A Simple and Approximately Optimal Mechanism for an Additive Buyer

A Simple and Approximately Optimal Mechanism for an Additive Buyer

Moshe Babaioff111Microsoft Research, moshe@microsoft.com, Nicole Immorlica222Microsoft Research, nicimm@microsoft.com, Brendan Lucier333Microsoft Research, brlucier@microsoft.com, and S. Matthew Weinberg444MIT, smweinberg@csail.mit.edu. Supported by a Microsoft Graduate Research Fellowship and NSF CCF-6923736.
Abstract

We consider a monopolist seller with heterogeneous items, facing a single buyer. The buyer has a value for each item drawn independently according to (non-identical) distributions, and his value for a set of items is additive. The seller aims to maximize his revenue. It is known that an optimal mechanism in this setting may be quite complex, requiring randomization [19] and menus of infinite size [15]. Hart and Nisan [17] have initiated a study of two very simple pricing schemes for this setting: item pricing, in which each item is priced at its monopoly reserve; and bundle pricing, in which the entire set of items is priced and sold as one bundle. Hart and Nisan [17] have shown that neither scheme can guarantee more than a vanishingly small fraction of the optimal revenue. In sharp contrast, we show that for any distributions, the better of item and bundle pricing is a constant-factor approximation to the optimal revenue. We further discuss extensions to multiple buyers and to valuations that are correlated across items.

1 Introduction

A monopolist seller has a collection of items to sell. How should he sell the items to maximize revenue given that the buyers are strategic? When there is only a single item for sale, and a single buyer with value drawn from a distribution , Myerson [23] shows that the optimal sale protocol is straightforward: the seller should post a fixed take-it-or-leave-it price chosen to maximize , the expected revenue. The optimality of this simple auction format extends to the case of multiple buyers, as well.555This assumes regularity of the value distributions and that the buyers’ values are drawn independently. Despite the simplicity of the single-item case, extending this solution to handle multiple items remains the primary open challenge in mechanism design. While recent work in the computer science literature has made progress on this front [2, 3, 5, 6, 8, 9, 10, 13, 15, 17, 21], it is still the case that very little is known about optimal multi-item auctions, and what is known lacks the simplicity of Myerson’s single-item auction.

Consider even the simplest multi-item scenario [17]: there is a single buyer666Note that if the seller has unlimited copies of each item for sale, then an auction for a single buyer directly extends to the case of multiple buyers. with item values drawn independently from distributions , and whose value for a set of items is additive. Even when there are only two items for sale, it is known that the revenue-optimal mechanism may involve randomization [19], even to the extent of offering the buyer a choice among infinitely many lotteries [15, 18]. This is troubling not only from the perspective of analyzing optimal mechanisms, but also from the point of view of their usefulness. For an auction to be useful in practice, it should be simple to describe and transparent in its execution. Indeed, Myerson’s single-item auction is exciting not only for its optimality, but also its practicality.777This simplicity again assumes regularity and independence. The danger, then, is that revenue-optimal but complex mechanisms for multiple items may share the fate of other mathematically optimal designs, such as the Vickrey-Clarke-Groves mechanism, which are very rarely used in practice [4]. It is therefore crucial to pair the study of revenue optimization with an exploration of the power of simple auctions. In other words, what is the relative strength of simple versus complex mechanisms?

The above question was posed in general by Hartline and Roughgarden [20], and by Hart and Nisan specifically for the setting of a single additive buyer [17]. They proposed the following suggestion for a simple multi-item auction: sell each item separately, posting a fixed price on each one. The optimal price to set on item is then , mirroring the single-item scenario. At first glance, it appears that perhaps this simple approach should be optimal: the buyer’s value for each item is sampled independently, and her value for item doesn’t depend at all on what other items she receives due to additivity. There is absolutely no interaction between the items at all from the buyer’s perspective, so why not sell the items separately? Somewhat counter-intuitive, it turns out that this mechanism need not achieve the optimal revenue. For example, suppose that there are items, and that the buyer’s value for each item is distributed uniformly on . Then the optimal price to set on a single item is , with a per-item revenue of and hence a total revenue of . However, there is a different and equally straightforward mechanism that performs much better: offer only the set of all items at a take-it-or-leave-it price of for some small . As grows large, the probability that the sum of item values exceeds this price approaches , and hence the buyer is almost certain to buy. This leads to a revenue slightly less than , a significant improvement over . Hart and Nisan [17] show how to modify this example to exhibit a gap of by replacing the uniform distribution with an Equal-Revenue distribution.888The Equal-Revenue distribution has CDF for , and for .

