On Inﬁnite Separations Between Simple and Optimal Mechanisms
Abstract
We consider a revenue-maximizing seller with k heterogeneous items for sale to a single additive buyer, whose values are drawn from a known, possibly correlated prior D .
It is known that there exist priors D such that simple mechanisms — those with bounded menu complexity — extract an arbitrarily small fraction of the optimal revenue ([BCKW15, HN19]).
This paper considers the opposite direction: given a correlated distribution D witnessing an inﬁnite separation between simple and optimal mechanisms, what can be said about D ?
[HN19] provides a framework for constructing such D : it takes as input a sequence of k -dimensional vectors satisfying some geometric property, and produces a D witnessing an inﬁnite gap.
Our ﬁrst main result establishes that this framework is without loss: every D witnessing an inﬁnite separation could have resulted from this framework.
An earlier version of their work provided a more streamlined framework [HN13].
Our second main result establishes that this restrictive framework is not tight.
That is, we provide an instance D witnessing an inﬁnite gap, but which provably could not have resulted from the restrictive framework.
As a corollary, we discover a new kind of mechanism which can witness these inﬁnite separations on instances where the previous “aligned” mechanisms do not.
Introduction
Consider a revenue-maximizing seller with k items for sale to a single additive buyer, whose values for the k items are drawn from a known distribution D .
When k = 1 , Myerson’s seminal work provides a closed-form solution to the revenue-optimal mechanism and it has a particularly simple form: simply post a price p :
= arg max p { p · Pr v D [ v  p ] } and let the buyer purchase the item if they please ([Mye81]).
For k > 1 , however, this multi-dimensional mechanism design problem remains an active research agenda forty years later
While simple, constant-factor approximations are known in quite general settings when D is a product distribution ([CHK07, CHMS10, CMS15, KW12, LY13, BILW14, Yao15, RW15, CDW16, CM16,
CZ17]), there may be an inﬁnite gap between the revenue-optimal auction and any simple counterpart when values are correlated ([BCKW15, HN19]).
More speciﬁcally: one simple way to sell k items is to treat the grand bundle of all items as if it were a single item, and sell it using Myerson’s optimal auction (which sets price arg max p { p · Pr ~ v D [ P i v i  p ] } ).
Letting BRev( D ) denote the revenue achieved by this simple scheme, [HN19] further show a connection between BRev( D ) and any simple mechanism through the lens of menu complexity: any mechanism with menu complexity at most C generates expected revenue at most C · BRev( D ) .
These works establish a strong separation between simple and optimal auctions: even when k = 2 , there exist distributions D such that Rev( D )
= 1 (the optimal revenue) while BRev( D ) = 1 .
The fact that Rev( D ) = 1 does not on its own suggest that D must be “weird” (the one- dimensional distribution with CDF 1 / p x has this property).
The “weird” property is that
Rev( D ) / BRev( D ) = 1 , which can never occur for a one-dimensional distribution.
[BCKW15] and [HN19] establish sufﬁcient conditions for a distribution D to satisfy
Rev( D ) / BRev( D ) = 1 .
Simply put, the goal of this paper is to study necessary conditions for a distribution to satisfy Rev( D ) /
BRev( D ) = 1 .
We provide two main results.
The ﬁrst establishes that the sufﬁcient condition presented in [HN19] is in fact necessary for Rev( D ) / BRev( D ) =(Theorem 6).
The second establishes that the sufﬁcient condition used in an earlier version of that work [HN13] and [BCKW15] is not necessary (Theorem 9).
In establishing Theorem 9, we also construct a distribution D such that Rev( D ) / BRev( D ) =
1 yet the mechanism witnessing this provably falls outside the scope of any previous constructions (Corollary 11).
A recent line of work has borrowed tools from machine learning (more speciﬁcally, recommender systems) to approximate buyers with complex, high-dimensional valuations via low-dimensional topic models [CD22].
