Information Design in the Hold-up Problem

We analyze a bilateral trade model where the buyer can choose the probability distribution of her valuation for the good. The seller, after observing the buyer’s choice of the distribution but not the realized valuation, makes a take-it-or-leave-it offer. If the buyer’s choice of the distribution is costless, the price and the payoffs of both the buyer and the seller are shown to be 1/e in the unique equilibrium outcome. The equilibrium distribution of the buyer’s valuation generates a unit-elastic demand and trade is ex-post efficient. These two properties are shown to be preserved even when different distributions are differentially costly as long as the cost is monotone in the dispersion of the distribution.