Equilibria on a circular market when consumers do not always buy from the closest firm

We study spatial competition by firms which is often studied in the context of linear markets where customers always shop at the nearest firm. Here, customer behavior is determined by a probability vector p = (p(1), ..., p(n)) where p(i) is the probability that a customer visits the ith closest firm. At the same time, the market is circular a la Salop (Bell J Econ 10(1):141-156, 1979), which has the advantage of isolating the impact of customer shopping behavior from market boundary effects. We show that non-convergent Nash equilibria, in which firms cluster at distinct positions on the market, always exist for convex probability vectors as well as probability vectors exhibiting a certain symmetry. For concave probability vectors, on the other hand, we show that non-convergent Nash equilibria are unlikely to exist.