What is going on in this example? The inherent problem is that the buyer’s value for the set of all items concentrates tightly around its expectation. This is potentially helpful for revenue generation, but the strategy of selling items separately cannot exploit this property. On the other hand, the mechanism designed to target such concentration (selling only the grand bundle at a fixed price) does very poorly in settings where concentration doesn’t occur; Hart and Nisan show that this grand-bundle mechanism achieves only an approximation to the optimal revenue in general. We must conclude that neither of these two simple mechanisms approximate the optimal revenue to within a constant factor.

Our main result is that the maximum of the revenue generated by these two approaches — either selling all items separately or selling only the grand bundle — is a constant-factor approximation to the optimal revenue. In other words, for any distribution of buyer values, either selling items separately approximates the optimal revenue to within a constant factor, or else bundling all items together does. Since a good approximation to the expected revenue of each approach can be computed in polynomial time given an appropriate access to the distribution (see Appendix G for a discussion of this claim), our results furthermore imply the first polytime constant-factor approximation mechanism for the case of an additive buyer with independently (and non-identically) distributed values, even without the restriction of simplicity.999When the distributions are identical, and furthermore satisfy the Monotone Hazard Rate condition, [16] provides a PTAS. However, other recent results based on linear programming formulations ([1, 3, 2, 5, 8, 9, 10, 11]) all run in time polynomial in the support of . In many correlated settings, this is the right runtime to shoot for, or the best one could hope for. But in our independent setting, this runtime will be exponential in when ideally we would like to run in time polynomial in . We show that if we have meaningful access to the distributions in a way that allows us to compute the optimal per-item reserves efficiently, then our mechanism runs in polynomial time. Furthermore, prior to our work, it was not even known if any deterministic mechanism could achieve a constant-factor approximation to the optimal mechanism, even without regard for simplicity or computational efficiency.


Main Result (Informal). In any market with a single additive buyer and arbitrary independent item value distributions, either selling every item separately or selling all items together as a grand bundle generates at least a constant fraction of the optimal revenue.


Our result nicely complements an active research area aimed at characterizing distributions and valuations in which simple mechanisms are precisely optimal [2, 17, 24, 25]. In contrast to that literature, we show that a maximum over simple mechanisms is approximately optimal, for arbitrary distributions and additive valuations. Our result also echoes a similar line of investigation for markets with unit-demand valuations in which a buyer’s value for a set of items is his maximum value for an item in the set. In this setting, it is known [12, 13, 14] that selling items separately achieves a constant approximation to the optimal revenue. Our result illustrates that a similar approximation can be achieved for additive buyers, provided that we also consider selling all items together as a grand bundle.

To obtain some intuition into our result, recall the example above with items and uniformly-distributed values. This example illustrates that selling all items separately may be a poor choice when the value for the grand bundle concentrates around its expectation. What we show is that, in fact, this is the only scenario in which selling all items separately is a poor choice. We prove that if the total value for all items does not concentrate, then selling separately must generate a constant fraction of optimal revenue.

Our argument makes use of a core-tail decomposition technique introduced by Li and Yao [22] to study the revenue of selling items separately. Roughly speaking, the idea is to split the support of each item’s value distribution into a “tail” (those values that are sufficiently large), and a “core” (the remainder). One then attributes the revenue of the optimal mechanism to the revenue extracted from values in the tail, plus the expected sum of values in the core. To bound the optimal revenue, it then suffices to bound each of these two quantities separately. Li and Yao define the tail of a distribution so that each value is in the tail with probability at most ; they use this to prove that selling all items separately obtains a logarithmic approximation to the optimal revenue (which is tight).

We apply a similar approach, but we define the boundary between core and tail in a different way. We aim to strike a balance between two opposing goals: we want the boundaries to be high enough that the probability of being in the tail is low, which will imply that the revenue from the tail is small relative to selling items separately. At the same time, we want values in the core to be small enough that, subject to their sum being large, the sum must necessarily concentrate around its expectation (which would imply that bundling all items together achieves good revenue). To meet these two goals, we design thresholds that are adapted to the revenue contributions of different items, which makes the core smaller (relative to non-adaptive thresholds) when the value distributions are highly asymmetric. This gives us the extra flexibility needed to derive a constant-factor approximation.

