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22
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 678 (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 Impact of Oligopolistic Competition in Networks
, 2007
"... In the traffic assignment problem, commuters select the shortest available path to travel from their origins to their destinations. This system has been studied for over 50 years since Wardrop’s seminal work (1952). Motivated by freight companies, we study a generalization of the traffic assignment ..."
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Cited by 24 (0 self)
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In the traffic assignment problem, commuters select the shortest available path to travel from their origins to their destinations. This system has been studied for over 50 years since Wardrop’s seminal work (1952). Motivated by freight companies, we study a generalization of the traffic assignment problem in which competitors, who may control a nonnegligible fraction of the total flow, ship goods across a network. This type of games, usually referred to as atomic games, readily applies to situations in which some of the freight companies have market power. Other applications include intelligent transportation systems, competition of telecommunication network service providers, and scheduling with flexible machines. Our goal is to determine to what extent these systems can benefit from some form of coordination or regulation. We measure the quality of the outcome of the game without centralized control by computing the worstcase inefficiency of Nash equilibria. The main conclusion is that although selfinterested competitors will not achieve a fullyefficient solution from the system’s point of view, the loss is not too severe. We show how to compute several bounds for the worstcase inefficiency, which depend on the characteristics of cost functions and on the market structure in the game. In addition, building upon the work of Catoni and Pallotino (1991), we show examples in which market aggregation (or collusion) adversely impacts the aggregated competitors, even though their market power increases. For example, all Nash equilibria of an
Local Smoothness and the Price of Anarchy in Atomic Splittable Congestion Games
"... We resolve the worstcase price of anarchy (POA) of atomic splittable congestion games. Prior to this work, no tight bounds on the POA in such games were known, even for the simplest nontrivial special case of affine cost functions. We make two distinct contributions. On the upperbound side, we def ..."
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Cited by 20 (4 self)
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We resolve the worstcase price of anarchy (POA) of atomic splittable congestion games. Prior to this work, no tight bounds on the POA in such games were known, even for the simplest nontrivial special case of affine cost functions. We make two distinct contributions. On the upperbound side, we define the framework of “local smoothness”, which refines the standard smoothness framework for games with convex strategy sets. While standard smoothness arguments cannot establish tight bounds on the POA in atomic splittable congestion games, we prove that local smoothness arguments can. Further, we prove that every POA bound derived via local smoothness applies automatically to every correlated equilibrium of the game. Unlike standard smoothness arguments, bounds proved using local smoothness do not always apply to the coarse correlated equilibria of the game. Our second contribution is a very general lower bound: for every set L that satisfies mild technical conditions, the worstcase POA of pure Nash equilibria in atomic splittable congestion games with cost functions in L is exactly the smallest upper bound provable using local smoothness arguments. In particular, the worstcase POA of pure Nash equilibria, mixed Nash equilibria, and correlated equilibria coincide in such games. 1
Routing (Un) Splittable Flow in Games with PlayerSpecific Linear Latency Functions
 In Proceedings of the 33rd International Colloquium on Automata, Languages, and Programming (ICALP’06), LNCS 4051
, 2006
"... Abstract. In this work we study weighted network congestion games with playerspecific latency functions where selfish players wish to route their traffic through a shared network. We consider both the case of splittable and unsplittable traffic. Our main findings are as follows: – For routing games ..."
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Cited by 18 (1 self)
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Abstract. In this work we study weighted network congestion games with playerspecific latency functions where selfish players wish to route their traffic through a shared network. We consider both the case of splittable and unsplittable traffic. Our main findings are as follows: – For routing games on parallel links with linear latency functions without a constant term we introduce two new potential functions for unsplittable and for splittable traffic respectively. We use these functions to derive results on the convergence to pure Nash equilibria and the computation of equilibria. We also show for several generalizations of these routing games that such potential functions do not exist. – We prove upper and lower bounds on the price of anarchy for games with linear latency functions. For the case of unsplittable traffic the upper and lower bound are asymptotically tight. 1
Equilibria of Atomic Flow Games are not Unique
"... In routing games with infinitesimal players, it follows from wellknown convexity arguments that equilibria exist and are unique (up to induced delays, and under weak assumptions on delay functions). In routing games with players that control large amounts of flow, uniqueness has been demonstrated o ..."
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Cited by 16 (0 self)
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In routing games with infinitesimal players, it follows from wellknown convexity arguments that equilibria exist and are unique (up to induced delays, and under weak assumptions on delay functions). In routing games with players that control large amounts of flow, uniqueness has been demonstrated only in limited cases: in 2terminal, nearlyparallel graphs; when all players control exactly the same amount of flow; when latency functions are polynomials of degree at most three. In this work, we answer an open question posed by Cominetti, Correa, and StierMoses (ICALP 2006) and show that there may be multiple equilibria in atomic player routing games. We demonstrate this multiplicity via two specific examples. In addition, we show our examples are topologically minimal by giving a complete characterization of the class of network topologies for which unique equilibria exist. Our proofs and examples are based on a novel characterization of these topologies in terms of sets of circulations.
Routing Games
, 2007
"... This chapter studies the inefficiency of equilibria in noncooperative routing games, in which selfinterested players route traffic through a congested network. Our goals are threefold: to introduce the most important models and examples of routing games; to survey optimal bounds on the price of anar ..."
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Cited by 14 (4 self)
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This chapter studies the inefficiency of equilibria in noncooperative routing games, in which selfinterested players route traffic through a congested network. Our goals are threefold: to introduce the most important models and examples of routing games; to survey optimal bounds on the price of anarchy in these models; and to develop proof techniques that are useful for bounding the inefficiency of equilibria in a range of applications.
