Nonexclusive competition under adverse selection

A seller of a divisible good faces several identical buyers. The quality of the good may be low or high, and is the seller's private information. The seller has strictly convex preferences that satisfy a single-crossing property. Buyers compete by posting menus of nonexclusive contracts, so that the seller can simultaneously and privately trade with several buyers. We provide a necessary and sufficient condition for the existence of a pure-strategy equilibrium. Aggregate equilibrium trades are unique. Any traded contract must yield zero profit. If a quality is actually traded, then it is efficiently traded. Depending on parameters, both qualities may be traded, or only one of them, or the market may break down to a no-trade equilibrium.