We apply the same methodology to prove that when there are many buyers (with valuations that are not necessarily samples from identical distributions), selling all items separately yields an approximation to the optimal mechanism. This bound is asymptotically tight, as Hart and Nisan have presented a lower bound that matches this for just a single buyer. Prior to our work, no non-trivial bounds were known on the revenue of selling separately to many buyers, or even on the revenue on any class of mechanisms. Furthermore, the observation that selling separately fails only under concentration has implications in this setting as well: we further show that unless the maximum attainable welfare (of all buyers together) concentrates, that selling items separately again obtains a constant-factor approximation. However, with many buyers the concentration of welfare does not imply that selling the grand bundle together obtains a constant-factor approximation. Indeed, unlike in the single-buyer case, one cannot improve the approximation ratio by using bundling: we prove that the lower bound applies against the better of selling separately and together as well. This realization motivates our first open problem:

Open Problem 1.

Is there a “simple,” approximately optimal mechanism for many additive buyers with independent values?

In attempt to make progress on this problem, we turn to a subclass of deterministic mechanisms that we call “partition mechanisms.” A partition mechanism first partitions the items into disjoint bundles, then sells each bundle separately. This natural class of mechanisms clearly generalizes both selling separately and selling together, so we study the performance of the optimal mechanism in this class relative to that of others. On this front, we show that unfortunately the revenue of the optimal mechanism for many independent buyers can still be an factor larger than that of the optimal partition mechanism, and further that revenue of the optimal partition mechanism can be an factor larger than the better of selling separately and together.101010Clearly, no example can exhibit both gaps simultaneously as selling separately achieves an -approximation to the optimal revenue.

Finally, we study the performance of selling separately and together against partition mechanisms for a single buyer whose values for the items may be arbitrarily correlated. While neither class of mechanisms can guarantee any finite factor of the optimal revenue ([7, 17]), the question remains as to whether simple mechanisms can approximate more complex (though still suboptimal) mechanisms in the presence of correlation. To this end, we prove that selling items separately obtains an -approximation the optimal obtainable revenue by a partition mechanism, and that this is tight. In fact, we show a gap of between the better of selling separately and together versus the optimal partition mechanism. We include several tables in Appendix A displaying the relative power of the various classes of mechanisms studied in this paper, noting here that as of our work, all upper and lower bounds are (asymptotically) matching.

Our paper leaves several natural open problems for future work. The first was already stated and concerns extending our results to many buyers. A second problem concerns extending our results beyond additive valuations. As for both unit-demand and additive valuations a constant-factor approximation mechanism is now known, one could naturally ask if such a result is also achievable for valuations that generalize both unit-demand and additive. One potential instantiation is a buyer with a -demand valuation; i.e., additive, but wants at most items. A significantly more challenging instantiation is the class of gross-substitute valuations.

Open Problem 2.

Is there a “simple,” approximately optimal mechanism for single buyer with a -demand valuation? With a gross-substitute valuation?

Finally, a third problem concerns extending our results to settings with mild (but not aribtrary) correlation. This approach was fruitful in [14] for the “common base-value” model.111111In the common base-value model, the buyer has distributions , and samples from each . Her value for item is then , and is called the “base-value.”

Open Problem 3.

Is there a “simple,” approximately optimal mechanism for a single additive buyer whose value for items is sampled from a common base-value distribution? What about other models of limited correlation?

2 Preliminaries

The setting we consider is that of a single monopolist seller with heterogeneous and indivisible items for sale to additive, risk-neutral, quasi-linear consumers (buyers). That is, each consumer has a value for item . While our main results are for the setting of a single buyer, we will define our setting more generally; this will be useful when discussing extensions. If a randomized outcome awards consumer item with probability and charges him a price in expectation, then his utility for this outcome is . Each value is sampled independently from a known distribution . We make no assumptions on whatsoever. We refer to as the joint -dimensional distribution over all consumers’ values for all items, as the -dimensional distribution over all consumers’ values for item . Furthermore, we denote by a random sample from , a random sample from . We also denote the maximum value for item as .