Their work recognizes some settings where ﬁnding approximately optimal mechanisms for these buyers reduces to ﬁnding approximately optimal mechanisms for buyers whose valuations are approximated by the topic models.
In this context, our work helps clarify the limits where this modeling approach is effective: a model is tractable if and only if it steers clear of the [HN19] framework.
Beyond this speciﬁc direction, we note that there is a signiﬁcant community at the intersection of machine learning and multi-item auction design, and understanding the limits of tractability is important for this community.
Proper context and formal statements of our results require precise deﬁnitions, which we provide in
Section 2 immediately below.
Section 3 provides formal statements of our results, along with context alongside related work.
Subsequent sections provide proofs.
Preliminaries
We consider an auction design setting with a single buyer, single seller, and k heterogeneous items.
Note that our positive results hold for arbitrary k , while our constructions use only k = 2 (and k =is not possible).
We use D to denote a distribution over R k  0 , the (possibly correlated) distribution over the buyer’s values for the k items.
The buyer is additive, meaning that their value for a set of items S is equal to
P i 2 S v i .
We use Rev( D ) to denote the optimal expected revenue achievable by any incentive-compatible mechanism (formally, the supremum of expected revenues, or 1 if the supremum is undeﬁned), and let BRev( D ) denote the revenue achieved by selling the grand bundle as a single item using Myerson’s auction.
Finally, a mechanism M is a set { ( ~ q i , p i ) } i , where each ~ q i 2 [0 , 1] k denotes a vector of probabilities, and p i 2 R denotes a price.
When the buyer’s valuation is ~ v , they pay the auctioneer p i ( ~ v ) , where i ( ~ v ) :
= arg max i { ~ v · ~ q i  p i } .
5 Rev( D , M ) denotes the expected revenue achieved by a particular mechanism M on distribution D .
We will also use the shorthand ~ q M ( ~ v ) to denote the allocation vector purchased by a vector ~ v , and p M ( ~ v ) to denote the price paid (we may drop the superscript of M if the mechanism is clear from context).
Brief Overview of [HN19].
Below, we formally deﬁne two geometric properties of sequences of points, which are the focus of this paper.
Many of the ideas below appear in both [BCKW15] and [HN19], but we will use the formal deﬁnitions from [HN19] (and an earlier published ver- sion [HN13]).
Below, morally the vectors ~ x i correspond to possible (scaled) valuation vectors, and the vectors ~ q i correspond to possible vectors of allocation probabilities.
Deﬁnition 1.
Let X = ( ~ x i )
Ni
=1 be an ordered sequence of N points ( N may be ﬁnite, or equal to + 1 ), with each ~ x i 2 R k  0 .
Let Q = ( ~ q i )
Ni {+=
+} be another ordered sequence of N points, with each ~ q i 2 [0 , 1] k , and starting with ~ q 0 = (0 , ..., 0) .
Deﬁne the following:
[Equation]
We will also slightly abuse notation and deﬁne MenuGap( X ) :
= sup Q { MenuGap( X, Q ) } .
MenuGap( X ) is some measure of how distinct the angles of points in X are.
To get intuition for this, one might try to write a short proof that when k = 1 , MenuGap( X )
= 1 for all X (or that
MenuGap( X ) = [-
-]1 whenever all ~ x i 2 X are parallel).
We provide such a proof in Appendix A.
MenuGap( X, Q ) is just some geometric measure with no obvious intuition for why this quantity should be of interest to auction designers.
However, one key result of [HN19] shows that this quantity has connections to simplicity vs. optimality gaps.
Speciﬁcally, they show:
Theorem 2 ([HN19], Proposition 7.1) .
For every pair of sequences X = ( ~ x i ) Ni =1 , Q = ( ~ q i ) Ni
= starting with ~ q 0 = (0 , ..., 0) , and all " > 0 , there exists a distribution D and mechanism M s.t.:
[Equation]
Moreover, for all i 2 [ N ] , the support of D contains a single point of the form c i ~ x i , for some c
i 2 R + (and no other points).
