The Simple Economics of Approximately Optimal Auctions

TL;DR

This paper explores the conditions under which marginal revenue maximization leads to optimal or near-optimal auctions in complex environments with multi-dimensional, non-linear agent preferences, extending classical auction theory.

Contribution

It characterizes revenue linearity as the key condition for optimality, provides simple implementation procedures, and demonstrates that marginal revenue maximization is approximately optimal in general settings.

Findings

Identifies revenue linearity as the condition for optimal marginal revenue auctions.

Provides procedures for implementing marginal revenue maximization.

Shows approximation guarantees degrade smoothly with environment complexity.

Abstract

The intuition that profit is optimized by maximizing marginal revenue is a guiding principle in microeconomics. In the classical auction theory for agents with linear utility and single-dimensional preferences, Bulow and Roberts (1989) show that the optimal auction of Myerson (1981) is in fact optimizing marginal revenue. In particular Myerson's virtual values are exactly the derivative of an appropriate revenue curve. This paper considers mechanism design in environments where the agents have multi-dimensional and non-linear preferences. Understanding good auctions for these environments is considered to be the main challenge in Bayesian optimal mechanism design. In these environments maximizing marginal revenue may not be optimal and there is sometimes no direct way to implement the marginal revenue maximization. Our contributions are three fold: we characterize the settings for which marginal revenue maximization is optimal (by identifying an important condition that we call revenue linearity), we give simple procedures for implementing marginal revenue maximization in general, and we show that marginal revenue maximization is approximately optimal. Our approximation factor smoothly degrades in a term that quantifies how far the environment is from ideal (where marginal revenue maximization is optimal). Because the marginal revenue mechanism is optimal for single-dimensional agents, our generalization immediately approximately extends many results for single-dimensional agents. One of the biggest open questions in Bayesian algorithmic mechanism design is developing methodologies that are not brute-force in the size of the agent type space. Our methods identify a subproblem that, e.g., for unit-demand agents with values drawn from product distributions, enables approximation mechanisms that are polynomial in the dimension.