Radomizatio beats Secod Price as a Prior-Idepedet Auctio Hu Fu Nicole Immorlica Breda Lucier Philipp Strack Jue 7, 5 Abstract Desigig reveue optimal auctios for sellig a item to symmetric bidders is a fudametal problem i mechaism desig. Myerso (98) shows that the secod price auctio with a appropriate reserve price is optimal whe bidders values are draw i.i.d. from a kow regular distributio. A corerstoe i the prior-idepedet reveue maximizatio literature is a result by Bulow ad Klemperer (996) showig that the secod price auctio without a reserve achieves ( )/ of the optimal reveue i the worst case. We costruct a radomized mechaism that strictly outperforms the secod price auctio i this settig. Our mechaism iflates the secod highest bid with a probability that varies with. For two bidders we improve the performace guaratee from.5 to.5 of the optimal reveue. We also resolve a questio i the desig of reveue optimal mechaisms that have access to a sigle sample from a ukow distributio. We show that a radomized mechaism strictly outperforms all determiistic mechaisms i terms of worst case guaratee. Itroductio Desigig reveue optimal auctios to sell a item to symmetric bidders is oe of the most fudametal problems i optimal mechaism desig. This problem was solved by Myerso (98): if the buyers valuatios for the item are idepedetly draw from a distributio which is kow to the seller, the reveue-maximizig mechaism is a secod price auctio (SPA) with a reserve price. Notably, to determie the reserve price ad implemet the optimal auctio the seller eeds to kow the valuatio distributio. Sice settig the reserve price too high might leave the seller with o reveue, the desig of auctios with good reveue guaratee whe the prior iformatio is uavailable, or very costly to lear, is of high ecoomic relevace. A first aswer to this questio was give by Bulow ad Klemperer (996), who show that for regular distributios a secod price auctio with a reserve price of zero guaratees the seller at least a / fractio of the optimal reveue, where is the umber of bidders. I particular, this yields at least half of the optimal reveue wheever there are at least two bidders i the auctio. Note that this auctio is prior-idepedet, i the sese that it does ot deped o the prior distributio from which valuatios are draw. Moreover, it is clear that this is the best possible Microsoft Research New Eglad, hufu@microsoft.com Microsoft Research New Eglad, icimm@gmail.com Microsoft Research New Eglad, brlucier@microsoft.com Uiversity of Califoria Berkeley, pstrack@berkeley.edu A distributio is regular if the virtual valuatio is o-decreasig. This assumptio, which is stadard i the mechaism desig literature, will be discussed i Sectio.
way to set a determiistic reserve price: ay reserve greater tha zero yields zero reveue whe all bidders valuatios fall below it, so o determiistic positive reserve ca guaratee the seller a higher share of the optimal reveue. I other words, the fixed reserve price that guaratees the seller the highest share of the optimal reveue, agaist all regular distributios, equals zero. I fact, with the characterizatio developed by Myerso, it is ot hard to show that o symmetric determiistic auctio ca improve upo Bulow ad Klemperer s guaratee of half of the optimal reveue (for two bidders). This paper gives the first mechaism that outperforms the secod price auctio i Bulow ad Klemperer (996) s settig, by way of makig use of radomizatio. This mechaism, which we call bid iflatio, works as follows: with a fixed probability ɛ <, the mechaism rus a SPA with a reserve price of zero; i.e., each bidder gets the object if ad oly if his valuatio is above all other bidders valuatios. With the remaiig probability ɛ >, the mechaism allocates the object to the bidder with the highest valuatio if ad oly if his valuatio is greater tha ( + δ) > times the valuatio of ay other bidder, otherwise the object is uallocated. The idea behid bid iflatio is based upo a refied aalysis of the result by Bulow ad Klemperer (996). Our aalysis distiguishes two types of distributios: those for which the optimal reserve price is high, ad those for which it is