Additionally, ~ q M ( c i ~ x i ) =
~ q i .
The “Moreover,...” portion of Theorem 2 gives some insight to their construction.
Further insight can be deduced by observing that the constraint “ gap X,Qi  ~ x i · ( ~ q i  ~ q j ) ” looks similar (but far from identical) to an incentive compatibility constraint involving a valuation vector ~ x i and two allocation vectors ~ q i , ~ q j .
We refer the reader to [HN19] for further details and intuition for this connection.
Theorem 2 gives a framework for proving simplicity vs. optimality gaps, but leaves open the question of actually ﬁnding a pair of sequences X, Q .
They approach this through the following observation:
Deﬁnition 3.
Given a sequence of points X = ( ~ x i ) Ni =0 2 [0 , 1] k starting with ~ x 0 = (0 , ..., 0) , deﬁne
SupGap( X ) (read “support gap of X ") as SupGap( X ) :=
MenuGap( X, X ) .
Observation 4.
For all X , MenuGap( X )  SupGap( X ) .
Finally, [HN13] propose an explicit construction of a sequence X with inﬁnite SupGap.
Theorem 5 ([HN13]) .
There exists an inﬁnite sequence X of points in [0 , 1] 2 s.t.
SupGap( X ) = 1 .
Theorems 2 and 5 together yield a two-dimensional distribution with Rev( D ) / BRev( D ) = 1 .
It is also worth noting that all prior constructions follow this approach as well.
For example, [BCKW15] provides an inﬁnite sequence X of points in [0 , 1] 3 such that SupGap( X ) = 1 .
Our Results
Independent of Observation 4 and Theorem 5, Theorem 2 alone provides a framework for constructing distributions D so that Rev( D ) / BRev( D ) is large: ﬁnd sequences X so that MenuGap( X ) is large.
Our goal is to understand to what extent this framework is complete for constructing such instances.
Our ﬁrst main result establishes that any distribution with Rev( D ) / BRev( D ) = 1 could have resulted from the framework induced by Theorem 2 .
Speciﬁcally:
Theorem 6.
For any distribution D over k items, there exists a sequence of N points ( N can be ﬁnite, or equal to + 1 ) X
= ( ~ x i )
Ni
=1 , with each ~ x i 2 supp ( D ) , such that
[Equation]
In particular, if Rev( D ) / BRev( D ) = 1 , then MenuGap( X ) = 1 as well.
A complete proof of Theorem 6 appears in Section 4.
Observe also that because MenuGap( X ) is monotone (in the sense that adding points to X , anywhere, cannot possibly decrease MenuGap( X ) ), the fact that X is a subset of the support of D (rather than the entire support) is immaterial.
7 Put another way, the important aspect in constructing X is how elements in the support of D are ordered, rather than which points are included.
Observation 4 further provides a framework to construct sequences so that MenuGap( X ) is large: construct sequences X so that SupGap( X ) is large.
One may then wonder if SupGap( X ) and
MenuGap( X ) are approximately related, for all X .
For this speciﬁc question, the answer is trivially no, due to incompatibility with scaling (multiplying every point in X by 1 / 2 will decrease
SupGap( X ) by a factor of 2 , but not MenuGap( X ) ).
Therefore, not much insight is gained by studying this precise question.
Instead, we observe that the interesting aspect of constructions resulting through SupGap( X ) is that ~ x i and ~ q i are aligned (that is, ~ x i =
c i · ~ q i for some c i 2 R  0 ).
Speciﬁcally, even if ~ q i = ~ x i , this equality is not maintained by the construction of Theorem 2.
However, if ~ q i and ~ x i are aligned, this alignment property is maintained by the construction.
We therefore propose the following deﬁnition, which captures the maximum value achievable by MenuGap( X, Q ) when X, Q are aligned.
Deﬁnition 7.
