Tight Revenue Gaps among Multi-Unit Mechanisms
aa r X i v : . [ c s . G T ] F e b Tight Revenue Gaps among Multi-Unit Mechanisms
Yaonan Jin ∗ Columbia University [email protected]
Shunhua Jiang † Columbia University [email protected]
Pinyan Lu ‡ Shanghai University of Finance and Economics [email protected]
Hengjie Zhang † Columbia University [email protected]
Abstract
This paper considers Bayesian revenue maximization in the k -unit setting, where a mo-nopolist seller has k copies of an indivisible item and faces n unit-demand buyers (whosevalue distributions can be non-identical). Four basic mechanisms among others have beenwidely employed in practice and widely studied in the literature: Myerson Auction , Sequen-tial Posted-Pricing , ( k + 1) -th Price Auction with Anonymous Reserve , and Anonymous Pricing .Regarding a pair of mechanisms, we investigate the largest possible ratio between the tworevenues (a.k.a. the revenue gap), over all possible value distributions of the buyers.Divide these four mechanisms into two groups: (i) the discriminating mechanism group,
Myerson Auction and
Sequential Posted-Pricing , and (ii) the anonymous mechanism group,
Anonymous Reserve and
Anonymous Pricing . Within one group, the involved two mechanismshave an asymptotically tight revenue gap of 1 + Θ(1 / √ k ). In contrast, any two mechanismsfrom the different groups have an asymptotically tight revenue gap of Θ(log k ). ∗ Supported by NSF IIS-1838154, NSF CCF-1703925, NSF CCF-1814873 and NSF CCF-1563155. Work donein part while the author was a Research Assistant at ITCS, Shanghai University of Finance and Economics. † Supported by NSF CAREER award CCF-1844887. ‡ Supported by Science and Technology Innovation 2030 – “New Generation of Artificial Intelligence” MajorProject No.(2018AAA0100903), NSFC grant 61922052 and 61932002, Innovation Program of Shanghai MunicipalEducation Commission, Program for Innovative Research Team of Shanghai University of Finance and Economics(IRTSHUFE) and the Fundamental Research Funds for the Central Universities.
Introduction “Simple vs. optimal” is one of the central themes in Bayesian mechanism design. The revenue-optimal mechanisms are more of theoretical significance, but are usually complicated and hardto implement in practice. On the other hand, most of the commonly used mechanisms in reallife are much simpler, although sacrificing a (small) amount of revenue. This trade-off motivatesthe study on how well simple mechanisms can approximate the optimal mechanisms.Even in the most basic single-item model, the optimal mechanism is already complicated.This mechanism is proposed in the seminal work by Myerson [Mye81]. It needs full knowledge ofall buyers’ individual value distributions. Typically, the value distributions are estimated frommarket research and historical transaction records. Hence, the knowledge can only be “probablyapproximately correct” (especially in large markets) and the optimal mechanism is very fragileto the estimation errors. Also, Myerson’s auction involves price discrimination among buyers,which is not allowed in most real businesses.Simple mechanisms can address the above issues and are prevalent in both online and offlineshops. For example, consider selling a number of identical copies of some product on Amazon.(This is captured by the multi-unit model, see Section 1.1 for details.) The seller simply postsa price, and a buyer decides to buy one copy if the price is acceptable to him. This mechanismis called anonymous pricing . In order to find the optimal price, the seller only needs to knowthe overall demand, which is much easier to estimate than the individual value distributions asin Myerson’s auction. Once again, the question is how well the anonymous pricing mechanismcan approximate the optimal revenue.The last two decades have seen extensive progress on the “simple versus optimal” trade-off[BK +
96, GHW01, BHW02, GHK +
05, HR09, Ala14, CGL14, CGL15, FILS15, DFK16, CFH + +
19, JLTX20, JLQ + Let us first review the previous results. In the most basic single-item model, four fundamentalmechanisms among others are widely studied. Denote by F = { F j } j ∈ [ n ] the independent valuedistributions of buyers j ∈ [ n ]. These four mechanisms work as follows (see Section 2.2 for theformal definitions). • Anonymous Pricing ( AP ): This mechanism treats all buyers equally by posting a price p .Upon arrival, a buyer will pay this price p and take the item, when his value b j ∼ F j ishigher than p (and the item is still available). If the seller knows the value distributions { F j } j ∈ [ n ] , she would select a particular price p to maximize her expected revenue amongall Anonymous Pricing mechanisms. • Sequential Posted Pricing ( SPM ): This mechanism selects an array of prices { p j } j ∈ [ n ] and anordering σ : [ n ] [ n ]. The buyers join in the mechanism sequentially σ (1) , · · · , σ ( n ), andeach index- σ ( j ) buyer must pay the order-specific price p j if winning. This discriminationcan give better revenue than Anonymous Pricing . • Anonymous Reserve ( AR ): This is a variant of the Second-Price Auction . The seller ignoresthe buyers whose bids b j are below an anonymous reserve r . The winner (which existsonly if the highest bid b (1) is above the reserve r ) is the highest of the remaining buyers,and his payment is the bigger one between the second highest bid b (2) and the reserve r . • Myerson Auction ( OPT ): A generic auction A : { b j } j ∈ [ n ] ( x , π ) is a mapping from thebids/values to the allocations x = ( x j ) j ∈ [ n ] and the payments π = ( π j ) j ∈ [ n ] . In the single-1tem case, Myerson Auction is the optimal one among those mappings [Mye81]. (When thedistributions { F j } j ∈ [ n ] are identical, Myerson Auction degenerates to
Anonymous Reserve .) OPT : discriminating auction
SPM : discriminating pricing AR : anonymous auction AP : anonymous pricing[1 . , .
49] [2 . , C ∗ ] C ∗ ≈ . C ∗ ≈ . π / ≈ . Figure 1: Demonstration for the previous results in the single-item setting with asymmetricregular buyers, where an interval indicates the best known lower/upper bounds, and a numberindicates a tight bound. For the references of these results and further discussions, one can referto [JLQ + • Anonymity ( AP and AR ) vs. Discrimination ( SPM and
OPT ). We say a mechanism is discriminating if, when different buyers become the winner, the required payments canbe different. Otherwise we say the mechanism is anonymous . Intuitively, discriminationgives a mechanism more power to extract revenue. • Pricing ( AP and SPM ) vs. Auction ( AR and OPT ). In a pricing scheme, the buyers simplymake take-it-or-leave-it decisions based on the given prices. In contrast, an auction is anarbitrary mapping from the bids to the allocations and the payments. Auctions can gainhigher revenues than pricing schemes by further leveraging the competition among buyers.Because
SPM is a discriminating pricing scheme and AR is an anonymous auction, they havedifferent powers and are incomparable. Accordingly, there are five comparable mechanism pairs(i.e., the five arrows in Figure 1).To understand the relative powers of those mechanisms, the very first question is how largethe revenue gap between any two mechanisms can be. We characterize the revenue gap as the approximation ratio between the two revenues. Formally, for a more complicated mechanism M and a simpler mechanism M , their approximation ratio is given by ℜ M / M = sup (cid:26) Rev M ( F ) Rev M ( F ) (cid:12)(cid:12)(cid:12)(cid:12) F ∈ F (cid:27) , where Rev M ( F ) denotes the revenue from a mechanism M on an input instance F = { F j } j ∈ [ n ] ,and the supremum is taken over a certain family of distributions F ∈ F .For the single-item model, the known results are shown in Figure 1. Notice that all theserevenue gaps are universal constants, and most of them have matching lower and upper bounds. From Single Unit to Multiple Units.
In this work, we focus on the k -unit setting , wherethe seller has k ≥ n unit-demand buyers.This setting is much more realistic and common in real business. Further, it is of intermediatecomplexity in comparison with the (more restricted) single-item setting and the (more general) The earlier “mechanism design for digit goods” literature [GHW01, BHW02, GHK +
05, CGL14, CGL15], dueto technical reasons, often uses the term “competitive ratio” rather than “approximation ratio”. Nonetheless, the “simple vs. optimal” trade-offs are much less understoodin this setting than in the single-item setting.Since the k -unit setting is still a single-parameter setting, Myerson Auction remains revenue-optimal [Mye81]. In addition, both of
Anonymous Pricing and
Sequential Posted Pricing can benaturally extended to this setting. For
Anonymous Reserve , the counterpart auction is no longer“second-price-type”, but is the ( k + 1) -th Price Auction with Anonymous Reserve . In the k -unit setting, previously only the revenue gap ℜ OPT / SPM between
OPT and
SPM iswell understood [Yan11, Ala14], whereas the other four gaps are widely open. By exploring therelative power of those mechanisms systematically, in this work we establish the (asymptotically)tight ratios of all previously unknown revenue gaps. We formalize our new results as the nexttwo theorems and demonstrate them in Figure 2. (Therein, the regularity assumption is verystandard in the mechanism design literature [Mye81]; see Section 2.1 for its definition.)
OPT : discriminating auction
SPM : discriminating pricing AR : anonymous auction AP : anonymous pricing1 + Θ(1 / √ k ) ✿✿✿✿✿✿✿ Θ(log k ) ✿✿✿✿✿✿✿ Θ(log k ) ✿✿✿✿✿✿✿ Θ(log k ) ✿✿✿✿✿✿✿✿✿✿✿ / √ k ) Figure 2: Demonstration for the revenue gaps among basic mechanisms in the k -unit setting,given that the value distributions are regular. Our new results are ✿✿✿✿✿✿✿✿✿✿✿ underwaved. The 1+Θ(1 / √ k )approximation result between AR and AP is given in Theorem 1, and the other three results aregiven in Theorem 2. Theorem 1 (Anonymous Reserve vs. Anonymous Pricing) . For the unit-demand buyers j ∈ [ n ] ,in each of the following three settings, the revenue gap ℜ AR / AP ( k ) between Anonymous Reserve and
Anonymous Pricing is ℜ AR / AP ( k ) = 1 + Θ(1 / √ k ) : The asymmetric general setting, where the buyers have independent but not necessarilyidentical value distributions. The i.i.d. general setting, where the value distributions are identical. The asymmetric regular setting, where the value distributions are regular but not neces-sarily identical.
Theorem 2 (Discriminating Mechanisms vs. Anonymous Mechanisms) . When the unit-demandbuyers j ∈ [ n ] have independent and regular value distributions, each of the next three revenuegaps is of order Θ(log k ) : The revenue gap ℜ OPT / AP ( k ) between Myerson Auction and
Anonymous Pricing . The revenue gap ℜ SPM / AP ( k ) between Sequential Posted Pricing and
Anonymous Pricing . The revenue gap ℜ OPT / AR ( k ) between Myerson Auction and
Anonymous Reserve . In the k -unit setting, the k copies are identical . But in the multi-item setting, the items can be heterogeneous . In the i.i.d. regular setting, an asymptotically tight bound 1 / (1 − k k / ( e k k !)) ≈ / (1 − / √ πk ) is shown in[Yan11, Section 4.2] and [Har13, Section 4.5]. AR vs. AP revenue gap, the prior works [Yan11, Ala14] show that the OPT vs.
SPM revenue gap is also of order 1 + Θ(1 / √ k ). Consequently, regarding the discriminatingmechanism group ( OPT and
SPM ) and the anonymous mechanism group ( AR and AP ), eachrevenue gap across these two groups is Θ(log k ), but the revenue gap between the two mecha-nisms in one group tends to vanish (at the rate of 1 / √ k ) when the number of copies k ∈ N ≥ becomes large. These messages can be easily inferred from Figure 2.As mentioned, the revenue gaps identify the power and the limit of “discrimination vs.anonymity” and “auction vs. pricing” in revenue maximization. Different from the single-itemsetting, where all the revenue gaps are universal constants (see Figure 1), our new results in the k -unit setting are more informative. When the number of copies k ∈ N ≥ is large: • Auctions are not much more helpful than pricing schemes in extracting the revenue (i.e.,just an 1 + Θ(1 / √ k ) improvement), no matter whether discrimination is allowed or not. • Discrimination is always very useful, and can even give an unbounded improvement (upto a Θ(log k ) factor) on the revenue.These propositions meet what we observe in real business: auctions are rarely used in practice,whereas different kinds of price discrimination are rather common. In this section, we sketch the proof of our 1 + Θ(1 / √ k ) approximation result for the AR vs. AP revenue gap (Theorem 1). In fact, we can represent the exact ratio ℜ AR / AP as an explicitintegration formula, (although this formula in general does not admit an elementary expression).We acquire this formula by solving a mathematical programming generalized from [JLTX20,Program (4)], which resolves the same problem for the single-item case k = 1.However, many crucial properties of the single-item case do not preserve in the general case k ≥
1. In the single-item case,
Anonymous Reserve relies on the first/second order statistics b (1) and b (2) (i.e., the biggest and second biggest sampled bids/values), and Anonymous Pricing relieson the b (1) . Therefore, we only need to reason about these two random variables, b (1) and b (2) ,together with the correlation between them. In the k -unit case, however, up to ( k + 1) randomvariables b (1) , · · · , b ( k +1) must be taken into account, and the correlation among them becomesmuch more complicated.For the above reasons, we cannot modify and re-adopt the approach of the work [JLTX20]in a naive way. Instead, with the purpose of handling the highly correlated order statistics b ( i ) ’s,we will develop a new structural lemma about the Poisson binomial distributions (PBDs). Thisnew lemma mainly relies on the log-concavity of the PBDs.
Lemma (Bernoulli Sum Lemma) . Given two arrays of Bernoulli random variables: { X j } j ∈ [ n ] are i.i.d., while { Y j } j ∈ [ n ] are independent yet not necessarily identically distributed. For therandom sums X = P j ∈ [ n ] X j and Y = P j ∈ [ n ] Y j , there exists some threshold s ∈ R such that: Pr[ X ≤ t ] ≥ Pr[ Y ≤ t ] for any t < s . Pr[ X ≤ t ] ≤ Pr[ Y ≤ t ] for any t ≥ s . With the help of this lemma, we can characterize the worst-case instance of the mentionedmathematical programming, for k ≥ n ≥
1. To this end, let us formulate the AR and AP revenues. Denote by F j the cumulative distribution function (CDF) of buyer j ’s value, and D i the CDF of the i -th order statistic b ( i ) . The Anonymous Reserve revenue (Fact 3) is given by AR ( r ) = AP ( r ) + k · Z ∞ r (1 − D k +1 ( x )) · d x, ∀ r ≥ . AP ( r ) is the revenue by posting the price p = r in Anonymous Pricing . Further, the AP revenue (Fact 2) depends on the top- k CDF’s { D i ( r ) } i ∈ [ k ] at this reserve r ≥ Y = P j ∈ [ n ] Y j , for which the individual failure probabilitiesare Pr[ Y j = 0] = F j ( r ). This choice of the failure probabilities ensures Pr[ Y ≤ i −
1] = D i ( r ) forevery i ≥
1. Further, we can find another array of i.i.d. Bernoulli random variables { X j } j ∈ [ n ] so that the sum X = P j ∈ [ n ] X j satisfiesPr[ X ≤ k ] = Pr[ Y ≤ k ] = D k +1 ( r ) . (The existence of such { X j } j ∈ [ n ] is obvious.) Then our Bernoulli Sum Lemma shows thatPr[ X ≤ i − ≥ Pr[ Y ≤ i −
1] = D i ( r )for each i ∈ [ k ], where the equality holds when the { Y j } j ∈ [ n ] are also i.i.d.Informally speaking, the above inequalities and the equality condition imply that, the ratio AR ( r ) / AP ( r ) is maximized when the value CDF’s are equal F ( r ) = · · · = F n ( r ) at this reserve.Following this argument and with extra efforts, we have the next observation. Observation.
