Random Pricing: Bertrand Competition with Uncontested Consumers

Two firms offer a homogeneous product and compete in prices. Consumers are homogeneous yet differ in their access to the firms. Three groups exist: Consumers who have access to both firms and can compare prices; and consumers who can access only one of the two firms and are thus uncontested. The group sizes may differ: One firm's uncontested consumer base may be zero. No pure strategy equilibrium exists. In the mixed equilibrium, firms randomize on the same continuum of prices; in expectation, the firm with the larger consumer base plays a higher price and has a higher expected demand. We study: the firms' incentive to invest in demand; effects of price discrimination; and collusion. Moreover, we extend the model to more than two firms and discuss mergers.