  We consider optimal mechanism design for the case with one buyer and two
items. The buyer's valuations towards the two items are independent and
additive. In this setting, optimal mechanism is unknown for general valuation
distributions. We obtain two categories of structural results that shed light
on the optimal mechanisms.
  The first category of results state that, under certain mild condition, the
optimal mechanism has a monotone menu. In other words, in the menu that
represents the optimal mechanism, as payment increases, the allocation
probabilities for both items increase simultaneously. Applying this theorem, we
derive a version of revenue monotonicity theorem that states stochastically
superior distributions yield more revenue. Moreover, our theorem subsumes a
previous result regarding sufficient conditions under which bundling is
optimal. The second category of results state that, under certain conditions,
the optimal mechanisms have few menu items. Our first result in this category
says, for certain distributions, the optimal menu contains at most 4 items. The
condition admits power (including uniform) density functions. Based on a
similar proof of this result, we are able to obtain a wide class of
distributions where bundling is optimal. Our second result in this category
works for a weaker condition, under which the optimal menu contains at most 6
items. This condition includes exponential density functions. Our last result
in this category works for unit-demand setting. It states that, for uniform
distributions, the optimal menu contains at most 5 items. All these results are
in sharp contrast to Hart and Nisan's recent result that finite-sized menu
cannot guarantee any positive fraction of optimal revenue for correlated
valuation distributions.