For each k ≥ and n ≥ , the worst case for the ℜ AR / AP revenue gap happenswhen the value distributions are identical, i.e., F ∗ = { F ∗ } n , (although this worst-case commondistribution F ∗ is given by an implicit equation and does not admit an elementary expression). Furthermore, it is noteworthy that the above approach enables a unified constructive prooffor the upper-bound/lower-bound parts of the general case k ≥
1. In contrast, the former work[JLTX20] establishes these two parts of the single-item case separately, and their upper-boundproof is non-constructive.Our Bernoulli Sum Lemma can find its applications in related directions. As mentioned, weleverage it mainly to handle the order statistics. Apart from the “simple vs. optimal mechanismdesign” paradigm, on other topics such as “learning simple mechanisms from samples” [CGL15,MM16, MR16, CD17, JLX19], the order statistics are also of fundamental interests. Conceivably,our new lemma would be helpful for those topics, in a similar manner as this paper.
In this section we sketch the proof of Theorem 2, which claims that the revenue gaps ℜ OPT / AP , ℜ SPM / AP and ℜ OPT / AR are all of order Θ(log k ). In fact, any one bound implies the other two.This is because the revenue gaps within the discriminating/anonymous groups ( OPT vs.
SPM ,and AR vs. AP ) are both constants / √ k ) = Θ(1), and these constants are dominated bythe Θ(log k ) bound.For these reasons, it suffices to only prove the OPT vs. AP revenue gap ℜ OPT / AP = Θ(log k ).Actually, an Ω(log k ) lower bound for this revenue gap is already shown in [HR09, Example 5.4],so we only need to prove the O (log k ) upper bound.We actually prove the O (log k ) upper bound between Anonymous Pricing and a benchmarkcalled
Ex-Ante Relaxation ( EAR in short). It is known that this benchmark always exceeds the
Myerson Auction revenue [CHMS10]. To acquire the O (log k ) upper bound, we will start with amathematical programming generalized from [AHN +
19, Equations (1) and (2)].However, the general-case mathematical programming has a very different structure as itis in the single-item case. When k = 1, the worst-case instance (i.e., the optimal solution, see[AHN +
19, Section 4.3]) turns out to be a continuum of “small” buyers – any single buyer has aninfinitesimal contribution to the
EAR benchmark, but there are infinitely many buyers n → ∞ (in the sense of large markets [MSVV07, AGN14]). Accordingly, it is better to think about the“density” of different types of buyers, instead of the number of buyers.But in the general case, the Ω(log k ) lower-bound instance [HR09, Example 5.4] essentiallyis constituted by “big” buyers – a certain amount of buyers contribute at least 1 /k unit to the5 AR benchmark each, while every other buyer contributes strictly 0 unit and can be omitted.More importantly (see Remark 5), if we insist on a continuum of “small” buyers in the generalcase k ≥
1, then the
EAR vs. AP revenue gap turns out to be (at most) a universal constant forwhatever k ≥ k ≥
1, we will classify the buyers j ∈ [ n ] into groups,and then bound the individual contributions from these groups to the EAR benchmark.In more details, we can employ the technique developed in [AHN +
19, Lemma 4.1], and thustransform the mentioned mathematical programming into the following one.
Variables : • { v j } j ∈ [ n ] ∈ R n ≥ , where v j = arg max { p · (1 − F j ( p )) : p ≥ } for each j ∈ [ n ], are the monopoly prices of the distributions F = { F j } j ∈ [ n ] . • { q j } j ∈ [ n ] ∈ [0 , n , where q j = 1 − F j ( v j ) for each j ∈ [ n ], are the monopoly quantiles . • The resulting { v j q j } j ∈ [ n ] ∈ R n ≥ are the monopoly revenues . Constraints : • The capacity constraint, P j ∈ [ n ] q j ≤ k . • The feasibility constraint, AP ( p, F ) ≤ p ∈ R ≥ . Objective : Maximize the
Ex-Ante Relaxation benchmark
EAR ( F ) = P j ∈ [ n ] v j q j .Regarding the EAR benchmark, the monopoly revenues { v j q j } j ∈ [ n ] are precisely the individ-ual contributions from the distributions { F j } j ∈ [ n ] . Given the capacity constraint (in the senseof the Knapsack Problem), the monopoly quantiles { q j } j ∈ [ n ] can be viewed as the individualcapacities. Therefore, the monopoly prices { v j } j ∈ [ n ] can be viewed as the bang-per-buck ratios (i.e., the contribution to the EAR benchmark per unit of the capacity).To find the optimal solution, of course we prefer those distributions with higher bang-per-buck ratios { v j } j ∈ [ n ] , but also need to take the capacities { q j } j ∈ [ n ] into account. Informally, wewill classify the buyers into three groups [ n ] = L ∪ H S ∪ H B : • L = { j ∈ [ n ] : v j < /k } . Because these group- L distributions have lower bang-per-buckratios v j < /k , conceivably the total contribution by this group to the EAR benchmarkshall be small. Indeed, we will prove a constant upper bound P j ∈ L v j q j = O (1). • H S = { j ∈ [ n ] : v j ≥ /k and v j q j < / (2 k ) } . In other words, the group- H S distributionshave high enough bang-per-buck ratios v j ≥ /k but small capacities, i.e., v j q j < / (2 k ).It turns out that the total contribution by this group is also small, and we also will provea constant upper bound P j ∈ H S v j q j = O (1). • H B = { j ∈ [ n ] : v j ≥ /k and v j q j ≥ / (2 k ) } . That is, these group- H B distributionshave high enough bang-per-buck ratios and big enough capacities. Therefore, this groupshould contribute the most to the EAR benchmark. Taking into account the feasibilityconstraint, AP ( p, F ) ≤ p ∈ R ≥ , we will show P j ∈ H B v j q j = O (log k ).The actual grouping criteria in our proof are more complicated than the above ones, in orderto handle other technical issues.Finally, we notice that our grouping criteria borrow ideas from the “budget-feasible mecha-nism” literature [Sin10, CGL11, GJLZ20], where the target is to design approximately optimalmechanisms for the Knapsack Problem under the incentive concerns. We hope that these ideascan find more applications to the “simple vs. optimal mechanism design” research topic.6 .3 Further related works The revenue gaps among the mentioned mechanisms,
Myerson Auction , Sequential Posted Pricing , Anonymous Reserve , and
Anonymous Pricing , are extensively studied in the literature. Below weprovide an overview of the previous results (mainly in the single-item setting and in the k -unitsettings). As a supplement, the reader can refer to the surveys [Luc17, CFH +
18, JLQ + AR vs. AP.
This revenue gap studies the relative power between the auction schemes and thepricing schemes, when the price discrimination is not allowed. The previously known results inthe single-item case are shown in the next table.i.i.d. regular e/ ( e − ≈ .
58 [CHMS10, Thm 6] & [Har13, Thm 4.13]i.i.d. general π / ≈ .
64 [JLTX20, Thm 2]asymmetric regularasymmetric generalIn the k -unit case, an asymptotically tight bound 1 / (1 − k k / ( e k k !)) ≈ / (1 − / √ πk )for i.i.d. regular buyers is shown in [Yan11, Section 4.2] and [Har13, Section 4.5]. Our newresults settle the remaining pieces of the puzzle – even though the i.i.d. assumption and/or theregularity assumption are removed, this revenue gap is still of order 1 + Θ(1 / √ k ). SPM vs. AP.
This revenue gap investigates the power of price discrimination in the pricingschemes. Below we summarize the known results and our new results, in both the single-itemcase and the k -unit case.single-item case SPM vs.
AP OPT vs. AP i.i.d. regular e/ ( e − ≈ .
58 [CHMS10, Thm 6] & [Har13, Thm 4.13]i.i.d. general 2 − /n [DFK16, Thm 3] [Har13, Thm 4.9]asymmetric regular constant C ∗ ≈ .
62 [JLTX20, Thm 1] [JLQ + n [AHN +
19, Prop 6.1] k -unit case SPM vs.
AP OPT vs. AP i.i.d. regular 1 / (1 − k k / ( e k k !)) ⋄ [DFK16, Thm 1] [Yan11, Sec 4.2]i.i.d. general 2 − k/n [DFK16, Thm 3] [Har13, Sec 4.5]asymmetric regular Θ(log k ) this workasymmetric general n [AHN +
19, Prop 6.1] ⋄ this bound is just asymptotically tight OPT vs. AP.
This revenue gap is to illustrate that even the simplest mechanism,
AnonymousPricing , can approximate the optimal revenue in quite general settings. Actually, in each ofthe single-item/ k -unit, i.i.d./asymmetric, regular/general settings, this ratio “coincedentally”is equal to the SPM vs. AP revenue gap, namely ℜ OPT / AP = ℜ SPM / AP . (But the results respec-tively for ℜ OPT / AP and ℜ SPM / AP are credited to different works.) For brevity, we summarize theresults on the both revenue gaps together in the above tables.Instead of the regularity assumption, the stronger monotone-hazard-rate (MHR) distribu-tional assumption is also very standard in the mechanism design literature. The previous works The reader may wonder why the revenue gaps ℜ OPT / AP and ℜ SPM / AP are equal, in each of the single-item/ k -unit, i.i.d./asymmetric, regular/general settings. This is because, in each of these settings, the worst-case instance { F ∗ j } j ∈ [ n ] of the OPT vs. AP problem has a nice property: for each F ∗ j , the corresponding virtual-value distribution is supported on the non-positive semiaxis ( −∞ ,
0] plus a single positive number v ∗ j >
0. When an instance satisfiesthis property, we can adopt the arguments in [JLTX20, Lemma 1] to show that
OPT and
SPM extract the sameamount of revenue, which implies ℜ OPT / AP = ℜ SPM / AP . OPT vs. AP revenue gap in the single-item i.i.d. MHR setting. OPT vs. AR.
This ratio studies the power of price discrimination in the auction schemes.When the value distributions are i.i.d. and regular,
Myerson Auction and
Anonymous Reserve turn out to be identical [Mye81]. The results beyond the i.i.d. regular case are given below.single-item casei.i.d. general 2 − /n [Har13, Thm 4.9]asymmetric regular LB ≈ .
15 [HR09, Sec 5] & [JLTX20, Thm 3]UB = C ∗ ≈ .
62 [HR09, Sec 5] & [JLQ + n [AHN +
19, Prop 6.1] k -unit casei.i.d. general 2 − k/n [Har13, Sec 4.5]asymmetric regular Θ(log k ) this workasymmetric general n [AHN +
19, Prop 6.1]Notably, the tight ratio in the single-item asymmetric regular setting is still unknown. Hart-line and Roughgarden first prove that this ratio is between 2 and 4 [HR09, Section 5]. After-wards, the lower bound is improved to ≈ .
15 [JLTX20, Theorem 3]. But the best known upperbound just follows from the tight
OPT vs. AP revenue gap C ∗ ≈ .
62 by implication. We highlybelieve this factor- C ∗ barrier can be broken, for which new techniques tailored for AnonymousReserve rather than
Anonymous Pricing are required.Beyond the
Anonymous Reserve mechanism, other simple auctions with the more powerful personalized reserves are also extensively studied [HR09, BGL +
18, MS20].
OPT vs. SPM.
This revenue gap investigates the relative power between the auction schemesand the pricing schemes, when the price discrimination is allowed. Indeed, the previous works[HKS07, CFPV19] show that this problem is identical to the ordered prophet inequality problemin stopping theory. In each of the single-item/ k -unit i.i.d./asymmetric settings, the tight rev-enue gaps under/without the regularity assumption turn out to be the same (see, e.g., [Yan11,Section 3.1]). The previous results in the single-item/ k -unit cases are summarized below.single-item casei.i.d. constant β ≈ .
34 [CFH +
17, Thm 1.3]asymmetric LB = β ≈ .
34 [CFH +
17, Thm 1.3]UB = 1 / (1 − /e + 1 / ≈ .
49 [CSZ19, Thm 1.1] k -unit casei.i.d./asymmetric LB = 1 + Ω(1 / √ k ) [HKS07, Thm 7]UB = 1 / (1 − k k / ( e k k !)) ≈ / (1 − / √ πk ) [Yan11, Sec 4.2]Noticeably, the tight ratio in the single-item asymmetric setting is still unknown. The bestknown lower bound just follows from the tight “i.i.d.” revenue gap β ≈ .
34 by implication.Recently, there is an outburst of activity on the upper bound [ACK18, BGL +
18, CSZ19], andthe best known result is 1 / (1 − /e + 1 / ≈ .
49 [CSZ19, Theorem 1.1]. It remains aninteresting open question to further refine the upper bound.Beyond the k -unit setting, the OPT vs.
SPM revenue gap is also studied in the more general matroid setting. For this, the work [CHMS10, Theorem 5] first shows an upper bound of 2, andthen [Yan11, Section 4.1] improves it to e/ ( e − ≈ . Sequential Posted Pricing mechanism crucially leverages the order in which the buyersparticipate in the mechanism. Instead, the order-oblivious counterpart mechanisms are exten-8ively studied as well [CHMS10, Ala14, AW18, ACK18, BGL +
18, EHKS18, CSZ19].
Organization.
In Section 2 we introduce the notation and the requisite knowledge about theconsidered mechanisms. The
Anonymous Reserve vs.
Anonymous Pricing problem is investigatedin Section 3 (with some technical details deferred to Appendix A). The
Ex-Ante Relaxation vs.
Anonymous Pricing problem is investigated in Section 4.
This section includes the notation to be adopted in this paper, and the basic knowledge aboutprobability (e.g. the regular/triangle distributions) and the concerning mechanisms.
