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90
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
Competitive Routing in MultiUser Communication Networks
 IEEE/ACM Transactions on Networking
, 1993
"... We consider a communication network shared by several selfish users. Each user seeks to optimize its own performance by controlling the routing of its given flow demand, giving rise to a noncooperative game. We investigate the Nash equilibrium of such systems. For a twonode multiplelinks system, ..."
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Cited by 185 (21 self)
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We consider a communication network shared by several selfish users. Each user seeks to optimize its own performance by controlling the routing of its given flow demand, giving rise to a noncooperative game. We investigate the Nash equilibrium of such systems. For a twonode multiplelinks system, uniqueness of the Nash equilibrium is proved under reasonable convexity conditions. It is shown that this Nash equilibrium point possesses interesting monotonicity properties. For general networks, these convexity conditions are not sufficient for guaranteeing uniqueness, and a counter example is presented. Nonetheless, uniqueness of the Nash equilibrium for general topologies is established under various assumptions. Also with Sun Microsystems, Mountain View, CA 1 1 Introduction Traditional computer networks were designed with a single administrative domain in mind. That is, the network is designed and operated as a single entity with a single control objective. A single control object...
The price of anarchy is independent of the network topology
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 2002
"... We study the degradation in network performance caused by the selfish behavior of noncooperative network users. We consider a model of selfish routing in which the latency experienced by network traffic on an edge of the network is a function of the edge congestion, and network users are assumed to ..."
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Cited by 178 (14 self)
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We study the degradation in network performance caused by the selfish behavior of noncooperative network users. We consider a model of selfish routing in which the latency experienced by network traffic on an edge of the network is a function of the edge congestion, and network users are assumed to selfishly route traffic on minimumlatency paths. The quality of a routing of traffic is measured by the sum of travel times, also called the total latency. The outcome of selfish routing—a Nash equilibrium—does not in general minimize the total latency; hence, selfish behavior carries the cost of decreased network performance. We quantify this degradation in network performance via the price of anarchy, the worstpossible ratio between the total latency of a Nash equilibrium and of an optimal routing of the traffic. We show the price of anarchy is determined only by the simplest of networks. Specifically, we prove that under weak hypotheses on the class of allowable edge latency functions, the worstcase ratio between the total latency of a Nash equilibrium and of a minimumlatency routing for any multicommodity flow network is achieved by a singlecommodity
Stackelberg scheduling strategies
 In Proceedings of the 33rd Annual ACM Symposium on the Theory of Computing
, 2001
"... AbstractWe study the problem of optimizing the performance of a system shared by selfish, noncooperative users. We consider the concrete setting of scheduling jobs on a set of shared machines with loaddependent latency functions specifying the length of time necessary to complete a job; we measure ..."
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Cited by 110 (6 self)
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AbstractWe study the problem of optimizing the performance of a system shared by selfish, noncooperative users. We consider the concrete setting of scheduling jobs on a set of shared machines with loaddependent latency functions specifying the length of time necessary to complete a job; we measure system performance by the total latency of the system. Assigning jobs according to the selfish interests of individual users (who wish to minimize only the latency that their own jobs experience) typically results in suboptimal system performance. However, in many systems of this type there is a mixture of "selfishly controlled " and "centrally controlled " jobs; as the assignment of centrally controlled jobs will influence the subsequent actions by selfish users, we aspire to contain the degradation in system performance due to selfish behavior by scheduling the centrally controlled jobs in the best possible way. We formulate this goal as an optimization problem via Stackelberg games, games in which one player acts a leader (here, the centralized authority interested in optimizing system performance) and the rest as followers (the selfish users). The problem is then to compute a strategy for the leader (a Stackelberg strategy) that induces the followers to react in a way that (at least approximately) minimizes the total latency in the system. In this paper, we prove that it is NPhard to compute the optimal Stackelberg strategy and present simple strategies with provable performance guarantees. More precisely, we give a simple algorithm that computes a strategy inducing a job assignment with total latency no more than a constant times that of the optimal assignment of all of the jobs; in the absence of centrally controlled jobs and a Stackelberg strategy, no result of this type is possible. We also prove stronger performance guarantees in the special case where every machine latency function is linear in the machine load.
Pricing network edges for heterogeneous selfish users
 Proc. of STOC
, 2003
"... We study the negative consequences of selfish behavior in a congested network and economic means of influencing such behavior. We consider the model of selfish routing defined by Wardrop [30] and studied in a computer science context by Roughgarden and Tardos [26]. In this model, the latency experie ..."
