Patterns of Competition with Captive Customers

We study mixed-strategy equilibrium pricing in oligopoly settings where consumers vary in the set of suppliers they consider for their purchase---some being captive to a particular firm, some consider two particular firms, and so on. In the case of "nested reach" we find equilibria, unlike those in more standard models, in which firms are ranked in terms of the prices they might charge. We characterize equilibria in the three-firm case, and contrast them with equilibria in the parallel model with capacity constraints. A theme of the analysis is how patterns of consumer interaction with firms matter for competitive outcomes.