99% Revenue via Enhanced Competition

01/09/2018 ∙ by Michal Feldman, et al. ∙ Tel Aviv University Harvard University 0

A sequence of recent studies show that even in the simple setting of a single seller and a single buyer with additive, independent valuations over m items, the revenue-maximizing mechanism is prohibitively complex. This problem has been addressed using two main approaches: (i) Approximation: the best of two simple mechanisms (sell each item separately, or sell all the items as one bundle) gives 1/6 of the optimal revenue [BILW14]. (ii) Enhanced competition: running the simple VCG mechanism with additional m buyers extracts at least the optimal revenue in the original market [EFFTW17]. Both approaches, however, suffer from severe drawbacks: On the one hand, losing 83% of the revenue is hardly acceptable in any application. On the other hand, attracting a linear number of new buyers may be prohibitive. Our main result is that by combining the two approaches one can achieve the best of both worlds. Specifically, for any constant ϵ one can obtain a (1-ϵ) fraction of the optimal revenue by running simple mechanisms --- either selling each item separately or selling all items as a single bundle --- with substantially fewer additional buyers: logarithmic, constant, or even none in some cases.

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1 Introduction

The scenario of a buyer with an additive, independent valuation over items has become the paradigmatic setting for studying optimal (revenue-maximizing) mechanisms. In this setting, the buyer’s valuation is drawn from a distribution that is known to the seller, and the seller wishes to design a selling mechanism that extracts as much revenue as possible.

By now it is well known that the optimal mechanism requires randomization [HN13], infinite, uncountable menus [DDT16], is non-monotone [HR12], and computationally intractable [DDT14]111Similar undesired properties have been observed with respect to the optimal mechanism in additional related (multi-dimensional) models, e.g., unit-demand buyers [BCKW10, CDO15, CDP14, RC98]. Thus it is mostly interesting as a theoretic benchmark to which one can compare more plausible mechanisms (much like the way an offline optimum serves as a benchmark in online settings). In recent years, two main approaches have been taken with respect to this challenge, both of which proposed simple mechanisms and measured their performance against the theoretic optimum:

The first line of work approaches this problem through the lens of approximation [CHK07, CHMS10, KW12, HN12, LY13, BILW14, CMS15, Yao15, RW15, BDHS15, MS15, CDW16, CZ17, CM16, EFF17b, Yao17]. In particular, the breakthrough result of [BILW14] shows that the better of two simple mechanisms — selling each item separately or selling all items together in a grand bundle — obtains at least (but at most  [Rub16]) of the optimal revenue.

While a constant factor approximation algorithm may sound appealing to an algorithm designer, losing 83% (or even 50%) of the revenue is simply unacceptable. Indeed, it will be difficult to convince a merchant who hopes to make in revenue to sell her merchandise by a mechanism that would guarantee her .222There are still many great reasons to study constant-factor approximations in mechanism design; see, e.g., [Har13] for an excellent discussion. On the other hand, a seller might be willing to compromise on optimality if guaranteed 99% of the optimal revenue. (For example, merchants around the world pay small fees to credit card companies in return for simple selling mechanisms). We therefore believe that the most interesting agenda here should be obtaining 99% of the optimal revenue. (More generally, we are interested in mechanisms whose revenue is arbitrarily close to optimum; i.e., -fraction of the optimal revenue for any constant .)

The second approach is to enhance the competition for the merchandise by increasing the population of potential buyers [BK96, RTCY15, EFF17a, LP18]. The state of the art for additive buyers is by Eden et al. [EFF17a], who showed that adding additional buyers is sufficient to recover the original optimal revenue with a simple mechanism. It is also known that at least additional buyers are required to achieve this benchmark. This result from [EFF17a] generalizes the seminal work of Bulow and Klemperer [BK96] who showed that for a market with a single item, under a regularity assumption, running the second price auction with one additional buyer extracts at least as much revenue as the original optimal revenue. However, when is large, adding additional buyers may be prohibitive. We therefore believe that the most interesting question here is whether the linear dependence on is necessary.

