 
Summary: THE PRICE OF STABILITY FOR NETWORK DESIGN
WITH FAIR COST ALLOCATION
ELLIOT ANSHELEVICH, ANIRBAN DASGUPTA, JON KLEINBERG §, ŽEVA TARDOS¶,
TOM WEXLER , 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 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
