Abstract
Consider the revenue-maximizing problem in which a single seller wants to sell k different items to a single buyer, who has independently distributed values for the items with additive valuation. The 
case was completely resolved by Myerson’s classical work in 1981, whereas for larger k the problem has been the subject of much research efforts ever since. Recently, Hart and Nisan analyzed two simple mechanisms: selling the items separately, or selling them as a single bundle. They showed that selling separately guarantees at least a 
fraction of the optimal revenue; and for identically distributed items, bundling yields at least a 
fraction of the optimal revenue. In this paper, we prove that selling separately guarantees at least 
fraction of the optimal revenue, whereas for identically distributed items, bundling yields at least a constant fraction of the optimal revenue. These bounds are tight (up to a constant factor), settling the open questions raised by Hart and Nisan. The results are valid for arbitrary probability distributions without restrictions. Our results also have implications on other interesting issues, such as monotonicity and randomization of selling mechanisms.
Keywords: auction, mechanism design
In the multiple-items auction problem, a seller wants to sell k different items to n bidders who have private values, drawn from some probability distributions, for these items. Economists are interested in studying incentive-compatible mechanisms under which the bidders are incentivized to report their values truthfully. One central question is how to design such mechanisms that can yield the maximal expected revenue for the seller.
The single item case (
) was resolved by Myerson’s classic work (1) for independently distributed item values. The general case of multiple item (
) is much subtler and not yet completely solved. In recent years, it has been the subject of intensive studies by both economists (e.g., refs. 2–7) and computer scientists (e.g., refs. 8–12). In particular, when the inputs are discrete, much progress has been made on the efficient computation of the optimal mechanism. Another line of investigation is to design simple mechanisms for approximating optimal revenues (13–15) in various situations, such as in the unit-demand setting. For fuller reviews and discussions of the literature, we refer the reader to the papers by Hart and Nisan (16), and Cai et al. (12), and the references therein.
There remain important aspects of the multiple-item auction that are not yet well understood. Most of the known results put restrictions on the distributions (such as refs. 12, 15, and 17). As noted in ref. 16, a precise characterization of optimal mechanisms is still wanting for general distributions, even for simple cases such as for one bidder and two items. The subtlety can be appreciated by considering the following three examples. If the two items are independent and identically distributed over values 
uniformly, then selling them separately at price $1 each yields revenue 1, which is better than bundling (with optimal bundle price $1 and revenue 3/4). However, if the values are uniform over 
, then selling them separately (at an optimal price of $2 each) yields revenue 2, which is worse than bundling them together for price $3, with revenue 
. In the third example, let F be the distribution that takes values 
with probabilities 
. In this case, the unique optimal mechanism (see ref. 7) is to offer the buyer the choice between a 50% lottery for buying any single item for $1, and buying both surely as a bundle for $4. (The reader may refer to refs. 3 and 16 for more examples and discussions.)
It is also not known how to characterize the optimal revenue (up to a constant factor) when there is one bidder and k items for large k; it is unsolved even for ER, the equal-revenue distribution as defined in ref. 16 (i.e., when all items have independent cumulative distributions 
for 
).
In the hope of addressing the above issues, Hart and Nisan (16) investigated the case of one bidder (
) and 
. They obtained interesting structural results that are valid for all distributions, and applied them to obtain performance bounds on two natural mechanisms: selling each of the k items separately, and selling all of the k items as a single bundle. Specifically, they showed that selling separately guarantees at least a 
fraction of the optimal revenue; and for identically distributed items, the bundling mechanism yields at least a 
fraction of the optimal revenue.
In this paper, we prove that selling separately guarantees at least 
fraction of the optimal revenue, whereas for identically distributed items, bundling yields at least a constant fraction of the optimal revenue. These bounds are tight (up to a constant factor), settling the open questions raised (16). Our results also have implications on other interesting issues, such as monotonicity and randomization of the selling mechanisms.
It is worth emphasizing that our results are valid for arbitrary distributions without restrictions, in the same spirit as the results of ref. 16. We present a technique called the “core–tails (CT) decomposition” (3. CT Decomposition and the Core Lemma) for analyzing the revenue of general distributions, which may be useful for removing the restrictions in previous works such as refs. 12 and 15.
1. Notations and Preliminaries
We follow the notations in ref. 16. A mechanism for selling k items specifies a (possibly randomized) protocol between a seller and a buyer who has a private valuation 
(where 
) for the items. The outcome is an allocation specifying the probability 
of getting each of the k items and an (expected) payment 
from the buyer to the seller.
The buyer is assumed to act in his self-interest, behaving rationally (paying no more than his value for the goods received), and reporting his valuation 
of the k items so as to maximize his utility. Therefore, as is common in economics theory, we consider only mechanisms aligned with these two considerations, so that the buyer is willing to participate and has incentive to report truthfully (that is, 
, the true valuation of the items). Precisely, we require the mechanism to be individually rational (IR) so that the buyer utility 
is nonnegative for all x, and also incentive compatible (IC) so that for all 
.