To summarize, the “optimal” benchmark is intractable. The approximation approach is stuck at a -approximation (forfeiting 83% of potential revenue). And if we wish to follow the enhanced competition approach, the best known bound on the number of additional buyers is linear in the number of items. Have we reached a dead end?

1.1 Our Contribution

We show that one can combine the two approaches (of approximation and enhanced competition) in a way that achieves the best of both worlds. We establish a host of results for various settings, but they all convey one theme: in order to obtain revenue that is very close to optimum, there is no need to recruit a linear number (in ) of additional buyers; that is:

Main take away (informal): A seller can obtain 99% of the optimal revenue in a simple mechanism (selling each item separately or selling all items together in a singe bundle) with substantially fewer additional buyers — logarithmic (in ), constant, or even none in some cases.

All of our results apply to the paradigmatic scenario of a seller who sells items to a single buyer with additive, independent valuations over the items, with a known prior. Some of our results extend to the more general scenario of i.i.d. buyers, namely where buyers’ values are drawn independently according to the same distribution. The induced product distribution is then denoted by . Our first set of results consider the simple mechanism that sells each item separately.

Theorem 1.1.

For every constant , selling each item separately to i.i.d. buyers extracts at least -fraction of the optimal revenue achievable by a single buyer.

This result improves upon the bound shown in [EFF17a], at the loss of fraction in revenue. In fact, this result can be extended even to a setting with i.i.d. buyers and items, as follows:

Theorem 1.2.

(implies Theorem 1.1) For a setting with i.i.d. buyers and items, for every constant , selling each item separately achieves at least -fraction of the optimal revenue if we increase the number of buyers by a factor of , and this is tight up to constant factors. Moreover, if , then selling each item separately achieves -fraction of the optimal revenue even with no additional buyers.

Theorem 1.2 essentially fully characterizes (up to constant factors) the number of additional buyers necessary for achieving of the optimal revenue. In particular, we consider three different regimes of and : For we prove that increasing the number of buyers by a factor of is both necessary (Theorem 6.1) and sufficient (Theorem 5.1). For , our new lower bound implies that the previous results of [EFF17a] (who showed that a linear number of additional buyer suffice) are essentially tight. Finally, for , we show that no additional buyers are necessary (Theorem 7.1). We note that our lower bound generalizes the special case of -factor for the case of a single buyer in [EFF17a] 333The lower bound in [EFF17a] was proven for mechanisms that target 100% of the optimal revenue, but it can be easily extended to mechanisms that target 99% or any constant fraction..

Let us return to the single buyer setting. Theorem 1.2 implies that we can recover -fraction of the optimal revenue by adding buyers. However, one may argue that for a large value of , is still too large. We address this issue by showing that the better of selling items separately and selling the grand bundle requires only a constant number of additional buyers444 There is no contradiction to the lower bound, which applies only for selling items separately..

Theorem 1.3.

For every constant , the better of selling each item separately and selling the grand bundle, to a constant number of i.i.d. buyers, extracts at least -fraction of the optimal revenue achievable from one buyer.

Up until now, we concentrated on prior-dependent mechanisms; namely, mechanisms that use the knowledge of the distribution of values. Our work, however, contributes also to the literature on prior-independent mechanisms. As in previous literature on prior-independent mechanisms, to achieve any meaningful result, we assume that the underlying single-dimensional distributions are regular (note that this does not generally imply regularity of the grand bundle’s distribution). Since bidders are additive, the prior-independent VCG mechanism simply runs the second price auction for each item simultaneously. Thus, Theorem 1.2 combined with the original result of Bulow and Klemperer [BK96] immediately implies the following corollary:

Corollary 1.4.

