Supplemental Material: Buyer-Optimal Learning and Monopoly Pricing

There is a seller who has an object to sell to a single buyer. The seller’s value for the object is c. The buyer’s valuation, v, is distributed according to the CDF F supported on (−∞, 1]. Let μ denote the expected valuation, that is, ∫ 1 0 vdF (v) = μ. The buyer observes a signal s about v. The joint distribution of v and s is common knowledge. The seller then gives a take-it-or-leave-it price offer to the buyer, p. Finally, the buyer trades if and only if her expected valuation conditional on her signal weakly exceeds p. If trade occurs, the payoff of the seller is p and the payoff of the buyer is v−p; otherwise, both the buyer’s payoff is zero and the seller’s payoff is c. Both the seller and the buyer are von Neumann-Morgenstern expected payoff maximizers. In what follows we fix the CDF F and analyze those signal structures which maximize the buyer’s expected payoff. We may assume without loss of generality that each signal s provides the buyer with an unbiased estimate about her valuation, that is, E (v|s) = s. The reason is that the buyer only needs to know E (v|s) in order to decide whether to trade at a given price, so it does not matter whether she observes s or E (v|s). In what follows, we restrict attention to unbiased signal structures.