  The question of the minimum menu-size for approximate (i.e.,
up-to-$\varepsilon$) Bayesian revenue maximization when selling two goods to an
additive risk-neutral quasilinear buyer was introduced by Hart and Nisan
(2013), who give an upper bound of $O(\frac{1}{\varepsilon^4})$ for this
problem. Using the optimal-transport duality framework of Daskalakis et al.
(2013, 2015), we derive the first lower bound for this problem - of
$\Omega(\frac{1}{\sqrt[4]{\varepsilon}})$, even when the values for the two
goods are drawn i.i.d. from "nice" distributions, establishing how to reason
about approximately optimal mechanisms via this duality framework. This bound
implies, for any fixed number of goods, a tight bound of
$\Theta(\log\frac{1}{\varepsilon})$ on the minimum deterministic communication
complexity guaranteed to suffice for running some approximately
revenue-maximizing mechanism, thereby completely resolving this problem. As a
secondary result, we show that under standard economic assumptions on
distributions, the above upper bound of Hart and Nisan (2013) can be
strengthened to $O(\frac{1}{\varepsilon^2})$.
