Optimal Mediation Mechanism in Bilateral Trade

We study the problem of designing revenue-maximizing mechanisms for a selfish mediator who facilitates trade between a buyer and a seller. We consider a setting where the mediator does not have information advantage and the buyer's valuation is interdependent with the seller's private information. The mechanism may involve multi-round negotiations and flexible fee structures. We show that the mediator can restrict attention to a class of joint menu-selection mechanisms, where each mechanism can be represented as a two-dimensional menu. Each party privately selects an option from their own dimension and the two options together determine the menu entry. The mediator then recommends both parties whether to trade based on the jointly selected menu entry. We then establish an impossibility trilemma: no mechanism can simultaneously satisfy incentive compatibility, obedience, and informativeness. Motivated by this result, we characterize the optimal mechanisms under two relaxation conditions. First, when the seller's cost is constant, the optimal mechanism exhibits a threshold structure: trade occurs whenever the quality of the item exceeds a threshold that is decreasing in the buyer's type. Consequently, low-typed buyers receive more information, which in turn gives the mediator more power to charge from them. Second, when the mediator has veto power, the optimal mechanism also takes a threshold form, but in the opposite direction: trade occurs only if the quality falls below a threshold that is increasing in the buyer's type. As a result, items with lower qualities are more likely to be traded and the corresponding sellers benefit more, which discourages sellers of high qualities from participating and gives rise to a ``lemons market'' effect.