Notation.
Denote by R ≥ (resp. N ≥ ) the set of all non-negative real numbers (resp. positiveintegers). For any pair of integers b ≥ a ≥
0, define the sets [ a ] def = { , , · · · , a } and [ a : b ] def = { a, a + 1 , · · · , b } . Denote by {·} the indicator function. The function | · | + maps a real number z ∈ R to max { , z } . We use the bold letter F = { F j } j ∈ [ n ] to denote an instance (namely an n -dimensional productdistribution ), where F j is the bid distribution of the buyer j ∈ [ n ]. For ease of notation, F j alsorepresents the corresponding cumulative density function (CDF).We assume the CDF’s { F j } j ∈ [ n ] to be left-continuous , in the sense that when the j -th buyerhas a random bid b j ∼ F j for a price- p item, his willing-to-pay probability is Pr[ b j ≥ p ] ratherthan Pr[ b j > p ]. We also define the inverse CDF F − j ( y ) def = inf { x ∈ R ≥ : F j ( x ) ≥ y } for any y ∈ [0 , F − j (1) = ∞ . We say a distribution F j stochastically dominates another F j , when F j ( x ) ≤ F j ( x ) for all x ∈ R ≥ . Further, an instance F = { F j } j ∈ [ n ] dominatesanother instance F = { F j } j ∈ [ n ] , when F j dominates F j for each j ∈ [ n ].For a CDF F j , we are also interested in two associated parameters ( v j , q j ). The monopolyquantile q j ∈ [0 ,
1] and the monopoly price v j ∈ R ≥ are respectively given by q j def = arg max q ∈ [0 , { F − j (1 − q ) · q } and v j def = F − j (1 − q j ) . If there are multiple maximizers q j , we would choose the smallest q j among the alternatives;notice that possibly q j = 0 and v j = ∞ .Sampling a bid profile from the instance b = ( b j ) j ∈ [ n ] ∼ F , the i -th highest bids (for i ∈ [ n ]) b (1) ≥ · · · ≥ b ( i ) ≥ · · · ≥ b ( n ) will be of particular interest. We denote by D i the correspondingdistributions/CDF’s, namely D i ( x ) = Pr[ b ( i ) < x ] for all x ∈ R ≥ . Again, we assume { D i } i ∈ [ n ] to be left-continuous. The formulas for the i -th highest CDF’s are given below. Fact 1 (Order Statistics) . For each i ∈ [ n + 1] , the i -th highest CDF is given by D i ( x ) = X t ∈ [0: i − X | W | = t (cid:16) Y j / ∈ W Pr[ b j < x ] (cid:17) · (cid:16) Y j ∈ W Pr[ b j ≥ x ] (cid:17) = X t ∈ [0: i − X | W | = t (cid:16) Y j / ∈ W F j ( x ) (cid:17) · (cid:16) Y j ∈ W (1 − F j ( x )) (cid:17) , ∀ x ≥ . Regular distribution.
Denote by
Reg this distribution family. According to [Mye81], adistribution is regular F j ∈ Reg if and only if the virtual value function ϕ j ( x ) def = x − − F j ( x ) f j ( x ) isnon-decreasing on the support of F j , where f j is the probability density function (PDF). Sucha regular CDF F j is illustrated in Figure 3a. 9 j ( x ) xv j − q j (a) The CDF of a regular F j F j ( x ) xv j − q j (b) The CDF of Tri ( v j , q j ) Figure 3: Demonstration for the regular distribution and the triangle distribution.
Triangle distribution.
This distribution family, denoted by
Tri , is introduced in [AHN + Reg . Such a distribution
Tri ( v j , q j ) is deter-mined by the monopoly price v j ∈ R ≥ and the monopoly quantile q j ∈ [0 , F j ( x ) def = ( (1 − q j ) · x (1 − q j ) · x + v j q j , ∀ x ∈ [0 , v j ]1 , ∀ x ∈ ( v j , ∞ ) . We focus on such a revenue maximization scenario: the seller has k ∈ N ≥ homogeneous itemsand faces n ≥ k unit-demand buyers, and the buyers draw their bids b = { b j } j ∈ [ n ] ∼ F independently from a publicly known product distribution F = { F j } j ∈ [ n ] . For convenience, weinterchange buyer/bidder.In the bulk of the work, we will concern three mechanisms: Anonymous Pricing , AnonymousReserve , and
Ex-Ante Relaxation . Below we briefly introduce these mechanisms; for more details,the reader can refer to [Har13, Chapter 4].
Anonymous Pricing.
In such a mechanism, the seller posts an a priori price p ∈ R ≥ to anysingle item; then in an arbitrary coming order, each of the first k coming buyers that are willingto pay the price p ∈ R ≥ , will get an item by paying this price. Given any bid profile b ∼ F ,let b ( n +1) def = 0 and reorder the bids such that b (1) ≥ · · · ≥ b ( i ) ≥ · · · ≥ b ( n +1) .Depending on how many bids exceed the posted price, the mechanism gives a revenue of Rev ( AP ) = X i ∈ [ k ] i · p · { b ( i ) ≥ p > b ( i +1) } + k · p · { b ( k +1) ≥ p } = X i ∈ [ k ] p · { b ( i ) ≥ p } . Taking the randomness over b ∼ F into account results in the expected revenue. Fact 2 (Revenue Formula for
Anonymous Pricing ) . Under any posted price p ∈ R ≥ , the Anony-mous Pricing mechanism extracts an expected revenue of AP ( p, F ) def = p · X i ∈ [ k ] (1 − D i ( p )) . Let AP ( F ) def = max p ∈ R ≥ { AP ( p, F ) } denote the optimal Anonymous Pricing revenue. nonymous Reserve. In such a mechanism, the seller sets an a priori reserve r ∈ R ≥ onany single item. When at most k bidders are willing to pay the reserve r ∈ R ≥ , AnonymousReserve has the same allocation/payment rule as
Anonymous Pricing , thus the same revenue.But when at least ( k + 1) bidders are willing to pay this reserve, each of the top- k bidders (withan arbitrary tie-breaking rule) wins an item by paying the ( k + 1)-th highest bid b ( k +1) ≥ r .Running on a specific bid profile b ∼ F , the mechanism generates a revenue of Rev ( AR ) = X i ∈ [ k ] i · r · { b ( i ) ≥ r > b ( i +1) } + k · b ( k +1) · { b ( k +1) ≥ r } = X i ∈ [ k ] r · { b ( i ) ≥ r } + k · | b ( k +1) − r | + . Taking the randomness over b ∼ F into account gives the expected revenue. (Note that [CGM15,Fact 1] get the revenue formula below in the single-item case k = 1.) Fact 3 (Revenue Formula for
Anonymous Reserve [CGM15, Fact 1]) . Under any reserve r ∈ R ≥ ,the Anonymous Reserve mechanism extracts an expected revenue of AR ( r, F ) def = r · X i ∈ [ k ] (1 − D i ( r )) + k · Z ∞ r (1 − D k +1 ( x )) · d x. Let AR ( F ) def = max r ∈ R ≥ { AR ( r, F ) } denote the optimal Anonymous Reserve revenue.
Ex-Ante Relaxation.
This notion is introduced by [CHMS10]. Although just being a “fake”mechanism, Ex-Ante Relaxation is useful to upper bound the revenue from the optimal truthfulmechanism,
Myerson Auction .For a regular instance, an
Ex-Ante Relaxation mechanism is specified by an allocation rule q ′ = { q ′ j } j ∈ [ n ] ∈ [0 , n . Here, each q j ∈ [0 ,
1] represents the probability that the buyer j ∈ [ n ]wins an item. This allocation rule is feasible iff P j ∈ [ n ] q ′ j ≤ k , because we only have k items.The following fact characterizes the resulting “revenue”. Fact 4 (Revenue Formula for
Ex-Ante Relaxation [CHMS10, Lemma 2]) . Given a regular instance F = { F j } j ∈ [ n ] , under any feasible allocation rule q ′ = { q ′ j } j ∈ [ n ] ∈ [0 , n that P j ∈ [ n ] q ′ j ≤ k , the Ex-Ante Relaxation mechanism extracts an expected revenue of
EAR ( q ′ , F ) def = X j ∈ [ n ] F − j (1 − q ′ j ) · q ′ j Remark . We will study the
Ex-Ante Relaxation mechanism just for the regular instances. Therevenue formulas for the irregular instances are more complicated, for which the reader can referto [CHMS10, Lemma 2].
Revenue monotonicity.
Based on the revenue formulas given in Facts 2 to 4, one can easilycheck the following fact (a.k.a. the revenue monotonicity in the literature).
Fact 5 (Revenue Monotonicity) . Given that an instance F = { F j } j ∈ [ n ] stochastically dominatesanother instance F = { F j } j ∈ [ n ] , the following hold: AP ( p, F ) ≥ AP ( p, F ) for any posted price p ∈ R ≥ , and thus AP ( F ) ≥ AP ( F ) . AR ( r, F ) ≥ AR ( r, F ) for any reserve r ∈ R ≥ , and thus AR ( F ) ≥ AR ( F ) . EAR ( q ′ , F ) ≥ EAR ( q ′ , F ) for any allocation q ′ = { q ′ j } j ∈ [ n ] ∈ [0 , n with P j ∈ [ n ] q ′ j ≤ k . Namely, in the concerning Bayesian mechanism design setting,
Ex-Ante Relaxation is unimplementable. Anonymous Reserve vs. Anonymous Pricing
In this section, we investigate the
Anonymous Reserve vs.
Anonymous Pricing problem. Basedon the revenue formulas (see Section 2.2), the revenue gap between both mechanisms is char-acterized by the following mathematical program.sup AR ( r, F ) = r · X i ∈ [ k ] (1 − D i ( r )) + k · Z ∞ r (1 − D k +1 ( x )) · d x, ∀ r ∈ R ≥ , (P1)s.t. AP ( p, F ) = p · X i ∈ [ k ] (1 − D i ( p )) ≤ , ∀ p ∈ R ≥ , (C1) F = { F j } j ∈ [ n ] , ∀ n ∈ N ≥ . By finding the optimal solution to Program (P1), we will prove the next theorem.
Theorem 3 ( AR vs. AP ) . Given that the seller has k ∈ N ≥ homogeneous items and faces n ≥ k independent unit-demand buyers, the revenue gap ℜ AR / AP ( k, n ) between Anonymous Reserve and
Anonymous Pricing satisfies the following: The revenue gap ℜ AR / AP ( k, n ) is maximized when all the buyers have the same bid distri-bution { F ∗ } n , and their common CDF F ∗ is an implicit function given by F ∗ ( x ) = 0 forall x ∈ [0 , k ] and AP ( x, { F ∗ } n ) = 1 for all x ∈ ( k , ∞ ) . Over all n ≥ k , the supremum revenue gap ℜ AR / AP ( k ) def = sup n ≥ k ℜ AR / AP ( k, n ) is achievedby ℜ AR / AP ( k, ∞ ) = 1 + k · Z ∞ T k ( x ) · (1 − T k +1 ( x ))( k − P i ∈ [ k ] T i ( x )) · d x, where the functions T i ( x ) def = e − x · P t ∈ [0: i −
1] 1 t ! · x t for all i ∈ [ k + 1] . For each k ∈ N ≥ , the supremum revenue gap is bounded between . √ k ≤ ℜ AR / AP ( k ) ≤ √ k . For each k ∈ N ≥ , the ratio ℜ AR / AP ( k ) is tight not only in the asymmetric general setting,but (more restrictedly) also in the asymmetric regular setting and/or the i.i.d. generalsetting. We first outline our approach towards Theorem 3. Central to the upper-bound analysis isa basic result about the sum of independent Bernoulli random variables, which is formalizedbelow as Theorem 4 and can be of independent interest (see Figure 4 for a demonstration). Wewill prove this theorem in Section 3.1.
Theorem 4 (Bernoulli Sum Lemma) . Given two arrays of Bernoulli random variables: { X j } j ∈ [ n ] are independent and identically distributed (i.i.d. in short), while { Y j } j ∈ [ n ] independent yet notnecessarily identically distributed. For the random sums X = P j ∈ [ n ] X j and Y = P j ∈ [ n ] Y j ,there exists some threshold s ∈ R such that: Pr[ X ≤ t ] ≥ Pr[ Y ≤ t ] for any t < s . Pr[ X ≤ t ] ≤ Pr[ Y ≤ t ] for any t ≥ s . Based on Theorem 4 together with further optimization arguments, we acquire Part 1 andPart 2 of Theorem 3 respectively in Sections 3.2 and 3.3. Part 3 of Theorem 3 requires someadvanced tools from real analysis; its proof is technically involved and is deferred to Appendix A.12DF t s = 50 1 2 3 4 5 6 7 8 9 10Figure 4: Demonstration for Theorem 4 when n = 10. Note that “red ≥ blue” when t < ≤ blue” when t ≥
5, where the red line refers to the sum of i.i.d. Bernoulli variables, andthe blue line refers to the sum of independent yet not necessarily identical Bernoulli variables.(Notice that Parts 1 to 3 require no distributional assumption.) Eventually, we construct twomatching lower-bound examples in Section 3.4, one in the i.i.d. general setting and one in theasymmetric regular setting, hence Part 4 of Theorem 3.All the above results concern the
Anonymous Reserve vs.
Anonymous Pricing problem undera cardinality constraint, namely up to k ∈ N ≥ buyers can simultaneously win. In the literatureon mechanism design, many works also consider the more general constraint that the winningbuyers satisfy a matroid constraint (e.g., see [HR09]). For this setting, we will show in Section 3.5an lower bound Ω(log k ) for the counterpart revenue gap. The following well known result, that any independent Bernoulli sum is a log-concave randomvariable (e.g., see [JKM13]), is crucial to our proof of Theorem 4.
