Limited and Online Supply and the Bayesian Foundations of Prior-free Mechanism Design

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1 Limited ad Olie Supply ad the Bayesia Foudatios of Prior-free Mechaism Desig ABSTRACT Nikhil R. Devaur Microsoft Research Oe Microsoft Way Redmod, WA We study auctios for sellig a limited supply of a sigle commodity i the case where the supply is kow i advace ad the case it is ukow ad must be istead allocated i a olie fashio. The latter variat was proposed by Mahdia ad Saberi [12] as a model of a importat pheomea i auctios for sellig Iteret advertisig: advertisig impressios must be allocated as they arrive ad the total quatity available is ukow i advace. We describe the Bayesia optimal mechaism for these variats ad exted the radom samplig auctio of Goldberg et al. [8] to address the prior-free case. Categories ad Subject Descriptors F.2.m [ANALYSIS OF ALGORITHMS AND PROB- LEM COMPLEXITY]: Miscellaeous Geeral Terms Algorithms, Ecoomics Keywords Olie, Auctios, Prior-free, Mechaism desig, Limited supply 1. INTRODUCTION Cosider a prior-free mechaism desiger lookig for a mechaism with good profit. Does limited supply pose a additioal challege over ulimited supply? Does olie supply pose a challege over offlie supply? I attempt to aswer the first questio Fiat et al. [6] gave a simple approximatiopreservig reductio from limited to ulimited supply auctios. Their aswer: o. I attempt to aswer the secod Supported i part by NSF CAREER Award CCF Permissio to make digital or hard copies of all or part of this work for persoal or classroom use is grated without fee provided that copies are ot made or distributed for profit or commercial advatage ad that copies bear this otice ad the full citatio o the first page. To copy otherwise, to republish, to post o servers or to redistribute to lists, requires prior specific permissio ad/or a fee. EC 09, July 6 10, 2009, Staford, Califoria, USA. Copyright 2009 ACM /09/07...$ Jaso D. Hartlie Electrical Egieerig ad Computer Sciece Northwester Uiversity Evasto, IL questio Mahdia ad Saberi [12] solved a olie pricig problem ad with it adopted the auctio from [8] to solve the olie case, though, at sigificat loss i performace. Their aswer: yes. We recosider both of these questios through a prior-free methodology that is rigorously grouded i the Bayesia mechaism desig theory. Our aswers are the opposite! Cosider the same problems but from the perspective of a Bayesia desiger. Suppose the agets valuatios are draw i.i.d. from a kow distributio. What is the optimal mechaism for the three cases: ulimited supply, (offlie) limited supply, ad olie limited supply? For ulimited supply, the optimal mechaism is simply to post the price that maximizes the price times the probability that a aget would buy (recall, all agets are idetically distributed). Notice that this price optimizatio (i.e., reveue curve) may ot be cocave. For limited supply ad a cocave reveue curve, Myerso showed that the optimal mechaism is the Vickrey auctio with a reservatio price set to the same price as i the ulimited supply case [13]. I the o-cocave case Myerso gives a techique that obtais the same reveue as if the reveue curve was the cocave hull of the actual reveue curve. To do this the auctio applies a distributio depedet weakly mootoe trasformatio of agets valuatios to get iroed virtual valuatios ad the optimal auctio allocates to the agets with the highest virtual valuatios (breakig ties arbitrarily or radomly, but ot by the agets valuatios). Notice that trasformatio is distributio depedet but ot supply depedet. Furthermore, the breakig of valuatio space ito equivalece classes (of equal virtual valuatio) makes the limited supply problem coceptually differet from the ulimited supply problem. For olie supply, the desiger ca make the same trasformatio from valuatios to virtual valuatios ad the assig the uits as they arrive to the agets with the highest virtual valuatios (agai breakig ties i virtual valuatios arbitrarily or radomly, but ot by the agets valuatios). Proceedig thusly, the olie Bayesia desiger will make the exact same allocatio as the offlie Bayesia desiger. By reveue equivalece [13], the paymet rule is implied by the allocatio rule ad the two scearios are equivalet. Thus, our olie desiger faces o complicatio risig from the olie ature of the supply. Our Bayesia desiger must coclude: ulimited supply is coceptually easier tha (offlie) limited supply, but there is

