A Facility Location Problem under Competition

We consider a generalization of the discrete Voronoi game, in which consumers with positive purchase power who are located on the vertices of a network wish to connect to the nearest facility. Knowing this, competitive players locate their facilities on vertices, trying to capture the largest possible market share. 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, focusing in the two-player case although some results can be extended to a larger number of players. We find that equilibria in cycles exist when there is at least one dominant vertex with a sufficiently big demand. In the case of trees, equilibria always exist and consist of players selecting a centroid, defined as a solution to a centralized facility location game. For the case of a general graph, we construct a tree of maximal bi-connected components and apply the results derived for the simpler classes to get sufficient conditions for the existence of an equilibrium. This is shown to provide a complete and efficient characterization of equilibria in a broad class of structures, that includes cactus graphs. Finally, we provide sufficient conditions that guarantee that removing edges from the network increases consumer cost, which precludes situations like the Braess paradox, whereby removing an edge can increase the performance for all players. We show these conditions hold in a broad class of structures. 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 we show that equilibria can be arbitrary inefficient in general, we provide parametric upper bounds that depend on the topology of the network.