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126
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 657 (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 stability for network design with fair cost allocation
 In Proceedings of the 45th Annual Symposium on Foundations of Computer Science (FOCS
, 2004
"... Abstract. Network design is a fundamental problem for which it is important to understand the effects of strategic behavior. Given a collection of selfinterested agents who want to form a network connecting certain endpoints, the set of stable solutions — the Nash equilibria — may look quite differ ..."
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Cited by 281 (30 self)
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Abstract. Network design is a fundamental problem for which it is important to understand the effects of strategic behavior. Given a collection of selfinterested agents who want to form a network connecting certain endpoints, the set of stable solutions — the Nash equilibria — may look quite different from the centrally enforced optimum. We study the quality of the best Nash equilibrium, and refer to the ratio of its cost to the optimum network cost as the price of stability. The best Nash equilibrium solution has a natural meaning of stability in this context — it is the optimal solution that can be proposed from which no user will defect. We consider the price of stability for network design with respect to one of the most widelystudied protocols for network cost allocation, in which the cost of each edge is divided equally between users whose connections make use of it; this fairdivision scheme can be derived from the Shapley value, and has a number of basic economic motivations. We show that the price of stability for network design with respect to this fair cost allocation is O(log k), where k is the number of users, and that a good Nash equilibrium can be achieved via bestresponse dynamics in which users iteratively defect from a starting solution. This establishes that the fair cost allocation protocol is in fact a useful mechanism for inducing strategic behavior to form nearoptimal equilibria. We discuss connections to the class of potential games defined by Monderer and Shapley, and extend our results to cases in which users are seeking to balance network design costs with latencies in the constructed network, with stronger results when the network has only delays and no construction costs. We also present bounds on the convergence time of bestresponse dynamics, and discuss extensions to a weighted game.
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 255 (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 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 217 (17 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
Nearoptimal network design with selfish agents
, 2003
"... We introduce a simple network design game that models how independent selfish agents can build or maintain a large network. In our game every agent has a specific connectivity requirement, i.e. each agent has a set of terminals and wants to build a network in which his terminals are connected. Possi ..."
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Cited by 151 (19 self)
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We introduce a simple network design game that models how independent selfish agents can build or maintain a large network. In our game every agent has a specific connectivity requirement, i.e. each agent has a set of terminals and wants to build a network in which his terminals are connected. Possible edges in the network have costs and each agent’s goal is to pay as little as possible. Determining whether or not a Nash equilibrium exists in this game is NPcomplete. However, when the goal of each player is to connect a terminal to a common source, we prove that there is a Nash equilibrium as cheap as the optimal network, and give a polynomial time algorithmtofinda(1+ε)approximate Nash equilibrium that does not cost much more. For the general connection game we prove that there is a 3approximate Nash equilibrium that is as cheap as the optimal network, and give an algorithm to find a (4.65 +ε)approximate Nash equilibrium that does not cost much more.
Computing Optimal Randomized Resource Allocations for Massive Security Games
 IN AAMAS09
, 2009
"... Predictable allocations of security resources such as police officers, canine units, or checkpoints are vulnerable to exploitation by attackers. Recent work has applied gametheoretic methods to find optimal randomized security policies, including a fielded application at the Los Angeles Internation ..."
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Cited by 117 (64 self)
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Predictable allocations of security resources such as police officers, canine units, or checkpoints are vulnerable to exploitation by attackers. Recent work has applied gametheoretic methods to find optimal randomized security policies, including a fielded application at the Los Angeles International Airport (LAX). This approach has promising applications in many similar domains, including police patrolling for subway and bus systems, randomized baggage screening, and scheduling for the Federal Air Marshal Service (FAMS) on commercial flights. However, the existing methods scale poorly when the security policy requires coordination of many resources, which is central to many of these potential applications. We develop new models and algorithms that scale to much more complex instances of security games. The key idea is to use a compact model of security games, which allows exponential improvements in both memory and runtime relative to the best known algorithms for solving general Stackelberg games. We develop even faster algorithms for security games under payoff restrictions that are natural in many security domains. Finally, introduce additional realistic scheduling constraints while retaining comparable performance improvements. The empirical evaluation comprises both random data and realistic instances of the FAMS and LAX problems. Our new methods scale to problems several orders of magnitude larger than the fastest known algorithm.
Selfish Traffic Allocation for Server Farms
, 2003
"... We study the price of selfish routing in noncooperative networks like the Internet. In particular, we investigate the price... ..."
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Cited by 76 (5 self)
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We study the price of selfish routing in noncooperative networks like the Internet. In particular, we investigate the price...
Designing networks for selfish users is hard
 In Proceedings of the 42nd Annual Symposium on Foundations of Computer Science
, 2001
"... Abstract We consider a directed network in which every edge possesses a latency function specifying the time needed to traverse the edge given its congestion. Selfish, noncooperative agents constitute the network traffic and wish to travel from a source s to a sink t as quickly as possible. Since th ..."
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Cited by 65 (8 self)
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Abstract We consider a directed network in which every edge possesses a latency function specifying the time needed to traverse the edge given its congestion. Selfish, noncooperative agents constitute the network traffic and wish to travel from a source s to a sink t as quickly as possible. Since the route chosen by one network user affects the congestion (and hence the latency) experienced by others, we model the problem as a noncooperative game. Assuming each agent controls only a negligible portion of the overall traffic, Nash equilibria in this noncooperative game correspond to st flows in which all flow paths have equal latency. A natural measure for the performance of a network used by selfish agents is the common latency experienced by each user in a Nash equilibrium. It is a counterintuitive but wellknown fact that removing edges from a network may improve its performance; the most famous example of this phenomenon is the socalled Braess's Paradox. This fact motivates the following network design problem: given such a network, which edges should be removed to obtain the best possible flow at Nash equilibrium? Equivalently, given a large network of candidate edges to be built, which subnetwork will exhibit the best performance when used selfishly? We give optimal inapproximability results and approximation algorithms for several network design problems of this type. For example, we prove that for networks with n vertices and continuous, nondecreasing latency functions, there is no approximation algorithm for this problem with approximation ratio less than n/2 (unless P = N P). We also prove this hardness result to be best possible by exhibiting an n/2approximation algorithm. For networks in which the latency of each edge is a linear function of the congestion, we prove that there is no ( 43 ffl)approximation algorithm for the problem (for any ffl> 0, unless P = N P); the existence of a 43approximation algorithm follows easily from existing work, proving this hardness result sharp. Moreover, we prove that an optimal approximation algorithm for these problems is what we call the trivial algorithm: given a network of candidate edges, build the entire network. A consequence of this result is that Braess's Paradox (even in its worstpossible manifestation) is impossible to detect efficiently.
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'06)
, 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 48 (5 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