low. A reserve price is cosidered high (low) if the probability that the valuatio exceeds it is low (high). We first show that wheever the optimal reserve price is sufficietly low, the secod price auctio without reserve obtais strictly more tha a ( /) fractio of the optimal reveue. This suggests that oe should desig a mechaism which works well for distributios with high reserve prices. A aalysis à la Myerso suggests that oe should try ot to allocate to bids below the optimal reserve price ad oly allocate to the bids above it (eve without kowig the optimal reserve). Whereas iflatig the secod highest bid will correctly keep the item usold whe the highest bid is ot high eough, this will also ievitably lose reveue whe the highest bid is high but the secod highest bid is ot far below. The key to our aalysis is to show that for distributios with high reserves, the loss from the latter sceario does ot offset the gai from the former oe. This i tur relies o a techical lemma which cotrols the quatile, i.e., the relative stadig i a distributio, as a valuatio chages i a regular distributio. With this aalysis for the iflated secod price auctio i had, oe ca the show via a cotiuity argumet that a probabilistic mixture of the iflated auctio ad the secod price auctio without a reserve price guaratees a better approximatio for all distributios. While the mai poit of this paper is to demostrate that there exist radomized mechaisms that outperform the secod-price auctio i a worst case aalysis over regular distributios, we also quatify the improvemet i the two bidder case. For two bidders, a secod price auctio with zero reserve price guaratees the seller half of the optimal reveue. We show that the bid iflatio mechaism, where the higher valuatio bidder receives the object with probability 85% if his valuatio is below twice the other bidder s valuatio, guaratees a reveue of at least 5.% of the optimal reveue. Thus, for two players the bid iflatio mechaism geerates a.4% higher reveue guaratee i the worst case sceario. Fially, we also cosider a variat of the prior-idepedet settig, i which the seller gais a small amout of iformatio about the prior distributio i the form of a sample. This samplig model has gaied recet attetio i the computer sciece literature. I a spirit similar to the result of Bulow ad Klemperer, it is kow that ruig a secod-price auctio, with a reserve set equal to the sampled value, guaratees at least + fractio of the optimal reveue for
arbitrary regular distributios; whe there is oly a sigle bidder, postig the sampled value as a take-it-or-leave-it price gives therefore half of the optimal reveue (Dhagwatotai et al., 4). It is also kow that o determiistic pricig method ca improve upo this guaratee for a sigle bidder (Huag et al., 4). However, we show how to costruct a radomized mechaism which improves upo this reveue guaretee ad obtais more tha half of the optimal reveue for a sigle bidder. The costructio is of a flavour similar to the bid-iflatio mechaism: it offers the sampled value as a take-it-or-leave-it price, but sometimes either iflates the value or shades it dow before usig it. Dhagwatotai et al. (4) uses the sigle sample techique to costruct a prior-idepedet auctio for settigs where bidders values are idepedet draws from o-idetical regular distributios but each distributio gives rise to at least two bidders values. Our improvemet for the sigle bidder sigle sample sceario immediately implies a better prior-idepedet auctio i this settig. Related Literature As the bid iflatio mechaism is idepedet of the distributio of valuatios it follows the doctrie propagated by Wilso (989) that ecoomic mechaisms should ot rely o precise details of the eviromet. Thereby, this paper falls i a recet literature i ecoomics ad computer sciece that aims at costructig mechaisms that work well uder differet distributios of valuatios (e.g. Bergema ad Schlag, 8, ; Dhagwatotai et al., 4; Roughgarde et al., ; Devaur et al., ; Fu et al., 3, 4). The secod part of our paper is also closely related to a recet literature that aalyzes the optimal use of a sigle sample i this cotext (Dhagwatotai et al., 4; Huag