Fact 6 (Log-Concavity of Bernoulli Sum) . Let Z = P j ∈ [ n ] Z j be the sum of n ∈ N ≥ independentyet not necessarily identical Bernoulli random variables, then for any integer t ∈ Z , Pr[ Z = t ] ≥ Pr[ Z = t − · Pr[ Z = t + 1] . For ease of presentation, we will only justify Part 1 of Theorem 4, and Part 2 follows fromsimilar arguments. Further, it suffices to consider the case that Pr[ X ≤ s ] = Pr[ Y ≤ s ] (i.e. theother case that Pr[ X ≤ s ] > Pr[ Y ≤ s ] can be accommodated by properly scaling the failureprobability of the i.i.d. random variables { X j } j ∈ [ n ] ). Further, since we concern about Bernoullirandom variables and their sums, we safely assume t ≤ s to be integers between [0 : n ]. Weobtain Part 1 of Theorem 4 in two steps: • Local transformation. In Lemma 1 and Corollary 1, we will pick a pair of non-identicallydistributed variables Y j and Y j (for some j = j ∈ [ n ]), and replace them by anotherpair of i.i.d. variables Y j and Y j . The new pair is carefully constructed, so as to ensurecertain properties. 13 Global transformation. We conduct the local transformation on the variables { Y j } j ∈ [ n ] round by round (in a nontrivial way), which preserves the mentioned properties by induc-tion. Together with extra arguments from real analysis (see Claim 1), these propertieswill lead to Part 1 of Theorem 4. Lemma 1 (Averaging Two Variables.) . Assume w.l.o.g. that two variables Y j and Y j givenin Theorem 4 (for some j = j ∈ [ n ] ) are not identically distributed, then there exists anotherpair of i.i.d. Bernoulli random variables Y j and Y j such that: Pr[ Y j + Y j + P j / ∈{ j ,j } Y j ≤ s ] = Pr[ Y ≤ s ] . Pr[ Y j + Y j + P j / ∈{ j ,j } Y j ≤ t ] ≥ Pr[ Y ≤ s ] for any t ∈ [0 : s − .Proof of Lemma 1. For simplicity, we reindex the variables { Y j } j ∈ [ n ] such that j = 1 and j = 2. We adopt the following notations: • Let q j def = Pr[ Y j = 0] ∈ [0 ,
1] for all j ∈ [ n ] and q def = Pr[ Y = 0] = Pr[ Y = 0] ∈ [0 , q < q , given that Y and Y are not identically distributed. • Let Y ′ def = P j ∈ [3: n ] Y j and a i def = Pr[ Y ′ = i ] for all i ∈ Z . Because Y ′ is the sum of ( n − a i = 0 only if i ∈ [0 : n − a t ≥ a t − · a t +1 for all t ∈ Z . By induction, one can easily see that a t · a s − ≥ a t − · a s (1)for all integers t ≤ s ∈ Z , which is more convenient for our later use.For any integer i ∈ Z , by considering all of the four possibilities ( Y , Y ) ∈ { , } , we canreformulate the probability Pr[ Y = i ] ∈ [0 ,
1] as follows:Pr[ Y = i ] = Pr[ Y + Y + Y ′ = i ]= Pr[ Y ′ = i ] · Pr[ Y = Y = 0] + Pr[ Y ′ = i − · Pr[ Y = Y = 1]+ Pr[ Y ′ = i − · Pr[ Y = 0 , Y = 1] + Pr[ Y ′ = i − · Pr[ Y = 1 , = Y = 0]= a i · q · q + a i − · (1 − q ) · (1 − q )+ a i − · q · (1 − q ) + a i − · (1 − q ) · q = (( a i − a i − ) − ( a i − − a i − )) · q · q + ( a i − − a i − ) · ( q + q ) + a i − . Thus, we can rewrite the telescoping sum Pr[ Y ≤ t ] = P i ∈ [0: t ] Pr[ Y = i ] as follows:Pr[ Y ≤ t ] = (( a t − a t − ) − ( a − − a − )) · q · q + ( a t − − a − ) · ( q + q ) + X i ∈ [0: t ] a i − = ( a t − a t − ) · q · q + a t − · ( q + q ) + const , (2)where the last step follows because a − = a − = 0 (recall that a i = 0 only if i ∈ [0 : n − def = P i ∈ [0: t − a i = Pr[ Y ′ ≤ t − multilinear function of { q j } j ∈ [ n ] ∈ [0 , n , and the lastsummand const is irrelevant to both q = Pr[ Y = 0] and q = Pr[ Y = 0]. That is, suppose that q and { q j } nj =3 are held constant, we can regard Equation (2) as a linear function of q ∈ [0 , a t − a t − ) · q + a t − = a t · q + a t − · (1 − q )must be non-negative , because the probabilities q , a t , a t − ∈ [0 , q ∈ [0 , q ∈ [0 ,
1] in place of q and q , we also havePr[ Y + Y + Y ′ ≤ t ] = ( a t − a t − ) · q + a t − · q + const , (3)Again, Equation (3) is a non-decreasing function in q ∈ [0 , q < q , we can easily check thatEquation (3) | q = q ≤ Equation (2) , Equation (3) | q = q ≥ Equation (2) . In the case that t = s , the equality Pr[ Y + Y + Y ′ ≤ s ] = Pr[ Y ≤ s ] holds for at least one q ∈ [ q , q ] ⊆ [0 , a s − a s − ) · q + a s − · q = ( a s − a s − ) · q · q + a s − · ( q + q ) . (4)This accomplishes Part 1 of Lemma 1. We next show that the above particular q ∈ [ q , q ](for which Pr[ Y + Y + Y ′ ≤ s ] = Pr[ Y ≤ s ]) guarantees Part 2:Pr[ Y + Y + Y ′ ≤ t ] ≥ Pr[ Y ≤ t ] , for all integers t ∈ [0 : s − Case I ( a s = a s − ). Based on Equation (1), i.e., a t · a s − ≥ a t − · a s , we easily infer a t ≥ a t − (note that a t − , a t , a s − , a s ∈ [0 ,
1] are probabilities). Moreover, Equation (4) degenerates into a s − · q = a s − · ( q + q ), by which we can safely choose q = · ( q + q ). (Particularly, when a s − = 0, the probabilities Pr[ Y + Y + Y ′ ≤ s ] = Pr[ Y ≤ s ] = const depend not on q ∈ [0 , q ∈ [0 ,
1] can be arbitrary, and we just choose q = · ( q + q ).) As a consequence,Pr[ Y + Y + Y ′ ≤ t ] − Pr[ Y ≤ t ] = ( a t − a t − ) · ( q − q · q )= 14 · ( a t − a t − ) · ( q − q ) ≥ , where the last step follows because a t ≥ a t − . Case II ( a s = a s − ). In this case, we can reformulate Equation (2) as follows:Pr[ Y ≤ t ] = a t − a t − a s − a s − · (( a s − a s − ) · q · q + a s − · ( q + q ))+ (cid:16) a t − − a t − a t − a s − a s − · a s − (cid:17) · ( q + q ) + const= a t − a t − a s − a s − · (( a s − a s − ) · q + a s − · q )+ (cid:16) a t − − a t − a t − a s − a s − · a s − (cid:17) · ( q + q ) + const= Pr[ Y + Y + Y ′ ≤ t ] − (cid:16) a t − − a t − a t − a s − a s − · a s − (cid:17) · q + (cid:16) a t − − a t − a t − a s − a s − · a s − (cid:17) · ( q + q )= Pr[ Y + Y + Y ′ ≤ t ] − (cid:0) a t − − a t − a t − a s − a s − · a s − (cid:1) · (2 q − q − q ) | {z } ♥ , where the first step rearranges Equation (2); the second step applies Equation (4) to the firstsummand; the third step applies Equation (3); and the last step is by elementary calculation.15o accomplish the lemma for a certain t ∈ [0 : s − ♥ ≥ Case II.A ( a s > a s − ). To see ♥ ≥ a t − ≤ a t − a t − a s − a s − · a s − , (5) q ≤ · ( q + q ) . (6)Because a s > a s − , Equation (5) is equivalent to a t − · a s ≤ a t · a s − , i.e. what we have shownin Equation (1). And for Equation (6), assume on the opposite that q > · ( q + q ), thenLHS of (4) = ( a s − a s − ) · q + a s − · q ≥ ( a s − a s − ) · q + a s − · ( q + q ) > ( a s − a s − ) · q · q + a s − · ( q + q )= RHS of (4) , where the third step is strict because a s > a s − and q > · ( q + q ) ≥ √ q · q . This gives acontradiction. By refuting our assumption, we confirm Equation (6) and thus ♥ ≥ Case II.B ( a s < a s − ). Via similar arguments as in Case II.A, we have a t − ≥ a t − a t − a s − a s − · a s − and q ≥ · ( q + q ). Given these, we also have ♥ ≥ • Define Φ j ,j as the operator specified by Lemma 1, i.e., replacing a pair of non-identicalfailure probabilities q j = q j by another pair of identical probabilities { q } . • Define the operation composition Φ( q ) def = Φ ,n ◦ · · · ◦ Φ , ◦ Φ , ( q ), i.e. modifying ( q , q )first, then the new ( q , q ) second, and so on. Note that both Φ j ,j and Φ are continuousmappings from [0 , n to [0 , n .Accessing the proof of Lemma 1, we can easily conclude the following corollary. Corollary 1 (Averaging Two Variables.) . Given that q j = q j , the operation Φ j ,j specifiedby Lemma 1 guarantees the strict inequalities min { q j , q j } < q < max { q j , q j } . We are ready to prove Part 1 of Theorem 4, namely the existence of a desired array of i.i.d.Bernoulli random variables { X j } j ∈ [ n ] . Claim 1 (Part 1 of Theorem 4) . Pr[ X ≤ t ] ≥ Pr[ Y ≤ t ] for any t < s .Proof of Claim 1. Indeed, when not all the failure probabilities q = { q j } j ∈ [ n ] ∈ [0 , n of thegiven variables { Y j } j ∈ [ n ] are the same, we can infer from Lemma 1 and Corollary 1 an iterativealgorithm, that computes the common failure probability q ∗ = Pr[ X j = 0] ∈ [0 ,
1] of theidentically distributed variables { X j } j ∈ [ n ] . This algorithm is shown in Figure 5.In a specific round τ ∈ N ≥ , because the interim probabilities q ( τ − = { q ( τ − j } j ∈ [ n ] arereindexed in increasing order, we can infer from Corollary 1 (together with the definition of theoperation composition Φ : [0 , n [0 , n ) thatmin j ∈ [ n ] q ( τ ) j ≥ min j ∈ [ n ] q ( τ − j and max j ∈ [ n ] q ( τ ) j ≤ max j ∈ [ n ] q ( τ − j . In particular, the second inequality above is strict, as long as not all the interim probabilities q ( τ − = { q ( τ − j } j ∈ [ n ] are identical. 16 nput: failure probabilities q = { q j } j ∈ [ n ] ∈ [0 , n of { Y j } j ∈ [ n ] .1. Initialize q (0) ← q .2. for τ ∈ N ≥ do
3. Reindex q ( τ − so that q ( τ − ≤ q ( τ − ≤ · · · ≤ q ( τ − n .4. Update q ( τ ) ← Φ( q ( τ − ).5. end for Figure 5: An algorithm for Part 1 of Theorem 4.We consider the distance ℓ ( q ) def = max j ,j ∈ [ n ] ( q j − q j ) ≥
0; notice that this is a continuousfunction from [0 , n to [0 , τ ∈ N ≥ , ℓ ( q ( τ ) ) = ℓ (Φ( q ( τ − )) ≤ ℓ ( q ( τ − ) , (7)where the inequality is strictly as long as not all { q ( τ − j } j ∈ [ n ] are identical. Due to the squeezetheorem, the sequence { q ( τ ) } ∞ τ =1 converges to some limit q ∗ = lim τ →∞ q ( τ ) ∈ [0 , n . Further,since both ℓ and Φ are continuous functions, we deduce that ℓ ( q ∗ ) = lim τ →∞ ℓ ( q ( τ ) ) = lim τ →∞ ℓ (Φ( q ( τ − )) = ℓ (Φ( lim τ →∞ q ( τ − )) = ℓ (Φ( q ∗ )) . As a result, it follows from Equation (7) that all coordinates of q ∗ must be the same, namely q ∗ = { q ∗ } n for some common failure probability q ∗ ∈ [0 ,
1] of the i.i.d. { X j } j ∈ [ n ] .We conclude with the existence of the desired i.i.d. Bernoulli random variables { X j } j ∈ [ n ] and the sum X = P j ∈ [ n ] X j . In particular, applying Lemma 1 over all rounds τ ∈ N ≥ givesPr[ X ≤ t ] ≥ Pr[ Y ≤ t ] , ∀ t ∈ [0 : s ] . This completes the proof of Claim 1.
This section is to certify Part 1 of Theorem 3. Concretely, for any given n ≥ k ≥
1, the worst-case instance F = { F j } j ∈ [ n ] of Program (P1) is achieved when the distributions { F j } j ∈ [ n ] areidentical and, for any posted price p ≥
0, make the most of the
Anonymous Pricing revenue. Weformalize this statement as the following claim.