If is a product of regular distributions, then for every constant , running the VCG mechanism with a factor increase in the number of buyers (and with no additional buyers in the case of ) extracts at least -fraction of the original optimal revenue.

It would be highly desirable to obtain such an analog to Theorem 1.3. An immediate barrier, however, is that the seller must know in advance whether to sell the items separately or sell the grand bundle. How can the seller determine which one of these mechanisms to run in the absence of a prior? This barrier is overcome by the surprising result that when is a product of regular distributions, the seller never needs to sell items separately; selling the grand bundle is always the “correct” strategy:

Theorem 1.5.

If is a product of regular distributions, then for every constant , selling the grand bundle in a second price auction to a constant number of i.i.d. additive buyers extracts at least -fraction of the optimal revenue achievable from one buyer.

1.2 Relation to work on approximate revenue maximization

As already mentioned, our work is related to and inspired by a long line of research that aims at understanding what fraction of the optimal revenue can be guaranteed with simple mechanisms (without adding buyers), including [LY13, BILW14, GK16] and references therein. It is interesting to note that obtaining a -approximation to the optimal revenue with additional buyers has implications on approximation results without additional buyers. In particular, if one can obtain a -approximation to the optimal revenue using mechanism with additional buyers, then, by revenue submodularity, this immediately implies a -approximation of the optimal revenue with a single buyer. Hence, our Theorem 1.1 implies the result of [LY13] that selling each item separately is an -approximation of the optimal revenue. Similarly, our Theorem 1.3 implies the main result of [BILW14] that the better of selling separately and selling the grand bundle yields a constant fraction of the optimal revenue. Of course, the reverse implication is not true, since the seller only has a single copy of each item to allocate to all buyers. Therefore, while our analysis builds on the techniques of [LY13, BILW14], it is significantly more challenging.

Recently, Goldner and Karlin [GK16] built on the result of Babaioff et al. [BILW14], and showed that for regular distributions, the prior-independent mechanism which sells either each item separately or the grand bundle, and sets the price according to a single sample from the valuations distribution, obtains a constant fraction of the optimal revenue. As a corollary of our Theorem 1.5, we can obtain the following qualitative strengthening of [GK16, Corollary 2] for the case of a single buyer:

Corollary 1.6.

If is a product of regular distributions, taking a single sample of the value of the grand bundle and selling the grand bundle for that price (to a single buyer) extracts at least fraction of the optimal revenue.

For the special case where the single-dimensional distributions satisfy the monotone hazard rate (MHR) condition 555

Roughly speaking, MHR distribution (a special case of regular distributions) has a tail that is thinner than that of an exponential distribution.

, Cai and Huang [CH13] gave a PTAS for the revenue maximizing auction. En route to obtaining their computational result, they prove the following structural lemma, which holds even for heterogeneous buyers. 99% of the optimal revenue can be obtained by one of the following simple mechanisms: (i) selling every item separately; or (ii) selling all but a constant number of items via a VCG-like mechanism. In other words, they show that when the single-dimensional distributions satisfy the MHR condition, no additional buyers are necessary to obtain 99% of the optimal revenue with mechanisms that are simple (in the above sense). Moreover, inspired by [CD11], they show that for MHR distributions, when the number of buyers is larger than some constant, selling items separately obtains 99% of the social welfare. (In contrast, note that for regular distributions the ratio of social welfare to revenue may be unbounded, even when adding any number of buyers.)

2 Model

We consider a setting in which a monopolist seller sells a set of heterogeneous items to additive buyers. Buyer ’s value is additive if there exist values such that buyer ’s value for a set of items is . The seller does not know the buyers’ values, but knows the distribution from which they are sampled. For every item , the value of each agent for item is independently sampled from a single-dimensional distribution .

A mechanism is given by a pair of an allocation function , and a payment function . The mechanism receives a valuation profile as input. Based on , the allocation function determines the (possibly random) allocation of items to buyers, and the payment function determines the payment of every buyer . Buyers are quasi-linear; namely, buyer ’s utility from a mechanism that allocates her each item

with probability

and charges her a payment is .