2 o coceptual difficulty i olie supply over offlie supply. The primary focus of this paper is i derivig the same result i the prior-free case (which is the opposite coclusio of Fiat et al. [6] ad Mahdia ad Saberi [12]). I prior-free mechaism desig the performace of a mechaism is compared to a (distributio idepedet) performace bechmark. The mechaism that performs best relative to, i.e., miimizes the maximum ratio to, this bechmark is the prior-free optimal mechaism (for the give bechmark). Hartlie ad Roughgarde [11] propose as a bechmark the performace of the optimal Bayesia optimal mechaism. This bechmark has the strog cosequece that a mechaism that approximates it is guarateed to simultaeously approximate, for all i.i.d. distributios, the performace of the Bayesia optimal mechaism for that distributio. For ulimited supply where the Bayesia optimal mechaism is a posted price, this bechmark is simply the optimal posted-price profit. Coicidetally this is exactly the bechmark chose for prior-free ulimited supply auctios (modulo oe small techical detail: the restrictio to prices for which there are at least two uits sold) [6]. Fiat et al. geeralized the ulimited supply bechmark to limited supply settigs i a atural way by cosiderig the optimal posted-price profit sellig at least two uits ad at most the full supply. Though atural, this defiitio does ot coicide with the bechmark of the optimal Bayesia optimal profit; i fact, it ca be up to but at most a factor of two off (as we prove). Precisely for this reaso the lossless reductio from the limited to ulimited supply proposed i [6] fails. Thus, Fiat et al. s results ca be iterpreted as givig four ad eight approximatios for the ulimited ad limited supply problems, respectively (agaist the optimal Bayesia optimal bechmark). Hartlie ad McGrew s 3.25 approximatio for the ulimited supply problem [10] gives a 6.5 approximatio to the limited supply problem. These prior-free auctios caot be easily adapted to the case of olie supply because the umber of uits they allocate is a discotiuous fuctio of the supply. Yet, as we already described, the Bayesia optimal auctio ca easily be adapted to olie supply. Thus it seems like these prior-free mechaisms are doig somethig that is ituitively wrog. To rectify this, we cosider the limited supply geeralizatio of the radom samplig optimal price (RSOP) auctio [8]: we radomly partitio the agets, the for each partitio we sell half the uits usig the Bayesia optimal mechaism for the distributio give by the agets i the opposite partitio. Notice that because this mechaism is derived from a Bayesia optimal auctio for limited supply, it iherits the property that it does ot eed to kow the supply limit i advace. We exted the aalysis approach of Alaei et al. [1] for RSOP to this limited supply auctio. Our aalysis is ecessarily more complicated, ad we lose slightly i the approximatio factor we are able to prove. We prove a 25- approximatio which is the best kow factor for the olie case. Furthermore, we have o reaso to believe that the approximatio factor is ot four as is cojectured for RSOP. Notice that a boud of four would be better tha the best kow (offlie) limited-supply approximatio factor of 6.5. Related Work. Prior-free optimal mechaism desig was iitiated by Goldberg et al. s desig of the radom samplig optimal price (RSOP) auctio for sellig a sigle commodity i ulimited supply [8]. The aalysis framework was refied by Fiat et al. who showed that RSOP is a costat approximatio i worst case relative to a atural bechmark, the optimal posted price profit sellig at least two uits [6]. Feige et al. refie the aalysis to improve the costat approximatio factor to 15 [5], which was later improved to 4.68 by Alaei et al. [1]. Fiat et al. exted the aalysis framework to the case of limited supply i a atural way; though i hidsight this geeralizatio is ot as well motivated as the origial ulimited supply framework. Hartlie ad Roughgarde give a methodology for prior-free mechaism desig that is based o Bayesia mechaism desig that suggests a alterative ad well motivated approach to limited supply [11]. We reexamie limited supply profit maximizatio from this ew perspective. (Note that [11] does ot cosider the objective of profit maximizatio ad their mechaism is simpler to aalyze because they do ot allow the possibility of olie supply.) Mahdia ad Saberi [12] adapt the limited supply auctios from Goldberg et al. [8] to the case