Claim 2 (Part 1 of Theorem 3) . The revenue gap ℜ AR / AP ( k, n ) is maximized when all the buyershave the same bid distribution { F ∗ } n , and their common CDF F ∗ is an implicit function givenby F ∗ ( x ) = 0 for all x ∈ [0 , k ] and AP ( x, { F ∗ } n ) = 1 for all x ∈ ( k , ∞ ) .Proof of Claim 2. Recall Program (P1), an implicit constraint is that the input must be
CDF’s ,i.e. each F j : R ≥ [0 ,
1] is a non-decreasing mapping with F j (0) = 0 and F j ( ∞ ) = 1. Werelax this constraint and consider all mappings from the domain R ≥ to the codomain [0 , F = { F j } j ∈ [ n ] the given mappings. Even though { F j } j ∈ [ n ] may not be CDF’s, we can still write down the corresponding “ i -th highest CDF” D i ( x ) andthe “revenue” formulas AP ( p, F ) and AR ( r, F ): D i ( x ) = X t ∈ [0: i − X | W | = t (cid:16) Y j / ∈ W F j ( x ) (cid:17) · (cid:16) Y j ∈ W (1 − F j ( x )) (cid:17) , ∀ x ≥ , i ∈ [ n + 1];17 P ( p, F ) = p · X i ∈ [ k ] (1 − D i ( p )) , ∀ p ≥
0; (8) AR ( r, F ) = AP ( r, F ) + k · Z ∞ r (1 − D k +1 ( x )) · d x, ∀ r ≥ . Notice that both “revenue” formulas satisfy the monotonicity given in Fact 5. For a bunch ofmappings { F j } j ∈ [ n ] that are feasible to Program (P1), we consider a two-step reduction:(i) Pointwise convert F = { F j } j ∈ [ n ] , according to Theorem 4, into a bunch of identical map-pings F = { F } n .(ii) Pointwise scale F = { F } n into another bunch of identical mappings F ∗ = { F ∗ } n , forwhich AP ( p, F ∗ ) = 1 for any p ≥ k and AP ( p, F ∗ ) = k · p for any p < k .Clearly, constraint (C1) holds for F ∗ . Below we show that for any reserve r ∈ R ≥ , the Anony-mous Reserve revenue increases, namely AR ( r, F ∗ ) ≥ AR ( r, F ).Given any x ≥
0, let us consider the independent Bernoulli random variables { Y ( x ) j } j ∈ [ n ] with the failure probabilities Pr[ Y ( x ) j = 0] def = F j ( x ). We denote their sum Y ( x ) def = P j ∈ [ n ] Y ( x ) j .According to Part 1 of Theorem 4, there exists a particular bunch of i.i.d. variables { X ( x ) j } j ∈ [ n ] ,for which the sum X ( x ) def = P j ∈ [ n ] X ( x ) j satisfiesPr[ X ( x ) ≤ k ] = Pr[ Y ( x ) ≤ k ];Pr[ X ( x ) ≤ i ] ≥ Pr[ Y ( x ) ≤ i ] , ∀ i ∈ [0 : k − . For each i ∈ [ n + 1], one can easily see that Pr[ Y ( x ) ≤ i −
1] = D i ( x ), and we further denote F ( x ) def = Pr[ X ( x ) j = 0] and D i ( x ) def = Pr[ X ( x ) ≤ i − x ≥ D k +1 ( x ) = D k +1 ( x ) , ∀ x ∈ R ≥ ; D i ( x ) ≥ D i ( x ) , ∀ x ∈ R ≥ , i ∈ [ k ] . In view of Equation (8), for any price p ∈ R ≥ we have AP ( p, F ) ≤ AP ( p, F ) ≤ AP ( p, F ∗ ) , (9)where the last inequality holds by the construction of F ∗ = { F ∗ } n given in Step (ii).Given Equation (9) and since either F ∗ = { F ∗ } n or F = { F } n involves identical mappings,we can infer from Fact 5 that the scaled common mapping F ∗ pointwise dominates F . In termsof the “( k + 1)-th highest CDF”, we have D ∗ k +1 ( x ) ≤ D k +1 ( x ) = D k +1 ( x ) for all x ≥
0. Wethus deduce that for any reserve r ∈ R ≥ , k · Z ∞ r (1 − D ∗ k +1 ( x )) · d x ≥ k · Z ∞ r (1 − D k +1 ( x )) · d x. (10)Combining Equations (9) and (10) together, we conclude that F ∗ gives a better AnonymousReserve revenue than F : for any reserve r ∈ R ≥ , AR ( p, F ∗ ) = AP ( p, F ∗ ) + k · Z ∞ r (cid:0) − D ∗ k +1 ( x ) (cid:1) · d x ≥ AP ( p, F ) + k · Z ∞ r (cid:0) − D k +1 ( x ) (cid:1) · d x = AR ( p, F ) .
18o complete the proof, it remains to show that the mapping F ∗ is indeed a CDF, namelythat F ∗ is non-decreasing, F ∗ (0) = 0 and F ∗ ( ∞ ) = 1. Under the construction given in Step (ii),we know from Equation (8) that P i ∈ [ k ] D ∗ i ( x ) = | k − x | + is an increasing function. Particularly, D ∗ i ( x ) = 0 for any x ≤ k and each i ∈ [ k ], and P i ∈ [ k ] D i ( ∞ ) = k .Indeed, suppose we regard q def = F ∗ ( x ) as a single variable, then each summand D ∗ i ( q ) := i − X t =0 (cid:18) nt (cid:19) · q n − t · (1 − q ) t is an increasing function on q ∈ [0 , D ∗ i ( q ) | q =0 = 0 and the maximum D ∗ i ( q ) | q =1 = 1. To meet all the promised properties of P i ∈ [ k ] D ∗ i (as a function of x ≥ F ∗ must be a CDF, namely an increasing function supported on x ∈ ( k , ∞ ) so that F ∗ ( k ) = 0 and F ∗ ( ∞ ) = 1.This completes the proof of Claim 2. In Section 3.2, we characterize the worst-case instance for any given population n ≥ k . To avoidambiguity, below we denote that instance by F ∗ ( n ) . In the next claim, we study the worst-casepopulation and the resulting supremum revenue gap ℜ AR / AP ( k ) = sup n ≥ k ℜ AR / AP ( k, n ). Claim 3 (Part 2 of Theorem 3) . Over all n ≥ k , the supremum revenue gap ℜ AR / AP ( k ) =sup n ≥ k ℜ AR / AP ( k, n ) is achieved by ℜ AR / AP ( k, ∞ ) = lim n →∞ AR ( F ∗ ( n ) ) = 1 + k · Z ∞ T k ( x ) · (1 − T k +1 ( x ))( k − P i ∈ [ k ] T i ( x )) · d x, where the functions T i ( x ) def = e − x · P t ∈ [0: i −
1] 1 t ! · x t for all i ∈ [ k + 1] .Proof of Claim 3. We first show that {ℜ AR / AP ( k, n ) } n ≥ k is an increasing sequence, which byinduction guarantees that ℜ AR / AP ( k ) = ℜ AR / AP ( k, ∞ ).Indeed, the worst-case n -buyer instance F ∗ ( n ) = { F ∗ ( n ) } n with the common CDF F ∗ ( n ) (spec-ified by Claim 2) can be regarded as such a ( n + 1)-buyer instance: the index-( n + 1) buyerhas a deterministic bid of zero, while every other buyer i ∈ [ n ] still has the bid CDF F ∗ ( n ) .This ( n + 1)-buyer instance is feasible to Program (P1), and gives a less Anonymous Reserve revenue than the worst-case ( n + 1)-buyer instance (due to Claim 2). That is, the n -buyer and( n + 1)-buyer revenue gaps satisfy that ℜ AR / AP ( k, n ) ≤ ℜ AR / AP ( k, n + 1), as desired.It remains to prove the promised revenue formula for the limit instance F ∗ ( ∞ ) . To this end,we first show the optimal Anonymous Reserve revenue from a specific n -buyer instance: AR ( F ∗ ( n ) ) = 1 + k · Z ∞ /k (1 − D k +1 ( x )) · d x. (11)Indeed, this optimal revenue can be achieved by any reserve r ∈ [0 , k ]: AR ( r, F ∗ ( n ) ) = r · X i ∈ [ k ] (1 − D i ( r )) + k · Z ∞ r (1 − D k +1 ( x )) · d x = k · r + k · Z /kr d x + k · Z ∞ /k (1 − D k +1 ( x )) · d x = RHS of (11) , i -th highest CDF D i ( x ) = 0 for all x ∈ [0 , k ], dueto Claim 2 that the common CDF F ∗ ( n ) is supported on x ∈ ( k , ∞ ).Moreover, any reserve r ∈ ( k , ∞ ) cannot generate a higher Anonymous Reserve revenue: AR ( r, F ∗ ( n ) ) = AP ( r, F ∗ ( n ) ) + k · Z ∞ r (1 − D k +1 ( x )) · d x = 1 + k · Z ∞ r (1 − D k +1 ( x )) · d x ≤ RHS of (11) , where the second step holds since AP ( r, F ∗ ( n ) ) = 1 (see Claim 2); and the last step holds sincethe ( k + 1)-th highest CDF D k +1 is pointwise bounded within [0 , k + 1)-th highest CDF D k +1 . Below,we consider a specific bid x ∈ ( k , ∞ ) and, for each n ≥ k and all i ∈ [ k + 1], use the shorthand F ∗ ( n ) = F ∗ ( n ) ( x ) and D i = D i ( x ). In addition, we denote by b D i def = lim n →∞ D i the i -th highestCDF resulted from the limit instance F ∗ ( ∞ ) = lim n →∞ F ∗ ( n ) .It turns out that F ∗ ( ∞ ) = 1. Otherwise, any individual buyer is willing to pay with a constant probability (1 − F ∗ ( ∞ ) ) >
0. This means the limit i -th highest CDF is b D i = lim n →∞ D i = 0 for all i ∈ [ k + 1], since there are infinite buyers n → ∞ . This incurs a contradiction to constraint (C2),namely that the Anonymous Pricing revenue exceeds one (note that x > k is given): AP ( x, F ∗ ( ∞ ) ) = x · X i ∈ [ k ] (1 − b D i ) = x · k > . Given that F ∗ ( ∞ ) = lim n →∞ F ∗ ( n ) = 1, for a sufficiently large n ≥ k we have n · (cid:16) F ∗ ( n ) − (cid:17) = (1 + o n (1)) · n · ln (cid:16) F ∗ ( n ) − (cid:17) = − (1 + o n (1)) · n · ln F ∗ ( n ) = − (1 + o n (1)) · ln D , (12)where the first step uses the Maclaurin series of ln(1 + w ) in the neighborhood of w = 0; andthe last step follows because the highest CDF D = ( F ∗ ( n ) ) n .Based on Equation (12), for any given i ∈ [ k + 1] and a sufficiently large n ≥ k , we canreformulate the i -th highest CDF D i as follows: D i = X t ∈ [0: i − (cid:18) nt (cid:19) · ( F ∗ ( n ) ) n − t · (1 − F ∗ ( n ) ) t = X t ∈ [0: i − (cid:18) nt (cid:19) · D · (cid:16) F ∗ ( n ) − (cid:17) t = (1 + o n (1)) · X t ∈ [0: i − (cid:18) nt (cid:19) · n t · D · ( − ln D ) t = (1 + o n (1)) · X t ∈ [0: i − t ! · D · ( − ln D ) t , where the first step applies Fact 1 (note that { F ∗ ( n ) } n are i.i.d.); the second step follows sincethe highest CDF D = ( F ∗ ( n ) ) n ; the third step applies Equation (12); and the last step uses thefact that (cid:0) nt (cid:1) · n t = (1 + o n (1)) · t ! . 20ollowing the above equation, the limit i -th highest CDF b D i = lim n →∞ D i satisfies that b D i = X t ∈ [0: i − t ! · ( lim n →∞ D ) · ( − ln( lim n →∞ D )) t = X t ∈ [0: i − t ! · b D · ( − ln b D ) t . (13)Note that this is an identity in the range x ∈ ( k , ∞ ). By taking the derivative, we also haved b D i d b D = X t ∈ [0: i − t ! · ( − ln b D ) t − X t ∈ [1: i − t − · ( − ln b D ) t − = 1( i − · ( − ln b D ) i − . (14)We actually have one more identity 1 = AP ( x, F ∗ ( ∞ ) ) = x · ( k − P i ∈ [ k ] b D i ) for x ∈ ( k , ∞ ),due to Claim 2 (in the case that n → ∞ ). Rearrange this identity and take the derivative:d x d b D = dd b D (cid:16) k − P i ∈ [ k ] b D i (cid:17) = 1( k − P i ∈ [ k ] b D i ) · X i ∈ [ k ] d b D i d b D = 1( k − P i ∈ [ k ] b D i ) · X i ∈ [0: k − i ! · ( − ln b D ) i = 1( k − P i ∈ [ k ] b D i ) · b D k b D , (15)where the third step applies Equation (14); and the last step applies Equation (13).Combining everything together, we deduce that AR ( F ∗ ( ∞ ) ) = 1 + k · Z ∞ /k (1 − b D k +1 ( x )) · d x = 1 + k · Z ∞ /k − b D k +1 ( x )( k − P i ∈ [ k ] b D i ( x )) · b D k ( x ) b D ( x ) · d b D ( x ) , (16)where the first step applies Equation (11) for the limit instance F ∗ ( ∞ ) = lim n →∞ F ∗ ( n ) ; and thelast step follows from Equation (15).For the above revenue formula AR ( F ∗ ( ∞ ) ), note that when the bid x ranges from k to ∞ , thehighest CDF b D ( x ) ranges from 0 to 1. Moreover, Equation (13) characterizes, as a formula of b D ( x ), the i -th highest CDF b D i ( x ). Thus, if we instead regard b D ∈ (0 ,
1) as the variable, AR ( F ∗ ( ∞ ) ) = 1 + k · Z − b D k +1 ( k − P i ∈ [ k ] b D i ) · b D k b D · d b D , where { b D i } i ∈ [2: k +1] once again are given by Equation (13).Under the substitution z def = − ln b D ∈ (0 , ∞ ), we can check via elementary calculation that AR ( F ∗ ( ∞ ) ) = 1 + k · Z ∞ T k ( z ) · (1 − T k +1 ( z )) · ( k − X i ∈ [ k ] T i ( z )) − · d z, for the functions { T i } i ∈ [ k +1] defined in the statement of the claim.This completes the proof of Claim 3. 21 emark . In the single-item case k = 1, we can deduce from Claim 3 that ℜ AR / AP (1) = 1 + Z ∞ e x − (1 + x )( e x − · d x = π ≈ . , which recovers the known result [JLTX20, Theorem 2]. In the multi-unit case k ≥
2, however,the supremum revenue gap ℜ AR / AP ( k ) does not have an elementary expression. We will showin Appendix A that ℜ AR / AP ( k ) = 1 + Θ(1 / √ k ). Associated with numeric calculation, it turnsout that the worst case arg max {ℜ AR / AP ( k ) : k ∈ N ≥ } happens when k = 1. We emphasize that all upper-bound results given in Sections 3.2 and 3.3 just require the inputdistributions { F j } j ∈ [ n ] to be independent. In this section, we construct matching lower-boundinstances respectively in the i.i.d. general setting and the asymmetric regular setting. Forconvenience, we reuse the notations introduced before. I.I.D. general setting.
Let us revisit the i.i.d. instance F ∗ ( n ) = { F ∗ ( n ) } n specified by Claim 2.As mentioned, the revenue gap {ℜ AR / AP ( k, n ) } n ≥ k is an increasing sequence in the population n ≥ k , and the limit/supremum revenue gap ℜ AR / AP ( k ) = lim n →∞ ℜ AR / AP ( k, n )is finite for any k ∈ N ≥ (see Appendix A). Accordingly, for a given ε >
0, there is a thresholdpopulation N ( ε ) ≥ k so that ℜ AR / AP ( k, n ) ≥ ℜ AR / AP ( k ) − ε , for any n ≥ N ( ε ). Clearly, suchinstances F ∗ ( n ) = { F ∗ ( n ) } n give the matching lower bound.The common CDF F ∗ ( n ) specified in Claim 2 turns out to be the equal-revenue distribution(i.e. a “boundary-case” regular distribution) when k = n = 1, but is an irregular distributionotherwise. For example, when k = 1 and n ≥
2, we have F ∗ ( n ) ( x ) = n r(cid:12)(cid:12)(cid:12) − x (cid:12)(cid:12)(cid:12) + , and the irregularity is shown in [JLTX20, Lemma 12]. In the other cases n ≥ k ≥
2, theirregularity can be seen via similar but more technical arguments. For ease of presentation,here we omit the formal proof.
Asymmetric regular setting.