Let X = ( ~ x i )
Ni
=1 be an ordered sequence of N points in [0 , 1] k
( N may be ﬁnite, or equal to + 1 ).
Let also C =
( c i )
Ni =0 be an ordered sequence of numbers, with each c i 2 [0 , 1 / || ~ x i || 1 ] , starting with c 0 = 0 .
Deﬁne: sgap X,Ci :=
min j<i ~ x i · ( c i ~ x i  c j ~ x j ) , and
AlignGap( X, C ) := N
X i = max { 0 , sgap X,Ci } || ~ x i ||.
We will also slightly abuse notation and denote by AlignGap( X ) :
= sup C { AlignGap( X, C ) } .
Recall that we have chosen to let c i range in [0 , 1 / || ~ x i || 1 ] (rather than be ﬁxed at 1 , or 1 / || ~ x i || 1 ) to give potential constructions ﬂexibility in scaling ~ q i .
Additionally, by ensuring that the contribution of each sgap X,Ci is non-negative, we give potential constructions ﬂexibility to ignore points in the sequence.
That is, any construction using MenuGap directly can always set ~ q i := arg max j<i { ~ q j · ~ x i } , which effectively just drops ~ x i from the sequence.
Counting max { 0 , sgap X,Ci } towards the objective (rather than just sgap X,Ci ) gives constructions that arise through AlignGap the same ﬂexibility.
Lemma 8.
For all X , AlignGap( X )  MenuGap( X ) .
The proof of Lemma 8 is in Appendix B. Lemma 8 induces a framework to design sequences with large MenuGap: design sequences with large AlignGap.
Our second main result establishes that this framework is not without loss of generality, even for k = 2 .
Speciﬁcally:
Theorem 9.
There exist sequences X =
( ~ x i ) {+
+}
i =1 2 [0 , 1] 2 such that:
AlignGap( X ) but
MenuGap [Equation]
A complete proof of Theorem 9 appears in Section 5.
By the discussion following Deﬁnition 7, the source of this gap is entirely due to the requirement that the sequence Q be aligned with X (and is not due to inability to scale, or inability to ignore difﬁcult points in X ).
We make this crisp with the following deﬁnition and corollary, which construct a novel distribution witnessing
Rev( D ) / BRev( D ) = 1 that is provably distinct from all previous approaches.
Deﬁnition 10.
For a distribution D and mechanism M , deﬁne the Aligned Revenue of M on D :
[Equation]
is parallel to [Equation]
[Equation]
Corollary 11.
There exist distributions D over two items such that Rev( D ) /
ARev( D ) = 1 .
A proof of Corollary 11 appears in Appendix C. It is worth noting that all previous constructions establishing Rev( D ) / BRev( D ) = 1 proceeded by producing an X such that SupGap( X ) = 1 .
Indeed, the [BCKW15] construction provides such an X when k = 3 , the [HN13] construction provides an
X when k = 2 , and the [PSW19] construction adapts parameters in that of [HN13].
By Theorem 2, this implies not only that Rev( D ) / BRev( D ) = 1 , but also that ARev( D ) / BRev( D ) = 1 .
Corol- lary 11 establishes the existence of a fundamentally different construction, 8 as our D has a ﬁnite ratio between ARev( D ) / BRev( D ) , yet still maintains an inﬁnite ratio between Rev( D ) / BRev( D ) .
Additional Related Work.
We’ve already discussed the most related work to ours, which is that of [HN19, BCKW15].
There is also a large body of work studying product distributions speciﬁcally, and establishes that simple mechanisms can achieve constant factor approximations in quite general settings [CHK07, CHMS10, CMS15, KW12, LY13, BILW14, Yao15, RW15, CDW16,
CM16, CZ17]).
Recent works have made progress in obtaining arbitrary approximations ([BGN17,
KMS + 19]), which again rely on the assumption that D is a product distribution.
Three recent lines of work address the [BCKW15, HN19] constructions in a different manner.
