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Unformatted text preview: THE PRICE OF STABILITY FOR NETWORK DESIGN WITH FAIR COST ALLOCATION * ELLIOT ANSHELEVICH † , ANIRBAN DASGUPTA ‡ , JON KLEINBERG § , ´ EVA TARDOS ¶ , TOM WEXLER k , AND TIM ROUGHGARDEN ** 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. Key words. network design, price of stability, Shapley cost-sharing AMS subject classifications. 68Q99, 90B18, 91A43 1. Introduction. In many network settings, the system behavior arises from the actions of a large number of independent agents, each motivated by self-interest and optimizing an individual objective function. As a result, the global performance of the system may not be as good as in a case where a central authority can simply dictate a solution; rather, we need to understand the quality of solutions that are consistent with self-interested behavior. Recent theoretical work has framed this type of question in the following general form: how much worse is the solution quality of a Nash * A preliminary version of this paper appeared in Proc. 45th Annual Symposium on Foundations of Computer Science, 2004....
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This note was uploaded on 12/08/2011 for the course CIS 677 taught by Professor Michaelkearns during the Fall '09 term at Penn State.
- Fall '09