The Competitive Facility Location Problem in a Duopoly

We consider a competitive facility location game on a network in which consumers who are located on the vertices wish to connect to the nearest facility. Knowing this, each competitor locates a facility on a vertex, trying to capture the largest-possible market share. Focusing in the two-player case, we study conditions that guarantee the existence of a pure-strategy Nash equilibrium in this finite non-cooperative game for progressively more complicated classes of networks. The case of trees, which extends the classic Hotelling model, is well-studied: equilibrium locations are characterized by centroids of the tree, which always exist because they are solutions to the (centralized) 1-median problem. We find that equilibria in cycles exist when there is at least one vertex with a sufficiently big demand, in which case it must also be a median of the cycle. For a general graph, we construct a tree of maximal bi-connected components and apply the results for trees and cycles to get sufficient conditions for the existence of an equilibrium. This provides a complete and efficient characterization of equilibria for simple-core graphs, a class of network topologies where the central bi-connected component is a vertex or a cycle (e.g., cactus graphs). We show that on simple-core graphs, at equilibrium both competitors locate their facilities in a solution to the 1-median problem, generalizing the classical insight by hotelling from a line to more complicated networks. We also show that removing edges from simple-core graphs increases the consumer cost. This precludes situations like the Braess paradox, whereby removing an edge can increase the performance for all players. These results imply that the networks with the worst-possible equilibria are achieved in trees because they are minimal instances with respect to inclusion. While equilibria can be arbitrary inefficient with respect to centralized solutions to the location problem, we quantify the inefficiency with parametric upper bounds that depend on topological parameters of the network.