  We study mechanisms for selling a single item when buyers have private values
for their outside options, which they forego by participating in the mechanism.
This substantially changes the revenue maximization problem. For example, the
seller can strictly benefit from selling lotteries already in the single-buyer
setting. We bound the menu size and the sample complexity for the optimal
single-buyer mechanism. We then show that posting a single price is in fact
optimal under the assumption that the item value distribution has decreasing
marginal revenue or monotone hazard rate. Moreover, when there are multiple
buyers, we show that sequential posted pricing guarantees a large fraction of
the optimal revenue when the item value distribution has decreasing marginal
revenue.
