On revenue maximization with sharp multi-unit demands

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.

1. This phenomenon does occur in our real life. For example, in most travel packages offered by travel agencies, they could lose money for some specific programs; but their overall net profit could always be positive.

2. By the nature of the solution concepts considered in the paper, it can be assumed without loss of generality that \(i\) will not get more than \(d_i\) items.

The authors thanks anonymous referees for their constructive review comments, helping to improve the readability of the paper. Ning Chen’s research was supported by the AcRF Tier 2 Grant of Singapore (No. MOE2012-T2-2-071). Xiaotie Deng was supported by Shanghai Jiaotong University with a 985 Project Grant and by an NSFC Grant No. 61173011. Paul W. Goldberg was supported by EPSRC Grant EP/G069239/1 “Efficient Decentralised Approaches in Algorithmic Game Theory” and EPSRC grant EP/K01000X/1 “Efficient Algorithms for Mechanism Design Without Monetary Transfer. Jinshan Zhang was supported by EPSRC Grant EP/K01000X/1 “Efficient Algorithms for Mechanism Design Without Monetary Transfer.

Authors and Affiliations

1. Division of Mathematical Sciences, Nanyang Technological University, Singapore, Singapore

   Ning Chen

2. Department of Computer Science, Shanghai Jiao Tong University, Shanghai, China

   Xiaotie Deng

3. Department of Computer Science, University of Oxford, Oxford, UK

   Paul W. Goldberg

4. Department of Computer Science, University of Liverpool, Liverpool, UK

   Jinshan Zhang

Corresponding author

Correspondence to Jinshan Zhang.

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

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. \(\square \)

Reprints and permissions