Screening Under Competition
We study competition among multiple firms that offer differentiated varieties of the same good to a unit-demand agent. The agent has heterogeneous valuations for goods from different firms. Firms do not observe the agent's exact valuations, but they know their distribution. Firms simultaneously post menus of contracts, after which the agent chooses a firm and one of its contracts to maximize her utility. This defines a game in which firms aim to maximize expected revenue. We introduce a sufficient condition, density-regularity, under which each firm's best response to any arbitrary menu profile posted by its opponents is equivalent to posting a menu that contains only a posted-price contract. Our result is not a direct extension of the canonical Myersonian model with a single seller. The standard argument in the literature breaks down once heterogeneous preferences and competition are introduced. We therefore adopt an optimal-control approach, in which the density-regularity condition is essential for establishing the optimality of posted prices. When this condition fails, posted prices may fail to be a best response.