Bounding the Menu-Size of Approximately Optimal Auctions via Optimal-Transport Duality

TL;DR

This paper establishes tight bounds on the menu-size and communication complexity of approximately optimal auctions for two goods, using optimal-transport duality, and improves existing upper bounds under standard assumptions.

Contribution

It provides the first lower bound on menu-size for approximate revenue maximization and tight bounds on communication complexity, resolving key open questions in auction design.

Findings

Lower bound of (rac{1}{ ext{} ext{} ext{} ext{}}) for menu-size.

Tight bound of (\lograc{1}{ ext{} ext{} ext{} ext{}}) on communication complexity.

Improved upper bound of (rac{1}{ ext{}^2}) under standard assumptions.

Abstract

The question of the minimum menu-size for approximate (i.e., up-to-$\varepsilon$) Bayesian revenue maximization when selling two goods to an additive risk-neutral quasilinear buyer was introduced by Hart and Nisan (2013), who give an upper bound of $O(\frac{1}{\varepsilon^4})$ for this problem. Using the optimal-transport duality framework of Daskalakis et al. (2013, 2015), we derive the first lower bound for this problem - of $\Omega(\frac{1}{\sqrt[4]{\varepsilon}})$, even when the values for the two goods are drawn i.i.d. from "nice" distributions, establishing how to reason about approximately optimal mechanisms via this duality framework. This bound implies, for any fixed number of goods, a tight bound of $\Theta(\log\frac{1}{\varepsilon})$ on the minimum deterministic communication complexity guaranteed to suffice for running some approximately revenue-maximizing mechanism, thereby completely resolving this problem. As a secondary result, we show that under standard economic assumptions on distributions, the above upper bound of Hart and Nisan (2013) can be strengthened to $O(\frac{1}{\varepsilon^2})$.