We consider a non-cooperative facility location game by two competitive companies on a linear and a planar market with discrete demand points. The leader and the follower locate their branch shops in the market one by one in order to maximize their own profit, which is assumed to be proportional to the amount of demand taken in their territories. The feature of this paper is that the demand is assumed to be vague and different according to each facility. We consider two types of problems, i.e., to find the optimal location for the follower and to find that for the leader. We formulate these problems with fuzzy number and solve them by using some properties derived in this paper.