Ex-Post Constrained-Efficient Bilateral Trade with Risk-Averse Traders

We address robust mechanism design for bilateral trade of an indivisible commodity, under incomplete information on traders’ private reservation values. Under ex-post individual rationality and ex-post incentive compatibility, we define the notion of ex-post constrained-efficiency. It is weaker than interim constrained efficiency, and it is a notion of constrained optimality that is independent of the details of the distribution of types. When traders are risk neutral the class of ex-post constrained-efficient mechanisms is equivalent to probability distributions over posted prices. In general, among mechanisms satisfying incentive and participation constraints, a sufficient condition for constrained efficiency is simplicity: for each draw of types the outcome is a lottery between trade at one type-contingent price and no trade. For environments with constant relative risk aversion, we characterize simple mechanisms. Generically, simple mechanisms converge to full efficiency as agents’ risk aversion goes to infinity. Under risk neutrality, ex-ante optimal mechanisms are deterministic, and under risk aversion, they are not. Our results are suitable for applications.