First, [CTT19, CTT20] consider the related buy-many model (where the auctioneer cannot prevent the buyer from interacting multiple times with the auction).
[CTT19] establishes that selling separately achieves an O (log k ) -approximation to the optimal buy-many mechanism in quite general settings (including the settings considered in this paper).
In a different direction, [PSW19] uses the lens of smoothed analysis ([ST04]) to reason about the robustness of the [HN19] constructions.
Finally, [Car17] considers a correlation robust framework in which the valuation proﬁle is drawn from a correlated distribution that is not completely known to the seller; the goal is to design a mechanism that maximizes the worst-case (over correlations) seller revenue, when only the items’ marginal distributions are known.
[Car17] shows that selling each item separately is optimal; see [BGLT19,
GL18] for further work in this model.
A Converse to Theorem 2: The [HN19] Framework is WLOG
In this section, we prove Theorem 6.
Our proof has two main parts.
First, we will take the optimal auction for D (or one that is arbitrarily close to optimal) and repeatedly simplify it through a sequence of lemmas, at the cost of small constant-factors of revenue.
The second part takes this simple menu and draws a connection to MenuGap.
Missing proofs can be found in Appendix B.
Simplifying the Optimal Mechanism.
We show that for every D , there exists an approximatelyoptimal mechanism which satisﬁes some useful properties.
First, we argue that we may ignore menu options with low prices.
Deﬁnition 12.
We say that a mechanism M is c -expensive if every option has price at least c .
Claim 13.
For all c 2 R  0 , all distributions D , and all mechanisms M , there exists a c -expensive mechanism M 0 satisfying Rev( D , M 0 )
Rev( D , M )  c .
Our next step will show that we can assume further structure on the prices charged, at the cost of a factor of two.
Deﬁnition 14.
A c -expensive mechanism M is oddly-priced (respectively, evenly-priced ) if for all ~ v , there exists an odd (respectively, even) integer i such that p M ( ~ v ) 2 [ c · 2 i , c · 2 i +1 ) .
Claim 15.
For all c -expensive mechanisms M , and all D , there exists either an oddly-priced or evenly-priced c -expensive mechanism M 0 satisfying Rev( D , M 0 )
Rev( D , M ) / 2 .
This concludes our simpliﬁcation of the mechanism.
In the subsequent section, we draw a connection between MenuGap and the revenue of oddly- or evenly-price c -expensive mechanisms.
Connecting Structured Mechanisms to MenuGap.
We begin with the following deﬁnition, which describes our proposed X, Q based on a structured mechanism for D .
Deﬁnition 16 (Representative Sequences) .
Let M be a c -expensive mechanism which is oddly-priced or evenly-priced, and let D be any distribution.
An " - representative sequence for M, D is the following:
• Deﬁne ~ q 0 ( D , M ) :
= (0 , . .
• Deﬁne offset a to be 1 if M is oddly-priced, and 0 if M is evenly-priced.
• For all j 2 N + , deﬁne B j := { ~ v 2 supp ( D ) : p M ( ~ v ) 2 [ c · 2 2( j  1)+ a , c · 2 2( j  1)+ a +1 ) } .
• For all j 2 N + , let ~ x j 2 B j be such that ||
~ x j || 1  || ~ v || 1 · (1 + " ) for all ~ v 2 B j .•
For all j 2 N + , let ~ q j :=
~ q M ( ~ x j ) .
Proposition 17.
Let M be a c -expensive, and oddly- or evenly-priced.
Let ( X, Q ) be an " - representative sequence for M, D .
Then: MenuGap( X, Q )
[Equation]
Proof.
The proof will follow from two technical claims.
The ﬁrst claim relates || ~ x i || 1 and BRev( D ) .
The second claim relates Rev( D , M ) and MenuGap( X, Q ) .
Crucially, this claim uses the fact that the mechanism is either oddly-priced or evenly-priced (and therefore p M ( ~ x i ) , p M ( ~ x j ) differ by at least a factor of 2 , for any i 6 = j ).