A mechanism is Bayesian Individually Rational (BIR) if every buyer’s expected utility (over the randomness of the mechanism and other buyers’ values, assuming they are drawn from ) is non-negative. A mechanism is Bayesian Incentive Compatible (BIC) if every buyer’s expected utility is maximized when the buyer reports her valuation truthfully. Throughout the paper, a BIR-BIC mechanism is termed a truthful mechanism. The expected revenue of a truthful mechanism is , where for every buyer and item buyer ’s value for item is drawn independently from . The optimal revenue is the optimal666In general for distributions of infinite support, the optimum revenue may not be achieved by any mechanism (i.e. it is the supremum of all revenues achievable by truthful mechanisms). This is not so important for our purposes since we are only trying to approximate the optimum. expected revenue among all truthful mechanisms.

For any single dimensional distribution and any , we assume w.l.o.g. that there exists such that . When the distribution is continuous this is true by the intermediate value theorem. When the distribution has point masses, we can smooth it with an infinitesimal perturbation; see e.g., [RW15] for a formal discussion. We note that when is regular, the perturbed distribution may not be regular. However, it will not be hard to see that our proof in Section 9 will also work with distributions that are infinitesimally-close to regular distributions.

3 Overview of Proofs

Our techniques build upon the now-standard approach for approximately optimal revenue analysis: Separately reason about revenue contribution from rare events of extremely high value (“tail” events), and revenue contribution from lower values (“core” events). This approach is made formal via the core-tail decomposition framework of Li and Yao [LY13]. Separation between core and tail events is done by setting a cutoff for each item, and then proving a “core-tail decomposition” lemma that upper bounds the optimal revenue by the sum of total value from core events in which items are below their cutoffs (a.k.a. contribution from the core), and the total revenue from tail events in which items are above their cutoffs (a.k.a. contribution from the tail).

For approximation results, the next step would typically be to show that simple mechanisms approximate both the contribution from the core and the tail, hence the best simple mechanism approximates the optimal revenue. However, when targeting 99% of the optimal revenue, even if one shows that simple mechanisms fully recover the contribution from the core and the tail, it does not yet imply that simple mechanisms recover the sum of the contributions. Hence this approach alone cannot yield better than a 1/2-approximation. Instead, we carefully set the cutoffs so that the contribution from the core is almost fully recovered using simple mechanisms with either buyers (Theorems 1.3 and 1.5) or buyers (Theorem 1.2), and the contribution from the tail is only a tiny fraction of the revenue from simple mechanisms with buyers.

Before delving into the specificities of each result, we provide a few general notes about the tail and the core.

Tail

Since tail events are rare, they are approximately “separable” across items and across buyers. First, since the probability of two or more items in the tail of any buyer is low, the revenue contribution from tail items is roughly “separable” across items, i.e., approximating revenue contribution of tail events by the revenue contribution from selling items separately loses only a moderate factor. Furthermore, each buyer is likely to have tail valuations for disjoint subsets of items (“separability across buyers”). Thus in an enhanced competition setting with more buyers we can simultaneously serve all of them, quickly increasing the revenue.

Core

Since the value of core items is bounded, a concentration bound typically suggests a near-optimal grand bundle price whenever the sum of item values is significantly larger than the revenue from selling items separately.

In what follows we elaborate on cases where interesting artifacts arise in the analysis. We use Rev, SRev, and BRev to denote the optimal revenue from any mechanism, the optimal revenue from selling each item separately, and the optimal revenue from selling the grand bundle, respectively.