We are interested in analyzing mechanisms at Bayes-Nash equilibrium of buyer behavior, with an eye toward maximizing revenue at equilibrium. By the revelation principle, we can restrict attention to mechanisms that are Bayesian Incentive Compatible (i.e., truthful).121212As it turns out, all of the mechanisms we describe will also satisfy the stronger property of dominant strategy truthfulness. As usual, we also impose the individual rationality constraint, saying that every buyer’s utility is non-negative when truthful.

We use the following terminology to discuss the revenue obtainable by various types of mechanisms, where the first three are taken from [17].

  • : The optimal revenue obtained by any (possibly randomized) truthful mechanism when the consumer profile is drawn from .

  • : The optimal revenue obtained by auctioning items separately when the consumer profile is drawn from . That is, the revenue obtained by running Myerson’s optimal auction separately for each item.

  • : The optimal revenue obtained by auctioning the grand bundle when the consumer profile is drawn from . That is, the revenue obtained by running Myerson’s optimal auction when treating the grand bundle as a single item.

  • : The optimal revenue obtained by any partition mechanism when the consumer profile is drawn from . That is, the maximal revenue obtained by first partitioning the items into disjoint bundles, and then running Myerson’s optimal auction separately for each bundle, treating each bundle as a single item.

Given a distribution over profiles, we will often consider the welfare of a consumer profile drawn from . We will write for the expected welfare, so that . We will also write for the variance of the welfare.

We will make use of some results from [17] that provide useful bounds on . We include proofs in Appendix B for completeness. Lemma 1 is stated and proved directly in [17]. Lemma 2 is not directly stated nor proved, but is similar to an implicit result from [17].

In the lemma below, we think of and as being distributions over values for disjoint sets of items, for the same set of consumers. The distribution then draws values for those two sets of items, independently, from and respectively.

Lemma 1.

([17]) .

The next result establishes a weak bound on with respect to .

Lemma 2.

.

3 The Core Decomposition

We make use of an idea developed by Li and Yao [22] called the “core” of a value distribution for a single consumer. In order to obtain our stronger results for a single consumer and also extend to many consumers, we define the core differently but in the same spirit. The idea is to separate each -dimensional value distribution for each item into the core and the tail, the tail being the part where some consumer has an unusually high value for the item. Then the core of the entire -dimensional distribution is the product of all the cores, and the tail is everything else.

3.1 Defining the Core and Prior Results

Below we formalize the notion of the core. We introduce some notation that will be used throughout the paper. By the “null” distribution, we mean a distribution whose product with any other distribution is also a null distribution, and that outputs with probability .

  • : The optimal revenue obtainable by selling just item (using Myerson’s optimal auction).

  • : . The same as but cleaner to write in formulas.

  • : A profile of parameters, one per item, to define the separation between the core and tail of distribution . We will think of as a multiplier applied to . The core for item will be supported on the interval , and the tail for item will be supported on . Different results throughout the paper will specify different choices for .

  • : , the probability that the highest value on item lies in the tail. Note that this may be .

  • : The core of , the conditional distribution of conditioned on . Note that this may be the null distribution if .

  • : The tail of , the conditional distribution of conditioned on . Note that this may be the null distribution if .

  • : Throughout our notation, we will use to represent a subset of items. We often think of as the items whose values lie in the tail of their respective distributions.

  • : is a subset of items, and is a product distribution equal to .

  • : is a subset of items, and is a product distribution equal to .

  • : . Note that this product is taken over the tail of items in and the core of items not in . In other words, is the distribution , conditioned on if and conditioned on if .

  • : . This is equal to .

Before stating our core decomposition lemma, we present some known results about the core. The lemmas below were either stated explicitly in [22] or [17], or use ideas from one of those papers. We put a citation in the statement of such lemmas, but include all proofs in Appendix C for completeness.

Lemma 3.

([22]) for all .

Lemma 4.

([22]) and .

Lemma 5.

([17]) .

3.2 The Core Decomposition Lemma

In this section we state our Core Decomposition Lemma, which relates the optimal revenue from a distribution to the revenue and welfare that can be extracted from the tail and core of . This result is similar in spirit to the core lemma of [22].

