On revenue maximization with sharp multi-unit demands

Abstract

We consider markets consisting of a set of indivisible items, and buyers that have sharp multi-unit demand. This means that each buyer $i$ wants a specific number $d_i$ of items; a bundle of size less than $d_i$ has no value. We consider the objective of setting prices and allocations in order to maximize the total revenue of the market maker. The pricing problem with sharp multi-unit demand buyers has a number of properties that the unit-demand model does not possess, and is an important question in algorithmic pricing. We consider the problem of computing a revenue maximizing solution for two solution concepts: competitive equilibrium and envy-free pricing. For unrestricted valuations, these problems are NP-complete; we focus on a realistic special case of “correlated values” where each buyer $i$ has a valuation $v_iq_j$ for item $j$, where $v_i$ and $q_j$ are positive quantities associated with buyer $i$ and item $j$ respectively. We present a polynomial time algorithm to solve the revenue-maximizing competitive equilibrium problem. For envy-free pricing, if the demand of each buyer is bounded by a constant, a revenue maximizing solution can be found efficiently; the general demand case is shown to be NP-hard.

Appendix: Hardness for general valuations

Theorem 5.1

It is NP-complete to determine the existence of a competitive equilibrium for general valuations in the sharp demand model (even when all demands are 3, and valuations are 0/1).

Proof

We reduce from exact cover by 3-sets (X3C): Given a ground set $A=\{a_1,\ldots ,a_{3n}\}$ and a collection of subsets $S_1,\ldots ,S_m\subset A$ where $|S_i|=3$ for each $i$, we are asked whether there are $n$ subsets that cover all elements in $A$. Given an instance of X3C, we construct a market with $3n+3$ items and $9n+m+1$ buyers as follows. Every element in $A$ corresponds to an item; further, we introduce another three items $B=\{b_1,b_2,b_3\}$. We use index $j$ to denote one item. For each subset $S_i$, there is a buyer with value $v_{ij}=1$ if $j\in S_i$ and $v_{ij}=0$ otherwise; further, for every possible subset $\{x,y,z\}$ where $x\in A$ and $y,z\in B$, there is a buyer with value $v_{ij}=1$ if $j\in \{x,y,z\}$ and $v_{ij}=0$ otherwise; finally, there is a buyer with value $v_{ij}=1$ if $j\in B$ and $v_{ij}=0$ otherwise. The demand of every buyer is 3.

We claim that there is a positive answer to the X3C instance if and only if there is a competitive equilibrium in the constructed market. Assume that there is $T\in \{S_1,\ldots ,S_m\}$ with $|T|=n$ that covers all elements in $A$. Then we allocate items in $A$ to the buyers in $T$ and allocate $B$ to the buyer who desires $B$, and set all prices to be 1. It can be seen that this defines a competitive equilibrium.

On the other hand, assume that there is a competitive equilibrium $(\mathbf {p},\mathbf {X})$. We first claim that all the items in $B$ must be allocated (Cl). Suppose the claim C1 is not true, there are two cases: Case 1, there is only one unallocated items, since if the items in $B$ are allocated, either two items are allocated to some buyer or all three items are allocated (because there only exist buyers who desire two items in $B$ or three items in $B$). W.l.o.g. suppose $b_1$ and $b_2$ together with some $x\in A$ are allocated to a buyer and $b_3$ is unallocated, then we know $p_{b_1}+p_{b_2}+p_x\le 3$. If $p_x=3$, it holds that $p_{b_1}=p_{b_2}=0$, then buyer who values $B$ will not be envy-free. If $p_x<3$, then we either have $p_{b_1}+p_x<3$ or $p_{b_2}+p_x<3$. W.l.o.g. suppose $p_{b_1}+p_x<3$, then the buyer who values the set $\{b_1,x,b_3\}$ will not be envy-free. Case 2: all three items in $B$ would be unallocated, contradicting envy-freeness of the buyer who values $B$. Second, we claim all the items are allocated (C2). Otherwise, by C1, there must exist an item $a_j\in A$ that is not allocated to any buyer. Then we have $p_{a_{j}}=0$. Consider the buyers who desire subsets $\{a_j,b_1,b_2\}, \{a_j,b_1,b_3\}, \{a_j,b_2,b_3\}$. They do not win since $a_j$ is not sold. Due to envy-freeness, we have

$$\begin{aligned} p_{b_1}+p_{b_2}&\ge 3 \\ p_{b_1}+p_{b_3}&\ge 3 \\ p_{b_2}+p_{b_3}&\ge 3 \end{aligned}$$

This implies that $p_{b_1}+p_{b_2}+p_{b_3}\ge 4.5$. Hence, the buyer who desires $B$ cannot afford the price of $B$ and at least one item in $B$, say $b_1$, is not allocated out, which contradicts with C1.

Now since all items in $A$ are allocated out, because of the construction of the market, we have to allocate all items in $A$ to $n$ buyers and allocate $B$ to one buyer; the former gives a solution to the X3C instance. $\square$