where the supply arrives olie ad must be allocated immediately while paymets may be determied offlie. They cosider a ogame-theoretic olie pricig problem where the seller is costraied to sell at a sigle price to all wiers, ad gave a algorithm for it with a costat competitive ratio [12]. Devaur ad Chakraborty gave aother algorithm for the same olie pricig problem that improved the competitive ratio to a factor of two [4]. It is crucial i our problem of olie supply that the mechaism is free to defer aget paymets util all supply has bee realized. As show by Babaioff et al. [2] whe paymets must be calculated olie, icetive costraits must icorporate the aget s beliefs over the supply or sigificat loss i performace is ievitable. Deferred paymets are reasoable for the motivatig applicatio of advertisemet auctios, sice the advertisers are typically charged at the ed of the billig cycle. 2. PRELIMINARIES We cosider the problem of a moopolistic seller attemptig to maximize their profit whe sellig k idivisible uits of a sigle item to uit-demad agets. We cosider ex post icetive compatible ad idividually ratioal auctios for solvig this problem. I such a mechaism each aget has a (weakly) domiat strategy of participatig i the auctio ad reportig their true valuatio as their bid. Such a auctio selects at most k wiers ad demads a paymet from each. From aget i s perspective, v i is their valuatio upo wiig, x i is a idicator for their wiig a uit or ot, ad p i is their paymet. Feasibility for k uits requires that i xi k. We cosider two variats of the limited supply problem. I the stadard variat the supply limitatio k is kow i advace. We refer to this as the offlie variat. I the olie variat the supply is ot kow i advace, istead the mechaism is give uits to allocate oe at a time. Upo receivig a uit the mechaism must either choose a aget to whom to allocate it or to throw it away (the desiger has free disposal). Evetually, perhaps adversarially, the desiger is told that there are o more uits. At this poit the desiger calculates paymets for agets who received uits. Notice that with both olie ad offlie supply the

3 mechaism is sigle-roud ad sealed-bid with respect to the agets: the agets bid, uits are allocated olie or offlie, paymets are determied. Formally, a auctio maps the valuatio profile v = (v 1,..., v ) ito a allocatio x(v) ad paymets p(v). It is ofte useful to look at the probability of allocatio ad expected paymet of a aget with a specific value (radomizatio take over aget valuatios ad cois flipped by a radomized auctio protocol); defie p i(v i) = E v i [p i(v i, v i)] ad x i(v i) = E v i [x i(v i, v i)] where v i = (v 1,..., v i 1,?, v i+1,..., v ). The lemma below that characterizes icetive compatible auctios allows us to igore the paymet rule (it is uique) ad focus o choosig a good allocatio rule. Lemma 1. [13] Ay icetive compatible auctio i which losers pay othig satisfies (for all agets i): 1. allocatio mootoicity: x i(v i) is o-decreasig i v i. 2. paymet idetity: p i(v i) = v ix i(v i) v i 0 xi(z)dz. Our results for the prior-free desiger are best discussed i the cotext of the followig stadard results for the Bayesia desiger [13] (for a survey see [9]). Assume the valuatios of the agets are draw idepedetly ad idetically from a distributio with distributio fuctio F ad desity fuctio f. For this settig the Bayesia optimal auctio specifies a distributio-depedet partitioig valuatio space ito itervals of equal priority ad allocates the uits to the (at most) k aget with the highest positive priority with ties broke radomly. Clearly such a allocatio rule satisfies the mootoicity coditio of Lemma 1. The paymets ca be calculated simply usig the paymet idetity. Notice that it does ot matter whether the supply is olie or offlie. Defiitio 2. The Bayesia optimal auctio for distributio F is M F. Our Bayesia desiger obtais a auctio which is optimal i a absolute sese: for valuatios from the give distributio, the expected performace of his chose (i.e., the Bayesia optimal) auctio is at least that of ay other auctio. Our prior-free desiger is ot edowed with prior kowledge of the distributio. For ay auctio our priorfree desiger cosiders, there is always a distributio for which some other auctio strictly out performs the cosidered auctio. Thus, the prior-free auctio desig literature has tured to a relative otio of optimality that bears close resemblace to the competitive aalysis of olie algorithms. Here a auctio