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 either (1) the outside option value is independent of the item value, and the item value distribution has decreasing marginal revenue or monotone hazard rate; or (2) the outside option value is a concave function of the item value. Moreover, when there are multiple buyers, we show that sequential posted pricing guarantees a large fraction of the optimal revenue under similar conditions.
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