Theorem 1.3 ()

An optimistic approach for proving Theorem 1.3 would use the same cutoff for all items, and apply a concentration bound (Chebyshev’s inequality) to suggest a bundle price that almost fully recovers the contribution from the core with constant probability, then improve this probability to almost

by considering additional buyers. Unfortunately, such strong concentration does not hold in general: a small number of items may still exhibit a large variance, even inside the core. To overcome this challenge, we separate the set of items to items that exceed their cutoffs with constant probability, which we call “high items”, and the remaining we call “low items”. We then use a concentration bound only for the sum of low items to get a good bundle price for the low items. We set the cutoff so that the number of high items is a constant, hence with probability close to

, one of our additional buyers simultaneously exceeds the cutoff of all the “high items”. We conclude that BRev with buyers recovers almost all of the contribution from the core with probability almost . The proof appears in Section 8.

Theorem 1.5 (BRev, regular distributions)

Single dimensional regular distributions are appealing since they have a “small tail” property. While the grand bundle distribution need not be regular even when the individual item distributions are, we show that the underlying regularity of the individual items still maintains some well behaved properties that we can exploit. Specifically, we can set cutoffs so that in the resulting core-tail decomposition, the contribution from the tail is significantly smaller than the contribution from the core. This is important since we are only allowed to sell the grand bundle, while the tail is typically covered by selling items separately.

It is now left to guarantee that the core is almost fully recovered. The challenge is that a small number of outlier items may have very large variance, ruining the naive concentration bound argument. (In Theorem 

1.3 we sometimes sold the outliers separately, but here we are only allowed to sell the grand bundle.) We show that given a large (but still constant) number of additional buyers, two of them are likely to have high values simultaneously for all of the outlier items. We can then use a concentration bound to suggest a good price for the remaining items, so that VCG on the grand bundle gives high revenue. The proof appears in Section 9.

Theorem 1.2 (SRev), regime

Instead of a concentration bound for the grand bundle (which is irrelevant when considering SRev) we further separate the contribution from the core to the contributions from lower and higher values. We then show that the contributions of lower values to the core can be almost fully recovered with probability almost by SRev with buyers, while the contributions of higher values to the core form only a tiny fraction of the revenue that can be extracted by SRev with buyers. The proof appears in Section 5.

Theorem 1.2 (SRev), regime

We show that selling each item separately achieves of the optimal auction without adding any buyers at all. Proving this result requires yet new ideas. The intuition is simple: with so many buyers, we can afford to set the cutoff much higher – so high that most buyers have at most one item in the tail – while the contribution from the core can be almost fully recovered by selling each item to a tiny fraction of the buyers. Therefore, we can set aside a small subset of special buyers, and offer the items at high (tail) prices to all other buyers; we can then recover the rest of the revenue by auctioning the items not previously sold to the special buyers. (Note that this mechanism is for analysis purposes only — once we establish guarantees for any mechanism for selling items separately, we can use Myerson’s optimal mechanism for selling items separately.) This part of the analysis also takes into account the probability that an item was sold in tail events.

Formalizing the above intuition is quite subtle. With high probability, there are still some buyers with multiple items in the tail – and each of those items has many other buyers whose valuations are in the tail, etc. Thus even after the core-tail decomposition, we have to reason about a multiple buyer, multiple item setting. To cope with this difficulty, we consider a bipartite graph of buyers and items, where we draw an edge whenever buyer ’s value of item is in the tail. We then argue that we can analyze the revenue from each connected component separately. In particular, while there are, w.h.p., large components in the graph (with buyers and items), we use simple ideas from percolation theory to bound the expected size of each connected component. The proof appears in Section 7.

4 Preliminaries

A monopolist seller sells a set of heterogeneous items to additive buyers. We assume the seller is able to attract more buyers. An additive buyer that values each item at values a set of items at . For a set of items , let . We use , therefore . Also, we use , and more generally, for a set , we use and . Every buyer is quasi-linear, i.e., in a randomized outcome that allocates item to buyer with probability , and charges a payment , ’s utility is . The seller does not know , but has a prior-distribution with density for each item , i.e., is drawn from . Let . Therefore, in a setting with i.i.d. buyers, each drawn from , the prior distribution is 777By we mean the product distribution ..

