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204
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 504 (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
Fair endtoend windowbased congestion control
 IEEE/ACM TRANS. ON NETWORKING
, 2000
"... In this paper, we demonstrate the existence of fair endtoend windowbased congestion control protocols for packetswitched networks with first comefirst served routers. Our definition of fairness generalizes proportional fairness and includes arbitrarily close approximations of maxmin fairness. T ..."
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Cited by 461 (3 self)
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In this paper, we demonstrate the existence of fair endtoend windowbased congestion control protocols for packetswitched networks with first comefirst served routers. Our definition of fairness generalizes proportional fairness and includes arbitrarily close approximations of maxmin fairness. The protocols use only information that is available to end hosts and are designed to converge reasonably fast. Our study is based on a multiclass fluid model of the network. The convergence of the protocols is proved using a Lyapunov function. The technical challenge is in the practical implementation of the protocols.
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 180 (20 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...
Tsitsiklis. Efficiency loss in a network resource allocation game
 Mathematics of Operations Research
"... We consider a resource allocation problem where individual users wish to send data across a network to maximize their utility, and a cost is incurred at each link that depends on the total rate sent through the link. It is known that as long as users do not anticipate the effect of their actions on ..."
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Cited by 141 (10 self)
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We consider a resource allocation problem where individual users wish to send data across a network to maximize their utility, and a cost is incurred at each link that depends on the total rate sent through the link. It is known that as long as users do not anticipate the effect of their actions on prices, a simple proportional pricing mechanism can maximize the sum of users’ utilities minus the cost (called aggregate surplus). Continuing previous efforts to quantify the effects of selfish behavior in network pricing mechanisms, we consider the possibility that users anticipate the effect of their actions on link prices. Under the assumption that the links’ marginal cost functions are convex, we establish existence of a Nash equilibrium. We show that the aggregate surplus at a Nash equilibrium is no worse than a factor of 4 √ 2 − 5 times the optimal aggregate surplus; thus, the efficiency loss when users are selfish is no more than approximately 34%. The current Internet is used by a widely heterogeneous population of users; not only are different types of traffic sharing the same network, but different end users place different values on their perceived network performance. This has led to a surge of interest in congestion pricing, where
COMPUTATION OF EQUILIBRIA in Finite Games
, 1996
"... We review the current state of the art of methods for numerical computation of Nash equilibria for nitenperson games. Classical path following methods, such as the LemkeHowson algorithm for two person games, and Scarftype fixed point algorithms for nperson games provide globally convergent metho ..."
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Cited by 118 (1 self)
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We review the current state of the art of methods for numerical computation of Nash equilibria for nitenperson games. Classical path following methods, such as the LemkeHowson algorithm for two person games, and Scarftype fixed point algorithms for nperson games provide globally convergent methods for finding a sample equilibrium. For large problems, methods which are not globally convergent, such as sequential linear complementarity methods may be preferred on the grounds of speed. None of these methods are capable of characterizing the entire set of Nash equilibria. More computationally intensive methods, which derive from the theory of semialgebraic sets are required for finding all equilibria. These methods can also be applied to compute various equilibrium refinements.
On the existence of equilibria in noncooperative optimal flow control
 Journal of the ACM
, 1995
"... Abstract. The existence of Nash equilibria in noncooperative flow control in a general productform network shared by K users is investigated. The performance objective of each user is to maximize its average throughput subject to an upper bound on its average timedelay. Previous attempts to study e ..."
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Cited by 71 (10 self)
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Abstract. The existence of Nash equilibria in noncooperative flow control in a general productform network shared by K users is investigated. The performance objective of each user is to maximize its average throughput subject to an upper bound on its average timedelay. Previous attempts to study existence of equilibria for this flow control model were not successful, partly because the timedelay constraints couple the strategy spaces of the individual users in a way that does not allow the application of standard equilibrmm existence theorems from the game theory literature. To overcome this difficulty, a more general approach to study the existence of Nash equilibria for decentralized control schemes is introduced. This approach is based on directly proving the existence of a fixed point of the best reply correspondence of the underlying game. For the investigated flow control model, the best reply correspondence is shown to be a function, implicitly defined by means of K interdependent linear programs. Employing an appropriate definition for continuity of the set of optimal solutions of parametrized linear programs, it is shown that, under appropriate conditions, the best reply function is continuous. Brouwer’s theorem implies, then, that the best reply function has a fixed point.
Competitive Routing in Networks with Polynomial Costs
, 2000
"... We study a class of noncooperative general topology networks shared by N users. Each user has a given flow which it has to ship from a source to a destination. We consider a class of polynomial link cost functions adopted originally in the context of road traffic modeling, and show that these costs ..."
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Cited by 56 (26 self)
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We study a class of noncooperative general topology networks shared by N users. Each user has a given flow which it has to ship from a source to a destination. We consider a class of polynomial link cost functions adopted originally in the context of road traffic modeling, and show that these costs have appealing properties that lead to predictable and efficient network flows. In particular, we show that the Nash equilibrium is unique, and is moreover efficient. These properties make the polynomial cost structure attractive for traffic regulation and link pricing in telecommunication networks. We nally discuss the computation of the equilibrium in the special case of the affine cost structure for a topology of parallel links.
Quality of Service Provision in Noncooperative Networks: Heterogenous Preferences, MultiDimensional QoS Vectors, and Burstiness
 In Proc. 1st International Conference on Information and Computation Economies
, 1998
"... This paper studies the quality of service (QoS) provision problem in noncooperative networks where applications or users are selfish and routers implement generalized processor sharing (GPS)based packet scheduling. First, we formulate a model of QoS provision in noncooperative networks where users ..."
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Cited by 44 (8 self)
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This paper studies the quality of service (QoS) provision problem in noncooperative networks where applications or users are selfish and routers implement generalized processor sharing (GPS)based packet scheduling. First, we formulate a model of QoS provision in noncooperative networks where users are given the freedom to choose both the service classes and traffic volume allocated, and heterogenous QoS preferences are captured by individual utility functions. We present a comprehensive analysis of the noncooperative multiclass QoS provision game, giving a complete characterization of Nash equilibria and their existence criteria, and show under what conditions they are Pareto and system optimal. We show that, in general, Nash equilibria need not exist, and when they do exist, they need not be Pareto nor system optimal. However, we show that for certain "resourceplentiful" systems, the world indeed can be nice with Nash equilibria, Pareto optima, and system optima collapsing into a s...
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 42 (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.