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Selfish Routing and the Price of Anarchy
 MATHEMATICAL PROGRAMMING SOCIETY NEWSLETTER
, 2007
"... Selfish routing is a classical mathematical model of how selfinterested users might route traffic through a congested network. The outcome of selfish routing is generally inefficient, in that it fails to optimize natural objective functions. The price of anarchy is a quantitative measure of this in ..."
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Cited by 252 (11 self)
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Selfish routing is a classical mathematical model of how selfinterested users might route traffic through a congested network. The outcome of selfish routing is generally inefficient, in that it fails to optimize natural objective functions. The price of anarchy is a quantitative measure of this inefficiency. We survey recent work that analyzes the price of anarchy of selfish routing. We also describe related results on bounding the worstpossible severity of a phenomenon called Braess’s Paradox, and on three techniques for reducing the price of anarchy of selfish routing. This survey concentrates on the contributions of the author’s PhD thesis, but also discusses several more recent results in the area.
The price of routing unsplittable flow
 In Proc. 37th Symp. Theory of Computing (STOC
, 2005
"... The essence of the routing problem in real networks is that the traffic demand from a source to destination must be satisfied by choosing a single path between source and destination. The splittable version of this problem is when demand can be satisfied by many paths, namely a flow from source to d ..."
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Cited by 136 (3 self)
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The essence of the routing problem in real networks is that the traffic demand from a source to destination must be satisfied by choosing a single path between source and destination. The splittable version of this problem is when demand can be satisfied by many paths, namely a flow from source to destination. The unsplittable, or discrete version of the problem is more realistic yet is more complex from the algorithmic point of view; in some settings optimizing such unsplittable traffic flow is computationally intractable. In this paper, we assume this more realistic unsplittable model, and investigate the ”price of anarchy”, or deterioration of network performance measured in total traffic latency under the selfish user behavior. We show that for linear edge latency functions the price of anarchy is exactly 2.618 for weighted demand and exactly 2.5 for unweighted demand. These results are easily extended to (weighted or unweighted) atomic ”congestion games”, where paths are replaced by general subsets. We also show that for polynomials of degree d edge latency functions the price of anarchy is dΘ(d). Our results hold also for mixed strategies. Previous results of Roughgarden and Tardos showed that for linear edge latency functions the price of anarchy is exactly 4 3 under the assumption that each user controls only a negligible fraction of the overall traffic (this result also holds for the splittable case). Note that under the assumption of negligible traffic pure and mixed strategies are equivalent and also splittable and unsplittable models are equivalent. 1
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 77 (5 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
Tolls for heterogeneous selfish users in multicommodity networks and generalized congestion games
"... ..."
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 57 (6 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.
Edge pricing of multicommodity networks for heterogeneous selfish users
 In Proceedings of the 45th Annual Symposium on Foundations of Computer Science (FOCS
, 2004
"... We examine how the selfish behavior of heterogeneous users in a network can be regulated through economic disincentives, i.e., through the introduction of appropriate taxation. One wants to impose taxes on the edges so that any traffic equilibrium reached by the selfish users who are conscious of bo ..."
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Cited by 53 (7 self)
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We examine how the selfish behavior of heterogeneous users in a network can be regulated through economic disincentives, i.e., through the introduction of appropriate taxation. One wants to impose taxes on the edges so that any traffic equilibrium reached by the selfish users who are conscious of both the travel latencies and the taxes will minimize the social cost, i.e., will minimize the total latency. We generalize previous results of Cole, Dodis and Roughgarden that held for a single origindestination pair to the multicommodity setting. Our approach, which could be of independent interest, is based on the formulation of traffic equilibria as a nonlinear complementarity problem by Aashtiani and Magnanti [1]. We extend this formulation so that each of its solutions will give us a set of taxes that forces the network users to conform, at equilibrium, to a certain prescribed routing. We use the special nature of the prescribed minimumlatency flow in order to reduce the difficult nonlinear complementarity formulation to a pair of primaldual linear programs. LP duality is then enough to derive our results. 1.
A network pricing game for selfish traffic
 in Proc. of SIGACTSIGOPS Symposium on Principles of Distributed Computing (PODC
, 2005
"... The success of the Internet is remarkable in light of the decentralized manner in which it is designed and operated. Unlike small scale networks, the Internet is built and controlled by a large number of disperate service providers who are not interested in any global optimization. Instead, provider ..."