Let be a truthful mechanism, i.e., Bayesian Individually Rational (BIR)888A mechanism is Bayesian Individually Rational (BIR) if every buyer’s expected utility is non-negative. and Bayesian Incentive Compatible (BIC)999A mechanism is Bayesian Incentive Compatible (BIC) if every buyer’s expected utility (over the randomness of and the other buyers’ values according to the prior distribution) is maximized by truth-telling., with an allocation function and a payment function , i.e., given the submitted bids , each buyer is allocated item with probability and pays . Let be mechanism ’s expected revenue, e.g., is mechanism ’s expected revenue from buyers, where each buyer’s valuation is drawn i.i.d. from , i.e., . Let be the optimal expected revenue by any truthful mechanism, e.g., is the optimal expected revenue by any truthful mechanism for i.i.d. buyers drawn from . Similarly, is the optimal expected revenue when selling the items separately, and is the optimal expected revenue when selling all items in a bundle. Let be the expected value of the optimal allocation. Since we consider additive, independently drawn buyers, .

For a single dimensional distribution , and a number , let denote the revenue of an optimal mechanism, among all truthful mechanisms that sell with ex-ante probability of at most , i.e., mechanisms with allocation function that satisfies

Throughout the analysis, for that is not an integer, we slightly abuse notation and use, e.g., instead of to denote the product distribution

. For a random variable

drawn from distribution , we use and interchangeably (and similarly for , etc.). Also, let

be the indicator random variable that equals

when event occurs, and otherwise.

Finally, we will use the following previously shown lemma.

Lemma 4.1.

[BILW14] For any dimensional distribution (i.e., buyers and items), .

4.1 Lemmas for single dimensional distributions.

We recall and develop tools for our analysis in the remaining sections. Let be drawn from a single dimensional distribution (i.e., ), and consider some cutoff .

Lemma 4.2.

[HN12] .

Lemma 4.3.

[LY13] Let . Then .

Let denote the variance of .

Lemma 4.4.

[LY13] Let , and supposed that both and that the support of is in . Then .

The following lemma shows that for that is not too small, the optimal revenue from i.i.d. buyers drawn from is at least a factor larger than the optimal revenue that can be extracted from a buyer with value distributed according to . This lemma will be used to bound the contribution (both to the core and tail) of higher values.

Lemma 4.5.

Suppose that . Then .

Proof.

Since is single-dimensional, there exists a price that extracts the optimal revenue, i.e., . For every , let be drawn from . Therefore, . By Bonferroni inequalities (the inclusion-exclusion principle),

where the last inequality follows by the lemma’s condition: . Therefore, we conclude that . For ease of exposition, we relax the to . ∎

The following lemma will be used to cover the contribution of values from the core.

Lemma 4.6.

Let satisfy . For any ,

Proof.

The following lemma is a specified analog to Lemma 4.6 that bounds the contribution from values in the core using the second price auction, and its proof is similar to the proof of Lemma 4.6. Let denote the revenue from the second price auction.

Lemma 4.7.

Let satisfy . For any ,

Proof.

4.1.1 Regular distributions.

For ease of exposition, we assume has a density and is strictly increasing. Nevertheless, our results extend to arbitrary regular distributions. A distribution with density is regular if is non-decreasing in

. Our analysis is done in quantile space, which is defined below.

Definition 4.8.

(Quantiles, demand curve, and revenue curve.)

  • Let be the quantile of .

  • Let for which , i.e., is the demand curve of .

  • Let , i.e., the revenue from the posted price that sells w.p. .

Observe that is increasing in . By change of variables, the expected value can be computed as follows: 101010By , we get , so . The following lemma is a well known characterization of regular distributions.

Lemma 4.9.

[Mye81] A distribution is regular if and only if is concave, i.e., for every , it holds that .

