Large Spatial Competition

We consider spatial competition when consumers are arbitrarily distributed on a compact metric space. Retailers can choose one of finitely many locations in this space. We focus on symmetric mixed equilibria which exist for any number of retailers. We prove that the distribution of retailers tends to agree with the distribution of the consumers when the number of competitors is large enough. The results are shown to be robust to the introduction of (1) randomness in the number of retailers and (2) different ability of the retailers to attract consumers.