The MenuSize Complexity of Revenue Approximation
Abstract
We consider a monopolist that is selling items to a single additive buyer, where the buyer’s values for the items are drawn according to independent distributions that possibly have unbounded support. It is well known that — unlike in the single item case — the revenueoptimal auction (a pricing scheme) may be complex, sometimes requiring a continuum of menu entries. It is also known that simple auctions with a finite bounded number of menu entries can extract a constant fraction of the optimal revenue. Nonetheless, the question of the possibility of extracting an arbitrarily high fraction of the optimal revenue via a finite menu size remained open.
In this paper, we give an affirmative answer to this open question, showing that for every and for every , there exists a complexity bound such that auctions of menu size at most suffice for obtaining a fraction of the optimal revenue from any . We prove upper and lower bounds on the revenue approximation complexity , as well as on the deterministic communication complexity required to run an auction that achieves such an approximation.
1 Introduction
As familiar economic institutions move to computerized platforms, they are reaching unprecedented sizes and levels of complexity. These new levels of complexity often become the defining feature of the computerized economic scenario, as in the cases of spectrum auctions and ad auctions. The use of the word “complexity” here is intentionally vague, and can refer to a wide variety of computational, informational, or descriptive measures of complexity. A highlevel goal of the field of “Economics and Computation” is to analyze such measures of complexity and understand the degree to which they are indeed a bottleneck to achieving desired economic properties.
This paper studies exactly such a question in the recently wellstudied scenario of pricing multiple items. The scenario is that of a monopolist seller who is selling items to a single additive buyer. The buyer has a private value for each item , where each is distributed according to a commonly known prior distribution , independently of the values of the other items. The valuation of the buyer is assumed to be additive, so that her value for a subset of the items is simply , and the seller’s goal is to design an “auction” (really just a a pricing scheme) that maximizes her revenue. The classical economic analysis (Myerson, 1981) shows that for a single item, the optimal auction is simply to sell the item at some fixed price. On the other hand, when there is more than a single item, it is known that the optimal auction may be surprisingly complex, randomized, and unintuitive (McAfee and McMillan, 1988; Thanassoulis, 2004; Manelli and Vincent, 2006; Giannakopoulos and Koutsoupias, 2014, 2015; Hart and Reny, 2015; Daskalakis et al., 2013).
A significant amount of recent work has studied whether “simple” auctions may yield at least an approximately optimal revenue. Following a sequence of results (Chawla et al., 2010; Hart and Nisan, 2012; Li and Yao, 2013), it was shown by Babaioff et al. (2014) that one of the following two “simple” auctions always yields at least a constant fraction () of the optimal revenue: either sell all items as a single takeitorleaveit bundle (for some carefully chosen price) or sell each item separately for its Myerson price. This was further extended (with different constants) to the case of multiple buyers (Yao, 2015) and to buyers with subadditive valuations (Rubinstein and Weinberg, 2015), but is in contrast to the case where the item values are distributed according to a joint (correlated) distribution, a case for which no finite approximation is possible by finite auctions (Briest et al., 2010; Hart and Nisan, 2013).
In this work, we study the tradeoff between the complexity of an auction and the extent to which it can approximate the optimal revenue. One may choose various measures of auction complexity (Hart and Nisan, 2013; Dughmi et al., 2014; Morgenstern and Roughgarden, 2015), and we will focus on the simplest one, the menu size suggested in Hart and Nisan (2013). The menu size of an auction (for a single buyer, an auction is just a pricing scheme) is defined to be the number of different possible outcomes of the auction. More specifically, every singlebuyer auction is equivalent to one that offers a menu of options to the buyer, where each option — entry — in the menu specifies a probability of acquiring each item as well as a price to be paid for the combination , and the buyer chooses an entry that maximizes her own expected utility . The number of entries in the menu is defined to be the menusize complexity of the auction. The logarithm of the menu size is exactly equal to the deterministic communication complexity of the auction: the auction is considered common knowledge, and the buyer, who knows her private values, must send enough information (see Appendix A for a formal definition) to the seller so that the outcome (allocation probabilities and price) of the auction for these values can be determined.^{1}^{1}1Since the seller has no private information in our setting, an arbitrary interactive protocol between the buyer and the seller must use at least as many bits as this oneway communication. Thus, this notion captures general twoway deterministic complexity as well. See Appendix A for more details, as well as a discussion regarding randomized communication complexity. It is known that for some distributions, the optimal auction has infinite menu size (Daskalakis et al., 2013) but a constant fraction of the optimal revenue may be extracted by a finitecomplexity auction (Babaioff et al., 2014). Is it possible to extract an arbitrarily high fraction of the optimal revenue via a finite menu size?