is good if its maximum (worst case over iputs) ratio with a bechmark performace is small. The prior-free optimal auctio for a give bechmark is the oe with the smallest worst-case ratio. This framework for priorfree desig is give i [7] (for a survey see [9]). Defiitio 3. A performace bechmark maps valuatio profile v ad supply limit k to a target performace, otated, e.g., G(k, v). Where k or v is implicit i the cotext use short ad otatios G(k) ad G(v) respectively. A auctio maps a valuatio profile ad supply limit to a expected reveue, otated, e.g., A(k, v). Defiitio 4. The prior-free optimal auctio for bechmark G is auctio A that miimizes mi max G(v) A v A(v) Some care must be take i choosig performace bechmarks for the prior-free results to be ecoomically meaigful. Hartlie ad Roughgarde give a geeral theory for meaigful bechmarks: choose as a bechmark the performace of the optimal Bayesia optimal auctio. Defiitio 5. The optimal Bayesia optimal bechmark is G(v) = sup F M F (v). A prior-free approximatio to this bechmark is a very strog result as stated by the followig fact. Fact 6. [11] A prior-free auctio that β-approximates G o ay iput v also β-approximates the Bayesia optimal auctio o ay i.i.d. distributio F. I ulimited supply settigs, the optimal Bayesia optimal bechmark ca be characterized succictly. Let v (i) deote the ith largest aget valuatio. For ulimited supply (i.e., k = ) the optimal-bayesia-optimal bechmark coicides precisely with the optimal-posted-price bechmark, F(, v) = max i iv (i), proposed by [6]. For limited supply the optimal-bayesia-optimal bechmark does ot geerally coicide with the limited supply bechmark F(k, v) = max i k iv (i) from [6]. For techical reasos described by [7] we must adjust the bechmark to igore the case where there is a sigle aget with a extreme high valuatio. This restrictio gives the ulimited supply bechmark of F (2) (v) = max i 2 iv (i) ad the geeral bechmark of G (2) (v) = sup F M 2 F (v). where is the Bayesia optimal auctio for distributio F costraied to offer prices that are at most v (2). Note that G (2) ad G coicide whe it is optimal to sell at least two uits. We focus o the aalysis of a geeralizatio of Goldberg et al. s [8] Radom Samplig Optimal Price (RSOP) auctio to limited supply. This geeralizatio is based o Baliga ad Vohra [3] approach to prior-free mechaisms (referred to as the Radom Samplig Empirical Myerso (RSEM) auctio i Hartlie ad Karli s survey [9]). A importat costruct i the defiitio of this auctio is the empirical distributio M (2) F for a valuatio profile. This is simply the distributio F with F (z) equal to the factio of agets i v with v i < z. Defiitio 7. The k-uit Radom Samplig Empirical Myerso (RSEM) auctio works as follows: 1. Radomly partitio the agets ito two sets, v A ad v B. 2. Calculate the empirical distributios for each set, F A ad F B. 3. Ru M F A(v B ) ad M F B (v A ) with k/2 uits each. Our mai theorem is that RSEM is a 25-approximatio to G (2). Notice that it does ot matter whether the supply is olie or offlie. 3. ANALYSIS OF THE RANDOM SAMPLING AUCTION. I this sectio we give our aalysis of the radom samplig auctio RSEM. We do so first with a more detailed discussio of Bayesia optimal auctios for cotiuous distributios. This discussio will eable deeper uderstadig of our bechmark G which is the supremum over such auctios. With this uderstadig we will show how a lemma from [1] proves the approximatio boud.

4 R R i,j *'+',-'" R i!"#$"%&'()" j Figure 1: Reveue curves: R( ), R i,j( ), ad R( ). 3.1 Bayesia optimal auctios, revisited. Recall our settig for Bayesia optimal auctios where the aget valuatios are distributed i.i.d. from distributio F. Cosider the case all k uits are sold. I such a case, ay particular aget s ex ate probability of wiig is k. Bayesia optimal auctios ca be ituitively uderstood by cosiderig the reveue obtaiable as a fuctio of the probability that the aget wis. To allocate with probability k to a aget with value draw from distributio F, we could simply use a posted price of F 1 (1 k ), which the aget would accept with probability k. Our expected reveue is thus R(k) = k F 1 (1 k ). Notice that this reveue fuctio may ot be cocave i k (as depicted by Figure 1). 