Competing over a finite number of locations

We consider a Hotelling location game where retailers can choose one of a finite number of locations. Consumers have strict preferences over the possible available store locations and retailers aim to attract the maximum number of consumers. We prove that a pure strategy equilibrium exists if the number of retailers is large enough. Moreover, as the number of retailers grows large, in equilibrium the distribution of retailers over the locations converges to the distribution of consumers’ preferences.