  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.

Computer Science and Game Theory