Our first and main result shows that, in fact, finite complexity suffices to get arbitrarily close to the optimal revenue.
Definition 1.1 (; ).
For a distribution on items, we denote by the maximal (formally, the supremum) revenue obtainable by an individually rational incentivecompatible auction that has at most menu entries and sells the items to a single additive buyer whose values for the items are distributed according to . We denote by the maximal revenue obtainable without any complexity restrictions on the auction.
Formally, our result shows that uniformly across all product distributions . In other words:
Theorem 1.1 (Qualitative Version).
For every number of items and every , there exists a finite menu size such that for every , we have that .
Theorem 1.1 gives a positive answer to Open Problem 6 from Hart and Nisan (2014),^{2}^{2}2Hart and Nisan (2014) is a manuscript combining Hart and Nisan (2012) and Hart and Nisan (2013). which asks precisely whether the statement of Theorem 1.1 holds. It is natural to ask what is the rate of the uniform convergence of the sequence . In other words, how complex must a revenueapproximating auction be?
Definition 1.2 (Revenue Approximation Complexity).
For every number of items and every , we define the revenue approximation complexity to be the smallest value such that for every .
The construction used in the proof of Theorem 1.1 gives an upper bound on (i.e., a lower bound on the rate of uniform convergence of ).
Theorem 1.1 (Quantitative Version^{3}^{3}3A more careful analysis can in fact show that . See Section 2.7 for more details.).
.
Corollary 1.2.
For every number of items , every , and every , there exists an item auction with a deterministic communication complexity of only that approximates the optimal revenue from up to a multiplicative loss.
This bound on the menu size is exponential in , and so the next natural question is whether polynomial menu size suffices. At first glance the answer seems to be “obviously not”: the menusize complexity measure is quite weak, and even the auction that sells each item separately has exponential menu size (since, when presented as a menu, a menu entry is needed for each possible subset of the items). This answer, however, is premature; in fact, we show that polynomial menu size turns out to suffice for approximating the revenue obtainable from selling items separately. Let us denote by the revenue obtainable by selling each item separately for its optimal price.
Theorem 1.3.
For every , there exists such that for every number of items and , we have for that .
The same bound applies also to the revenue obtainable by selling the items after arbitrarily prepartitioning them into bundles. Using the result of Babaioff et al. (2014), this immediately implies that polynomial menu size suffices for extracting a constant fraction of the optimal revenue.
Corollary 1.4.
There exist a fixed constant number and a fixed constant fraction such that .
The above reasoning shows that Corollary 1.4 holds for every arbitrarily close to , which is the constant fraction of the optimal revenue shown by Babaioff et al. (2014) to be obtainable by the better of bundled selling and separate selling.
Does polynomial menu size suffice for extracting revenue arbitrarily close to the optimal revenue? We prove that this is not the case, at least for that is polynomially small in .
Theorem 1.5.
.
The proof of Theorem 1.5 shows, in fact, that polynomial dependence on is impossible even for approximating the revenue from selling the items separately.
At this point, we leave two main problems open. The first one is whether for every fixed , polynomial (or at least quasipolynomial) menu size suffices for approximating the optimal revenue to within a multiplicative . In terms of communication complexity, this translates to whether logarithmic or polylogarithmic deterministic communication suffices^{4}^{4}4The authors have opposing conjectures regarding the answer to this open problem. for every fixed value of .
Open Problem 1.6.
Is it true that for every , there exists such that ? How about ?
The second open problem (or rather, class of open problems) is whether stronger notions of auction description complexity may allow for better revenue in polynomial complexity. This may be asked with respect to any complexity measure, and it is not clear which specific natural choice to consider, so an identification of such a measure is part of what is left open.^{5}^{5}5One may be tempted to consider the additive menu size complexity defined by Hart and Nisan (2013), which allows the seller to present menus from which the buyer is allowed to take any combination of menu entries for the sum of their prices. However, this notion is not well defined for lottery pricing except when we disallow overlaps in the specification of menu entries, a restriction that brings us back to a prepartitioning of the items, a lower bound for which is already implied by Theorem 1.3. There are several possible more general definitions, but we have not found a truly satisfactory one.
