Optimal Mechanism for Selling Two Items to a Single Buyer Having Uniformly Distributed Valuations

We consider the design of an optimal mechanism for a seller selling two items to a single buyer so that the expected revenue to the seller is maximized. The buyer’s valuation of the two items is assumed to be the uniform distribution over an arbitrary rectangle [c1, c1 + b1] × [c2, c2 + b2] in the positive quadrant. The solution to the case when (c1, c2) = (0, 0) was already known. We provide an explicit solution for arbitrary nonnegative values of (c1, c2, b1, b2). We prove that the optimal mechanism is to sell the two items according to one of eight simple menus. We also prove that the solution is deterministic when either c1 or c2 is beyond a threshold. Finally, we conjecture that our methodology can be extended to a wider class of distributions. We also provide some preliminary results to support the conjecture.