1 Aother way we could allocate with probability k is to pick ay i ad j such that i < k < j, ad allocate to the aget always if their valuatio is at least price F 1 (1 i ) ad with probability (k i)/(j i) if their valuatio is betwee prices F 1 (1 j ) ad F 1 (1 i ). By the paymet idetity of Lemma 1, the reveue of such a allocatio strategy is R i,j(k) = (k i)r(i) + (j k)r(j), j i i.e., R i,j(k) is the fuctio that is equal to R(k) for k < i ad k > j, ad for k (i, j) it is the lie coectig R(i) to R(j) (See Figure 1). Notice that whe R(k) is o- 1 As poited out by Baliga ad Vohra [3], radom samplig based auctios caot assume cocave reveue fuctios as these reveue fuctios are give the empirical distributio of a sample of the bidders. cocave this radom allocatio ca give more reveue tha the aforemetioed posted price. It is clear that the optimal reveue possible form this kid of approach is obtaied from cosiderig the cocave hull of R(k) which we deote by R(k). Ituitively i the above example the aget has the same priority whe their value is aywhere i the iterval betwee F 1 (1 i ) ad F 1 (1 j ). To achieve the optimal reveue possible, we break valuatio space ito itervals of equal priority that correspod with the lie segmets of R( ). For istace, the stadard approach i the literature is to cosider the derivative (a.k.a., slope) of R( ) as the priority. (Sice R( ) is cocave, priority is a weakly mootoe fuctio of valuatio; this particular priority fuctio is kow i the literature as the iroed virtual valuatio of the aget [13].) 3.2 The optimal Bayesia optimal bechmark Now cosider repeatig the above discussio but with the empirical distributio F for valuatio profile v. We could sell k uits at price v (k). Deote this reveue by R(k) = kv (k). Agai, R(k) is the cocave hull of R(k). We could the pick a i ad j with i < k < j ad make the followig offer to this particular set of agets with profile v: high-valued agets with values i [v (i), ) wi for sure ad low-valued agets with values i [v (j), v (i) ) wi with probability give by the ratio betwee the umber of uits left ad the umber of agets o the iterval (i.e., by lottery). The paymets for such a allocatio rule are give by Lemma 1: low-valued agets pay v (j) ad high-valued agets pay v (i) (v (i) v (j) )(k i+1). Notice that to calculate the high-valued aget paymets oe must ote that a high-

5 %&'()("*"+,$'-$."//"/0$ 1 k i +1 j i +1 v (j) v (i)!"#$ Figure 2: The allocatio fuctio x i(v i) for v i fixed. The area of the shaded regio is the paymet p i(v i) specified by Lemma 1. valued aget might try to bid as if they were low-valued, this would result i oe fewer uits beig claimed by highvalued agets which meas there are k i + 1 uits to be radomly allocated amog the remaiig j i + 1 agets (see Figure 2). Let ˆR i,j(k) deote the total resultig reveue from this approach. Notice that ˆR i,j(k) R i,j(k). As described i the precedig sectio the Bayesia optimal auctio divides valuatio space ito itervals of equal priority. For the purpose of maximizig profit oe would oly wat to use itervals bouded by aget valuatios, i.e., of the form [v (j), v (i) ), as this would have all wiers payig the maximum possible. This ad the above discussio gives ituitio for the followig lemma. Lemma 8. [11] G(k, v) = sup i,j ˆRi,j(k, v) ad G (2) (k, v) = sup i,j 2 ˆRi,j(k, v). G will be tough for us to work with so we use the followig defiitio which (as the theorem below shows) provides a good upper boud. Defiitio 9. F(k, v) = maxi k R(i, v) ad F (2) (k, v) = max 2 i k R(i, v). The followig theorem relates G, F, ad F. This will be used later i our proof to relate coceptually easier bouds i terms of F with our desired boud i terms of G. Also, the argumet v i the above bechmarks will be dropped whe it is clear from the cotext. Theorem 10. For all valuatios, v, 1. F(k, v) G(k, v) F(k, v). Similarly F (2) (k, v) G (2) (k, v) F (2) (k, v). 2. F(k, v) 2F(k, v). There exists v such that G(k, v) = (2 1 )F(k, v). k 3. F(k, v) mi{ 4, }G(k, v). 3 k Proof. We prove each part separately. 1. The first part of the theorem follows almost immediately from the defiitios ad Lemma For the secod part, recall that R is cocave. If R attais its maximum before the supply rus out, that is F(k) = R(l) for some l k, the the lemma follows trivially sice F(k) = F(k). So suppose ot. The R is mootoically o-decreasig i the iterval [1, k] ad hece F(k) = R(k). Suppose R(k) is o the lie joiig R(i) ad R(j), where i < k < j. The R(k) = 1 ((k i)r(j) + (j k)r(i)) j i = R(i) + k i (R(j) R(i)) j i (The 2d term is segmet AB i Figure 3) R(i) + k R(j) (segmet CD i Figure 3) j R(i) + R(k) 2F(k). The first iequality o the last lie follows sice k j R(j) = k j jv (j) = kv (j) kv (k) = R(k). The above boud is almost tight. Let v 1 = k, v 2 = v 3 = = v = 1. The F(k) = k, where as G(k) = ˆR 1,k (k) = k + (k 1)(1 k ) which teds to 2k 1 as teds to ifiity. 3. For the third part, agai, the iterestig case is whe F(k) = R(k), ad R(k) = 1 ((k i)r(j) + (j k)r(i)). j i