For a cumulative distribution 
on 
, let 
denote the maximal (expected) revenue 
obtainable by any incentive compatible and individually rational mechanism. We consider two simple mechanisms: (i) selling each item separately, where each item i has a posted price 
so that the buyer can decide whether or not to take the item; and (ii) selling all of the items as a bundle with a fixed price, so that the buyer gets the whole bundle or nothing. It is easy to see that both mechanisms are IR and IC. Let 
be the maximal revenue obtainable by selling each item separately, and 
be the maximal revenue obtainable by selling all of the items as a bundle.
Let 
with values in 
distributed according to 
. The notation 
, 
, 
will be used interchangeably with 
, 
, 
. Note that we have 
and 
.
In this paper, we consider only independently distributed item values, i.e., 
, where 
is the cumulative distribution of item i and the 
values are not necessarily identical. Clearly, 
. For a one-dimensional distribution F, Myerson’s characterization of the optimal values gives the following:
 |
For 
, characterizing the optimal value 
is a subtler issue and has been the subject of extensive studies. Here, we only list some results from ref. 16 that will be needed for our paper.
Theorem 0.
[See Hart and Nisan (16).] There exists a constant
such that, for all
and
where each
is a one-dimensional distribution,
 |
also when all
are identical (
for all i),
 |
Hart and Nisan (16) raised the question whether the above two bounds can be replaced by 
and c, respectively. Such bounds would be tight, as the choice of 
for 
(
) achieves these lower bounds. [It was shown in ref. 16 that, for this 
, 
and 
.] Our main result is to answer this open question affirmatively.
We need the following structural results from ref. 16 as useful tools. First, it is obvious that
 |
for any
.
Lemma A.
(See ref. 16.) Let X and Y be multidimensional random variables. If X, Y are independent, then
.
Notation.
Let Z be a k-dimensional random variable. For any (measurable) subset S of 
, let 
be the indicator random variable that takes on the value 1 if 
and 0 otherwise. We sometimes write 
as 
for brevity when there is no confusion.
Lemma B.
(See ref. 16.) (Subdomain Stitching) Let
be (measurable) subsets of
such that
contains the support of Z. Then
 |
Lemma C.
(See ref. 16.) For every
and
where
are independent and identical distributions, we have
.
2. Main Results
For any one-dimensional distribution F, let 
. Myersons’ classic result says that 
. For a k-dimensional distribution 
, we introduce the concept of the core of 
and prove that it plays an essential role in determining 
.
Definition:
Let 
be a k-dimensional distribution, where 
are independent one-dimensional distributions (not necessarily identical). Define the core of 
to be the finite k-dimensional interval as follows:
 |
Let 
be the random variable distributed according to 
. Our first result is a structural theorem, identifying 
as the critical domain that determines how much revenue can be extracted beyond simply selling each item separately.
Remarks:
Without loss of generality, we can assume that 
for all i; otherwise, Theorems 1–3 will be trivially true as all revenues will be infinite.
Theorem 1.
There exist constants
such that for every integer
and every
,
 |
Theorem 1 reduces the original problem dealing with distributions over infinite range into a problem over a finite range, making it possible to use the law of large numbers for our analysis.
Theorem 2.
There exists a constant
such that for every integer
and every
,
 |
For identically distributed 
, we show that bundling achieves optimality to within a constant factor.
Theorem 3.
There exists a constant
such that for every integer
and every
where
are independent and identical distributions, we have the following:
 |
Theorems 2 and 3 answer an open question raised in Hart and Nisan (16); and by the examples given there, these bounds are the best possible.
Theorem 3 also gives insight into the issues of nonmonotonicity and randomization. Hart and Reny (7) observed a counter intuitive phenomenon: there exist one-dimensional distributions
, 
where
stochastically dominates
(i.e., 
for all x), yet
. It raised an interesting open question how large this nonmonotonicity difference can get. Theorem 3 yields as easy corollary that the above anomalous ratio of the revenues is bounded by a constant.
Corollary 1.
There exists a constant
such that for any
and any one-dimensional distributions
, 
where
stochastically dominates
, we have
where
(k times) and
(k times).
The corollary is true because BREV is monotonic in the sense that 
if 
stochastically dominates 
. Theorem 3 also implies a constant factor between the revenues of deterministic auctions versus randomized ones. Let 
denote the maximum revenue derivable by any deterministic IC and IR mechanism. Noting that bundling is a deterministic mechanism, we have the following.
Corollary 2.
There exists a constant
such that for any
and
(k times) where F is any one-dimensional distribution, 
.
In the ensuing sections, we will prove Theorems 1 first, followed by Theorems 2 and 3. Theorem 1 provides the basis for focusing attention on 
only in analyzing mechanisms such as SREV and BREV, while ignoring contributions involving any tail components. We first introduce this CT decomposition and study its properties in the next section, before proving Theorem 1 in Section 4.