2 Upper Bound on
RevenueApproximation Complexity
In this Section, we prove our main result, Theorem 1.1, which states that is finite for every number of items and , and moreover, that . The proof proceeds in four steps. Section 2.1 provides a rough overview of the proof strategy, Sections 2.6, 2.5, 2.4 and 2.3 provide the formal details of each of the four steps of the proof, and Section 2.7 connects the dots by combining the four steps. Finally, Section 2.8 concludes with a short discussion of the application of the proof steps to obtain uniform approximation results for correlated distributions over a restricted valuation space, which generalize bounded distributions.
2.1 Proof Overview
Let be the respective distributions of the values of the items. We will construct an auction with (finite) menu size that guarantees a multiplicative approximation to the optimal revenue.
Limitations of Existing Techniques
One possible approach to approximate revenue maximization, taken by Li and Yao (2013), Babaioff et al. (2014), and Rubinstein and Weinberg (2015), is to use a core/tail decomposition and bound the revenue from (or the welfare of) the core and the revenue from the tail. Unfortunately, such a decomposition inherently entails a nonnegligible revenue loss (it only guarantees a constant fraction of the optimal revenue), either due to bounding the welfare of the core instead of the revenue from it, or due to estimating the total revenue using the revenues obtained by selling to the core and to the tail separately. Therefore, while this approach makes no assumptions regarding the valuation space (beyond independence), this technique, as used in the literature so far, is unsuitable for guaranteeing negligible loss in revenue as in the result that we seek.
Another possible approach, taken by Daskalakis and Weinberg (2012), Hart and Nisan (2013), and Dughmi et al. (2014), is to round all possible menu entries onto a discrete grid via “nudge and round” operations. Unfortunately, for the grid (and thus the menu size) to be finite and for the revenue loss to indeed be negligible, the above papers all require that the valuation space be bounded. Therefore, while this approach can guarantee negligible loss in revenue, this technique, as used in the literature so far, is unsuitable for the analysis of unbounded valuation spaces as in as in our setting.
To overcome the abovedescribed limitations of the core/tail decomposition technique, we take a more subtle approach, by analyzing core and tail regions together. We first show (in Step 1 below) that one does not lose much revenue by disregarding what can be described as “secondorder” tails, i.e., valuations where two or more of the item prices lie in the tail. Then, we show how to gradually simplify an optimal auction (which may be arbitrarily complex, even infinite in size) for the valuation space consisting of the core plus all firstorder tails while losing only a tiny fraction of the revenue in each step. For every modification that we perform to the menu, we must “simultaneously” check that we do not significantly hurt the revenue from either core or (firstorder) tail buyers. To gradually simplify the menu, we first carefully modify the menu so that only a small number of menu entries have a high price (this is the most technically elaborate part of the proof, performed in Steps 2 and 3 below), and then (in Step 4 below) round the menu entries with low prices to a finite grid using “nudge and round” operations. At this point, the use of “nudge and round” onto a finite grid is possible without significant revenue loss since the price of the menu entries that we round is bounded. Nonetheless, care still has to be taken beyond previous “nudge and round” uses, to ensure that this rounding does not incentivize buyers in the (firstorder) tails to switch to buying a lowerpriced rounded entry. Before moving on to the definitions and formal statements and proof, we first give a somewhat more detailed, yet still highlevel, overview of each of the four steps of the proof.
Step 1: Move to an “almost bounded” valuation space
This step, taken in Section 2.3, simplifies the valuation space by showing that since item prices are independent, finding an approximately optimal auction under the assumption that at most one item has a price higher than (i.e., has a price that lies in the tail), for some , entails a very small loss compared to doing so without this assumption. This is possible, very roughly speaking, because the probability of two item prices lying in the tail, for as above, can be thought of as being of order , while the revenue conditioned upon being in this “secondorder” tail (i.e., conditioned upon the prices of both of these items lying in the tail) is of order . Therefore, it is enough to construct our finite approximation for the distribution conditioned upon being in the valuation space comprised of the core and the firstorder tail, i.e., the valuation space where at most one item price lies in the tail. We call distributions over this valuation space exclusively unbounded distributions. We note that this is the only step in which the independence of the item prices is used; indeed, combining the remaining steps shows that the revenue from all exclusive unbounded distributions (even highly correlated distributions not originating from a product distribution over ) can be uniformly approximated using finitesize menus (see Proposition 2.6 in Section 2.8).
