Revenue Maximization Under Unknown Private Values with Non-Obligatory Inspection

We consider the problem of selling a single item to $n$ unit-demand buyers to maximize revenue, where the buyers' values are independently distributed (not necessarily identical) according to publicly known distributions but unknown to the buyers themselves, with the option of allowing buyers to inspect the item at a cost. This problem can be interpreted as a revenue-maximizing variant of Weitzman's Pandora's problem with non-obligatory inspection. We present an approximation mechanism that achieves $1/2$. The proposed mechanism generalizes to the case of selling $k$ units of an item to unit-demand buyers, obtaining $1-1/\sqrt{k+3}$ of the optimal revenue in expectation. The mechanism is sequential and has a simple implementation that works in an online setting where buyers arrive in an arbitrary unknown order, yet achieving the aforementioned approximation with respect to the optimal offline mechanism