  Facility location games have been a topic of major interest in economics,
operations research and computer science, starting from the seminal work by
Hotelling. In the classical pure location Hotelling game businesses compete for
maximizing customer attraction by strategically locating their facilities,
assuming no price competition, while customers are attracted to their closest
facilities. Surprisingly, very little rigorous work has been presented on
multi-unit facility location games, where the classical pure location Hotelling
games are extended to contexts where each player is to locate several
facilities. In this paper we present two major contributions to the study of
multi-unit pure location Hotelling games. In the first part of this paper we
deal with the two-player multi-unit setting, and fully characterize its
equilibria. In the second part of this paper we deal with multi-unit facility
location games, with $N\geq 3$ players; our main result in this part is the
full characterization of the settings where pure strategy equilibria exist. Our
results also extend classical results on necessary and sufficient conditions
for a strategy profile to be in equilibrium in a pure location (single-unit)
Hotelling game to multi-unit pure location games.
