Selling Two Goods Optimally

TL;DR

This paper characterizes optimal selling mechanisms for two goods with independent buyer valuations, providing explicit formulas under certain conditions and introducing the first non-deterministic optimal mechanism in an i.i.d. setting.

Contribution

It offers new sufficient conditions for revenue maximization, explicit formulas for optimal prices and allocations, and demonstrates the use of duality techniques for approximation.

Findings

Exact formulas for optimal prices and allocations under certain conditions

First example of a non-deterministic optimal mechanism in i.i.d. setting

Duality-based methods for mechanism approximation

Abstract

We provide sufficient conditions for revenue maximization in a two-good monopoly where the buyer's values for the items come from independent (but not necessarily identical) distributions over bounded intervals. Under certain distributional assumptions, we give exact, closed-form formulas for the prices and allocation rule of the optimal selling mechanism. As a side result we give the first example of an optimal mechanism in an i.i.d. setting over a support of the form $[0,b]$ which is not deterministic. Since our framework is based on duality techniques, we were also able to demonstrate how slightly relaxed versions of it can still be used to design mechanisms that have very good approximation ratios with respect to the optimal revenue, through a "convexification" process.