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.

本文提出一种框架，使用最优输运理论和对偶理论设计最优机制，得出了在多项目景下仅定价整个 bundle 时是最优的条件，同时描述了在多种两项目景下的最优机制。

通过最优输运实现机制设计

Mechanism Design via Optimal Transport

We characterize optimal mechanisms for the multiple-good monopoly problem and
provide a framework to find them. We show that a mechanism is optimal if and
only if a measure $\mu$ derived from the buyer's type distribution satisfies
certain stochastic dominance conditions. This measure expresses the marginal
change in the seller's revenue under marginal changes in the rent paid to
subsets of buyer types. As a corollary, we characterize the optimality of
grand-bundling mechanisms, strengthening several results in the literature,
where only sufficient optimality conditions have been derived. As an
application, we show that the optimal mechanism for $n$ independent uniform
items each supported on $[c,c+1]$ is a grand-bundling mechanism, as long as $c$
is sufficiently large, extending Pavlov's result for $2$ items [Pavlov'11]. At
the same time, our characterization also implies that, for all $c$ and for all
sufficiently large $n$, the optimal mechanism for $n$ independent uniform items
supported on $[c,c+1]$ is not a grand bundling mechanism.

该研究研究了多商品垄断问题的最优机制，表明 CD（Cohen-Dehez）中的 mu 测度可以用于确定最优机制，带来了强化了的 Grand-Bundling 机制。