Step 2: Modify expensive menu items to behave “almost like” singleitem auctions
This step, taken in Section 2.4, starts with an optimal (possibly arbitrarily complex) revenuemaximizing auction for some exclusively unbounded distribution. In this step, we simplify the “expensive” part of the menu, i.e., the part of the menu consisting of all menu entries that cost more than , for some , so that each expensive menu entry allocates only a single item with nonzero probability. This means that while in the “cheap” part of the menu we can allocate arbitrary combinations of items, once the price increases beyond , our auction must act like a unitdemand one and never allocate more than a single item. We call such an auction exclusive. This is possible since, roughly speaking, due to the assumption of exclusive unboundedness, most of the value from an expensive menu entry chosen by some buyer type comes only from the unique item whose price lies in the tail for the valuation of that buyer type. Thus, instead of offering that (nonexclusive) menu entry, we offer an (exclusive) entry with only the corresponding winning probability of that item, for a slightly discounted price. While in most natural cases, this step in fact increases the size of the expensive part of the menu (as each expensive menu entry possibly becomes exclusive menu entries, each allocating a distinct item with nonzero probability), this simplification allows the next step to significantly reduce the size of this part of the menu.
Step 3: Apply Myerson’s result to obtain “almost one” expensive entry per item
This step, taken in Section 2.5, reduces the size of the expensive part of the menu to at most menu entries. This is the most technically elaborate step. Since exclusivity means that the expensive auction entries “look like” separate auctions for each of the items, we show that we are able to carefully use the analysis of Myerson (1981) to replace each of these separate expensive auctions with a simple “almost deterministic” one. In contrast to Myerson’s single (nonzero) menu entry, we require two menu entries for each item: a deterministic one analogous to Myerson’s “optimal price” entry, and an additional randomized one analogous to the “opt out” zero entry in Myerson’s auction. The function of the latter entry is to make sure that buyers are not incentivized to “jump” from the expensive part to the cheap part of the menu following the reduction of the size of the former.
Step 4: Discretize cheap menu entries “almost to a grid”
This final step, taken in Section 2.6, simplifies the cheap part of the menu by “rounding” the menu entries into a discrete set. We note that even at this point in the proof, the “nudge and round” techniques that allowed this rounding to be done with only negligible loss of revenue for bounded valuations in previous papers (Daskalakis and Weinberg, 2012; Hart and Nisan, 2013; Dughmi et al., 2014) cannot just be used “out of the box” in this step. Indeed, slight changes in allocation probabilities may result in large revenue changes, since the valuation space is not bounded but only exclusively unbounded. Nonetheless, these techniques can be carefully extended to be used here as well. Roughly speaking, we construct discretizations of each cheap menu entry, where each discretization rounds the price and all but one allocation; rounding in the right direction guarantees that at least one of these discretizations is still a leading candidate for any buyer type that previously chose the corresponding original (nondiscretized) menu entry. As all but one coordinate of each of the discretized menu entries lie on a grid, we show that only finitely many of the menu entries are in fact chosen by any buyer type.
While the second and third (and first) steps each entail a slight multiplicative revenue drop, the fourth step entails also a slight additive revenue drop. Recall, however, that we aim to achieve only a slight multiplicative drop (with no additional additive drop) in overall revenue. To obtain this result, when combining all of the above steps in Section 2.7 we assume w.l.o.g. that is normalized^{6}^{6}6The cases in which cannot be normalized, i.e., when it is or infinite, are easy to handle separately. In the former case, there is nothing to show. In the latter case, for some , and so by the theorem of Myerson (1981), an arbitrarily high revenue can be extracted using a takeitorleaveit offer for item . (by scaling the currency) to a suitable value such that the additive drop in the fourth step can be quantified to be less than a slight multiplicative drop. Clearly, as the obtained bound on the overall cumulative revenue drop for normalized auctions is purely multiplicative, the proof also implies the same multiplicative bound for all (even nonnormalized) distributions.
2.2 Preliminaries
Definition 2.1 (Notation).

