Supportability of network cost functions

Abstract

This paper considers a network supply problem in which flows between any pair of nodes are possible. It is assumed that users place a value on connection to other users in the network, and (possibly) on access to an external source. Cost on each link is an arbitrary concave function of link capacity. The objective is to study coalitional stability in this situation, when collections of flows can be served by competing suppliers. In contrast to other network games, this approach focuses on the cost of serving flows rather than the cost of attaching nodes to the network. The network is said to be stable if the derived cost function is supportable. Supportable cost functions, defined by Sharkey and Telser [9], are cost functions for which there exists a price vector which covers total cost, and simultaneously deters entry at any lower output by a rival firm with the same cost function. If the minimal cost network includes a link between every pair of nodes, then the cost function is shown to be supportable. In the special case in which link cost is independent of capacity, the cost function is also supportable. The paper also considers “Steiner” networks in which new nodes may be created in order to minimize total cost, or in which access may be obtained at more than one source location. When link costs are independent of capacity in such a network, it is argued that the cost function is approximately supportable in a well defined sense.