  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.