(Naturals). We denote the strictly positive natural numbers by .

([n]). For every , we define .

(Nonnegative Reals). We denote the nonnegative reals by .
Definition 2.2 (Outcome; Type; Utility).
Let be a number of items.

An outcome is an tuple , denoting an allocation (to the buyer) of every item with probability , for a total price (paid by the buyer) of .

We denote the (expected) utility of a (riskneutral additive) buyer with type (respective item valuations) from an outcome by
Definition 2.3 (IC Auction as Menu).
Let be a number of items. By the taxation principle, we identify any incentivecompatible (IC) item auction with a (possibly infinitely large) menu of outcomes (the entries in the menu are all the possible outcomes of the auction), where by IC the buyer chooses an entry that maximizes her utility.^{7}^{7}7If the menu is infinite, then the fact that it corresponds to an IC auction guarantees that some menu entry maximizes the utility of each buyer type. See Section B.1.1 for more details. If the auction is individually rational (IR), then we assume w.l.o.g. that the menu includes the entry that allocates no item and costs nothing. (Conversely, if the menu includes the entry , then the auction is IR.) Following Hart and Nisan (2013), we define the menu size of an IC and IR auction as the number of entries, except , in the menu of that auction.
Definition 2.4 (; ; ).
Let be a number of items and let be a distribution over .

Given an IC and IR item auction , we denote the (expected) revenue obtainable by from (a single riskneutral additive buyer with type distributed according to) , by
where is the price of the entry from that maximizes the utility of , with ties broken in favor of higher prices.^{8}^{8}8The results of this paper hold regardless of the tiebreaking rule chosen. See Section B.2 for more details.^{,}^{9}^{9}9If the menu is infinite, then the fact that a utilitymaximizing menu entry exists for every buyer type does not guarantee that a utilitymaximizing entry with maximal price (among all utilitymaximizing entries) exists for every buyer type. (I.e., it is not guaranteed that the supremum price over all utilitymaximizing entries is attained as a maximum.) Indeed, to be completely general, a more subtle definition of the revenue obtainable by an IC auction would have been needed. Nonetheless, for the auctions considered in this paper, this subtle definition is not required as we make sure that they all possess, for each buyer type, a utilitymaximizing entry with maximal price. See Section B.1.2 for more details.

Given , we denote the highest revenue (more accurately, the supremum of the revenues) obtainable from by an IC and IR item auction with at most menu entries by

We denote the highest revenue (more accurately, the supremum of the revenues) obtainable from by an IC and IR item auction by
Theorem 2.1 (Hart and Nisan, 2012).
, for every , , and .
2.3 At Most One High Price
As outlined above, our first step toward proving Theorem 1.1, which we take in this Section, simplifies the valuation space by showing that since item prices are independent, any auction that extracts most of the revenue under the assumption that the valuation space is restricted to some exclusively unbounded valuation space, i.e., to a valuation space where for each buyer type at most one item has price higher than some , also extracts most of the revenue without this assumption. This step is formalized by Lemma 2.2.
Definition 2.5 (; Exclusively Unbounded Type Distribution).
Let and .

We denote the subset of where at most one coordinate is greater than by

We say that a type distribution is ()exclusively unbounded if .
Definition 2.6 ().
For a set and a distribution defined over some superset of s.t. is measurable and , we denote the conditional distribution of conditioned upon by . Formally, for every measurable set , we define .
Lemma 2.2.
Let s.t. , let , let , and let . For every s.t. , all of the following hold.

(hence the exclusively unbounded conditioned distribution is well defined).

.

For every and for every IC and IR item auction , if , then .
Proof.
For every , we denote the probability of being greater than by . We first note that
(1) 
for every . Indeed, the revenue from of the auction selling (item ) for a takeitorleaveit price of is at least , and by definition of we therefore have and so , as claimed. In particular, since , we note that this implies that and so, since , we obtain that , proving Item 1. (Thus, is well defined.) For the proof of Items 3 and 2, we will need the following Sublemma.
Sublemma 2.2.1.
Let and let .

for every^{10}^{10}10If for some , then even though is not defined, we henceforth define to equal . .

