Bilateral trade in discrete type spaces by linear and convex optimization with application to supply chain coordination

We consider the bilateral trade of an object between a seller and a buyer through an intermediary who aims to maximize his expected gains as in Myerson and Satterthwaite [1], in a Bayes-Nash equilibrium framework where the seller and buyer have private, discrete valuations for the object. Using duality of linear network optimization, the intermediary's initial problem is transformed into an equivalent linear programming problem with explicit formulae of expected revenues of the seller and the expected payments of the buyer, from which the optimal mechanism is immediately obtained. Then, an extension due to Spulber [2] is considered where the seller is also a producer with a cost parameter that is private information. Assuming suitable utility and production functions for the respective parties, the resulting non-convex mechanism design problem of a utility maximizing broker is revealed to have hidden convexity and transformed into an equivalent (almost) unconstrained convex optimization problem over output variables, which, in many cases, can be solved easily by calculus. A contracting game under asymmetric information specific to two-echelon supply chain coordination between a retailer of unknown type and a supplier, akin to the bilateral trade game of the paper, is also studied. Special cases leading to closed-form solution with increasing information rent for higher types are identified along with the requisite conditions for their validity.