et al., 4). Huag et al. (4) showed that postig the sampled value as a take-it-or-leave-it price, which guaratees half of optimal reveue, is the optimal determiistic mechaism. We show i Sectio 4 that radomly iflatig ad shadig the sample value ca strictly improve the guaratee. I cotrast to studyig the use of a sigle sample, Cole ad Roughgarde (4) give the asymptotic umber of samples eeded for a give approximatio factor, i the presece of asymmetric bidders. Fially, this aveue of study is related to a literature o parametric auctios, i which a mechaism ca deped o limited statistical iformatio about the valuatio distributios (such as the media, mea, ad/or variace) (Azar ad Micali, ; Azar et al., 3). The best-kow parametric auctios for our settig are determiistic, but it is atural to ask whether radomizatio ca help i this cotext. Prelimiaries Sigle Item Auctios I a classical sigle item auctio, the auctioeer has a sigle item to sell to bidders. Each bidder i has a private value v i for receivig the item. We will cosider the symmetric Bayesia settig, where each bidder s value is draw idepedetly from a distributio F with desity fuctio f = F. We write v = (v,..., v ) for the profile of values. A auctio cosists of allocatio rules x i : R + [, ] ad paymet rules p i : R + R, meaig that, whe the bid profile is v, each bidder i gets the item with probability x i (v) ad makes a paymet of p i (v). A bidder s utility is the x i (v)v i p i (v). The allocatio rules have to satisfy the feasibility costrait i x i(v), for all v. We also require a auctio to be idividually ratioal, that is, bidders always get oegative returs. 3
Formally, for each bidder i ad all bid profiles v, v i x i (v) p i (v). A auctio is Bayesia icetive compatible if, for each bidder i ad value v i, the bidder s expected utility (over radomess i other bidders values) is maximized by biddig her true valuatio. Formally, for each i, value v i, ad deviatio v i, [ E v i xi (v i, v i )v i p i (v i, v i ) ] ] E v i [x i (v i, v i )v i p i (v i, v i ). For each bidder i, a allocatio rule geerates a iterim allocatio rule which maps her value v i to a wiig probability, i expectatio over the other bidders bids. We abuse otatio ad use x i (v i ) = E v i [x i (v i, v i )] to deote the iterim allocatio rule. The iterim paymet rule p i (v i ) is similarly defied. Bayesia icetive compability is therefore easily expressed by the iterim allocatio rules ad iterim paymet rules: for each bidder i, each value v i ad possible deviatio v i, we have x i (v i )v i p i (v i ) x i (v i )v i p i (v i ). The expected reveue of a auctio is E v [ i p i(v)]. Give a value distributio F, a reveue optimal auctio is oe whose expected reveue is optimal amog all idividually ratioal ad Bayesia icetive compatible auctios. We refer to the reveue of this auctio as the optimal reveue. All auctios i this paper will satisfy the stroger (domiat strategy) icetive compatibility property, i.e., o matter what the the other bidders bid, it is always i a bidder s best iterest to truthfully bid her value. Sice we will oly cosider icetive compatible auctios (which is without loss of geerality by the revelatio priciple (Myerso, 98)), we have used the same symbol v for values ad bids. I geeral, a auctio will be icetive compatible if each bidder faces a take-it-or-leave-it price that does ot deped o her ow bid. All auctios cosidered i this paper will obviously satisfy this coditio. Reveue-Optimal Auctios I his semial work, Myerso (98) laid the foudatio for the study of reveue-optimal auctios. The followig theorem summarizes the part of his results that will be used i this work. Theorem (Myerso, 98). I a sigle item auctio where each bidder s value is draw i.i.d. from a distributio F, for ay Bayesia icetive compatible auctio with iterim allocatio rules x i s, (i) The expected reveue from each bidder is equal to the virtual surplus, defied as ( ) E vi F x i (v i ) v i F (v i). f(v i ) The term v i F (v i) f(v i is called the virtual value of the value v i. I other words, the virtual surplus is the wiig virtual value i expectatio. (ii) Whe the distributio F is regular, i.e., whe the virtual value is mootoe odecreasig with the value, the optimal auctio is the secod price auctio with a reserve price v. I this auctio, the item is oly sold whe at least oe bid is above v, ad the wier pays the higher of v ad the secod highest bid. We itroduced the otio of Bayesia icetive compatibility here because our bechmark optimal reveue mechaism eeds oly to satisfy Bayesia icetive compatibility. As Myerso showed, this is i fact equal to the reveue of the optimal domiat strategy icetive compatible auctio. 4