The following corollary is the only way we use regularity (and is only used to prove Lemma 4.11).

Corollary 4.10.

If is regular and , then .

Proof.

The following lemma shows that if the probability of exceeding the cutoff is , then is a price that provides a approximation to the optimal revenue from the random variable .

Lemma 4.11.

For a cutoff , it holds that .

Proof.

Since is single-dimensional, there exists a price that achieves the optimal expected revenue for this random variable. Let satisfy , then we have . Since , we get that . By monotonicity of we have , and by Corollary 4.10 , as required. ∎

The following lemma relates the contribution from the core to the contribution from the tail.

Lemma 4.12.

For a cutoff , for every and such that it holds that:

Proof.

First observe that . By segmenting the expectation over we get

In the first inequality we reduce values in to . The second inequality follows by monotonicity of . The fact that for every , combined with Lemma 4.11 implies that , which completes the proof:

4.1.2 Mechanisms with restricted probability of sale.

We describe optimal single item mechanisms with restricted probability of sale. The lemmas established in this section are used in Section 7. In particular, we formally show that a naive generalization of Myerson’s optimal auction for the unrestricted case is also optimal for the restricted case.

Fix a number and i.i.d. buyers, each buyer with value drawn from a single dimensional distribution . A mechanism with allocation function that satisfies is said to sell with ex-ante probability of at most .

Recall that by Myerson’s theory, a mechanism with allocation rule and payment is truthful if and only if each buyer ’s allocation rule is monotone, and buyer ’s payment function is fully determined by via the equality . Furthermore, Myerson defines virtual value functions111111When has a density , its virtual value function is defined by . See e.g., [Elk07, CDW16] for the general case. for single dimensional distributions (we henceforth drop the superscript ). The virtual value function is particularly useful because one can re-amortize the expected payment to a term that uses the virtual value functions, and this term can be optimized point wise. This becomes apparent in Myerson’s payment identity:

Lemma 4.13.

[Mye81] (Payment identity) In every truthful mechanism with allocation function and payment function , for every buyer it holds that

A regular distribution is a distribution whose corresponding virtual value function is monotone. For irregular distributions, Myerson defines yet another transformation that transforms the virtual value function to an ironed virtual value function (the details are not so important for our application; for more details see e.g., [Mye81, Har13]). The key point is that ironed virtual value functions are always monotone, and furthermore, maintain the following property:

Theorem 4.14.

([Har13, Theorem 3.12]) For every truthful mechanism with allocation function , for every buyer , with equality if is constant on ironed intervals.

In order to construct the optimal mechanism with restricted probability of sale, we will need the following definitions. Let , and let and

Note that if , then it must be that or that . 121212Otherwise but for every we have . Otherwise, it must be that , 131313Otherwise but for every we have . but , i.e,

Consider the mechanism that sells a single item to buyers as follows. If , the mechanism allocates the item to a random buyer among those with whenever , and otherwise does not allocate.

If , then the mechanism allocates the item to a random buyer among those with whenever , and whenever the mechanism draws a Bernoulli random variable that equals w.p. , and if the variable equals the mechanism allocates to a random buyer among those with .

Lemma 4.15.

For i.i.d. buyers drawn from a single dimensional distribution , and as above, the mechanism described above sells with ex-ante probability of at most , and extracts expected revenue of:

Moreover, every buyer contributes the same amount of expected revenue.

Proof.

By definition of the mechanism , all the allocation rules depend on values of , which implies that they are constant on ironed intervals, and since buyers are drawn from the same distribution, every buyer contributes the same amount of expected ironed virtual surplus which, by Theorem 4.14 and Myerson’s payment identity, 141414For more details, see e.g. [Har13, Equation (3.3)]. is also the contribution to the mechanism’s expected revenue.

Whenever , we saw that or . The mechanism’s ironed virtual surplus in this case is exactly

If , the statement holds, and if , then

as required.