6 2/3/45/" )" R(i) %"!" R R(j) (" ˆR $" '" i &" k #" j *"+,"-./01" Figure 3: Illustratio for proof of Theorem 10. Let p = v (i) ad q = v (j) ad defie ˆR(k) = ˆR ( i,j(k) = (k i)q + i p (p q)(k i+1) ). I the iterval ( [i, j], ) ˆR is a liear fuctio of k with ˆR(i) = i p p q ad slope q(j+1) ip. I compariso, i the same iterval, R is a liear fuctio of k with R(i) = ip ad slope qj ip. R(i) j i > ˆR(i) where as the slope of ˆR is higher tha R. Hece the miimum ratio of ˆR(k) to R(k) occurs at k = i, ad R(i) ˆR(i) = i(p q) which is clearly less tha ip if 4 j i 3. (I Figure 3, ratio of segmet EF to EG is 1:j-i+1.) Also, j i is at least 2, sice i < k < j. So we are oly left with the case whe j i = 2. I this case we use the fact that R(j) R(i), which follows from the defiitio of R. This implies (i + 2)q ip, which i tur implies that i(p q) is less tha ip by simple 4 algebra. I ay case, we have that R(i) 4 ˆR(i) ad 3 i tur F(k) 4 G(k). 3 Agai, usig the fact that R(j) R(i), we get that i(p q) q(j i) is less tha q. Further ote that qk G(k) sice qk is the reveue of the posted price auctio with price q. Thus R(i) ˆR(i), ad i fact, R(k) ˆR(k) ( is less ) tha G(k)/k for all k [i, j]. Thus, F(k) k G(k). Remark. It immediately follows from the above theorem that ay algorithm that is a α-approximatio to F is a 2αapproximatio to F. Curretly the best kow auctio is from Hartlie ad McGrew [10] ad achieves a approximatio factor of 3.25 to F. Thus, it is a 6.5 approximatio to F. We do ot maage to prove that RSEM is better tha this; however, there is o reaso to believe that it is ot. 3.3 The Myerso auctio for empirical distributios The radom samplig auctio we wish to aalyze rus the Myerso auctio o a partitio of the agets with distributio give empirically by the opposite partitio. We ow explicitly describe what M F does whe F is the empirical distributio for valuatio profile v. Recall that Myerso s auctio for a distributio simply allocates the k uits to the agets with highest priority breakig ties i priority radomly. Of course, the key property of Myerso s auctio is that it partitios valuatio space ito itervals of equal priority i the optimal way. From our precedig discussio this partitioig is give by lookig at the reveue fuctio R( ) ad its cocave hull R( ). Itervals where theses fuctios are ot equal form a priority class. Specifically, let i 1 i T idex the T agets i satisfyig R(i) = R(i) o the risig slope of R( ) (i.e., it = arg max i R(i)). The equal-priority itervals are [v (it ), v (it 1 )], (v (it 1 ), v (it 2 )],..., (v (2), v (1) ], (v (1), ). Agets with values below the smallest iterval, i.e., strictly less tha v (it ) are rejected. The set of bidders is partitioed ito sets A ad B. Let the bids i A be v A (1) v A (2)... ad so o. Let F A be the empirical distributio from bidders i A ad cosider ruig the Myerso auctio with this distributio o the bidders i B, i.e., M F A(v B ). Myerso allocates as follows. Let i A 1 < i A 2 < < i A T be the idices of agets o the cocave hull of the reveue curve for F A (as described i the precedig paragraph). Let p A t = v A (i A t ). The equal priority itervals are [p A T, p A T 1], (p A T 1, p A T 2],..., (p A 2, p 1], (p A 1, ). The bidders i B are allocated odd-umbered items. Let

7 j B t = {l B : v B (l) p A t }. Now if k 2 [jb t, j B t+1), the the top jt B always get the item ad pay p A t pa t pa t+1. Bidders j t+1 B jb t +1 with values i the iterval (p A t+1, p A t ] wi remaiig uits at price p A t+1 with ties broke radomly. 3.4 The performace of the radom samplig auctio Our aalysis of the performace of the RSEM auctio uses a lemma from the recet paper of [1] that aalyses the RSOP auctio for ulimited supply. They prove a lemma about the expectatio of a certai radom variable, let s call it X, that is used to lower boud the reveue of the RSOP auctio. We show that the same quatity X ca be used to boud the reveue of the RSEM auctio for limited supply as well. While the fact that X lower bouds the reveue of the RSOP auctio for ulimited supply is more or less straightforward, the aalogous statemet for the RSEM auctio for limited supply is more complicated because the reveue of lotteries is more