Abstract

We give an introduction to the design of mechanisms for profit maximization with a focus on single parameter settings.

Introduction

In previous chapters, we have studied the design of truthful mechanisms that implement social choice functions, such as social welfare maximization. Another fundamental objective, and the focus of this chapter, is the design of mechanisms in which the goal of the mechanism designer is profit maximization. In economics, this topic is referred to as optimal mechanism design.

Our focus will be on the design of profit-maximizing auctions in settings in which an auctioneer is selling (respectively, buying) a set of goods/services. Formally, there are n agents, each of whom desires some particular service. We assume that agents are single-parameter; i.e., agent i's valuation for receiving service is vi and their valuation for no service is normalized to zero. A mechanism takes as input sealed bids from the agents, where agent i's bid bi represents his valuation vi, and computes an outcome consisting of an allocation x = (x1, …, xn) and prices p = (p1, …, pn). Setting xi = 1 represents agent i being allocated service whereas xi = 0 is for no service, and pi is the amount agent i is required to pay the auctioneer. We assume that agents have quasi-linear utility expressed by ui = vixi – pi.