We next use the triangle distributions to construct an instancewhose revenue gap matches the bound ℜ AR / AP ( k ) given in Claim 3. (Recall Section 2.1 that atriangle distribution must be regular.) To this end, we would reuse the notations introduced inthe proof of Claim 3. The following claim is useful. Claim 4 (Threshold for Lower Bound) . Consider the limit instance F ∗ ( ∞ ) as well as its i -th highest CDF’s { b D i } i ∈ [ k +1] given in Claim 3. For any ε > , there exists a large enough N ( ε ) ∈ N ≥ so that k · Z ba (1 − b D k +1 ( x )) · d x ≥ ℜ AR / AP ( k ) − − ε, where a def = k + N ( ε ) and b def = k + N ( ε ) ; note that a ≤ b .Proof. According to Equation (16), in the limit case N ( ε ) → ∞ we have k · Z ∞ /k (1 − b D k +1 ( x )) · d x = AR ( F ∗ ( ∞ ) ) − ℜ AR / AP ( k ) − . We know from Claim 6 (see Appendix A) that the above improper integral =
Θ(1) √ k is finite. Inaddition, the integrand is a non-negative function. Given these, we can easily see Claim 4.22 P ( p, F ) p − σ /k v n,l v n − ,l v n − ,l · · · · · · v ,l v ,l v ,l b = v ,l Figure 6: Demonstration for the triangle instance F given in Example 1, where σ > δ > n ∈ N ≥ is large enough).Based on the above parameters b ≥ a , we now construct a desired lower-bound instance. Example 1 (Lower-Bound Instance in Asymmetric Regular Setting) . Denote δ def = b − an > n ≥ k will be determined later. As Figure 6 shows, consider such an ( n + nk )-buyer triangle instance F def = { Tri ( v ,l , q ,l ) } l ∈ [ n ] ∪ { Tri ( v j,l , q j,l ) } j ∈ [ n ] ,l ∈ [ k ] : • In the 0-th group, the involved monopoly prices v ,l def = b for l ∈ [ n ] are identical. In eachgroup j ∈ [ n ], the involved monopoly prices v j,l def = b − j · δ for l ∈ [ k ] are identical. • In the 0-th group, the involved monopoly quantiles { q ,l } l ∈ [ n ] are identical, which togethergive a unit Anonymous Pricing revenue AP ( p, F ) = 1 under the posted price p = v ,l = b .The remaining monopoly quantiles { q j,l } j ∈ [ n ] ,l ∈ [ k ] are defined recursively. In each group j ∈ [ n ], the involved { q j,l } l ∈ [ k ] are identical, which give a unit Anonymous Pricing revenue AP ( p, F ) = 1 under the posted price p = v j,l = b − j · δ . Claim 5 (Part 4 of Theorem 3 in Asymmetric Regular Setting) . The ( n + nk ) -buyer triangleinstance F in Example 1 is well defined, and satisfies the following: AP ( p, F ) ≤ for any posted price p ∈ R ≥ . There exists a threshold N ( ε ) ∈ N ≥ such that for any n ≥ N ( ε ) , AR ( F ) ≥ ℜ AR / AP ( k ) − · ε. Proof of Claim 5.
We first show that F is well defined or more precisely, the monopoly quantilesare well defined. Because the monopoly prices/quantiles in an individual group are identical,without ambiguity we denote v = v ,l = b and q = q ,l for l ∈ [ n ] and v j = v j,l = b − j · δ and q j = q j,l for l ∈ [ k ]. Recall Section 2.1 that a triangle distribution Tri ( v j , q j ) has the CDF F j ( x ) = ( (1 − q j ) · x (1 − q j ) · x + v j q j , when x ∈ [0 , v j ]1 , when x ∈ ( v j , ∞ ) . Under the posted price p = v = b , we have F ( p ) = 1 − q for the 0-th group and F j ( p ) = 1for any other group j ∈ [ n ]. Thus, only the group-0 buyers contribute to the Anonymous Pricing revenue AP ( v , F ). This revenue formula AP ( v , F ) can be regarded as a continuous function in q ∈ [0 , p = v = b with probability q = 0, then AP ( v , F ) = 0.(ii) If a group-0 buyer is willing to pay p = v = b with probability q = 1, then AP ( v , F ) = v · min { n, k } = b · k > k · k = 1 (given that n ≥ k and b > k ).Given these and due to the intermediate value theorem, AP ( v , F ) = 1 for some q ∈ [0 , m ∈ [0 : n − q j ∈ [0 ,
1] in the groups j ∈ [0 : m ] are well defined, below we justify the existence of the group-( m + 1) monopolyquantiles q m +1 ∈ [0 , p ∈ ( v m +1 , v m ], the revenue AP ( p, F ) is contributedonly by the buyers in the groups j ∈ [0 : m ]. In particular, when p = v m , by construction wehave AP ( v m , F ) = 1. Within the support x ∈ [0 , v j ], a triangle distribution Tri ( v j , q j ) has thevirtual value function ϕ j ( x ) = x − − F j ( x ) f j ( x ) = ( − v j q j − q j , when x ∈ [0 , v j ) v j , when x = v j . Hence, any allocation under any posted price p ∈ ( v m +1 , v m ) gives a negative virtual welfare.Due to the revenue-equivalence theorem [Mye81], the revenue formula AP ( p, F ) is a strictlyincreasing function in p ∈ ( v m +1 , v m ].When p = v m +1 , we shall incorporate the contribution from the group-( m + 1) buyers intothe revenue AP ( v m +1 , F ) as well. Once again, this revenue formula AP ( v m +1 , F ) can be regardedas a continuous function in q m +1 ∈ [0 , m + 1) buyer is willing to pay p = v m +1 with probability q m +1 = 0, then wehave AP ( v m +1 , F ) = lim p → v + m +1 AP ( p, F ) <
1, where the inequality holds because AP ( p, F )is a strictly increasing function when p ∈ ( v m +1 , v m ].(ii) If a group-( m + 1) buyer is willing to pay p = v m +1 with probability q = 1, since thereare k such buyers, we have AP ( v m +1 , F ) = v m +1 · k > k · k = 1, where the inequality holdsbecause v m +1 ≥ a > k (by construction).Once again, we deduce from the intermediate value theorem that AP ( v m +1 , F ) = 1 for some q m +1 ∈ [0 , m + 1) monopoly quantiles are well defined. By induction, thetriangle instance F is well defined.From the above arguments, we also conclude Part 1 that AP ( p, F ) ≤ p ∈ R ≥ .We next justify Part 2 that the optimal Anonymous Reserve revenue AR ( F ) ≥ ℜ AR / AP ( k ) − · ε when the n ∈ N ≥ is large enough. To this end, let us consider the specific reserve r = v n = a .Indeed, when n is large enough, the ( k + 1)-th highest D k +1 resulted from F satisfy that k · Z ba (1 − D k +1 ( x )) · d x ≥ k · Z ba (1 − b D k +1 ( x )) · d x − ε, (17)Assume Equation (17) to be true, then Part 2 follows immediately: AR ( a, F ) = AP ( a, F ) + k · Z ba (1 − D k +1 ( x )) · d x ≥ k · Z ba (1 − D k +1 ( x )) · d x ≥ k · Z ba (1 − b D k +1 ( x )) · d x − ε = ℜ AR / AP ( k ) − · ε, AP ( a, F ) = AP ( v n , F ) = 1; the third stepapplies Equation (17); and the last step applies Claim 4.We are left to prove Equation (17). By construction, one can easily see that in the limit case n → ∞ , every individual monopoly quantile q j involved in F approaches to 0 + . Namely, theCDF lim n →∞ F j ( x ) → − for any x ∈ R ≥ . Given this, reusing the arguments for Equation (13),it can be seen that for each i ∈ [ k + 1], the following holds for the limit i -th highest CDF:lim n →∞ D i ( x ) = X t ∈ [0: i − t ! · ( lim n →∞ D ( x )) · ( − ln( lim n →∞ D ( x ))) t , for all x ∈ ( k , ∞ ). Accessing the proof of Claim 3, for the limit instance F ∗ ( ∞ ) therein, we havethe counterpart identities b D i ( x ) = P t ∈ [0: i −
1] 1 t ! · b D ( x ) · ( − ln b D ( x )) t for all x ∈ ( k , ∞ ).By construction (as Figure 6 suggests), in the limit case n → ∞ , we have another identity lim n →∞ AP ( x, F ) = 1 , for all x ∈ [ a, b ]. Accessing the proof of Claim 3, for the limit instance F ∗ ( ∞ ) therein, we havethe counterpart identity AP ( x, F ∗ ( ∞ ) ) = 1 for all x ∈ ( k , ∞ ). Recall that [ a, b ] ⊆ ( k , ∞ ).Based on the above identities, we can reapply the arguments for Claim 3 and deduce thatlim n →∞ D k +1 ( x ) = b D k +1 ( x ) , for all x ∈ [ a, b ]. Given this, and since both integrals R ba (1 − D k +1 ( x )) · d x and R ba (1 − b D k +1 ( x )) · d x in Equation (17) are definite integrals, and both integrands y = 1 − D k +1 ( x ) and b y = 1 − b D k +1 ( x )are bounded between [0 , n ∈ N ≥ .This completes the proof of Claim 5. In Sections 3.2 to 3.4 we assume that any subset of up to k ∈ N ≥ willing-to-pay buyers canwin simultaneously, i.e. the winners meet a rank- k uniform matroid constraint. To model someparticular markets, many past works on Bayesian mechanism design also consider the generalmatroid constraints.In this new scenario, the revenue gap between Anonymous Reserve and
Anonymous Pricing is no longer a constant, or precisely, ℜ AR / AP = Ω(log k ). In the rest of this section, we assumebasic knowledge about matroid, for which the reader can turn to [Oxl06].Regarding a general rank- k matroid constraint, Anonymous Pricing runs almost in the sameway: a certain buyer i ∈ [ n ], upon arriving, gets a copy of the item iff (i) he together with thepast winning buyers form an independent set of the matroid; and (ii) he is willing to pay theposted price p ≥
0. No ambiguity would arise throughout the conduct of
Anonymous Pricing ,due to the greedy structure of matroids.To implement
Anonymous Reserve , the seller should use
VCG Auction instead of ( k + 1) -thPrice Auction : (i) the seller runs VCG Auction only on the buyers whose bids { b j } j ∈ [ n ] are atleast the reserve r ≥
0, by taking the matroid constraint into account; and (ii) each winner paysthe threshold bid for him to keep winning.Our lower-bound example with the Ω(log k ) revenue gap is constructed below. More precisely, by construction we have AP ( v j , F ) = 1 for every j ∈ [0 : n ]. Concerning the revenue formula AP ( p, F ) = p · P i ∈ [ k ] (1 − D i ( p )), we notice that the i -th highest CDF’s { D i } i ∈ [ k ] are increasing functions. Giventhese, for any j ∈ [0 : n −
1] and any posted price p ∈ ( v j +1 , v j ] we have AP ( p, F ) ≥ pv j · AP ( v j , F ) = pv j ≥ v j +1 v j .Under our construction that v j = b − j · δ for all j ∈ [0 : n ], where δ = b − an , the minimum v j +1 v j is equalto v n v n − = aa +( b − a ) /n ≥ /k /k +( b − a ) /n ≥ − ( b − a ) · kn . Thus, for any p ∈ [ a, b ] we have lim n →∞ AP ( p, F ) ≥ lim n →∞ (1 − ( b − a ) · kn ) = 1. On the other hand, we have shown that AP ( p, F ) ≤ p ∈ [ a, b ] (see Part 1of the claim). heorem 5 ( AR vs. AP under a Matroid Constraint) . When the seller faces n ≥ independentunit-demand buyers and the winners satisfy a rank- k matroid constraint, the revenue gap ℜ AR / AP between Anonymous Reserve and
Anonymous Pricing is lower bounded by
Ω(log k ) .Proof of Theorem 5. For simplicity, we assume that n = 2 m is an even integer and that k ≤ m ;the lower-bound instance for the general case is very similar. The buyers are divided into m pairs, and each pair i ∈ [ m ] involves the (2 i − i -th buyers. We consider a specificrank- k matroid M in terms of the collection B of its bases:Any base B ∈ B contains exactly one buyer from each chosen pair, for some choiceof k pairs. In total, there are |B| = (cid:0) mk (cid:1) · k bases.One can easily justify the augmentation property, thus showing M to be a matroid (or moreprecisely, a laminar matroid with the laminar family { [1 : m ] , [ m + 1 : 2 m ] , [1 : 2 m ] } and thecapacity function c ([1 : m ]) = c ([ m + 1 : 2 m ]) = c ([1 : 2 m ]) = k ). Further, both buyers of each i -th pair have a deterministic bid max { i , k +1 } .In Anonymous Pricing , when the seller posts a price p > k , exactly ⌊ p ⌋ pairs would pay thisprice, hence a revenue p · ⌊ p ⌋ ≤
1. When the price p ≤ k , although all the k copies will be soldout, the revenue p · k is still at most 1. But when the seller instead employs VCG Auction (evenwithout a reserve), either buyer in each of the top- k pairs will get an item by paying i , thus arevenue of P j ∈ [ k ] 1 i = Ω(log k ).This completes the proof of Theorem 5. Remark . We notice that a deterministic bid satisfies the regularity distributional assumption,as well as the stronger monotone-hazard-rate assumption. Thus, the Ω(log k ) lower bound stillholds for the revenue gap ℜ AR / AP in these restricted settings. In this section, we investigate the
Ex-Ante Relaxation ( EAR ) vs.
Anonymous Pricing ( AP ) prob-lem, under the regularity assumption that F = { F j } j ∈ [ n ] ⊆ Reg . Based on the revenue formulas(see Section 2.2), the revenue gap between both mechanisms is given by the optimal solutionto the following mathematical program. Recall that D i is the i -th highest bid distribution, and Reg is the family of all regular distributions.sup
EAR ( q ′ , F ) = X j ∈ [ n ] F − j (1 − q ′ j ) · q ′ j (P2)s.t. AP ( p, F ) = p · X i ∈ [ k ] (1 − D i ( p )) ≤ , ∀ p ∈ R ≥ , X j ∈ [ n ] q ′ j ≤ k, q ′ = { q ′ j } j ∈ [ n ] ∈ [0 , n , F = { F j } j ∈ [ n ] ⊆ Reg , ∀ n ∈ N ≥ . We will establish an O (log k ) upper bound for the optimal solution to Program (P2), whichis formalized as Theorem 6. Combine this result with the matching lower bound by [HR09,Example 5.4], then the revenue gap gets understood. Theorem 6 ( EAR vs. AP ) . Given that the seller has k ∈ N ≥ homogeneous items and faces n ≥ k independent unit-demand buyers, who have regular value distributions F = { F j } j ∈ [ n ] ⊆ Reg ,the revenue gap between
Ex-Ante Relaxation and
Anonymous Pricing is ℜ EAR / AP ( k ) = Θ(log k ) . A distribution F j has monotone hazard rate if y = ln(1 − F j ( x )) is a concave function, e.g., see [JLX19].