complicated that the reveue of posted price auctios. We ow defie the radom variable X. Without loss of geerality, let the highest bidder be i B. Let s i = {j A : vj v (i) } be the umber of bidders i A amog the top i bidders. Note that the distributio of s i is idepedet of the actual values, ad has the same distributio as the followig discrete radom walk o itegers: for each i, s i is either s i 1, or s i 1 + 1, with probability half each, with s 1 = 0. Let z = mi i (i s i)/s i. Suppose that the optimum sigle price auctio sells λ uits, i.e. F (2) = λv (λ). The X = z sλ λ. Note that for every possible value of λ we get a differet radom variable X. The mai lemma that we eed from [1] lower bouds E[X] for all possible values of λ. Lemma 11. [1] For all positive itegers λ, E[X] As is usual i the aalysis of radom samplig auctios, we relate the reveue of the RSEM auctio from side B to some fuctio of the values from side A. Here, that fuctio is the cocave hull of the reveue curve from side A. Let R A be the reveue curve restricted to the bidders i A; more precisely R A (i) = iv A (i). Let R A be its cocave hull. However, for i > i A T, we defie R A (i) = R A (i A T ). Note that this makes R A a mootoically o-decreasig cocave fuctio. Let RSEM(k) deote the profit of the RSEM Auctio with k items. Lemma 12. RSEM(k) z R A ( k 2 )/ mi{ 4 3, k }. Proof. Recall that the highest bidder is i B. Defie RSEM B (i) to be the reveue obtaied by the RSEM auctio from side B, as a fuctio of the umber of uits allocated to B. I the proof we will oly cosider the reveue obtaied from side B, i.e., we will use that RSEM(k) RSEM B (k). This is tight i the worst case, because if v (1) is high eough, the the optimal Myerso auctio for F B (the empirical distributio for B) is to ru a Vickrey Auctio with reserve price equal to v (1). This auctio would fetch o reveue from side A. Suppose k 2 [jb t, jt+1). B Cosider the followig fuctio defied o the iterval [jt B, jt+1): B let L(i) be the liear fuctio such that L(jt B ) is equal to jt B p A t ad L(jt+1) B = jt+1p B A t+1. As i the proof of part 3 of Theorem 10, i the iterval [jt B, jt+1), B the fuctio RSEM B (i) is withi a factor mi{ 4, } of L(i). From the defiitio of z, we get 3 k that jt B zi A t. These two facts poit to the coclusio we eed. However, the proof eeds more argumet because of the followig: It could be that L(j B t+1) < L(j B t ). We used the fact that this does ot happe i the proof of Theorem 10. It could be that k / 2 [ia t, i A t+1). Thus we caot directly compare L( k ) to R A ( k ). 2 2 We get aroud these difficulties by comparig L to the followig fuctio istead. Let R A (zi A t ) = zi A t p A t. Exted R A to all i [1, zi A T ] by a liear iterpolatio. For i > zi A T, let R A (i) = R A (zi A T ). We ote the followig easy observatios without proof (see Figure 4): For all i, R A (zi) = z R A (i). R A is a icreasig cocave fuctio. For all i [j B t, j B t+1], L(i) R A (i). We ow argue that L( k ) z R A ( k ) via the followig sequece of iequalities; this sequece of iequalities corre- 2 2 spods to the poits A B C = zd i Figure 4. L( k 2 ) RA ( k 2 ) RA (z k 2 ) = z R A ( k 2 ). Therefore it is eough to show that RSEM B ( k 2 ) L( k 2 )/ mi{ 4 3, k }. The proof of this is idetical 2 to that i the proof of part 3 of Theorem 10, uless L(j B t+1) < L(j B t ). I case L(j B t+1) < L(j B t ), we use the observatio that the reveue of the auctio actually decreases i the iterval [j B t, j B t+1) ad is i fact at least L(j B t+1). The coclusio holds from the followig sequece of iequalities, each of which is easy to see give the discussio so far. RSEM B ( k 2 ) L(jB t+1) R A (j B t+1) R A ( k 2 ) z R A ( k 2 ). This completes the proof of the lemma. Fially, we relate R A (k/2) to the bechmark F (2) (k) as i the followig lemma. Lemma 13. RA (k/2) s λ 2λ F (2) (k). Proof. I ulimited supply, if the optimum sigle price sells λ uits, i.e. F (2) = λv (λ), the oe ca offer v (λ) to the bidders i A ad get a reveue of v (λ) s λ, so oe would coclude that F A v (λ) s λ = s λ λ F (2). However the same argumet does ot work i case of limited supply, because we split the supply also i half for each side. Thus offerig v (λ) to the bidders i A is guarateed to get a reveue of v (λ) mi{s λ, k }. Sice s 2 λ k, we ca use that mi{s λ, k } s λ 2 2 to get the required coclusio for the lemma. 2 We also have to cosider a additioal case, whe j B t+1 j B t = 1, which is similar to the case j B t+1 j B t = 2.