Whenever , the mechanism’s ironed virtual surplus is exactly

As required. ∎

We are now ready to prove the optimality of among mechanisms with restricted probability of sale.

Lemma 4.16.

For as above, the mechanism above is a revenue maximizing mechanism among all truthful mechanisms that sell to i.i.d. buyers drawn from with ex-ante probability of at most , i.e.,

Proof.

Let for all . Note that since the buyers are all drawn from , they all have the same virtual function and ironed virtual function.

Fix a truthful mechanism that sells w.p. at most . Let be its allocation function for buyer . By Myerson’s theory, is truthful if and only if is monotone w.r.t. for every buyer . Clearly, if for some , setting whenever does not increase the ex-ante probability of sale, and does not decrease the expected virtual surplus, and maintains monotonicity of . Therefore, we may consider w.l.o.g. only mechanisms with for all , i.e., mechanisms that count only non-negative virtual surplus.

Where the first equality is Myerson’s payment identity, and the first inequality holds by Theorem 4.14. Let , i.e., the total probability of sale of mechanism . We know that . Recall that by definition of , it holds that . Observe that

because the former counts non-negative values of that accumulate to at most mass, while the latter counts all the highest values of , that accumulate to mass. The proof follows by Lemma 4.15. ∎

Corollary 4.17.

Let . Then

Proof.

Let be the mechanism considered in Lemma 4.15 for buyers drawn i.i.d. from . Let be the expected payment of a buyer in . Fix some , and consider the mechanism that sells a single item to an arbitrary set of buyers by sampling the remaining buyers i.i.d. from , and by executing with the buyers’ bids and the samples. Truthfulness of follows by truthfulness of . The revenue from is , and its ex-ante probability of sale is at most . Therefore,

Therefore by rearranging we get that

As required. ∎

Lemma 4.18.

is non-decreasing in .

Proof.

Fix some , and consider the mechanism for buyers that selects a set of buyers, and executes the mechanism considered in Lemma 4.15 for buyers drawn i.i.d. from , only on these buyers, and completely ignores the rest. By independence across buyers, extracts exactly the same revenue as , and sells with the same probability, i.e., Therefore, as required. ∎

Lemma 4.19.

Fix a cutoff . Let . Let denote the distribution of for every buyer . Then for every and .

Proof.

Consider the following buyers mechanism . The mechanism first receives bids, and in the first step removes all bids that are at most , and remains with some buyers. In the second step executes the mechanism from Lemma 4.15 for buyers drawn i.i.d. from . For every buyer, the participation rule (value more than ) is monotone, and every mechanism from Lemma 4.15 has a monotone allocation rule, therefore has a monotone allocation rule, and thus is truthful. Also, in the second step always executes a mechanism that sells with ex-ante probability at most , hence also sells with ex-ante probability at most .

Let . Denote by the revenue of where every is drawn from , conditioned on the event . Conditioned on , the revenue of the mechanism is by definition exactly that of from Lemma 4.15 for buyers drawn i.i.d. from , which, by Lemma 4.16 equals . Therefore by law of total expectation,

Where the inequality follows by monotonicity w.r.t. the number of buyers (Lemma 4.18). Finally, observe that

The last inequality follows by that sells with probability at most . This completes the proof. ∎

4.2 Partition of the support.

Recall that for every buyer and item , , and that . Every item is assigned a cutoff . If the value of an item is greater than its cutoff (i.e., ) then it is said to be in the tail w.r.t. , and otherwise it is in the core w.r.t .

For a set of items , let be the distribution conditioned on being exactly the set of items in the tail. Let be the probability of being exactly the set of items in the tail, i.e., . Let (resp., ) be the distribution restricted to just items in the core (resp., tail).

For every buyer consider some , and let . Let represent the event that for every it holds that is exactly the set of items that are in the tail with respect to buyer ’s valuation, i.e., for every item it holds that , and for every