Our first result, Lemma 6, is our main decomposition lemma. The lemma states that the optimal revenue from distribution can be split into a contribution from the core of and a contribution from the tail of . One might hope for a bound of the form “the optimal revenue from is at most the optimal revenue from the tail plus the optimal revenue from the core.” Indeed, such a bound is attainable for a single buyer [22], but is problematic for many buyers, see Section 4.4 and Appendix 3 in [17] for a discussion. We will therefore settle for a weaker bound: the optimal revenue from the tail plus the expected welfare from the core. We also note that the approach of Li and Yao eventually upper bounds the optimal revenue of the core with the expected welfare anyway.

Lemma 6 (Core Decomposition).

Proof.

By Lemma 1,

for all . Also, since is the expected sum of values for items not in , we have

By Lemma 5,

As the desired result follows. ∎

4 Revenue Bounds for a Single Buyer

In this section we focus on the case of a single buyer, . We will work toward proving our main result, which is that is a constant-factor approximation to in this setting. Our argument will make use of the core decomposition, described in the previous section. We will begin with a simpler result that illustrates our techniques: that is at most times . A logarithmic approximation was already established in [22]; we obtain a slightly tighter bound, but the primary purpose of presenting this result is as a warm-up to introduce our techniques and those of [22]. We will then show how this bound can be improved to a constant by considering the maximum of and .

4.1 Warm-up:

We first give a simple application of our approach to provide a bound on SRev vs. Rev, which is slightly improved relative to the bound obtained in [22].

Theorem 1.

For a single buyer, and any , . This is minimized at , yielding .

The idea of the proof is to consider the core decomposition of , choosing for each item . By the Core Decomposition Lemma (Lemma 6), Theorem 1 follows if we can bound the optimal revenue from the tail and the expected welfare from the core, given this choice of .

We begin with Proposition 1, which effectively shows that for constant , the revenue from the tail is at most a constant times . The intuition behind this result is that each item lies in the tail with probability , and hence a large fraction of the time there will be at most a single item whose value lies in the tail. In this case, the revenue from the values in the tail is certainly no more than , since the optimal mechanism can do no better than setting the optimal price for the single item present. To bound the revenue contribution when many values lie in the tail, the relatively weak bound in Lemma 2 will suffice.

Proposition 1.

For a single buyer, and any , if for all , then .

Proof.

By Lemma 2 and Lemma 4, . Therefore, we may rewrite the sum by first summing over item , and then sets containing , obtaining:

We now wish to interpret the term . Observe that is exactly the probability that the set of items are in the tail, conditioned on being in the tail, and is just the size of . Summing over all therefore yields the expected size of the set of items in the tail, conditioned on being on the tail.131313This observation is due to Aviad Rubinstein, and we thank him for allowing us to include it. An earlier version of this paper presented a -approximation in Theorem 1 and a -approximation in Theorem 2. This observation improved those factors to and 6, respectively. Clearly this expectation is just , which is at most by Lemma 3. As we have just observed that , we have now shown that , which is exactly . ∎

Having established a bound on the revenue of the tail, we turn to the welfare of the core. For this, we use the definition of to directly bound for all , and then take an expectation over the range of the core.

Proposition 2.

For a single buyer, and any , if for all , then .

Proof.

Note that . The last inequality would be equality if we replaced with a random variable drawn from , but since stochastically dominates such a random variable, we get an inequality instead. As the optimal revenue of is , this means that . So we have

Summing this guarantee over all yields the proposition. ∎

Combining Propositions 1 and 2 with Lemma 6 yields Theorem 1.

4.2 Main Result:

In this section we prove our main result, showing that the best of selling items separately and bundling all of them together is a constant-factor approximation to the optimal mechanism. The proof will follow a similar skeleton to that of Section 4.1, by proving propositions similar to Propositions 1 and 2. The notable difference is that we will need to be more careful in defining the core.

When all are identical, the approach in Section 4.1 (setting each ) can be leveraged to yield the bound ([22]), but fails in the case that a small number of items contributes the majority of the optimal revenue. To see the problem, note that the definition of the core depends on the number of items , but this can be made arbitrarily large by adding extra items of negligible value. The effect is that the core is potentially larger than necessary when value distributions are asymmetric. What we need instead is for to depend on the value distribution . We let scale inverse proportionally to , so that high-revenue items are more likely to occur in the tail. This allows us to capture scenarios in which revenue comes primarily from one heavy item (by analyzing the tail), as well as instances driven by the combined contribution of many light items (by analyzing the core). Indeed, note that if we set , then the boundary between core and tail becomes for each item. This turns out to be precisely the threshold that we need to attain constant-factor approximation bounds for both the core and the tail, simultaneously.