8 R A.+/+01+" #" %" L $"!" R A z k 2 jt B k/2 jt+1 B &"'(")*+,-" Figure 4: Reveue curves illustrated for the proof of Lemma 12. Our mai theorem ow follows easily from Lemmas 11, 12, 13 ad Theorem 10. Theorem 14. RSEM is at most a mi{8/3, 2(1 + 1 )} k 4.68-approximatio with respect to F (2) ad at most a mi{16/3, 4(1+ 1 )} 4.68-approximatio with respect to F (2). k Remark. I usig Lemma 11 from [1] as a blackbox, we lose a factor of 2 i the proof of Lemma 13. It is atural to woder if the radom variable X is defied as z mi{ s λ λ, 1 2 }, the what is the lower boud o E[X]? Naively, the boud is 1/(2 4.68) (as used i the preset aalysis). A improvemet i this boud would improve the approximatio factor i our aalysis. 4. CONCLUSIONS I this paper we show that the stadard prior-free mechaism desig approach to radom samplig gives a approximately optimal auctio for multi-uit sigle-item settigs of limited supply. Furthermore, the supply limitatio may be determied olie ad this does ot affect the auctio protocol or its performace. We obtai a boud o the approximatio factor of 25 which implies that the auctio (RSEM) is the best kow auctio for the olie supply problem. Furthermore we have o reaso to believe that the actual approximatio factor does ot i fact match the cojectured 4-approximatio for the ulimited supply case. Thus, the radom samplig auctio could potetially outperform the best kow auctio for (offlie) limited supply [10] which is a 6.5 approximatio. Ope Questio. What is RSEM s approximatio factor to G (2)? Oe directio for future study is i whether the bechmark G(k) has a profit extractor, i.e., is there a (icetive compatible ad idividually ratioal) auctio that whe give target profit R ca extract profit R o ay valuatio profile v with R G(k, v). This questio is especially iterestig because the best prior-free auctios for ulimited supply are based o profit extractors for F. As we argued, F is the wrog bechmark for limited supply. The challege i developig a profit extractor for G(k) is that the bechmark is iheretly two dimesioal where as all kow profit extractors work by searchig liearly i a sigle dimesio. Ope Questio. Is there a profit extractor for G(k)? 5. REFERENCES [1] S. Alaei, A. Malekia, ad A. Sriivasa. O radom samplig auctios for digital goods. I Proc. 11th ACM Cof. o Electroic Commerce, [2] M. Babaioff, L. Blumrose, ad A. Roth. Auctios with olie supply. Workig paper. [3] S. Baliga ad R. Vohra. Market research ad market desig. Advaces i Theoretical Ecoomics, 3, [4] S. Chakraborty ad N. Devaur. A Olie Multi-uit Auctio with Improved Competitive Ratio. ArXiv e-prits, Jauary [5] U. Feige, A. Flaxma, J. Hartlie, ad R. Kleiberg. O the Competitive Ratio of the Radom Samplig Auctio. I Proc. 1st Workshop o Iteret ad Network Ecoomics, pages , [6] A. Fiat, A. Goldberg, J. Hartlie, ad A. Karli. Geeralized competitive auctios. I Proc. 34th ACM

9 Symp. o Theory of Computig, pages ACM Press, [7] A. V. Goldberg, J. D. Hartlie, A. Karli, M. Saks, ad A. Wright. Competitive auctios. Games ad Ecoomic Behavior, 55: , [8] A. V. Goldberg, J. D. Hartlie, ad A. Wright. Competitive auctios ad digital goods. I Proc. 12th ACM Symp. o Discrete Algorithms, pages ACM/SIAM, [9] J. Hartlie ad A. Karli. Profit maximizatio i mechaism desig. I N. Nisa, T. Roughgarde, É. Tardos, ad V. Vazirai, editors, Algorithmic Game Theory, chapter 13, pages Cambridge Uiversity Press, [10] J. Hartlie ad R. McGrew. From optimal limited to ulimited supply auctios. I Proceedigs of the 7th ACM Coferece o Electroic Commerce, pages , [11] J. Hartlie ad T. Roughgarde. Optimal mechaism desig ad moey burig. I Proc. 39th ACM Symp. o Theory of Computig, [12] M. Mahdia ad A. Saberi. Multi-uit auctios with ukow supply. I Proc. 8th ACM Cof. o Electroic Commerce, [13] R. Myerso. Optimal auctio desig. Mathematics of Operatios Research, 6:58 73, 1981.

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