The (l, k) Facility Location Duel

Motivated by applications in clustering and information retrieval, we extend the classical Hotelling setting to deal with multi-facility location duels. In the classical Hotelling setting, customers' locations are taken from the uniform distribution on the $[0,1]$ segment, and there are two facility owners, each needs to decide on the location of her (single) facility, aiming to maximize the proportion of customers closer to it. We extend this duel to competition among the owner of $k$ facilities to the owner of $l$ facilities, for arbitrary $k,l$, where $l\leq k$. Our main message is quite striking: in no equilibrium a facility will materialize in a location which is not part of the social welfare maximizing locations of the player who has $k$ facilities, if she were to locate her facilities under no competition. This is obtained despite the lack of pure strategy equilibrium in many of these settings. We also study two sets of other natural families of distributions; for one of the families we show the above findings extend beyond the uniform distribution; for the other we show that selecting only among the stronger-player social welfare maximizing locations will not materialize in equilibrium.