Competitive location on a network

Consider a network with a given number of customers at fixed locations (vertices) and where each customer will purchase a commodity from the facility closer to his location more frequently than from a remote one. As a generalization of the Condorcet concept the authors define an optimal point as a location such that there exists no competitor with higher expected value. They show that the set of optimal points consists entirely of vertices. In general the authors provide polynomial algorithms to answer the question as to: What is the maximum percentage of customers located on the network prefering some rival point to an existing location? Suboptimal points where the maximal relative rejection by a rival point is minimal are determined in polynomial time.