This paper by Roger B. Myerson and Mark A. Satterthwaite examines efficient mechanisms for bilateral trading between a buyer and a seller of a single object. The seller and buyer have independent, private valuations of the object, and the goal is to design a mechanism that is both incentive-compatible and individually rational. The authors show that it is generally impossible to achieve ex post efficiency without outside subsidies. They also demonstrate how to construct mechanisms that maximize expected total gains from trade and mechanisms that maximize a broker's expected profit. The paper begins by discussing the impossibility of designing a mechanism that ensures honest revelation of valuations, no outside subsidies, and ex post efficiency. It then introduces the concept of Bayesian incentive compatibility, which allows for efficient allocations without outside subsidies. However, these mechanisms may result in negative expected gains for some individuals. The authors analyze a wide class of trading problems and show how to compute mechanisms that maximize expected total gains from trade. They also consider the case where a broker is involved, who may subsidize or exploit the traders. The paper characterizes incentive-compatible, individually rational trading mechanisms that can be implemented with a broker and shows how to construct mechanisms that maximize the broker's expected profit. The paper concludes by demonstrating that, in the case of symmetric, uniform valuations, the optimal mechanism for a broker involves transferring the object only if the buyer's valuation exceeds the seller's by at least 1/4. This mechanism is shown to maximize expected total gains from trade and is equivalent to the equilibrium strategies of a bargaining game studied by Chatterjee and Samuelson. The paper also highlights the importance of the revelation principle, which allows for the construction of equivalent incentive-compatible direct mechanisms from any Bayesian equilibrium.This paper by Roger B. Myerson and Mark A. Satterthwaite examines efficient mechanisms for bilateral trading between a buyer and a seller of a single object. The seller and buyer have independent, private valuations of the object, and the goal is to design a mechanism that is both incentive-compatible and individually rational. The authors show that it is generally impossible to achieve ex post efficiency without outside subsidies. They also demonstrate how to construct mechanisms that maximize expected total gains from trade and mechanisms that maximize a broker's expected profit. The paper begins by discussing the impossibility of designing a mechanism that ensures honest revelation of valuations, no outside subsidies, and ex post efficiency. It then introduces the concept of Bayesian incentive compatibility, which allows for efficient allocations without outside subsidies. However, these mechanisms may result in negative expected gains for some individuals. The authors analyze a wide class of trading problems and show how to compute mechanisms that maximize expected total gains from trade. They also consider the case where a broker is involved, who may subsidize or exploit the traders. The paper characterizes incentive-compatible, individually rational trading mechanisms that can be implemented with a broker and shows how to construct mechanisms that maximize the broker's expected profit. The paper concludes by demonstrating that, in the case of symmetric, uniform valuations, the optimal mechanism for a broker involves transferring the object only if the buyer's valuation exceeds the seller's by at least 1/4. This mechanism is shown to maximize expected total gains from trade and is equivalent to the equilibrium strategies of a bargaining game studied by Chatterjee and Samuelson. The paper also highlights the importance of the revelation principle, which allows for the construction of equivalent incentive-compatible direct mechanisms from any Bayesian equilibrium.