Approximate revenue maximization with multiple items

Maximizing the revenue from selling more than one good (or item) to a single buyer is a notoriously difficult problem, in stark contrast to the one-good case. For two goods, we show that simple “one-dimensional” mechanisms, such as selling the goods separately, guarantee at least 73% of the optimal revenue when the valuations of the two goods are independent and identically distributed, and at least 50% when they are independent. For the case of k>2 independent goods, we show that selling them separately guarantees at least a c/log²⁡k fraction of the optimal revenue; and, for independent and identically distributed goods, we show that selling them as one bundle guarantees at least a c/log⁡k fraction of the optimal revenue. Additional results compare the revenues from the two simple mechanisms of selling the goods separately and bundled, identify situations where bundling is optimal, and extend the analysis to multiple buyers.