Equilibrium opacity in ultimatum-offer bargaining

Abstract

We consider ultimatum bargaining between a seller and a buyer of an asset. They know each other's valuation of the asset. Both can defer their decisions to delegates. These delegates have opaque preferences. Seller and buyer choose the opacity of their delegate. For the seller's delegate this choice is restricted to a random reservation price drawn from the set of symmetric two-point distributions around the seller's true reservation price. The opacity choice of the buyer's delegate is restricted to a random willingness-to-pay drawn from the set of symmetric two-point distribution around the buyer's true willingness-to-pay. We characterize the set of pure-strategy equilibria in their delegation choices. Multiple equilibria arise. Except for two corner solutions, both players will exploit the strategy of opacity. A large set of efficient equilibria exist. For these, opacity choices do not reduce the probability of transacting, but benefit the buyer compared with the no-delegation equilibrium. We also study the robustness of the results with respect to the player's ability to also resort to a tougher delegate in addition to the opacity choice.