Location of Competing Facilities in a User-Optimizing Environment with Market Externalities

In many location models, it is assumed that customers select the facility they will patronize based on the distance to that facility, or perhaps on a function of distances between the customer and the different available facilities. We consider a location problem in which customers select a facility based not only on travel time or distance to the facility, but also on negative externalities associated with the market share of the facility. A customer from any particular location frequents the facility that minimizes his travel time to that facility plus an externality cost that depends on the aggregated actions of all customers in the system. We consider the case of two competing facilities, each of which wishes to locate to maximize its market share. We specialize our analysis to the case of a tree network. We characterize the optimal facility locations, and develop an O(n²) algorithm for finding them.