Revenue-Maximizing Auctions

Abstract

Lectures 2-4 focused on the design of mechanisms that maximize, exactly or approximately, the social welfare of the outcome. Revenue is generated in such mechanisms only as a side effect, a necessary evil to incentivize agents to report truthfully their private information. This lecture studies mechanisms that are designed to raise as much revenue as possible, and characterizes the expected revenue-maximizing mechanism with respect to a prior distribution over agents’ valuations. Section 5.1 explains why reasoning about revenue maximization is harder than welfare maximization, and introduces Bayesian envi-ronments. Section 5.2 is the heart of this lecture, and it characterizes expected revenue-maximizing mechanisms as “virtual welfare maxi-mizers.” Section 5.3 describes how this theory was used to boost sponsored search revenue at Yahoo. Section 5.4 proves a technical lemma needed for the characterization in Section 5.2. The Challenge of Revenue Maximization Spoiled by Social Welfare Maximization There are several reasons to begin the study of mechanism design with the objective of maximizing social welfare. The first reason is that this objective is relevant to many real-world scenarios. For instance, in government auctions (e.g., to sell wireless spectrum; see Lecture 8), the primary objective is welfare maximization. Revenue is also a consideration in such auctions, but it is usually not the first-order objective. Also, in competitive markets, a rule of thumb is that a seller should focus on welfare maximization, since otherwise a competitor will (thereby stealing their customers). The second reason to start with social welfare maximization is pedagogical: social welfare is special. In every single-parameter environment, there is a DSIC mechanism that, for every profile of private valuations, assuming truthful bids, computes the welfare-maximizing outcome (cf., Exercise 4.1). Such a mechanism optimizes the social welfare as effectively as if all of the private information was known in advance—the DSIC constraint is satisfied for free. This amazingly strong performance guarantee, called an “ex post” guarantee, cannot generally be achieved for other objective functions.