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Cited by 88 (10 self)
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We study the negative consequences of selfish behavior in a congested network and economic means of influencing such behavior. We consider the model of selfish routing defined by Wardrop [30] and studied in a computer science context by Roughgarden and Tardos [26]. In this model, the latency experienced by network traffic on an edge of the network is a function of the edge congestion, and network users are assumed to selfishly route traffic on minimumlatency paths. The quality of a routing of traffic is measured by the sum of travel times (the total latency). It is well known that the outcome of selfish routing (a Nash equilibrium) does not minimize the total latency and can be improved upon with coordination. An ancient strategy for improving the selfish solution is the principle of marginal cost pricing, which asserts that on each edge of the network, each network user on the edge should pay a tax offsetting the congestion effects caused by its presence. By pricing network edges according to this principle, the inefficiency of selfish routing can always be eradicated. This result, while fundamental, assumes a very strong homogeneity property: all network users are assumed to trade off time and money in an identical way. The guarantee also ignores both the algorithmic
Selfish Routing In Capacitated Networks
 MATHEMATICS OF OPERATIONS RESEARCH
, 2003
"... According to Wardrop's first principle, agents in a congested network choose their routes selfishly, a behavior that is captured by the Nash equilibrium of the underlying noncooperative game. A Nash equilibrium does not optimize any global criterion per se, and so there is no apparent reason why it ..."
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Cited by 77 (4 self)
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According to Wardrop's first principle, agents in a congested network choose their routes selfishly, a behavior that is captured by the Nash equilibrium of the underlying noncooperative game. A Nash equilibrium does not optimize any global criterion per se, and so there is no apparent reason why it should be close to a solution of minimal total travel time, i.e. the system optimum. In this paper, we offer extensions of recent positive results on the efficiency of Nash equilibria in traffic networks. In contrast to prior work, we present results for networks with capacities and for latency functions that are nonconvex, nondifferentiable and even discontinuous. The inclusion of upper bounds on arc flows has early been recognized as an important means to provide a more accurate description of traffic flows. In this more general model, multiple Nash equilibria may exist and an arbitrary equilibrium does not need to be nearly efficient. Nonetheless, our main result shows that the best equilibrium is as efficient as in the model without capacities. Moreover, this holds true for broader classes of travel cost functions than considered hitherto.
Potential games with continuous player sets
 Journal of Economic Theory
"... Chapter 4 of my doctoral dissertation (Sandholm [29]). I thank an ..."
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Cited by 76 (9 self)
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Chapter 4 of my doctoral dissertation (Sandholm [29]). I thank an
Potential Function Methods for Approximately Solving Linear Programming Problems: Theory and Practice
, 2001
"... After several decades of sustained research and testing, linear programming has evolved into a remarkably reliable, accurate and useful tool for handling industrial optimization problems. Yet, large problems arising from several concrete applications routinely defeat the very best linear programming ..."
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Cited by 69 (4 self)
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After several decades of sustained research and testing, linear programming has evolved into a remarkably reliable, accurate and useful tool for handling industrial optimization problems. Yet, large problems arising from several concrete applications routinely defeat the very best linear programming codes, running on the fastest computing hardware. Moreover, this is a trend that may well continue and intensify, as problem sizes escalate and the need for fast algorithms becomes more stringent. Traditionally, the focus in optimization algorithms, and in particular, in algorithms for linear programming, has been to solve problems "to optimality." In concrete implementations, this has always meant the solution ofproblems to some finite accuracy (for example, eight digits). An alternative approach would be to explicitly, and rigorously, trade o# accuracy for speed. One motivating factor is that in many practical applications, quickly obtaining a partially accurate solution is much preferable to obtaining a very accurate solution very slowly. A secondary (and independent) consideration is that the input data in many practical applications has limited accuracy to begin with. During the last ten years, a new body ofresearch has emerged, which seeks to develop provably good approximation algorithms for classes of linear programming problems. This work both has roots in fundamental areas of mathematical programming and is also framed in the context ofthe modern theory ofalgorithms. The result ofthis work has been a family ofalgorithms with solid theoretical foundations and with growing experimental success. In this manuscript we will study these algorithms, starting with some ofthe very earliest examples, and through the latest theoretical and computational developments.
How Much Can Taxes Help Selfish Routing?
 EC'03
, 2003
"... ... in networks. We consider a model of selfish routing in which the latency experienced by network tra#c on an edge of the network is a function of the edge congestion, and network users are assumed to selfishly route tra#c on minimumlatency paths. The quality of a routing of tra#c is historically ..."
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Cited by 62 (6 self)
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... in networks. We consider a model of selfish routing in which the latency experienced by network tra#c on an edge of the network is a function of the edge congestion, and network users are assumed to selfishly route tra#c on minimumlatency paths. The quality of a routing of tra#c is historically measured by the sum of all travel times, also called the total latency. It is well known
Bounding the Inefficiency of Equilibria in Nonatomic Congestion Games
 Games and Economic Behavior
, 2002
"... Equilibria in noncooperative games are typically inefficient, as illustrated by the Prisoner's Dilemma. In this paper, we quantify this inefficiency by comparing the payoffs of equilibria to the payoffs of a "best possible" outcome. We study a nonatomic version of the congestion games defined by Ros ..."
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Cited by 61 (8 self)
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Equilibria in noncooperative games are typically inefficient, as illustrated by the Prisoner's Dilemma. In this paper, we quantify this inefficiency by comparing the payoffs of equilibria to the payoffs of a "best possible" outcome. We study a nonatomic version of the congestion games defined by Rosenthal [15], and identify games in which equilibria are approximately optimal in the sense that no other outcome achieves a significantly larger total payoff to the players  games in which optimization by individuals approximately optimizes the social good, in spite of the lack of coordination between players. Our results extend previous work on traffic routing games [16, 17, 18].