For every and every s.t. , we have that .
The proof of Sublemma 2.2.1 is given after the proof of Lemma 2.2. We now proceed to prove Items 3 and 2 of Lemma 2.2. We note that for every , if , then
(2) 
Indeed, by Sublemma 2.2.1(1), we have that , and so , as required.
For every , we let (the “doubletail” w.r.t. and ) and . We claim that
(3) 
for every . Since the claim trivially holds when , we need only prove it when and . In this case, by Theorem 2.1 (applied twice) and by Eqs. 1 and 2, we have
as claimed.
We define . As , by Sublemma 2.2.1(2) we have that . Therefore, by Eq. 3, we have that
(4) 
In particular, , proving Item 2.
Proof of Sublemma 2.2.1.
For Item 1, if then there is nothing to prove, so we assume henceforth that . Therefore, also and thus and are well defined. We begin by noting that . Indeed, this inequality holds since for any auction (in particular, any auction obtaining close to optimal revenue from ), we have ; by definition of , the inequality follows. Therefore, we have that , as required.
For Item 2, we start by defining . By definition, is a partition of . We first claim that . Indeed, for any auction (in particular, any auction obtaining close to optimal revenue from ), we have that ;^{11}^{11}11Similarly, if for some , then we define to equal . by definition of , the inequality follows. By Item 1, we have that for every . Combining both of these, we obtain that , as required. ∎
2.4 Exclusivity at Expensive Menu Entries
Having proven Lemma 2.2, we phrase and prove the next steps for arbitrary exclusively unbounded distributions, i.e., not necessarily product distributions conditioned upon . As outlined above, our second step toward proving Theorem 1.1, which we take in this Section, shows that in any auction over some exclusively unbounded distribution, the “expensive” part of the menu, i.e., the part of the menu consisting of all menu entries that cost more than some , can be simplified without significant loss in revenue to make the auction exclusive, i.e., to make each expensive menu entry only allocate a single item with nonzero probability. This step is formalized by Lemma 2.3.
Definition 2.7 (Exclusive Auction).
Let and let . We say that an item auction is exclusive if it allocates (with positive probability) at most one item whenever it charges strictly more than .^{12}^{12}12While this also implies that the allocated item is quite expensive, and therefore some may say “exclusive,” the exclusivity discussed in the definition is that of solely this specific item being sold.
Lemma 2.3.
Let s.t. , let , let , and set . For every and for every IC and IR item auction , there exists an exclusive IC and IR item auction such that .
Proof.
Set . We construct a new IC and IR auction as follows:

For every menu entry with , we add the menu entry , unmodified, to .

For every menu entry with , we add the following^{13}^{13}13Following standard notation, we use , for , , , and , to denote the outcome , i.e., an outcome that is identical to in price and all winning probabilities, except the winning probability of item , which is set to . menu entries to : , ,,. Each of these menu entries is a modified version of that completely “unallocates” all but one of the items, while giving a slight multiplicative price discount of .
Finally, we define to be the closure of the set of menu entries added above to .^{14}^{14}14Taking the closure ensures that a utilitymaximizing entry with maximal price exists for every buyer type. See Section B.1 for more details.
By definition, is exclusive. We note that since is IR, it contains the menu entry . Therefore, also contains this menu entry, and hence is IR as well. It remains to show that obtains a revenue of at least from . Let us compare the payments that and extract from a buyer of each type . We reason by cases according to the menu entry of choice^{15}^{15}15If more than one utilitymaximizing menu entry with maximal price exists, then here and whenever we henceforth refer to the “menu entry of choice” of some buyer type, we choose one such entry arbitrarily. of buyer type from , which we denote by , and show that in either case, the payment extracted from a buyer of type decreases by at most a multiplicative factor of in compared to .

If , then by definition . We claim that weakly prefers to all menu entries with^{16}^{16}16As we show weak preference and as is defined via a strict inequality, by continuity of the utility function the correctness of the claim for all before taking the closure of implies its correctness for all in the closure as well. . Indeed, for every such menu entry , we have that , and so by definition we have that , and so by definition of we have that weakly prefers to . Therefore, the price of the menu entry chosen by from is at least , and so the payment extracted from a buyer of type decreases by at most a multiplicative factor of in compared to .