Theorem 2.

For a single buyer, .

As in Theorem 1, our approach will be to apply the Core Decomposition Lemma (Lemma 6) with an appropriate choice of values , then bound separately the revenue from the tail and the welfare from the core. As discussed above, we will make the non-uniform choice for each .

Proposition 3.

For a single buyer, when for each , .

Proof.

We begin similarly to the proof of Proposition 1, using Lemma 2 and Lemma 4 to write . Again, summing this over all yields:

Just like in Proposition 1, is exactly the expected number of items in the tail, conditioned on being in the tail. It’s again clear that this sum is exactly . By Lemma 3, this is at most . By our choice of , the second term is upper bounded by , as and . Therefore, , and . ∎

We now turn to bounding the welfare from the core. We will use the small range of the core to derive an upper bound on the variance of its welfare. This will allow us to conclude that the welfare is highly concentrated whenever it is sufficiently large relative to . Thus, if the welfare is “small” compared to , then selling separately extracts most of the welfare (within the core); otherwise the welfare concentrates and so bundling extracts most of the welfare (within the core). The following lemma of [22] will be helpful for this approach; its proof appears in Appendix D for completeness.

Lemma 7.

([22]) Let be a one-dimensional distribution with optimal revenue at most supported on . Then .

Corollary 1.

For a single buyer, and any choice of , .

Proof.

, and the distribution is supported on . Therefore, plugging into Lemma 7 (and relaxing) yields the desired bound. ∎

Proposition 4.

For a single buyer, when all ,

Proof.

There are two cases to consider. If , then we have that as required.

On the other hand, if , then Corollary 1 tells us that . Summing over all and recalling that we get

So and . By Chebyshev’s inequality, we get

Since is at least the revenue obtained by setting price on the grand bundle, this implies . As , as required. ∎

Combining Propositions 3 and 4 with Lemma 6 yields Theorem 2. To our knowledge, the best known lower bound on vs. Rev is , provided by an example in [15]. The example has two items with and , with , , and . It is an interesting open question to close the gap between and , either by tightening our analysis or providing better lower bounds.

5 Revenue Bounds for Multiple Buyers

Here we extend our results to multiple buyers with valuations sampled independently (but not necessarily identically). We will refer to this as the independent setting, as the buyers’ valuations are independent and furthermore each buyer’s item values are also drawn independently. We first show in Theorem 3 that for the independent setting, selling items separately achieves a logarithmic (in ) approximation to the optimal revenue. We next show in Theorem 5 that like in the single buyer case, the only case in which selling items separately fails to achieve a good approximation, is the case that welfare is highly concentrated. Unfortunately, such concentration is no longer sufficient to achieve a constant approximation by selling all items together. This is so because even though the welfare is concentrated, the partition that provides such welfare can change dramatically between realizations. Indeed, in Proposition 8 we show not only that fails to provide a constant approximation to the optimal mechanism, but even fails, and this is so even when item values are sampled i.i.d. for all items and buyers. Finally, in Proposition 9 we show that in the independent setting, cannot be approximated well by .

5.1 An Upper Bound:

We first show that selling items separately achieves a logarithmic (in ) approximation to the optimal revenue.

Theorem 3.

For arbitrarily many buyers, in the independent setting, . (Note that .)

Our proof will proceed via amplification. We will begin with the (awful) bound on SRev vs. Rev from Lemma 2, then show in Theorem 4 how to amplify any such bound into an improved bound. We will then iterate this amplification process over and over, until we reach the desired logarithmic approximation (which will be a fixed point of the amplification process). To prove the amplification theorem, we use an approach similar to the single-buyer analysis from Section 4.1. That is, we will apply the Core Decomposition Lemma (Lemma 6), then bound the revenue of the tail and the welfare of the core with respect to .

Theorem 4 (Amplification).

For arbitrarily many buyers in the independent setting, assume that for some it holds that . Then, for any , as well. Setting yields .

To prove Theorem 4, we will apply the Core Decomposition Lemma (Lemma 6), using for each . Theorem 4 will then follow from bounds on the revenue from the tail and the expected welfare from the core, which we establish in the following propositions.

Proposition 5.

For arbitrarily many buyers in the independent setting, if for all and , then .

Proof.

