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217
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 minimum-latency 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 self-interested 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 self-interested 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 widely-studied protocols for network cost allocation, in which the cost of each edge is divided equally between users whose connections make use of it; this fair-division 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 best-response 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 near-optimal 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 best-response 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 self-interested 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 self-interested 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 worst-possible 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.
Tight bounds for worst-case equilibria
- Proc. 13th SODA
, 2002
"... We study the problem of traffic routing in non-cooperative networks. In such networks, users may follow selfish strategies to optimize their own performance measure and therefore their behavior does not have to lead to optimal performance of the entire network. In this paper we investigate the worst ..."
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Cited by 172 (5 self)
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We study the problem of traffic routing in non-cooperative networks. In such networks, users may follow selfish strategies to optimize their own performance measure and therefore their behavior does not have to lead to optimal performance of the entire network. In this paper we investigate the worst-case coordination ratio, which is a game theoretic measure aiming to reflect the price of selfish routing. Following a line of previous work, we focus on the most basic networks consisting of parallel links with linear latency functions. Our main result is that the worst-case coordination ratio on m parallel links of possibly different speeds is logm Θ log log logm In fact, we are able to give an exact description of the worst-case coordination ratio depending on the number of links and the ratio of the speed of the fastest link over the speed of the slowest link. For example, for the special case in which all m parallel links have the same speed, we can prove that the worst-case coordination ratio is Γ (−1) (m) + Θ(1) with Γ denoting the Gamma (factorial) function. Our bounds entirely resolve an open problem posed recently by Koutsoupias and Papadimitriou [KP99].
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 165 (6 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.
Near-optimal 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 NP-complete. 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 3-approximate 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.
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 140 (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
Pricing network edges for heterogeneous selfish users
- Proc. of STOC
, 2003
"... We study the negative consequences of selfish behavior in a congested network and economic means of influencing such behavior. We consider the model of selfish routing defined by Wardrop [30] and studied in a computer science context by Roughgarden and Tardos [26]. In this model, the latency experie ..."
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Cited by 113 (9 self)
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We study the negative consequences of selfish behavior in a congested network and economic means of influencing such behavior. We consider the model of selfish routing defined by Wardrop [30] and studied in a computer science context by Roughgarden and Tardos [26]. In this model, 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 minimum-latency paths. The quality of a routing of traffic is measured by the sum of travel times (the total latency). It is well known that the outcome of selfish routing (a Nash equilibrium) does not minimize the total latency and can be improved upon with coordination. An ancient strategy for improving the selfish solution is the principle of marginal cost pricing, which asserts that on each edge of the network, each network user on the edge should pay a tax offsetting the congestion effects caused by its presence. By pricing network edges according to this principle, the inefficiency of selfish routing can always be eradicated. This result, while fundamental, assumes a very strong homogeneity property: all network users are assumed to trade off time and money in an identical way. The guarantee also ignores both the algorithmic
Intrinsic Robustness of the Price of Anarchy
- STOC'09
, 2009
"... The price of anarchy (POA) is a worst-case measure of the inefficiency of selfish behavior, defined as the ratio of the objective function value of a worst Nash equilibrium of a game and that of an optimal outcome. This measure implicitly assumes that players successfully reach some Nash equilibrium ..."
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Cited by 101 (12 self)
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The price of anarchy (POA) is a worst-case measure of the inefficiency of selfish behavior, defined as the ratio of the objective function value of a worst Nash equilibrium of a game and that of an optimal outcome. This measure implicitly assumes that players successfully reach some Nash equilibrium. This drawback motivates the search for inefficiency bounds that apply more generally to weaker notions of equilibria, such as mixed Nash and correlated equilibria; or to sequences of outcomes generated by natural experimentation strategies, such as successive best responses or simultaneous regret-minimization. We prove a general and fundamental connection between the price of anarchy and its seemingly stronger relatives in classes of games with a sum objective. First, we identify a “canonical sufficient condition ” for an upper bound of the POA for pure Nash equilibria, which we call a smoothness argument. Second, we show that every bound derived via a smoothness argument extends automatically, with no quantitative degradation in the bound, to mixed Nash equilibria, correlated equilibria, and the average objective function value of regret-minimizing players (or “price of total anarchy”). Smoothness arguments also have automatic implications for the inefficiency of approximate and Bayesian-Nash equilibria and, under mild additional assumptions, for bicriteria bounds and for polynomial-length best-response sequences. We also identify classes of games — most notably, congestion games with cost functions restricted to an arbitrary fixed set — that are tight, in the sense that smoothness arguments are guaranteed to produce an optimal worst-case upper bound on the POA, even for the smallest set of interest (pure Nash equilibria). Byproducts of our proof of this result include the first tight bounds on the POA in congestion games with non-polynomial cost functions, and the first