Otherwise, i.e., if , then since (since and ), by IR there must exist s.t. . Since , we have that for every . Let be the menu entry in corresponding to that unallocates all items except item . We claim that weakly prefers to all menu entries with^{17}^{17}17See Footnote 16. . Let be such a menu entry and denote the menu entry corresponding to in by (where either or for some ). Noting that , we indeed get
Therefore, the payment that extracts from a buyer of type is at least
and so the payment extracted from a buyer of type decreases by at most a multiplicative factor of in compared to .
To summarize, the revenue from each buyer type decreases by at most a multiplicative factor of in compared to , and so the (overall) revenue that obtains from is at least a fraction of the revenue that obtains from , as required. ∎
2.5 Trimming the Expensive Part of the Menu
As outlined above, our third step toward proving Theorem 1.1, which we take in this Section, shows that in any exclusive auction over some exclusively unbounded distribution, the expensive part of the menu can be simplified without significant loss in revenue, so that it contains at most menu entries. This step, which is the most technically elaborate of all steps, is formalized by Lemma 2.4.
Lemma 2.4.
Let s.t. , let , let , and let . For every and for every exclusive IC and IR item auction , there exists an exclusive IC and IR item auction , such that both of the following hold.

.

(The set of menu entries in) coincides with a subset of the set of menu entries in that cost at most , with the addition of at most menu entries.
Proof.
Throughout the proof, we assume w.l.o.g. that each menu entry in (with the possible exception of ) is chosen by at least one buyer type .^{18}^{18}18While is not necessarily closed after the removal of all menu entries that are not chosen by any buyer type, it does possess a utilitymaximizing entry with maximal price for every buyer type. See Section B.1 for a discussion. For every , we define
and
As is exclusive, we have that for every and . Thus, is the set of buyer types that choose to pay more than , and receive in return a positive probability for winning item and zero probability for winning any other item. Our goal is to apply the singledimensional analysis of Myerson (1981) in order to replace the plethora of menu entries in each with a small constant number of menu entries. Until noted otherwise, fix s.t. .
We define and , and set . One may intuitively think of (charging for an probability of winning item ) as the cheapest entry, which also allocates the least probability, in , although formally (since is not closed, and also due to the way in which is defined) need not necessarily be in .^{19}^{19}19Nonetheless, due to the assumption that every menu entry is chosen by some buyer type, we are able to show that is in the closure of ; see Sublemma 2.4.4(4) below. This property is heavily used throughout our proof. Our strategy is to show that, in a precise sense, behaves on as follows: first allocate the buyer (i.e., provide a “starting winning probability” of for item , and charge a “base price” of ), and then hold a “continuation auction” for possibly allocating some or all of the remaining probability of winning item . In this “continuation auction,” we allow the buyer to swap for a different entry from , paying the difference in costs and increasing the probability of getting item accordingly. We now make this statement precise. Until noted otherwise, fix s.t. (and so, by definition, ) and . As we will see below, we require that in order for this “continuation auction” to be well defined (and in fact make sense; otherwise there is no “remaining probability” to sell) and that for the valuation distribution of that auction to be well defined. As we will show, if then (see Sublemma 2.4.4(3) below), and so there will be no need to reduce the number of menu entries in in this case (and if , then the revenue of the original auction from is zero, and so we will be able to simply delete from the original auction, without replacing it with anything).
For every dimensional buyer type , we define the singledimensional valuation for the “remaining probability” of winning item by
We define the corresponding singledimensional buyer type space of our “continuation auction” as
and define a distribution over it by
for every measurable set . (Recall that ; therefore, is well defined.) Very roughly speaking, is defined such that its density at every can be informally thought of as the sum of the densities of at all s.t. .
Having defined the singledimensional buyer type space of the “continuation auction,” we now turn to defining the (menu of the) auction itself. For an outcome (not necessarily a menu entry in ) of the form , we define
i.e., an entry selling a fraction of the remaining probability so that (in addition to the starting winning probability ) the overall winning probability is