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118
Worstcase equilibria
 IN PROCEEDINGS OF THE 16TH ANNUAL SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE
, 1999
"... In a system in which noncooperative agents share a common resource, we propose the ratio between the worst possible Nash equilibrium and the social optimum as a measure of the effectiveness of the system. Deriving upper and lower bounds for this ratio in a model in which several agents share a ver ..."
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Cited by 631 (19 self)
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In a system in which noncooperative agents share a common resource, we propose the ratio between the worst possible Nash equilibrium and the social optimum as a measure of the effectiveness of the system. Deriving upper and lower bounds for this ratio in a model in which several agents share a very simple network leads to some interesting mathematics, results, and open problems.
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
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 208 (28 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.
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
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 144 (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
The Price of Anarchy of Finite Congestion Games
 In Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC
, 2005
"... Abstract We consider the price of anarchy of pure Nash equilibria in congestion games with linearlatency functions. For asymmetric games, the price of anarchy of maximum social cost is \Theta (p N),where N is the number of players. For all other cases of symmetric or asymmetric games andfor both max ..."
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Cited by 122 (7 self)
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Abstract We consider the price of anarchy of pure Nash equilibria in congestion games with linearlatency functions. For asymmetric games, the price of anarchy of maximum social cost is \Theta (p N),where N is the number of players. For all other cases of symmetric or asymmetric games andfor both maximum and average social cost, the price of anarchy is 5 /2. We extend the results tolatency functions that are polynomials of bounded degree. We also extend some of the results to mixed Nash equilibria.
Nearoptimal network design with selfish agents
 IN PROCEEDINGS OF THE 35TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING (STOC
, 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 121 (21 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 algorithm to find a (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.
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.
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 106 (4 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
The Structure and Complexity of Nash Equilibria for a Selfish Routing Game
, 2002
"... In this work, we study the combinatorial structure and the computational complexity of Nash equilibria for a certain game that models sel sh routing over a network consisting of m parallel links. We assume a collection of n users, each employing a mixed strategy, which is a probability distribu ..."
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Cited by 101 (22 self)
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In this work, we study the combinatorial structure and the computational complexity of Nash equilibria for a certain game that models sel sh routing over a network consisting of m parallel links. We assume a collection of n users, each employing a mixed strategy, which is a probability distribution over links, to control the routing of its own assigned trac. In a Nash equilibrium, each user sel shly routes its trac on those links that minimize its expected latency cost, given the network congestion caused by the other users. The social cost of a Nash equilibrium is the expectation, over all random choices of the users, of the maximum, over all links, latency through a link.