Robust selling mechanisms

We consider the problem of a seller who faces a privately informed buyer and only knows one moment of the distribution from which values are drawn. In face of this uncertainty, the seller maximizes his worst-case expected profits. We show that a robustness property of the optimal mechanism imposes restrictions on the seller's ex-post profit function. These restrictions are used to derive the optimal mechanism. The optimal mechanism entails distortions at the intensive margin, e.g., except for the highest value buyer, sales will take place with probability strictly smaller than one. The seller can implement such allocation by committing to post prices drawn from a non-degenerate distribution, so that randomizing over prices is an optimal robust selling mechanism. We extend the model to deal with the case in which: (i) M goods are sold and the buyer's private information is multidimensional and (ii) the seller and the buyer interact for several periods. In the case of multiple goods, there are several optimal mechanisms. With multiple goods full bundling is optimal, as well as selling the goods in a fully separable way. In the dynamic model, we show that repetition, period by period, of the static-optimal mechanism is optimal.