Optimal mechanisms with simple menus

TL;DR

This paper characterizes the structure of optimal mechanisms for a single buyer with two independent, additive valuations, revealing conditions for monotonicity and bounds on menu size, advancing understanding in auction design.

Contribution

It provides new structural results showing when optimal mechanisms have monotone menus and limited menu items, including conditions for bundling and specific distribution classes.

Findings

Optimal mechanisms have monotone menus under mild conditions.

Optimal menus contain at most 4, 5, or 6 items depending on distribution.

Bundling is optimal for certain distribution classes.

Abstract

We consider optimal mechanism design for the case with one buyer and two items. The buyer's valuations towards the two items are independent and additive. In this setting, optimal mechanism is unknown for general valuation distributions. We obtain two categories of structural results that shed light on the optimal mechanisms. The first category of results state that, under certain mild condition, the optimal mechanism has a monotone menu. In other words, in the menu that represents the optimal mechanism, as payment increases, the allocation probabilities for both items increase simultaneously. Applying this theorem, we derive a version of revenue monotonicity theorem that states stochastically superior distributions yield more revenue. Moreover, our theorem subsumes a previous result regarding sufficient conditions under which bundling is optimal. The second category of results state that, under certain conditions, the optimal mechanisms have few menu items. Our first result in this category says, for certain distributions, the optimal menu contains at most 4 items. The condition admits power (including uniform) density functions. Based on a similar proof of this result, we are able to obtain a wide class of distributions where bundling is optimal. Our second result in this category works for a weaker condition, under which the optimal menu contains at most 6 items. This condition includes exponential density functions. Our last result in this category works for unit-demand setting. It states that, for uniform distributions, the optimal menu contains at most 5 items. All these results are in sharp contrast to Hart and Nisan's recent result that finite-sized menu cannot guarantee any positive fraction of optimal revenue for correlated valuation distributions.