26e establish Theorem 6 in three steps. First, we give a reduction from a regular instanceto a triangle instance, which preserves the feasibility; then we just need to optimize n pairs ofmonopoly price and quantile { ( v j , q j ) } j ∈ [ n ] instead of n regular distributions { F j } j ∈ [ n ] . Second,we relax the constraint AP ( p, F ) ≤ { D i } i ∈ [ k ] . Afterwards, we divide all buyers into three careful groupsunder certain criteria for { ( v j , q j ) } j ∈ [ n ] , and separately bound the contribution from each groupto the EAR revenue. The total
EAR revenue turns out to be O (log k ). Reduction to triangle instances.
For the single-item case k = 1, [AHN +
19] show that theworst case of Program (P2) w.l.o.g. is achieved by a triangle instance. Indeed, their argumentswork in the general case k ∈ N ≥ as well. Formally, we have the following lemma (see Figure 7for a demonstration). Lemma 2 (Reduction for
EAR vs. AP [AHN +
19, Lemma 4.1]) . Given a feasible solution ( q ′ , F ) to Program (P2) , there exists another n -buyer feasible instance ( q ∗ , F ∗ ) such that: The distributions F ∗ = { F ∗ j } j ∈ [ n ] ⊆ Tri are triangle distributions, and q ∗ = { q ∗ j } j ∈ [ n ] ∈ [0 , n (such that P j ∈ [ n ] q ∗ j ≤ k ) are the monopoly quantiles thereof. The
Ex-Ante Relaxation revenue keeps the same, i.e.
EAR ( q ∗ , F ∗ ) = EAR ( q ′ , F ) . The distributions F ∗ = { F ∗ j } j ∈ [ n ] are stochastically dominated by F = { F j } j ∈ [ n ] and thus,for any price p ∈ R ≥ , the Anonymous Pricing revenue drops, i.e. AP ( p, F ∗ ) ≤ AP ( p, F ) . In view of Lemma 2, to establish Theorem 6 we can focus on Program (P3) in place of theprevious Program (P2). For a triangle distribution
Tri ( v j , q j ), where v j = F − j (1 − q j ) ≥ F j to denote its CDF. Recall Section 2.1 that F j ( x ) = (1 − q j ) · x (1 − q j ) · x + v j q j for all x ≤ v j and F j ( x ) = 1 for all x > v j .sup EAR ( F ) = X j ∈ [ n ] v j q j (P3)s.t. AP ( p, F ) = p · X i ∈ [ k ] (1 − D i ( p )) ≤ , ∀ p ∈ R ≥ , (C2) X j ∈ [ n ] q j ≤ k, (C3) F = { Tri ( v j , q j ) } j ∈ [ n ] ⊆ Reg , ∀ n ∈ N ≥ . For a single triangle distribution
Tri ( v j , q j ), the optimal Anonymous Pricing revenue from itequals AP ( Tri ( v j , q j )) = v j q j , which ≤ AP ( F ) ≤ v j q j ≤ , ∀ j ∈ [ n ] . (C4) Relaxing constraint (C2) . Given Program (P3), both the objective function
EAR ( F ) andconstraint (C3) are easy to deal with. However, constraint (C2) is rather complicated, becauseit involves the correlated top- k bids { b ( i ) } i ∈ [ k ] and the corresponding order CDF’s { D i } i ∈ [ k ] (asformulas of the individual CDF’s { F j } j ∈ [ n ] ) are cumbersome.The following Lemma 3 relaxes constraint (C2) to another constraint. The resulting con-straint is much easier to reason about. Namely, it avoids the correlation among the top- k bids { b ( i ) } i ∈ [ k ] and admits a clean formula of the individual CDF’s { F j } j ∈ [ n ] . Later we will see thatafter this relaxation, the optimal objective value of Program (P3) blows up just by a constantmultiplicative factor. Denote m def = ⌊ k ⌋ ≥ ( q ) q R j ( q ) 1 q ∗ j = 1 − F j ( v ∗ j ) R j ( q ∗ j ) (a) A concave revenue-quantile curve R ( q ) q R ∗ j ( q ) 1 q ∗ j = 1 − F j ( v ∗ j ) v ∗ j q ∗ j (b) A triangular revenue-quantile curve Figure 7: Demonstration for the reduction in Lemma 2, in terms of the revenue-quantile curves .For a distribution F j , its revenue-quantile curve is given by R j ( q ) = q · F − j (1 − q ) for q ∈ [0 , F j is regular iff the R j is a concave function (as Figure 7a suggests). And therevenue-quantile curve of a triangle distribution is basically a triangle (i.e., a 2-piecewise linearfunction, as Figure 7b suggests); in particular, the two base angles have the tangent values v ∗ j and v ∗ j q ∗ j / (1 − q ∗ j ), respectively. Lemma 3 (Relaxed Constraint) . The following is a necessary condition for constraint (C2) : X j ∈ [ n ] (1 − F j ( p )) ≤ p , ∀ p ∈ h m , i . Proof of Lemma 3.
Let us consider a specific price p ∈ [ m , ] for constraint (C2). For any j ∈ [ n ], let the independent Bernoulli random variable X j ∈ { , } denote whether the j -thbuyer is willing to pay the price p , with the failure probability Pr[ X j = 0] = F j ( p ). Then X def = P j ∈ [ n ] X j denotes how many buyers are willing to pay, and Y def = min { k, X } denotes howmany items are sold out in Anonymous Pricing .We have the revenue AP ( p, F ) = p · E [ Y ], and constraint (C2) is identical to E [ Y ] ≤ p . Forthe equation given in Lemma 3, the LHS = P j ∈ [ n ] Pr[ X j = 1] = P j ∈ [ n ] E [ X j ] = E [ X ].On the opposite of Lemma 3, suppose that E [ X ] > p . We have E [ X ] >
8, given that theprice p ≤ . Since X is the sum of independent Bernoulli random variables, due to Chernoffbound, Pr[ X < (1 − δ ) · E [ X ]] < e − δ · E [ X ] (1 − δ ) (1 − δ ) · E [ X ] for any δ ∈ (0 , h X < · E [ X ] i < (cid:16) e (cid:17) · E [ X ] < (cid:16) e (cid:17) ≈ . < , (18)where the first step follows by setting δ = ; and the second step follows since E [ X ] > Y = min { k, X } , we further deduce thatPr h Y ≥ min n k, · E [ X ] oi = 1 − Pr h Y < min n k, · E [ X ] oi = 1 − Pr h X < min n k, · E [ X ] oi ≥ − Pr h X < · E [ X ] i > , (19)where the second step follows since Y < min { k, · E [ X ] } holds only if Y < k , and thus only if Y = X ; and the last step follows from Equation (18).28ased on the above arguments, we conclude a contradiction E [ Y ] > p as follows: E [ Y ] ≥ Pr h Y ≥ min n k, · E [ X ] oi · min n k, · E [ X ] o > · min n k, · E [ X ] o ≥ · min n k, p o ≥ p , where the second step applies Equation (19); the third step applies our assumption E [ X ] > p ;and the last step follows as p ≤ m ≤ k , given that p ∈ [ m , ] and m = ⌊ k ⌋ .By refuting the assumption, we get E [ X ] ≤ p for any price p ∈ [ m , ]. This completes theproof of Lemma 3.Given a triangle instance { Tri ( v j , q j ) } j ∈ [ n ] , by plugging the CDF formulas { F j } j ∈ [ n ] , we canreformulate Lemma 3 as follows: X j ∈ [ n ]: v j ≥ p v j q j (1 − q j ) · p + v j q j ≤ p , ∀ p ∈ h m , i . (C2 ′ ) Grouping the buyers.
To upper bound the objective function
EAR ( F ) = P j ∈ [ n ] v j q j , let uspartition all the buyers into three groups [ n ] = A ⊔ B ⊔ C , where A def = n j ∈ [ n ] : v j ≥ m and v j q j − q j ≥ m o ,B def = n j ∈ [ n ] : v j ≥ m and v j q j − q j < m o ,C def = n j ∈ [ n ] : v j < m o . Regarding the groups A , B and C given above, their individual contributions to the bench-mark EAR ( F ) actually admit the following bounds: X j ∈ A v j q j = O (log k ) , X j ∈ B v j q j ≤ , X j ∈ C v j q j ≤ . Suppose these bounds to be true, then combining them together immediately gives Theorem 6.Below we explain the intuitions of our grouping criteria (Remark 4), give an interesting ob-servation for the instances that are constituted by “small” distributions (Remark 5), and thenverify the above three bounds in the reverse order.
Remark . Recall the objective function of Program (P3), i.e.,
EAR ( F ) = P j ∈ [ n ] v j q j , and constraint (C3), i.e., P j ∈ [ n ] q j ≤ k . Here the monopoly revenues { v j q j } j ∈ [ n ] arethe individual contributions by the triangle distributions { Tri ( v j , q j ) } j ∈ [ n ] , and (in the senseof the Knapsack Problem ) the monopoly quantiles { q j } j ∈ [ n ] can be regarded as the individualcapacities. Thereby, the monopoly prices { v j } j ∈ [ n ] somehow are the bang-per-buck ratios (i.e.,the contribution to the EAR benchmark per unit of the capacity).Of course we prefer those distributions with higher bang-per-buck ratios { v j } j ∈ [ n ] , but alsoneed to take the capacities { q j } j ∈ [ n ] into account. In particular: • The group- C distributions have lower bang-per-buck ratios v j ≤ /m . So conceivably, thetotal contribution P j ∈ C v j q j by this group to the EAR benchmark shall be small, and wewill prove an upper bound of 3. 29
The group- B distributions have high enough bang-per-buck ratios v j ≥ /m but smallcapacities, namely v j q j / (1 − q j ) < /m . It turns out that the total contribution P j ∈ B v j q j by this group is also small, and we will prove an upper bound of 8. • The group- A distributions have high enough bang-per-buck ratios as well as big enoughcapacities. Thus, this group should contribute the most to the EAR benchmark, for whichwe will show P j ∈ A v j q j = O (log k ).Indeed, our grouping criteria borrow ideas from the “budget-feasible mechanism design” lit-erature [Sin10, CGL11, GJLZ20], where the primary goal is to design approximately optimalmechanisms for the Knapsack Problem under the incentive concerns. Remark . As argued in Section 1.2, regarding a continuum of “small”buyers (i.e., any single buyer has an infinitesimal contribution to the
EAR benchmark, but thereare infinitely many buyers n → ∞ ), the EAR vs. AP revenue gap would be (at most) a universalconstant for whatever k ≥
1. This is because every “small” buyer belongs to either group B orgroup C , and thus the EAR benchmark is at most P j ∈ B ∪ C v j q j ≤ Revenue from group C . Since such a buyer j ∈ C has a monopoly price v j < m , we have X j ∈ C v j q j ≤ m · X j ∈ C q j ≤ m · X j ∈ [ n ] q j ≤ m · k ≤ , where the second step follows since C ⊆ [ n ]; the third step follows from constraint (C3); andthe last step holds for m = ⌊ k ⌋ and k ≥
4. (We will deal with the cases k ∈ { , , } separately,at the end of this section.) Revenue from group B . Setting p = m for constraint (C2 ′ ), we deduce that4 m = RHS of (C2 ′ ) ≥ LHS of (C2 ′ ) = X j ∈ [ n ]: v j ≥ m v j q j (1 − q j ) · m + v j q j ≥ X j ∈ B v j q j (1 − q j ) · m + v j q j ≥ X j ∈ B v j q j (1 − q j ) · m + (1 − q j ) · m ≥ m · X j ∈ B v j q j , where the second line follows since { j ∈ [ n ] : v j ≥ m } ⊇ B (see the definition of B ); the thirdline follows since v j q j − q j < m for any j ∈ B ; and the last line drops the (1 − q j ) terms and thenrearranges the formula.Rearranging the above equation immediately gives P j ∈ B v j q j ≤
8, as desired.
Revenue from group A . To verify the upper bound about this group, we shall generalize thedefinition of A , and get a chain of subgroups A = A m ⊇ A m − ⊇ · · · ⊇ A : A t def = n j ∈ [ n ] : v j ≥ t and v j q j − q j ≥ t o , ∀ t ∈ [2 : m ] . t ∈ [2 : m ], by setting p = t ∈ [ m , ] for constraint (C2 ′ ), we deduce that4 t = RHS of (C2 ′ ) ≥ LHS of (C2 ′ ) = X j ∈ [ n ]: v j ≥ t v j q j (1 − q j ) · t + v j q j ≥ X j ∈ A t v j q j (1 − q j ) · t + v j q j ≥ X j ∈ A t v j q j v j q j + v j q j = 12 · | A t | , where the second step follows because { j ∈ [ n ] : v j ≥ m } ⊇ A t (see the definition of A t ); andthe third step follows because (1 − q j ) · t ≤ v j q j for each j ∈ A t .Based on the above equation, we easily bound the cardinality | A t | ≤ t for each t ∈ [2 : m ].Combining the above arguments together gives X j ∈ A v j q j = X j ∈ A m v j q j = X j ∈ A v j q j + X t ∈ [3: m ] X j ∈ A t \ A t − v j q j ≤ X j ∈ A v j q j + X t ∈ [3: m ] X j ∈ A t \ A t − t − · X j ∈ A v j q j + X t ∈ [3: m ] | A t | − | A t − | t − (cid:16) X j ∈ A v j q j − | A | (cid:17) + | A m | m − X t ∈ [3: m ] | A t | · (cid:16) t − − t (cid:17) ≤ (cid:16) X j ∈ A v j q j − | A | (cid:17) + 8 mm − X t ∈ [3: m ] t · (cid:16) t − − t (cid:17) ≤ (cid:16) X j ∈ A v j q j − | A | (cid:17) + 16 + X t ∈ [3: m ] t · (cid:16) t − − t (cid:17) = (cid:16) X j ∈ A v j q j − | A | (cid:17) + 8 + X t ∈ [ m − t , (20)where the second line follows because the monopoly price v j ∈ ( t − , t ] for each j ∈ A t \ A t − (see the definitions of A t and A t − ), and the monopoly quantiles q j ∈ [0 ,
1] are bounded; thefifth line applies the bounds | A t | ≤ t for each t ∈ [2 : m ]; the sixth line holds for m = ⌊ k ⌋ and k ≥
4; and the last line is by elementary calculation.Because v j q j ≤ j ∈ A (see constraint (C4)) and | A | ≤
16, we can bound the firstterm in Equation (20): P j ∈ A v j q j − | A | ≤ | A | − | A | ≤
8. Plug this into Equation (20): X j ∈ A v j q j ≤
16 + X t ∈ [ m − t = O (log k ) , where the last step holds for m = ⌊ k ⌋ . Upper bound when k ∈ { , , } . Clearly, the optimal value ℜ EAR / AP ( k ) of Program (P2),which involves k ∈ N ≥ items in both mechanisms, is at most the revenue gap between the k -item Ex-Ante Relaxation and the 1-item
Anonymous Pricing . The later revenue gap is given bythe next mathematical program.sup X j ∈ [ n ] F − j (1 − q ′ j ) · q ′ j (P4)31.t. p · (1 − D ( p )) ≤ , ∀ p ∈ R ≥ , X j ∈ [ n ] q ′ j ≤ k, q ′ = { q ′ j } j ∈ [ n ] ∈ [0 , n , F = { F j } j ∈ [ n ] ⊆ Reg , ∀ n ∈ N ≥ . The only difference between Program (P4) and the one in [AHN +
19, Section 4] is theconstraint P j ∈ [ n ] q ′ j ≤ k (rather than ≤ +
19, Section 4]. By doing so, we will get ℜ EAR / AP ( k ) ≤ optimal value of (P4) = 1 + V ( Q − ( k )) , where the functions V ( p ) def = p · ln( p p − ) and Q ( p ) def = ln( p p − ) − · P ∞ t =1 t − · p − t . Then we canderive Theorem 6 in the case k ∈ { , , } via numeric calculation, as the next table shows. k V ( Q − ( k )) ≈ . ≈ . ≈ . Acknowledgements.
