How Good Are Simple Mechanisms for Selling Multiple Goods?

Maximizing the revenue from selling two goods (or items) is a notoriously difficult problem, in stark contrast to the single-good case. We show that simple “one-dimensional” mechanisms, such as selling the two 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. However, in the general case where the valuations may be correlated, simple mechanisms cannot guarantee any positive fraction of the optimal revenue. We also introduce a “measure of complexity” for mechanisms—the menu size—and show that it is naturally related to the fraction of the optimal revenue that can be guaranteed. This paper is based on results from two working papers: Hart and Nisan (2012) and Hart and Nisan (2013). Research partially supported by a European Research Council Advanced Investigator grant (Hart) and by an Israel Science Foundation grant (Nisan). We thank Motty Perry and Phil Reny for introducing us to the subject and for many useful discussions. A presentation is available at http://www.ma.huji.ac.il/hart/abs/2good-p.html The Hebrew University of Jerusalem (Center for the Study of Rationality, Department of Economics, and Institute of Mathematics). E-mail : [email protected] Web site: http://www.ma.huji.ac.il/hart The Hebrew University of Jerusalem (Center for the Study of Rationality, and School of Computer Science and Engineering) and Microsoft Research. E-mail : [email protected] Web site: http://www.cs.huji.ac.il/~noam