Bulow ad Roberts (989) s Iterpretatio ad Reveue Curves I a classic work, Bulow ad Roberts (989) gave a iterpretatio of Myerso (98) s optimal aucito ad drew a coectio betwee the theory of optimal auctios ad the theory of moopolist price discrimiatio. This coectio reveals much ecoomic ituitio uderlyig Myerso s results, ad provides powerful techical tools. It is this viewpoit ad tools that we will heavily use i this work, ad we explai this coectio i some detail here. A bidder whose value is draw from a distributio F ca be see as a market where the customers values are distributed accordig to F, which gives rise to its demad curve. I particular, if the moopolizer sets the price of a good to sell at p, the oly customers whose value are above p will buy the good. The demad is therefore F (p). Each value v is i this way mapped to its quatile q(v) := F (v), its relative stadig i this market. The reveue collected whe the moopolizer sells a quatity q is give by R(q) := v(q)q, where v(q) := F ( q). This fuctio R : [, ] R + is called the reveue curve. Back from this aalogy to the bidder, the quatile q of a value v is the probability with which the buyer will buy at a take-it-or-leave-it price of v. Note that for ay distributio F, q is uiformly distributed o [, ]. Bulow ad Roberts (989) showed that the slope of the reveue curve at a quatile q is exactly equal to the virtual value of the value correspodig to q. Formally, R (q(v)) = dr(q) dq = v F (v). q=q(v) f(v) As a cosequece, regular distributios, i which virtual values are mootoe odecreasig with values, are exactly the distributios with cocave reveue curves. Also, for a auctio with iterim allocatio rule x i, the virtual surplus (equal to the expected reveue) from a bidder is give by R (q)x i (q(v)) dq. The highest poit of a reveue curve correspods to the optimal reveue oe could get by settig a take-it-or-leave-it price for a bidder. This price we call the moopoly reserve price ad deote by v. We deote the quatile of v by. Throughout the paper we will assume R() = R() =. This is a rather stadard assumptio i the literature ad is without loss of geerality; for completeess we justify it i Appedix A. Aalysis of Secod Price Auctios by Bulow ad Klemperer (996) Bulow ad Klemperer (996) showed that, for bidders whose values are draw from a regular distributio, the reveue of a + bidder secod price auctio without reserve price is always (weakly) better tha the reveue of the bidder reveue optimal reveue auctio. This immediately implies that for bidders, the reveue of the secod price auctio without a reserve price is at least of the optimal reveue. 3 Iflated Secod Price Auctio We will cosider a variat of the secod price auctio, which will sometimes iflate the bid of the secod-highest bidder before offerig that bid as a fixed price for the highest bidder. Defiitio. The δ-iflated secod price auctio offers the item to the highest bidder at a take-itor-leave-it price set as ( + δ) times the secod highest bid. The (ɛ, δ)-iflated secod price auctio rus the δ-iflated secod price auctio with probability ɛ, ad rus the secod price auctio with probability ɛ. 5
