revisit a classical load balancing problem in the modern context of decentralized systems andselfinterested clients. In particular, there is a set of clients, each of whom must choose a server from ..."
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Cited by 72 (3 self)
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Abstract We revisit a classical load balancing problem in the modern context of decentralized systems andselfinterested clients. In particular, there is a set of clients, each of whom must choose a server from
On the price of anarchy and stability of correlated equilibria of linear congestion games
, 2005
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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 54 (9 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 for selfish scheduling
 THEORETICAL COMPUTER SCIENCE
, 2009
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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 47 (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.
Tight bounds for selfish and greedy load balancing
 ICALP 2006. LNCS
, 2006
"... Abstract. We study the load balancing problem in the context of a set of clients each wishing to run a job on a server selected among a subset of permissible servers for the particular client. We consider two different scenarios. In selfish load balancing, each client is selfish in the sense that it ..."
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Cited by 41 (5 self)
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Abstract. We study the load balancing problem in the context of a set of clients each wishing to run a job on a server selected among a subset of permissible servers for the particular client. We consider two different scenarios. In selfish load balancing, each client is selfish in the sense that it selects to run its job to the server among its permissible servers having the smallest latency given the assignments of the jobs of other clients to servers. In online load balancing, clients appear online and, when a client appears, it has to make an irrevocable decision and assign its job to one of its permissible servers. Here, we assume that the clients aim to optimize some global criterion but in an online fashion. A natural local optimization criterion that can be used by each client when making its decision is to assign its job to that server that gives the minimum increase of the global objective. This gives rise to greedy online solutions. The aim of this paper is to determine how much the quality of load balancing is affected by selfishness and greediness. We characterize almost completely the impact of selfishness and greediness in load balancing by presenting new and improved, tight or almost tight bounds on the price of anarchy and price of stability of selfish load balancing as well as on the competitiveness of the greedy algorithm for online load balancing when the objective is to minimize the total latency of all clients on servers with linear latency functions. 1
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
On the price of stability for designing undirected networks with fair cost allocations
 IN PROCEEDINGS OF THE 33RD ANNUAL INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES, AND PROGRAMMING (ICALP
, 2006
"... In this paper we address the open problem of bounding the price of stability for network design with fair cost allocation for undirected graphs posed in [1]. We consider the case where there is an agent in every vertex. We show that the price of stability is O(log log n). We prove this by defining a ..."
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Cited by 35 (1 self)
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In this paper we address the open problem of bounding the price of stability for network design with fair cost allocation for undirected graphs posed in [1]. We consider the case where there is an agent in every vertex. We show that the price of stability is O(log log n). We prove this by defining a particular improving dynamics in a related graph. This proof technique may have other applications and is of independent interest.
(Almost) optimal coordination mechanisms for unrelated maching scheduling
 IN 18TH ACMSIAM SYMP. ON DISCRETE ALGORITHMS (SODA
, 2008
"... We investigate the influence of different algorithmic choices on the approximation ratio in selfish scheduling. Our goal is to design local policies that minimize the inefficiency of resulting equilibria. In particular, we design optimal coordination mechanisms for unrelated machine scheduling, and ..."
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Cited by 35 (6 self)
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We investigate the influence of different algorithmic choices on the approximation ratio in selfish scheduling. Our goal is to design local policies that minimize the inefficiency of resulting equilibria. In particular, we design optimal coordination mechanisms for unrelated machine scheduling, and improve the known approximation ratio from Θ(m) to Θ(log m), where m is the number of machines. A local policy for each machine orders the set of jobs assigned to it only based on parameters of those jobs. A strongly local policy only uses the processing time of jobs on the the same machine. We prove that the approximation ratio of any set of strongly local ordering policies in equilibria is at least Ω(m). In particular, it implies that the approximation ratio of a greedy shortestfirst algorithm for machine scheduling is at least Ω(m). This closes the gap between the known lower and upper bounds for this problem, and answers an open question raised by Ibarra and Kim [16], and Davis and Jaffe [10]. We then design a local ordering policy with the approximation ratio of Θ(log m) in equilibria, and prove that this policy is optimal among all local ordering policies. This policy orders the jobs in the nondecreasing order of their inefficiency, i.e, the ratio between the processing time on that machine over the minimum processing time. Finally, we show that best responses of players for the inefficiencybased policy may not converge to a pure Nash equilibrium, and present a Θ(log² m) policy for which we can prove fast convergence of best responses to pure Nash equilibria.