  We study a class of location games where players want to attract as many
resources as possible and pay a cost when deviating from an exogenous reference
location. This class of games includes political competitions between
policy-interested parties and firms' costly horizontal differentiation. We
provide a complete analysis of the duopoly competition: depending on the
reference locations, we observe a unique equilibrium with, or without
differentiation, or no equilibrium. We extend the analysis to a competition
between an arbitrary number of players and we show that there exists at most
one equilibrium which has a strong property: only the two most-left and
most-right players deviate from their reference locations.