Claim
[Equation]
Claim 19.
If X, Q is an " -representative sequence for M, D , then gap X,Qi
[Equation]
With these two claims, we can complete the proof of Proposition 17:
[Equation]
Above, the ﬁrst line is the deﬁnition of MenuGap.
The fourth line uses the fact that for all ~ v 2 B i , p M ( ~ v )  2 [-
 (-]2( i  1)+ a +1 = 2 · 2 2( i  1)+ a  2 p M ( ~ x i ) .
The ﬁnal line is just rewriting the deﬁnition of
Rev( D , M ) , and using the fact that every ~ v 2 supp ( D ) with p ( ~ v ) > 0 is in some B i .
Wrapping Up.
Proposition 17, together with the fact that an oddly- or evenly-priced mechanism is guaranteed to get a good approximation to the optimum, now sufﬁces to prove Theorem 6.
Proof of Theorem 6.
Set c =
Rev( D ) / 100 , and " = 1 / 100 .
Note that " -representative sequences are guaranteed to exist for any M, D , as " > 0 .
Let M 0 denote the c -expensive mechanism promised by Claim 13.
Let then M 00 denote the oddly- or evenly-priced mechanism promised by Claim 15.
Finally, let X, Q denote the " -representative sequence for M 00 , D .
We get:
[Equation]
as B j is non-empty). In particular, because M is c -expensive, we know that ||
~ v || 1  c > 0 for all ~ v who pay  c If is ~ both lists i. decrease future indices
[Figure]
No Converse to Lemma 8: Separating MenuGap and AlignGap
In this section we prove our second main result: Lemma 8 does not admit a converse, even approximately.
We brieﬂy remind the reader that all previous constructions witnessing Rev( D ) / BRev( D ) = 1 arose by establishing sequences X with AlignGap( X )
= 1 (in fact, even SupGap( X ) = 1 ).
Theorem 9 establishes that constructions exist outside of this more restrictive framework.
Missing proofs can be found in Appendix B.
Description of our construction.
Our inﬁnite sequence X consists of consecutive layers of points.
For ` = 2 to 1 , layer ` has n ` := ` d ln 2 ( ` ) e + 1 points.
These points/vectors have ` 2 norm equal to one, and are evenly spaced (in terms of their angle) between (1 , 0) and (0 , 1) .
If ` is even, they go counterclockwise from (1 , 0) to (0 , 1) .
If ` is odd, they go clockwise from (0 , 1) to (1 , 0) :
Deﬁne n ` := ` d ln 2 ( ` ) e + 1 .
Deﬁne ✓ ` := ⇡ 2( n `  1) .
• Point ~ x ` ,j is the j th point in the ` th layer, and is (cos( j ✓ ` ) , sin( j ✓ ` )) when ` is even, or (sin( j ✓ ` ) , cos( j ✓ ` )) when ` is odd.
• The inﬁnite sequence X is (( ~ x ` ,j ) n `
[Table] [Equation]
[Table]
[Equation]
Figure 1 demonstrates two layers of our construction.
In the remainder of this section, we may refer to the sequence of points ~ x ` ,j by a single indexed sequence ~ x i .
The latter is the same as the former where points are ordered lexicographically with respect to the original indexing.
Upper Bounding AlignGap( X ) via Lagrangian duality.
We ﬁrst
upper bound AlignGap( X ) and establish that it’s ﬁnite.
To this end, observe that for any sequence X , AlignGap( X ) is the solution to the following (inﬁnite, if X is inﬁnite) mathematical program, where the variables are sgap i ,
c i (the sequence ~ x i is ﬁxed, as we’re aiming to compute AlignGap( X ) ):
[Equation]
to:
[Equation]
We next proceed with a series of relaxations of this program.
Some steps are speciﬁc to our choice of
X from Section 5, while others hold for arbitrary X .
Our ﬁrst step is speciﬁc to this construction, and simply bounds || ~ x || 1 .
