# Optimal Multi-Unit Mechanisms with Private Demands

## TL;DR

This paper characterizes the optimal deterministic pricing mechanism for multi-unit sales with private demands under certain regularity conditions, showing it can be efficiently computed and providing detailed solutions for specific demand cases.

## Contribution

It proves that under decreasing marginal revenue, the optimal mechanism is a deterministic price curve, and it establishes polynomial-time computability of this optimal pricing.

## Key findings

- Optimal mechanism is a deterministic price curve under regularity conditions.
- Revenue function is concave in the prices, enabling polynomial-time optimization.
- Provides detailed solutions for cases with two possible demand levels.

## Abstract

In the multi-unit pricing problem, multiple units of a single item are for sale. A buyer's valuation for $n$ units of the item is $v \min \{ n, d\} $, where the per unit valuation $v$ and the capacity $d$ are private information of the buyer. We consider this problem in the Bayesian setting, where the pair $(v,d)$ is drawn jointly from a given probability distribution. In the \emph{unlimited supply} setting, the optimal (revenue maximizing) mechanism is a pricing problem, i.e., it is a menu of lotteries. In this paper we show that under a natural regularity condition on the probability distributions, which we call \emph{decreasing marginal revenue}, the optimal pricing is in fact \emph{deterministic}. It is a price curve, offering $i$ units of the item for a price of $p_i$, for every integer $i$. Further, we show that the revenue as a function of the prices $p_i$ is a \emph{concave} function, which implies that the optimum price curve can be found in polynomial time. This gives a rare example of a natural multi-parameter setting where we can show such a clean characterization of the optimal mechanism. We also give a more detailed characterization of the optimal prices for the case where there are only two possible demands.