# On Optimal Mechanisms in the Two-Item Single-Buyer Unit-Demand Setting

## TL;DR

This paper provides a complete explicit characterization of revenue-optimal mechanisms for a single buyer with two items, uniformly distributed valuations, identifying five simple structures and analyzing how the mechanism changes with parameters.

## Contribution

It introduces a complete solution for optimal mechanisms in the two-item, single-buyer setting with uniform valuations over arbitrary rectangles, including the first identification of mechanisms without exclusion regions.

## Key findings

- Optimal mechanisms have five simple structures.
- Mechanism form depends on the parameter c, with posted prices and exclusion regions.
- First demonstration of optimal mechanisms without exclusion regions.

## Abstract

We consider the problem of designing a revenue-optimal mechanism in the two-item, single-buyer, unit-demand setting when the buyer's valuations, $(z_1, z_2)$, are uniformly distributed in an arbitrary rectangle $[c,c+b_1]\times[c,c+b_2]$ in the positive quadrant. We provide a complete and explicit solution for arbitrary nonnegative values of $(c,b_1,b_2)$. We identify five simple structures, each with at most five (possibly stochastic) menu items, and prove that the optimal mechanism has one of the five structures. We also characterize the optimal mechanism as a function of $b_1, b_2$, and $c$. When $c$ is low, the optimal mechanism is a posted price mechanism with an exclusion region; when $c$ is high, it is a posted price mechanism without an exclusion region. Our results are the first to show the existence of optimal mechanisms with no exclusion region, to the best of our knowledge.