Consider the following mathematical program:
[Equation]
Observation 20.
For the sequence X deﬁned in Section 5, AlignGap 0 ( X )  AlignGap( X ) .
Proof.
Every ~ x i in the construction has || ~ x i || {+1  || ~ x i || 
+}
= 1 .
Therefore, the new objective function is only larger.
Moreover, every ~ x i in the construction has || ~ x i {+|| 
+}
1 / p 2 (because the ` 2 norm is 1 ), so this is relaxing the upper bound on c i .
We proceed to upper bound AlignGap {+
+} ( X ) via a Lagrangian relaxation of the formulation above.
Consider the following Lagrangian relaxation.
We put a Lagrangian multiplier of 1 on every constraint of the form sgap i  ~ x i · ( c i ~ x i  c i  1 ~ x i  1 ) , for all i > 1 , and a Lagrangian multiplier of 0 on all other constraints involving sgap .
We will not put a Lagrangian multiplier on constraints binding c i to [0 , p 2] , and keep those in the program.
This yields the following Lagrangian relaxation (for simplicity of notation below, deﬁne c 0 := 0 , and deﬁne ~ x 0 = ~ 0 ):
[Equation]
Observation 21.
For all X , AlignGap 0 ( X )  LagRel 1 ( X ) .
We now proceed to further simplify LagRel 1 ( X ) .
The next step is deﬁned below:
[Equation]
Observation 22.
For all X , LagRel 1 ( X ) =
LagRel 2 ( X ) .
Finally, we can rewrite the objective function to group all coefﬁcients of c i .
For ease of notation below, deﬁne ~ x N +1 :=[-(cid:126) 
-]
~ 0 (if N is ﬁnite.
If N = 1 , there are no notational issues).
[Equation]
Observation 23.
For all X , AlignGap 0 ( X )  LagRel 1 ( X ) =
LagRel 2 ( X ) =
LagRel( X ) .
Now, we move to analyze LagRel( X ) for our particular sequence X .
Claim 24.
For the sequence X deﬁned in Section 5, LagRel( X ) = p 2 · [Equation]
Proof.
For all i , ~ x i · ( ~ x i  ~ x i +1 )  0 , as each ~ x i has ` 2 norm exactly one.
Thus, the optimal solution for LagRel( X ) sets each c i := p 2 .
Recalling that ~ x i · ~ x i = 1 for all i concludes the claim.
Proposition 25.
For the sequence X deﬁned in Section 5, LagRel( X )  6 .
Proof.
Let us ﬁrst observe that if ~ x i is the last point in a layer, then in fact ~ x i +1 =
~ x i , and therefore 1  ~ x i · ~ x i [-+
-]+1 = 0 .
Therefore, these terms do not contribute to the sum.
We can then rewrite the term to sum over all layers as follows:
[Equation] [Equation]
Above, the ﬁrst line follows by the reasoning in the ﬁrst paragraph.
The second line follows by observing that the angle between any two points in layer ` is exactly ✓ ` (and the two points in question of ` 2 norm equal to one).
The third line follows as cos( ✓ ` )  1  ✓ 2 ` / 2 for any ✓ ` 2 [0 , ⇡ / 2] (and all ✓ ` are indeed in [0 , ⇡ / 2] ).
The fourth line follows by substituting the deﬁnition of ✓ ` as a function of n ` .
The ﬁfth line follows by deﬁnition of n ` .
The ﬁnal line is just calculation for this particular inﬁnite series (this follows as
[Equation]
Observation 20, Observation 23, and Proposition 25 yield the main result of this section:
Proposition 26.
For the sequence X deﬁned in Section 5, AlignGap( X )  6 .
Step 3: Picking a Q to Lower Bound MenuGap( X ) .
Finally, we propose a sequence Q and show that MenuGap( X, Q ) = 1 .
We describe the sequence again in layers, to match our description of
X (that is, the vector ~ q ` ,j corresponds to the vector ~ x ` ,j ).