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Cited by 52 (1 self)
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The success of the Internet is remarkable in light of the decentralized manner in which it is designed and operated. Unlike small scale networks, the Internet is built and controlled by a large number of disperate service providers who are not interested in any global optimization. Instead, providers simply seek to maximize their own profit by charging users for access to their service. Users themselves also behave selfishly, optimizing over price and quality of service. Game theory provides a natural framework for the study of such a situation. However, recent work in this area tends to focus on either the service providers or the network users, but not both. This paper introduces a new model for exploring the interaction of these two elements, in which network managers compete for users via prices and the quality of service provided. We study the extent to which competition between service providers hurts the overall social utility of the system.
Fast convergence to Wardrop equilibria by adaptive sampling methods
 in Proc. 38th Ann. ACM. Symp. on Theory of Comput. (STOC
, 2006
"... We study rerouting policies in a dynamic roundbased variant of a well known game theoretic traffic model due to Wardrop. Previous analyses (mostly in the context of selfish routing) based on Wardrop’s model focus mostly on the static analysis of equilibria. In this paper, we ask the question whethe ..."
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Cited by 50 (6 self)
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We study rerouting policies in a dynamic roundbased variant of a well known game theoretic traffic model due to Wardrop. Previous analyses (mostly in the context of selfish routing) based on Wardrop’s model focus mostly on the static analysis of equilibria. In this paper, we ask the question whether the population of agents responsible for routing the traffic can jointly compute or better learn a Wardrop equilibrium efficiently. The rerouting policies that we study are of the following kind. In each round, each agent samples an alternative routing path and compares the latency on this path with its current latency. If the agent observes that it can improve its latency then it switches with some probability depending on the possible improvement to the better path. We can show various positive results based on a rerouting policy using an adaptive sampling rule that implicitly amplifies paths that carry a large amount of traffic in the Wardrop equilibrium. For general asymmetric games, we show that a simple replication protocol in which agents adopt strategies of more successful agents reaches a certain kind of bicriteria equilibrium within a time bound that is independent of the size and the structure of the network but only depends on a parameter of the latency functions, that we call the relative slope. For symmetric games, this result has an intuitive interpretation: Replication approximately satisfies almost everyone very quickly. In order to achieve convergence to a Wardrop equilibrium besides replication one also needs an exploration component discovering possibly unused strategies. We present a
An Empirical Analysis of Network Externalities in PeertoPeer MusicSharing Networks
, 2004
"... Peertopeer (P2P) file sharing networks are an important medium for the distribution of information goods. However, there is little empirical research into the optimal design of these networks under realworld conditions. Early speculation about the behavior of P2P networks has focused on the role ..."
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Cited by 50 (3 self)
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Peertopeer (P2P) file sharing networks are an important medium for the distribution of information goods. However, there is little empirical research into the optimal design of these networks under realworld conditions. Early speculation about the behavior of P2P networks has focused on the role that positive network externalities play in improving performance as the network grows. However, negative network externalities also arise in P2P networks because of the consumption of scarce network resources or an increased propensity of users to free ride in larger networks, and the impact of these negative network externalities—while potentially important—has received far less attention. Our research addresses this gap in understanding by measuring the impact of both positive and negative network externalities on the optimal size of P2P networks. Our research uses a unique dataset collected from the six most popular OpenNap P2P networks between December 19, 2000, and April 22, 2001. We find that users contribute additional value to the network at a decreasing rate and impose costs on the network at an increasing rate, while the network increases in size. Our results also suggest that users are less likely to contribute resources to the network as the network size increases. Together, these results suggest that the optimal size of these centralized P2P networks is bounded—At some point the costs that a marginal user imposes on the network will exceed the value they provide to the network. This finding is in contrast to early predictions that larger
The Effectiveness of Stackelberg Strategies and Tolls for Network Congestion Games
 In Proc. Symposium on Discrete Algorithms (SODA
, 2007
"... Abstract It is well known that in a network with arbitrary(convex) latency functions that are a function of edge ..."
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Cited by 38 (1 self)
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Abstract It is well known that in a network with arbitrary(convex) latency functions that are a function of edge