  Optimal mechanisms have been provided in quite general multi-item settings,
as long as each bidder's type distribution is given explicitly by listing every
type in the support along with its associated probability. In the implicit
setting, e.g. when the bidders have additive valuations with independent and/or
continuous values for the items, these results do not apply, and it was
recently shown that exact revenue optimization is intractable, even when there
is only one bidder. Even for item distributions with special structure, optimal
mechanisms have been surprisingly rare and the problem is challenging even in
the two-item case. In this paper, we provide a framework for designing optimal
mechanisms using optimal transport theory and duality theory. We instantiate
our framework to obtain conditions under which only pricing the grand bundle is
optimal in multi-item settings (complementing the work of [Manelli and Vincent
2006], as well as to characterize optimal two-item mechanisms. We use our
results to derive closed-form descriptions of the optimal mechanism in several
two-item settings, exhibiting also a setting where a continuum of lotteries is
necessary for revenue optimization but a closed-form representation of the
mechanism can still be found efficiently using our framework.