We would like to thank Xi Chen, Eric Neyman, Tim Roughgarden,and Rocco Servedio for helpful comments on an earlier version of this work.
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A Asymptotic Formulas for the Revenue Gap ℜ AR / AP ( k ) Claim 6 (Part 3 of Theorem 3) . For each k ∈ N ≥ , the supremum revenue gap ℜ AR / AP ( k ) = 1 + k · Z ∞ T k ( x ) · (1 − T k +1 ( x ))( k − P i ∈ [ k ] T i ( x )) · d x, where the functions T i ( x ) def = e − x · P t ∈ [0: i −
1] 1 t ! · x s for all i ∈ [ k + 1] , is bounded between . √ k ≤ ℜ AR / AP ( k ) ≤ √ k . The value of ℜ AR / AP ( k ) when k ≤ is listed in Table 6. ℜ AR / AP ( k ) π / . . . . . . . c k . . . . . . . . k ℜ AR / AP ( k ) 1 . . . . . . . . c k . . . . . . . . k
17 18 19 20 21 22 23 24 ℜ AR / AP ( k ) 1 . . . . . . . . c k . . . . . . . . ℜ AR / AP ( k ) for k ≤
24, where c k means that ℜ AR / AP ( k ) = 1 + c k / √ k . Proof of Claim 6 (Lower Bound).
We define lb ( k ) def = 1 + k · R ∞ T k ( x ) · (1 − T k +1 ( x )) · k − · d x .Apparently lb ( k ) ≤ ℜ AR / AP ( k ) for all k ≥
1. Then we have lb ( k ) = 1 + 1 k Z ∞ T k ( x ) · (1 − T k +1 ( x )) · d x = 1 + 1 k Z ∞ k − X i =0 e − x x i /i ! · (1 − k X i =0 e − x x i /i !) · d x = 1 + (cid:16) k Z ∞ k − X i =0 e − x x i /i ! · d x (cid:17) − (cid:16) k Z ∞ k − X i =0 k X j =0 e − x x i + j /i ! /j !d x (cid:17) = 2 − k Z ∞ k − X i =0 k X j =0 e − x x i + j /i ! /j !d x = 2 − k Z ∞ k − X i =0 k X j =0 e − x x i + j /i ! /j ! / i + j d x = 2 − k k − X i =0 k X j =0 (cid:18) i + ji (cid:19) / i + j = 2 − k k − X i =0 i X j =0 (cid:18) ij (cid:19) / i − X i + j ≤ k − i ≥ k or j>k (cid:18) i + ji (cid:19) / i + j = 1 + 22 k · X i + j ≤ k (cid:18) i + j + kj (cid:19) / i + j + k = 1 + 1 k · k X m = k g ( m ) / m , where the second step is by definition of T k ( x ), the third step is by Fact 8 that R ∞ e − x x n d x = n !,the fifth step is by substitution, the sixth step is by Fact 8, and in the last step we define g ( m ) def = P m − ki =0 (cid:0) mi (cid:1) .For all m ∈ {⌈ k − √ k/ ⌉ , ⌈ k − √ k/ ⌉ + 1 , · · · , k } , g ( m ) = m − k X i =0 (cid:18) mi (cid:19) ⌈ m/ ⌉ X i =0 (cid:18) mi (cid:19) − ⌈ m/ ⌉ X i = m − k +1 (cid:18) mi (cid:19) ≥ m / − ( ⌈ m/ ⌉ − ( m − k )) · (cid:18) m ⌈ m/ ⌉ (cid:19) ≥ m − √ k · · m √ πm ≥ m − m √ π ≥ m / , where the fourth step is by Fact 7.Therefore, ℜ AR / AP ( k ) ≥ lb ( k )= 1 + 1 k · k X m = k g ( m ) / m ≥ k k X m = ⌈ k −√ k/ ⌉ g ( m ) / m ≥ / (10 √ k ) . This accomplishes the lower-bound part of Claim 6.
Proof of Claim 6 (Upper Bound).
We define a ( x ) def = k − X i =0 x i i ! ; b ( x ) def = ∞ X i = k +1 x i i ! . Then the integral part of ℜ AR / AP ( k ) can be written as h ( x ) def = T k ( x ) · (1 − T k +1 ( x )) · ( k − X i ∈ [ k ] T i ( x )) − = a ( x ) b ( x ) · (cid:0) X i ∈ [ k ] (1 − T i ( x )) (cid:1) − = a ( x ) b ( x ) · (cid:0) X i ∈ [ k ] ∞ X j = i x j j ! (cid:1) − = a ( x ) b ( x ) · (cid:0) X i ∈ [ k ] i · x i i ! + k · ∞ X i = k +1 x i i ! (cid:1) − = a ( x ) b ( x ) · ( x · a ( x ) + k · b ( x )) − . Then ℜ AR / AP ( k ) = 1 + k · Z ∞ h ( x )d x = 1 + k · (cid:16) Z k −√ k h ( x )d x | {z } h + Z kk −√ k h ( x )d x | {z } h + Z ∞ k h ( x )d x | {z } h (cid:17) .
37e are going to upper bound h , h , h separately. Case 1: Bound h h = Z k −√ k h ( x )d x ≤ Z k −√ k b ( x ) x · e x d x = Z k −√ k ∞ X i = k +1 x i − i ! e − x d x = Z k √ k ∞ X i = k +1 ( k − x ) i − i ! e − ( k − x ) | {z } G ( x ) d x, (21)where the second step is by Part 3 of Fact 9, the third step is by definition of b ( x ).We define G ( x ) def = P ∞ i = k +1 ( k − x ) i − i ! e − ( k − x ) . So G (0) = ∞ X i = k +1 k i − i ! e − k = 1 k ∞ X i = k +1 k i i ! e − k = 1 k (1 − Γ(1 + k, k ) / Γ(1 + k )) ≤ k , where the last step is by Lemma 4 that Γ(1 + k, k ) /k ! > / d ( i, x ) = ( k − x ) i − i ! e − ( k − x ) . So that G ( x ) = P ∞ i = k +1 d ( i, x ).we have ∀ i ≥ k + 1,ln( d ( i, x ) /d ( i, k − x ) i − e − ( k − x ) /i ! k i − e − k /i ! )= ln((1 − x/k ) i − e x )= x + ( i −
2) ln(1 − x/k )= x − ( i − ∞ X j =1 j ( xk ) j = x − i ∞ X j =1 j ( xk ) j + 2 ∞ X j =1 j ( xk ) j ≤ − i ∞ X j =2 j ( xk ) j + 2 ∞ X j =1 j ( xk ) j = − i xk ) − ∞ X j =1 (cid:18) ij + 2 x k − j (cid:19) ( xk ) j ≤ − i xk ) ≤ − x k , i ≥ k , and the eighth step follows from x ≥ √ k . So d ( i, x ) ≤ e − x k d ( i, G ( x ) = ∞ X i = k +1 d ( i, x ) ≤ ∞ X i = k +1 e − x k d ( i, ≤ e − x k G (0) . Thus, we have h ≤ Z k √ k G ( x )d x ≤ Z k √ k e − x k G (0)d x ≤ √ kG (0) Z √ k √ e − x d x ≤ √ kG (0) Z ∞√ e − x d x ≤ . √ kG (0) ≤ . k − . , where the first step is by Eq.(21), the fifth step is by R ∞√ e − x d x ≤ . G (0) ≤ k − / Case 2, Bound h h = Z kk −√ k a ( x ) b ( x )( x · a ( x ) + k · b ( x )) d x ≤ Z kk −√ k kx d x = ln( k/ ( k − √ k )) / (4 k ) ≤ ( k/ ( k − √ k ) − / (4 k )= ( √ / · k − . / ( k − √ k ) ≤ (2 √ / · k − . , where the first step follows from ( a + b ) ≥ ab , the last step is by k − √ k ≥ . k for k ≥ Case 3, Bound h . We have h = Z ∞ k a ( x ) b ( x )( xa ( x ) + kb ( x )) d x ≤ k − · Z ∞ k a ( x ) e − x d x ≤ a ( k ) · k − Z ∞ k x k − e − x k k − d x e k k !2 k · k k Z ∞ k x k − e − x / ( k − x = e k k !2 k · k k (cid:0) Γ( k, k ) / ( k − (cid:1) ≤ e k k !2 k · k k · (1 / ≤ e k k · k k · ( ek k +1 / e − k )= e/ · k − . , where the second step is by Part 4 of Fact 9 that for all x ≥ k + c √ k , b ( x ) / ( xa ( x ) + kb ( x )) ≤ e − x /k , the third step is by Part 5 of Fact 9, the fourth step is by Part 2 of Fact 9 and that ∀ x, a ( x ) + b ( x ) ≤ e x , the fifth step is by definition of incomplete gamma function, the sixthstep is by Lemma 4. the last step is by Lemma 5.Therefore, we can upper bound ℜ AR / AP ( k ) = 1 + k · Z ∞ h ( x )d x = 1 + k · ( h + h + h ) ≤ .
009 + 2 √ / e/ / √ k ≤ / √ k. This accomplishes the upper-bound part of Claim 6.
B Mathematical Tools
Lemma 4 (Incomplete gamma function [OLBC10, Chapter 8]) . Define the incomplete gammafunction Γ( n, x ) def = R ∞ x t n − e − t d t . Then for all positive integer n , we have Γ( n,n )( n − < < Γ( n,n − n − . Γ( n,x )( n − = e − x P n − i =0 x i i ! . Lemma 5 (Stirling’s approximation [Rob55]) . For all positive n , the following holds: √ πn n +1 / e − n ≤ n ! ≤ en n +1 / e − n . It’s easy to see the following facts:
Fact 7.
For all positive integer n , we have
12 4 n √ πn ≤ (cid:18) nn (cid:19) ≤ n √ πn Fact 8.
For all positive integer n , we have Z ∞ e − x x n d x = Γ( n + 1) = n ! . Fact 9. a ( x ) and b ( x ) satisfies the following facts: For all x ∈ [0 , k ] , b ( x ) · e − x ≤ / , For all x ∈ [ k, ∞ ) , a ( x ) · e − x ≤ / , . For all x ∈ [0 , k − , a ( x )( xa ( x ) + kb ( x )) ≤ x e x , For all x ∈ [ k, ∞ ) , b ( x )( xa ( x ) + kb ( x )) ≤ k e x , For all x ∈ [ k, ∞ ) , a ( k ) · x k − /k k − ≥ a ( x ) . Proof.
Part 1.
For all x ∈ [0 , k ], b ( x ) · e − x = 1 − Γ( k + 1 , x ) /k ! ≤ − Γ( k + 1 , k ) /k ! ≤ / , where the first step is by Part 2 of Lemma 4, the second step is because Γ( k +1 , x ) is a decreasingfunction on x ∈ [0 , ∞ ), the third step is by Part 1 of Lemma 4. Part 2.
By the same reason, for all x ∈ [ k, ∞ ), we have that a ( x ) · e − x = Γ( k, x ) / ( k − ≤ Γ( k, k ) / ( k − ≤ / . Part 3.
For all x ∈ [0 , k − x · a ( x ) + k · b ( x )) − x a ( x ) e x = x a ( x ) + 2 xk · a ( x ) b ( x ) + k b ( x ) − x a ( x ) (cid:0) a ( x ) + b ( x ) + x k /k ! (cid:1) ≥ xk · a ( x ) b ( x ) − x a ( x ) (cid:0) b ( x ) + x k /k ! (cid:1) ≥ (2 k − ( k − b ( x ) − x k +1 /k ! ≥ ( k + 1) · x k +1 / ( k + 1)! − x k +1 /k != 0 , where the fourth step is by b ( x ) ≥ x k +1 / ( k +1)!. Therefore, a ( x )( xa ( x )+ kb ( x )) ≤ x e x follows directly. Part 4.
For all x ∈ [ k, ∞ ), we have( x · a ( x ) + k · b ( x )) − k b ( x ) e x = x a ( x ) + 2 xk · a ( x ) b ( x ) + k b ( x ) − k b ( x ) (cid:0) a ( x ) + b ( x ) + x k /k ! (cid:1) ≥ xk · a ( x ) b ( x ) − k b ( x ) (cid:0) a ( x ) + x k /k ! (cid:1) ≥ (2 x − k ) · a ( x ) − x k / ( k − ≥ k · x k /k ! − x k / ( k − , where the fourth step is by a ( x ) ≥ x k /k !. Therefore, b ( x )( xa ( x )+ kb ( x )) ≤ k e x follows directly. Part 5.
For all x ≥ k , we have a ( k ) · x k − /k k − ≥ k − X i =0 k i i ! x k − k k − ≥ k − X i =0 k i i ! x i k i k − X i =0 x i i != a ( x ) ..