3. CT Decomposition and the Core Lemma
In this section, we discuss a technique called CT decomposition for partitioning a multidimensional distribution into “core” and “tails,” such that the optimal revenue can be effectively estimated by focusing only on the core part. This decomposition is key to our analysis of various mechanisms. We let k ≥ 2.
3.1. CT Decomposition.
Definition:
Let 
be a k-dimensional distribution. For any subset 
, let 
be defined as 
where 
if 
, and 
if 
.
Thus, 
and the entire region 
is decomposed into 
components:
 |
We are interested in the tail distributions obtained by restricting 
to 
where 
. In fact, we will only be interested in those nonempty A for which all of its tail sections have positive weights. We give precise definitions in the following.
Definition:
Define 
. Note that it is always the case that 
; otherwise, the revenue at price 
is positive and equal to 
, exceeding the maximum revenue 
, which is a contradiction.
Definition:
A subset 
is said to be proper (relative to 
), if 
and 
(and hence 
) for all 
. Denote the collection of all proper subsets by 
.
We next define formally the tail distribution obtained by restricting 
to 
, for any proper 
. To do so, we first split each one-dimensional distribution 
at 
into two distributions 
as follows.
Definition:
Let 
be a one-dimensional distribution with 
and 
. Let 
be two distributions obtained from 
by restricting the random variable 
to 
, and to 
, respectively, properly normalized. Define
 |
 |
Remarks:
To simplify the notation, we sometimes abbreviate 
as 
, and 
as 
when there is no confusion. It is also convenient to extend the definition of 
to include those 
for which 
by simply letting 
for all x (but still leaving 
undefined) in such cases.
It is easy to check that 
and 
are continuous from the right and monotone nondecreasing in their values ranging from 0 to 1, and thus are indeed valid distributions. We are now ready to define the family of tail distributions of 
.
Definition:
For any proper subset 
, let 
and 
. Define
 |
 |
Let 
be the resulting distribution defined over region 
. We refer to the family of distributions 
as the tail distributions of 
induced by the CT decomposition.
Note also that the probability that the value of 
falls in the region 
is equal to 
where
 |
 |
Finally, we finish this section with some bounds on the basic parameters of 
and 
.
Lemma 1.
and
.
Proof:
By definition of 
, we have 
.
□
Lemma 2.
, and if
then
.
Proof:
 |
 |
□
3.2. The Core Lemma.
We now prove a key structural result, which isolates the contributions of the tail regions from that of the core region.
Lemma 3. (Core Lemma)
 |
Proof:
By Subdomain Stitching (Lemma B) and Eq. 2, we have
 |
where we used the fact that 
if 
. For any 
, Lemma A of 1. Notations and Preliminaries implies the following:
 |
Note that, by Eq. 1,
 |
where we have used Lemma 1 to conclude 
. We have from Eqs. 3–5 that
 |
As all 
by Lemma 1,
 |
We have thus proved Lemma 3.
□
4. Proof of Theorem 1
To prove Theorem 1, we will bound the second term on the right-hand side (RHS) of Lemma 3 in terms of 
. By Lemma 2, 
. By Theorem 0 of 1. Notations and Preliminaries, we have for any 
 |
Noting 
from Lemma 1, we have
 |
It follows from Eq. 6 and Lemma 3 (the Core Lemma) that
 |
where 
and 
. This completes the proof of Theorem 1.
5. Proof of Theorem 2
Because of Theorem 1, it suffices to analyze 
. Note that
 |
Now, by standard argument,
 |
Note that
 |
Therefore,
 |
The integral on the RHS satisfies the following:
 |
Thus,
 |
By Theorem 1, this implies
 |
and hence Theorem 2.
6. Proof of Theorem 3
By assumption, 
and hence 
for 
.
 |
 |
where 
. Let Y be the random variable distributed according to 
, we consider two cases as follows.
Case 1:
Then
 |
Theorem 1 implies
 |
Because
 |
by Lemma C (from ref. 16; see 1. Notations and Preliminaries), it follows that
 |
Case 2:
 |
Because 
are all independent, we have
 |
and by Chebycheff’s inequality,
 |
It follows that, by selling the k items as a bundle at price 
, we get
 |
Also, observe that
 |
It follows from Theorem 1 and Eqs. 7 and 8 that
 |
This completes the proof of Theorem 3.
Remarks:
As pointed out by a reviewer, the proofs of Theorems 2 and 3 actually show stronger results in special cases: in Theorem 2, if 
is supported on 
, then 
; in Theorem 3, if F is supported on 
, then 
.
Acknowledgments
This work was supported in part by National Natural Science Foundation of China Grant 61033001, the Danish National Research Foundation, and the National Natural Science Foundation of China Grant 61061130540.
Footnotes
The authors declare no conflict of interest.
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