We consider a location problem of two servers called S1 and S2, which must provide an essential service to a set of users located at the nodes of a tree network. The servers belong to different organizations and can locate at any two different points on the tree. S1 locates first at a point x, then S2, knowing x, locates at a point y, y ≠ x. Thereafter, each user select its closest server to attend its demand; ties are broken by some given assignment rule. In this competitive situation, each server wishes to optimize a given globalizing function depending on the distance to that server and the level of demand of the users selecting that server. We analyze this problem as a two-stage location game for which Nash equilibria are obtained by backward induction. In particular, we consider market share and maximum travel time as globalizing functions.
