Essays on bilateral trade with discrete types

Bilateral trade is probably the most common market interaction problem and can be considered as the simplest form of two sided markets where a seller and a buyer bargain over an indivisible object subject to incomplete information on the reservation values of participants. We treat this problem as a combinatorial optimization problem and re-establish some results of economic theory that are well-known under continuous valuations assumptions for the case of discrete valuations using linear programming techniques. First, we propose mathematical formulation for the problem under dominant strategy incentive compatibility (DIC) and ex-post individual rationality (EIR) properties. Then we derive necessary and sufficient conditions under which ex-post efficiency can be obtained together with DIC and EIR. We also define a new property called Allocation Maximality and prove that the Posted Price mechanism is the only mechanism that satisfies DIC, EIR and allocation maximality. In the final part we consider ambiguity in the problem framework originating from different sets of priors for agents types and derive robust counterparts. Next, we study the bilateral trade problem with an intermediary who wants to maximize her expected gains. Using network programming we transform the initial linear program into one from which the structure of mechanism is transparent. We then relax the risk-neutrality assumption of the intermediary and consider the problem from the perspective of risk-averse intermediary. The effects of risk-averse approach are presented using computational experiments. Finally, we broaden the scope of the problem and discuss the case in which the seller is also a producer at the same time and consider benefit and cost functions for the respective parties. Starting by a non-convex optimization problem, we obtain an equivalent convex optimization problem from which the problem is solved easily. We also reconsider the same problem under dominant strategy incentive compatibility and ex-post individual rationality constraints to preserve the practicality of all obtained solutions.