Spatial competition with heterogeneous firms

I model endogenous horizontal and vertical product differentiation with arbitrarily many heterogeneous firms. Firms are asymmetric in that they differ in their marginal costs. I prove that under an equilibrium refinement, all economically relevant firm outcomes are uniquely determined across all strict subgame perfect Nash equilibria. There are two central results. First, a firm's price, market share, and profit are independent of its neighbors' marginal costs, conditional on the average marginal cost in the market. Second, more productive firms are more isolated, all else equal. In particular, the distance between two firms is strictly decreasing in their average marginal cost.