Stackelberg strategies and collusion in network games with splittable flow
 In Proc. of the 6th Workshop on Approximation and Online Algorithms (WAOA), volume 5426 of Lecture Notes in Computer Science
, 2008
"... We study the impact of collusion in network games with splittable flow and focus on the well established price of anarchy as a measure of this impact. We first investigate symmetric load balancing games and show that the price of anarchy is bounded from above by m, where m denotes the number of coal ..."
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Cited by 13 (1 self)
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We study the impact of collusion in network games with splittable flow and focus on the well established price of anarchy as a measure of this impact. We first investigate symmetric load balancing games and show that the price of anarchy is bounded from above by m, where m denotes the number of coalitions. For general networks, we present an instance showing that the price of anarchy is unbounded, even in the case of two coalitions. If latencies are restricted to polynomials, we prove upper bounds on the price of anarchy for general networks, which improve upon the current best ones except for affine latencies. In light of the negative results even for two coalitions, we analyze the effectiveness of Stackelberg strategies as a means to improve the quality of Nash equilibria. In this setting, an α fraction of the entire demand is first routed centrally by a Stackelberg leader according to a predefined Stackelberg strategy and the remaining demand is then routed selfishly by the coalitions (followers). For a single coalitional follower and parallel arcs, we develop an efficiently computable Stackelberg strategy that reduces the price of anarchy to one. For general networks and a single coalitional follower, we show that a simple strategy, called SCALE, reduces the price of anarchy to 1+α. Finally, we investigate SCALE for multiple coalitional followers, general networks, and affine latencies. We present the first known upper bound on the price of anarchy in this case. Our bound smoothly varies between 1.5 when α = 0 and full efficiency when α = 1. 1
Internalization of airport congestion: A network analysis
 International Journal of Industrial Organization
, 2005
"... The likely resurgence of air tra±c in the U.S. means that airport congestion is a problem that must soon be confronted by policy makers. As part of their policy response, it is probable that some form of congestion pricing will be imposed at selected U.S. airports in the relatively near future. The ..."
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Cited by 13 (2 self)
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The likely resurgence of air tra±c in the U.S. means that airport congestion is a problem that must soon be confronted by policy makers. As part of their policy response, it is probable that some form of congestion pricing will be imposed at selected U.S. airports in the relatively near future. The theory developed in this paper, which extends the results of Brueckner (2002), provides an important guide for the formulation of congestion pricing rules. In particular, the theory says that the congestion tolls levied on the various airlines at a particular airport should generally be di®erent, with the tolls being inversely related to a carrier's airport °ight share. Internalization of airport congestion is the reason for this inverse relationship. In operating another peak °ight, a carrier takes account of the congestion damage imposed on the other °ights it operates. If these °ights account for a large share of the airport's tra±c, then most of the congestion created by the additional °ight is internalized, justifying a low toll. By contrast, if the carrier operates only a few of the airport's °ights, then little internalization occurs, and a high toll is needed to force the carrier to take into account the congestion damage it causes. The resulting °ightshare rule is easy to implement, and it could help policymakers design proper toll systems at U.S. airports. Internalization of Airport Congestion: A Network Analysis by Jan K. Brueckner* 1.
Traffic Congestion and Congestion Pricing
 Handbook of Transport Systems and Traffic Control
, 2000
"... For several decades growth of traffic volumes has outstripped investments in road infrastructure. The result has been a relentless increase in traffic congestion. This paper reviews the economic principles behind congestion pricing in static and dynamic settings, which derive from the benefits of ch ..."
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Cited by 10 (2 self)
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For several decades growth of traffic volumes has outstripped investments in road infrastructure. The result has been a relentless increase in traffic congestion. This paper reviews the economic principles behind congestion pricing in static and dynamic settings, which derive from the benefits of charging travellers for the externalities they create. Special attention is paid to various complications that make simple textbook congestion pricing models of limited relevance, and dictate that congestion pricing schemes be studied from the perspective of the theory of the second best. These complications include pricing in networks, heterogeneity of users, stochastic congestion, interactions of the transport sector with the rest of the economy, and tolling on private roads. Also the implications of congestion pricing for optimal road capacity are considered, and finally some explanations for the longstanding social and political resistance to road pricing are offered. * The authors would...
Efficiency and Braess’ Paradox under Pricing in General Networks
 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATION
, 2002
"... We study the flow control and routing decisions of selfinterested users in a general congested network where a single profitmaximizing service provider sets prices for different paths in the network. We define an equilibrium of the user choices. We then define the monopoly equilibrium (ME) as the ..."
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Cited by 9 (4 self)
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We study the flow control and routing decisions of selfinterested users in a general congested network where a single profitmaximizing service provider sets prices for different paths in the network. We define an equilibrium of the user choices. We then define the monopoly equilibrium (ME) as the equilibrium prices set by the service provider and the corresponding user equilibrium. We analyze the networks containing different types of user utilities: elastic or inelastic. For a network containing inelastic user utilities, we show the flow allocations at the ME and the social optimum are the same. For a network containing elastic user utilities, we explicitly characterize the ME and study its performance relative to the user equilibrium at 0 prices and the social optimum that would result from centrally maximizing the aggregate system utility. We also define Braess’ Paradox for a network involving pricing and show that Braess’ Paradox does not occur under monopoly prices.