  A fundamental assumption in classical mechanism design is that buyers are
perfect optimizers. However, in practice, buyers may be limited by their
computational capabilities or a lack of information, and may not be able to
perfectly optimize. This has motivated the introduction of approximate
incentive compatibility (IC) as an appealing solution concept for practical
mechanism design. While most of the literature focuses on the analysis of
particular approximate IC mechanisms, this paper is the first to study the
design of optimal mechanisms in the space of approximate IC mechanisms and to
explore how much revenue can be garnered by moving from exact to approximate
incentive constraints. We study the problem of a seller facing one buyer with
private values and analyze optimal selling mechanisms under
$\varepsilon$-incentive compatibility. We establish that the gains that can be
garnered depend on the local curvature of the seller's revenue function around
the optimal posted price when the buyer is a perfect optimizer. If the revenue
function behaves locally like an $\alpha$-power for $\alpha \in (1,\infty)$,
then no mechanism can garner gains higher than order
$\varepsilon^{\alpha/(2\alpha-1)}$. This improves upon state-of-the-art results
which imply maximum gains of $\varepsilon^{1/2}$ by providing the first
parametric bounds that capture the impact of revenue function's curvature on
revenue gains. Furthermore, we establish that an optimal mechanism needs to
randomize as soon as $\varepsilon>0$ and construct a randomized mechanism that
is guaranteed to achieve order $\varepsilon^{\alpha/(2\alpha-1)}$ additional
revenues, leading to a tight characterization of the revenue implications of
approximate IC constraints. Our work brings forward the need to optimize not
only over allocations and payments but also over best responses, and we develop
a new framework to address this challenge.
