The Price of Stability for Network Design with Fair Cost Allocation

TLDR

Network design involves self‑interested agents forming connections, and the resulting Nash equilibria can differ markedly from centrally optimal solutions, with the best Nash equilibrium representing the most stable feasible network. The study investigates the price of stability, defined as the ratio between the cost of the best Nash equilibrium and the optimal network cost. The authors analyze the price of stability under a fair‑division cost allocation derived from the Shapley value, relate it to potential games, extend the analysis to cost‑latency trade‑offs—especially in delay‑only networks—and provide convergence bounds for best‑response dynamics and weighted extensions. The price of stability under fair cost allocation is bounded by O(log k), and best‑response dynamics converge to a near‑optimal Nash equilibrium, demonstrating that this protocol effectively induces strategic behavior toward near‑optimal networks.

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.