We will prove the followig mai theorem i this sectio. Theorem. For ay, there is a (ɛ, δ)-iflated secod price auctio, such that for ay bidders with values draw i.i.d. from a regular distributio, the reveue of the iflated secod price auctio is strictly larger tha fractio of the optimal reveue. At the ed of the sectio, we give the improved approximatio ratios of.5 for the case =. We will begi our aalysis by studyig the relatioship betwee the optimal reveue ad the reveue of the secod price auctio (SPA) without reserve. The goal of this aalysis is to refie the stadard -approximatio result ad to preset a approximatio that depeds o the quatile of the optimal reserve price,. We begi by derivig a boud o the additive reveue loss suffered by usig SPA rather tha the optimal auctio. Lemma 3. For bidders with values draw i.i.d. from a regular distributio, the differece betwee the reveue of the optimal auctio ad that of the secod price auctio is at most R( )( ). Proof. The optimal auctio is a secod price auctio with a reserve at v. The allocatio rules for the optimal auctio ad the secod price auctio therefore differ oly whe the highest value is below v, i.e., its correspodig quatile is larger tha. Such quatiles correspod to egative virtual values, ad make up the differece betwee the optimal auctio (where the cotributio is zero) ad the secod price auctio (where the cotributio is egative). For a bidder biddig at quatile q, the probability that it is the lowest quatile is ( q). The total egative virtual surplus geerated by oe bidder, over quatiles above, is therefore R (q)( q) dq =R(q)( q) + ( ) = R( )( ) + ( ) R(q)( q) dq R(q)( q) dq This is the egative of the differece betwee the optimal reveue ad the reveue of the secod price auctio. To get a upper boud of the differece, we eed to fid a lower boud for the itegral. Usig the cocavity of R(q), we kow that, for q, R(q) R( ) q. Therefore, we kow that the quatity above is at most R( )( ) + ( ) R( ) q ( q) dq = R()( ). This is the reveue differece due to each bidder. Multiplyig this by gives us the lemma. As a corollary, we obtai the followig boud o the ratio betwee the reveue of SPA ad the optimal reveue. Corollary. For i.i.d. bidders with a regular valuatio distributio, for ay (, ], the secod price auctio extracts at least a ( q ) ( ) fractio of the optimal reveue. 6
Proof. We give a lower boud o the optimal reveue, which, combied with Lemma 3, gives a lower boud o the approximatio ratio of the SPA. The optimal auctio could post v as a takeit-or-leave-it price to each bidder i tur, ad sell at that price to the first bidder who accepts. This gives a reveue of R( )[ + ( ) + ( ) + + ( ) ]. The ratio of the reveue of the secod price auctio to the optimal reveue is therefore at least ( ) + ( ) + + ( ) = + ( ) + + ( ) + ( ) + + ( ) = ( ) ( ). Note that the factor i Corollary is a strictly icreasig fuctio i, ad is equal to whe =. For the case =, we see that the approximatio ratio of the SPA is at least. We will ow aalyze the δ-iflated secod price auctio, ad show that its approximatio ratio is better whe is small. Before that, we first prove a techical lemma that gives us bouds o quatiles i a regular distributio for values that are apart by a multiplicative factor. Lemma 4. For ay regular distributio: ( ). q v +δ ( + δ), ad ( ). For ay q, q v( q) +δ +δ +δ q q. Proof. The first statemet is a direct cosequece of the optimality of v as a reserve price for a sigle bidder: v = R( ) R(q(v /( + δ))) = q(v /( + δ)) v /( + δ). For the secod statemet, let us deote by q the quatile of v( q)/( + δ). By regularity of the distributio, the reveue curve is cocave, ad because q is greater tha q ad both are greater tha, o the reveue curve the poit (q, R(q )) is above the straight lie coectig (, R() = ) ad ( q, R( q)). Therefore, Rearragig the terms we have that q +δ +δ q q. R( q) q R(q ) q, v( q) q q q v( q) ( + δ)( q ). The ext lemma lower bouds the approximatio ratio of δ-iflated SPA for distributios with a small. Lemma 5. For a regular distributio with < /, the reveue of the δ-iflated SPA for i.i.d. bidders is at least ( ) R( ) [ ( + δ) ] ( + δ)q ( )( + δ) ( q) + + δ ( + δq) dq. Proof. Let the iterim allocatio rule of the δ-iflated secod price auctio for a bidder i be x i. Recall that the reveue from a bidder i is give by R (q)x i (q) dq. O [, ], R (q) is positive, ad we eed to lower boud x i (q); o (, ], R (q) is egative, ad we eed to upper boud x i (q). 7