The proof is nearly identical to that of Proposition 1. The only difference is that we start with the fact that instead of just , and make use of the fact that when there is only one item, . We include the details in Appendix E for completeness. ∎

The following bound on the welfare from the core follows in a manner similar to Proposition 2. We defer its proof to Appendix E.

Proposition 6.

For arbitrarily many buyers in the independent setting, if for all , then .

Theorem 4 then follows from Propositions 5 and 6, together with Lemma 6. We now show how to prove Theorem 3 using Theorem 4.

Proof (of Theorem 3).

By Lemma 2, we may apply Theorem 4 starting with . This yields a bound of the form for some new . We can then apply Theorem 4 again, taking to be this new value . We can iteratively apply Theorem 4 over and over until we either reach a fixed point (with respect to the value of ) or reach . One can verify that, for all , no is a fixed point and that the function is continuous. Therefore, we can always iterate until and then apply Theorem 4 with , yielding the desired bound. ∎

5.2 A Concentration Result

We next present a characterization of when is a constant-factor approximation to for the independent setting with multiple buyers. We will show (in Theorem 5, below) that this occurs unless the welfare of is sufficiently well concentrated around its expectation.

We begin with a corollary of Theorem 3, which will be useful for our analysis.

Corollary 2.

For arbitrarily many buyers in the independent setting, .

Proof.

This is a direct application of Theorem 3 and noting that for all . ∎

We next prove an alternative bound on the revenue from the tail of the distribution , using a familiar choice of . The proof, which closely follows that of Proposition 3, appears in Appendix E.

Proposition 7.

For arbitrarily many buyers in the independent setting, if we choose for all , then .

We are now ready to establish the claimed bound between SRev and Rev, subject to the welfare of not being too concentrated around its expectation.

Definition 1.

We say that a one-dimensional distribution is -concentrated if there exists a value such that .

Theorem 5.

For arbitrarily many buyers in the independent setting, and any , either or the welfare of (the random variable with expectation ) is -concentrated.

Proof.

Let all . Then combining Proposition 7 and Lemma 6 yields

There are two cases to consider. Maybe . In this case, we have .

On the other hand, maybe . In this case, Corollary 1 tells us that . Summing over all and recalling that , we get

So and . By Chebyshev’s inequality, we get

meaning that the welfare of is -concentrated. The last step is observing that is sampled in the support of with probability exactly . As and each , this is minimized when exactly one is and the rest are , yielding . So with probability at least is in the support of . When this happens, the welfare is concentrated. So the welfare of is -concentrated. ∎

5.3 A Lower Bound: even for i.i.d. Item Values

We next show that there is a setting with many buyers with item valuations that are sampled i.i.d from the same distribution, for which (and thus also ) provides a poor approximation to .

Proposition 8.

There exists a setting with items and many buyers, with item valuations that are sampled i.i.d from the same distribution, for which .

Proof.

Consider a setting with items and buyers with the following value distributions. For every item and buyer , the distribution such that the value is 0 with probability , and with the remaining probability it is sampled from a distribution with CDF for and for (an Equal-Revenue distribution with all mass above moved to an atom at ). To prove the claim we show in Lemma 11 in Appendix E that while (actually, since it holds that ). ∎

5.4 A Lower Bound:

We next show that there is a setting with many buyers with item valuations that are sampled independently (but not identically), for which provides a poor approximation to .

Proposition 9.

There exists a independent setting with items and many buyers for which .

Proof.

Fix such that is an integer. Consider a setting with buyers, and a partition the items to disjoint sets of size . Buyer has value for every item that is not in the -th set of items, and for item in that set his value is sampled independently from an Equal-Revenue distribution.

Clearly, . is the same as the revenue that can get in a setting with buyers and only items for which each item value is sampled i.i.d. from an Equal-Revenue distribution. That revenue is . We conclude that
. on the other hand, can bundle each of the sets of size separately and sell it to the interested buyer, getting a total revenue of . ∎

6 One Buyer with Correlated Values

In this section, we study the relationship between SRev(D), Max{SRev(D), BRev(D)}, and PRev(D) for a single buyer with correlated values. The prior work of [17, 7] already shows that there is no hope of obtaining a finite bound between any of these quantities and Rev(D) because they are all deterministic, even when there are only two items. But it is still important to understand the relationship between these mechanisms of varying complexity even if their revenue cannot compare to that of the optimal mechanism. We show in Theorem 6 that for any distribution for a single buyer, possibly even correlated, is a approximation to , and thus also to Max{SRev(D), BRev(D)} and PRev(D).141414As SRev approximate BRev for any set of items, it can do so for any part in the partition in PRev separately, and thus also approximate PRev. We then show in Proposition 10 that this bound is tight, Max{SRev(D), BRev(D)} . In other words, SRev(D)  provides a logarithmic approximation to PRev(D), but taking can’t guarantee anything better.