In particular, for each even layer ` , the vectors ~ q ` ,j will have a ﬁxed x -coordinate, and the y -coordinate will increase with j .
For each odd layer, we will introduce no new vectors (i.e. we will just let gap X,Qi = 0 for all i in an odd layer).
Speciﬁcally, the construction is as follows:
• Deﬁne ↵ := P 1 ` = 1 ` ln 2 ( ` ) (and note that ↵ < 1 ).
• Deﬁne
z ` := 1 ↵ P ` j = [-
-]1 j ln 2 ( j ) .
• Deﬁne  ` :=
z `  z `  1 = 1 ↵` ln 2 ( ` ) .
• For j < n `  1 , deﬁne z ` ,j := 1   ` cot(( j + 1) ✓ ` ) .
For j = n `  1 , deﬁne z ` ,j := 1 .•
For all even ` , and all j , set ~ q ` ,j :=
( z ` , z ` ,j ) .
• For all odd ` , and all j , set ~ q ` ,j := arg max ` 0 < ` ,j 0 { ~ x ` ,j · ~ q ` 0 ,j 0 } .
Proposition 27.
MenuGap( X, Q ) = 1 .
In our proof (Appendix B) we ﬁrst analyze which point sets the gap for ~ x ` ,j , and observe that it must either be ~ q ` ,j  1 or ~ q `  2 ,n `  2  1 (that is, it must be the previous point in the same layer, or the ﬁnal point in the previous even layer).
We can then bound the gap of each layer exactly, and then take a sum over layers: for any even layers we have that the sum of gaps is at least
P n
[Equation]
) / 2 , where gap X,Q
[Equation]
,j :=
gap X
[Equation]
Qi , where ~ x i :=
~ x ` ,j ( ~ x i is the j th point on layer ` ).
Recall that  ` := 1 ↵ n ` = 1 ↵` ln 2 ( ` ) we conclude that:
[Equation]
This completes the proof of Theorem 9: Proposition 26 establishes that AlignGap( X )  6 , while
Proposition 27 establishes that MenuGap( X ) = 1 .
Conclusion
We study the nature of distributions D with Rev( D ) /
BRev( D ) = 1 .
Prior work established a framework to construct such distributions, and therefore established sufﬁcient conditions [BCKW15,
HN19].
Our ﬁrst main result establishes that the most general of these frameworks is in fact complete (Theorem 6).
Our second main result establishes that the more restrictive framework, through which all previous constructions arose, is not complete (Theorem 9).
Finally, we build upon our main construction to develop a novel distribution D witnessing Rev( D ) /
BRev( D ) = 1 , but for which none of the “aligned” mechanisms of prior work can possibly witness this (Corollary 11).
In terms of future work, it remains open as to whether there is an alternative deﬁnition for “sequences with MenuGap( X ) = 1 ” which is easier to parse.
Our work establishes that understanding such sequences is necessary and sufﬁcient to understand distributions with Rev( D ) / BRev( D ) = 1 , and establishes that “sequences with MenuGap( X ) = 1 ” is not equivalent to “sequences with
AlignGap( X ) = 1 .”
But it would be exciting for future work to better understand sequences with
MenuGap( X ) = 1 .
Interestingly, ongoing work by [AS22] uses the framework developed in this paper in order to prove approximation results for so-called ﬁne-grained buy-many mechanisms, a class of mechanisms which interpolates between buy-one and the recently introduced buy-many mechanisms [CTT19, CTT20].
In general, our techniques make it possible for future work to explore the gap between various simple versus optimal benchmarks without having to reason directly about the underlying mechanisms.
[-i || 
-]
[-Substituting max { 0 , sgap i } − sgap i = 
-]
[-Let D sample (cid:126)v i with probability 1 /B 
-]
[-Let D sample (cid:126) 0 with probability 1 − (cid:80) i ≥ 1 1 /B 
-]