Market with Bilateral Bargaining and Incomplete Information

We study the relationship between bargaining and competition with incomplete information. We consider a model with two uninformed and identical buyers and two sellers. One of the sellers has a privately-known reservation price, which can either be Low or High. The other seller’s reservation price is commonly known to be in between the Low and High values of the privately-informed seller. Buyers move in sequence, and make offers with the second buyer observing the offer made by the first buyer. The sellers respond simultaneously. We show that there are two types of (perfect Bayes) equilibrium. In one equilibrium, the buyer who moves second does better. In the second equilibrium, buyers’ expected payoffs are equalised, and the price received by the seller with the known reservation value is determined entirely by the equuilibrium of the two-player game between a single buyer and an informed seller. We also discuss extensions of the model to multiple buyers and sellers, and to the case where both sellers are privately informed.