We start by showing that is a approximation to . The proof of the theorem appears in Appendix F.

Theorem 6.

For any -dimensional value distribution for a single buyer (possibly correlated across items), . Therefore, as well.

Finally, we show that there is a setting with one buyer (and correlated item values) for which provides poor approximation not only to but even to . The proof of the proposition appears in Appendix F.

Proposition 10.

There exists a (correlated) distribution of the valuation of a single buyer over items for which .

7 Acknowledgments

In an earlier version of this paper, we proved a factor of 7.5 in Theorem 2. This factor was later improved by Aviad Rubinstein to a factor of 6. We thank Aviad for allowing us to include this improvement in our paper.

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Appendix A Summary of Known Results

Table 1 and Table 2 presents the best results known for one additive buyer with item values sampled independently or arbitrarily, respectively. Additionally, Table 3 presents the best results known for many additive buyers in the independent setting. In each cell there is the known upper and lower bounds of the ratio between the corresponding column quantity and row quantity, and the source of the result. For example, in table 1, table entry that corresponds to the row marked by and column marked by Rev there is the upper bound of that we have presented in Theorem 2 for the ratio which holds for every distribution . Results from this paper that are not implied by other results appear in bold. Results that are implied from other results, point to the results that imply them.

max{SRev,BRev} Rev
SRev  []  [22]
 [17]  []
1 6 [Thm 2]
1  [Folklore]
Table 1: One buyer, independent item values
max{SRev,BRev} PRev Rev
SRev  []  [Thm 6] 0
 [17]  []  []
1  [] 0
1  [Prop 10]  []
PRev 1 0
1  [17]
Table 2: One buyer, correlated item values
max{SRev,BRev} PRev Rev
SRev  []  []  [Thm 3]
 [17]  []  []
1  []  []
1  [Prop 9]  []
PRev 1  []
1  [Prop 8]
Table 3: Many buyers, independent item values

Appendix B Omitted Proofs from Section 2

The proofs of Lemmas 1 and 2 require some technical lemmas from [17]. We include them below with proofs for completeness. In Lemma 8 below, and are distributions over values for disjoint sets of items for the same consumers, and and may be dependent. By we mean the optimal revenue obtainable by selling to consumers whose values for items is sampled from the joint distribution according to and .

Lemma 8.

(“Marginal Mechanism” [17]) .

Proof.

We will establish a lower bound on by constructing a truthful mechanism for selling items in the support of , based on one for those in the support of . First, sample values from for each consumer. Then announce that whenever a consumer would have received an item in the support of , he will instead receive money equal to the (make-believe) value sampled from . Note that, due to this announcement, each consumer now has a value for each item in the support of corresponding to the sampled value profile. Then run the optimal truthful mechanism for selling items in the support of . Conditioned on , this mechanism is truthful. Therefore, conditioned on , the revenue of this mechanism is upper bounded by . Taking an expectation over all , we see that the total expected revenue obtained by this procedure is upper bounded by . Also taking an expectation over all , we see that the expected revenue obtained is exactly minus the expected amount of money given away via the reduction, which upper bouned by . These two observations together prove the lemma. ∎

Proof of Lemma 1: This is an immediate corollary of Lemma 8. as and are independent, , for all .

Lemma 9.

(“Sub-Domain Stitching” [17]) Let form a partition of and let . Then .

Proof.

Let be the optimal mechanism for , and denote the revenue of when consumers are sampled from . Then we have . We also clearly have , and for all , proving the lemma. ∎

In the lemma below, again think of and as independent distributions for the same consumers over disjoint sets of items.

Lemma 10.

(“Marginal Mechanism on Sub-Domain” [17]) be any subset of , and . Then .

Proof.

Let denote the indicator variable for event (that is when occurs or otherwise). Then when we write , we mean the distribution that first samples , and outputs if the event occurs, and otherwise. In particular, when we write