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

We consider the design of a revenue-optimal mechanism when two items are available to be sold to a single buyer whose valuation is uniformly distributed over an arbitrary rectangle $[c_1,c_1+b_1]\times[c_2,c_2+b_2]$ in the positive quadrant. We provide an explicit, complete solution for arbitrary nonnegative values of $(c_1,c_2,b_1,b_2)$. We identify eight simple structures, each with at most $4$ (possibly stochastic) menu items, and prove that the optimal mechanism has one of these eight structures. We also characterize the optimal mechanism as a function of $(c_1,c_2,b_1,b_2)$. The structures indicate that the optimal mechanism involves (a) an interplay of individual sale and a bundle sale when $c_1$ and $c_2$ are low, (b) a bundle sale when $c_1$ and $c_2$ are high, and (c) an individual sale when one of them is high and the other is low. To the best of our knowledge, our results are the first to show the existence of optimal mechanisms with no exclusion region. We further conjecture, based on promising preliminary results, that our methodology can be extended to a wider class of distributions.