  In this paper we study variations of the standard Hotelling-Downs model of
spatial competition, where each agent attracts the clients in a restricted
neighborhood, each client randomly picks one attractive agent for service.
  Two utility functions for agents are considered: support utility and winner
utility. We generalize the results by Feldman et al. to the case where the
clients are distributed arbitrarily. In the support utility setting, we show
that a pure Nash equilibrium always exists by formulating the game as a
potential game. In the winner utility setting, we show that there exists a Nash
equilibrium in two cases: when there are at most 3 agents and when the size of
attraction area is at least half of the entire space. We also consider the
price of anarchy and the fairness of equilibria and give tight bounds on these
criteria.
