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112
How bad is selfish routing?
 JOURNAL OF THE ACM
, 2002
"... We consider the problem of routing traffic to optimize the performance of a congested network. We are given a network, a rate of traffic between each pair of nodes, and a latency function for each edge specifying the time needed to traverse the edge given its congestion; the objective is to route t ..."
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Cited by 516 (27 self)
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We consider the problem of routing traffic to optimize the performance of a congested network. We are given a network, a rate of traffic between each pair of nodes, and a latency function for each edge specifying the time needed to traverse the edge given its congestion; the objective is to route traffic such that the sum of all travel times—the total latency—is minimized. In many settings, it may be expensive or impossible to regulate network traffic so as to implement an optimal assignment of routes. In the absence of regulation by some central authority, we assume that each network user routes its traffic on the minimumlatency path available to it, given the network congestion caused by the other users. In general such a “selfishly motivated ” assignment of traffic to paths will not minimize the total latency; hence, this lack of regulation carries the cost of decreased network performance. In this article, we quantify the degradation in network performance due to unregulated traffic. We prove that if the latency of each edge is a linear function of its congestion, then the total latency of the routes chosen by selfish network users is at most 4/3 times the minimum possible total latency (subject to the condition that all traffic must be routed). We also consider the more general setting in which edge latency functions are assumed only to be continuous and nondecreasing in the edge congestion. Here, the total
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 122 (7 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.
Convergence time to nash equilibria
 In ICALP
, 2003
"... Abstract. We study the number of steps required to reach a pure Nash Equilibrium in a load balancing scenario where each job behaves selfishly and attempts to migrate to a machine which will minimize its cost. We consider a variety of load balancing models, including identical, restricted, related a ..."
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Cited by 82 (5 self)
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Abstract. We study the number of steps required to reach a pure Nash Equilibrium in a load balancing scenario where each job behaves selfishly and attempts to migrate to a machine which will minimize its cost. We consider a variety of load balancing models, including identical, restricted, related and unrelated machines. Our results have a crucial dependence on the weights assigned to jobs. We consider arbitrary weights, integer weights, K distinct weights and identical (unit) weights. We look both at an arbitrary schedule (where the only restriction is that a job migrates to a machine which lowers its cost) and specific efficient schedulers (such as allowing the largest weight job to move first). 1
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 67 (7 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.
Market Sharing Games Applied to Content Distribution in AdHoc Networks
 MOBIHOC'04
, 2004
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On the price of anarchy and stability of correlated equilibria of linear congestion games
, 2005
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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 53 (3 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.
Routing without regret: On convergence to nash equilibria of regretminimizing algorithms in routing games
 In PODC
, 2006
"... Abstract There has been substantial work developing simple, efficient noregret algorithms for a wideclass of repeated decisionmaking problems including online routing. These are adaptive strategies an individual can use that give strong guarantees on performance even in adversariallychanging envi ..."
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Cited by 47 (6 self)
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Abstract There has been substantial work developing simple, efficient noregret algorithms for a wideclass of repeated decisionmaking problems including online routing. These are adaptive strategies an individual can use that give strong guarantees on performance even in adversariallychanging environments. There has also been substantial work on analyzing properties of Nash equilibria in routing games. In this paper, we consider the question: if each player in a routing game uses a noregret strategy, will behavior converge to a Nash equilibrium? In general games the answer to this question is known to be no in a strong sense, but routing games havesubstantially more structure. In this paper we show that in the Wardrop setting of multicommodity flow and infinitesimalagents, behavior will approach Nash equilibrium (formally, on most days, the cost of the flow will be close to the cost of the cheapest paths possible given that flow) at a rate that dependspolynomially on the players ' regret bounds and the maximum slope of any latency function. We also show that priceofanarchy results may be applied to these approximate equilibria, and alsoconsider the finitesize (noninfinitesimal) loadbalancing model of Azar [2].
Competition and Efficiency in Congested Markets
"... We study the efficiency of oligopoly equilibria in congested markets. The motivating examples are the allocation of network flows in a communication network or of traffic in a transportation network. We show that increasing competition among oligopolists can reduce efficiency, measured as the differ ..."
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Cited by 43 (7 self)
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We study the efficiency of oligopoly equilibria in congested markets. The motivating examples are the allocation of network flows in a communication network or of traffic in a transportation network. We show that increasing competition among oligopolists can reduce efficiency, measured as the difference between users ’ willingness to pay and delay costs. We characterize a tight bound of 5/6 on efficiency in pure strategy equilibria when there is zero latency at zero flow and a tight bound of 2 √ 2 − 2 with positive latency at zero flow. These bounds are tight even when the numbers of routes and oligopolists are arbitrarily large.