We first lower boud R (q)x i (q) dq. For such quatiles, the correspodig value is larger tha v. Obviously, if ay bidder has a value v > v, as log as all other bidders values are below v +δ, the bidder biddig v will wi. By Lemma 4, the quatile of v +δ is at most ( + δ), ad therefore the probability that a bidder s value is below v +δ is at least ( + δ). Hece, the expected reveue collected from each bidder for values larger tha the moopoly reserve is at least R (q)[ ( + δ) ] dq = [ ( + δ) ] R( ). We the upper boud the egative of R (q)x i (q) dq. For such quatiles, the correspodig value is smaller tha v. Recall that we would like to upper boud x i (q). By the defiitio of δ-iflated SPA, such a value wis the auctio if ad oly if all other bidders bid below v/( + δ). By Lemma 4, the quatile of v/( + δ) is at least +δ +δq q. I other words, the probability that a idepedet draw has value less tha v +δ +δ is at most +δq q = q +δq. Therefore, the egative cotributio to the virtual surplus by each bidder is lower bouded by ( ) ( ) R q q ( ) q (q) dq =R(q) + δq + δq + ( ) R(q) + δq ( ) q = R( ) + ( )( + δ) R(q) + δ + δ ( + δq) dq ( q) ( + δq) dq, where i the first step we did a itegratio by part, ad i the secod step we used the fact R() =. Sice we aim to lower boud this quatity, we use the fact that R(q) to the right of is poitwise lower bouded by R( ) q because of its cocavity. Substitutig this, we have that R(q) ( q) ( + δq) dq R( ) q ( q) ( + δq) dq = R(q ) ( q) dq. () ( + δq) Combiig everythig together, the reveue of the δ-iflated secod price auctio is at least ( ) R( ) [ ( + δ) ] ( + δ)q ( )( + δ) ( q) + + δ ( + δq) dq. () Lemma 6. For δ = ad <, the ratio of the -iflated secod price auctio reveue to the optimal reveue is at least [ ( + δ) ] ( ( + δ) + δ [ ( + ) ( ) We relegate the proof to Appedix B. ( q ) ) + ( )( ) ( + ) ] ( ). 8
Proof of Theorem. By Lemma 6, for =, the approximatio ratio of the -iflated secod price auctio is at least [ + ( + ) ], which is strictly greater tha for ay. Sice the boud give by Lemma 6 is a cotiuous fuctio, for a sufficietly small q >, for all < q, the -iflated SPA has a approximatio ratio at least [ + ( + ) ]. Recall that the approximatio ratio we derived i Corollary is a strictly icreasig fuctio i that equals to at =. Therefore at q, the approximatio ratio of the SPA is ( + γ) for some γ >. Takig ɛ > small eough such that (+γ)( ɛ) > +η for some η >, we will esure that for all > q, the approximatio of the (ɛ, δ)-iflated SPA is at least ( + η) (this pessimistically does ot assume ay reveue comig from the -iflated SPA). O the other had, for q, sice the approximatio ratio of the SPA is always at least, the (ɛ, δ)-iflated SPA has a approximatio ratio at least ( + ɛ ( + ) ). This proves the theorem. Remark. Lemma 6 is i place because the itegral i () is ot easy for geeral values of. The argumet i the proof of Theorem is rather pessimistic. However, for cocrete values of, oe ca compute the reveue lower boud i Lemma 5 for ay δ >, without goig through further losses i the aalysis of Lemma 6 ad Theorem, ad get better approximatio ratios. For example, for = ad δ =, ( q) ( + δq) dq = q ( + q) dq = + + log ( q + ). Oe ca substitute this ito () ad combie it with Corollary ; umerical computatio shows that the (.5, )-iflated secod price auctio gives at least.5 fractio of the optimal reveue. 4 Reveue Maximizatio with a Sigle Sample I this sectio we show that, for ay buyer with value draw from a regular distributio F, with oe sample from the same distributio, oe ca extract strictly more tha half of the the optimal reveue, by itroducig radomizatio i the use of the sample. We deote the sample by s, ad the buyer s value by v. Note that i this settig, the optimal reveue is simply R( ). Defiitio. The post-the-sample algorithm posts the sample s as a take-it-or-leave-it price. The ρ-shaded post-the-sample algorithm posts ( ρ)s as a take-it-or-leave-it price. The δ-iflated post-the-sample algorithm posts ( + δ)s as the take-it-or-leave-it price. The (ζ, ρ, ɛ, δ)-radomized post-the-sample algorithm rus ρ-shaded post-the-sample with probaiblity ζ, δ-iflated post-thesample with probability ɛ, ad (ormal) post-the-sample with probability ζ ɛ. Theorem 7. There exists ζ, ρ, ɛ, δ such that for ay regular distributio, with a sigle sample, the (ζ, ρ, ɛ, δ)-radomized post-the-sample algorithm extracts strictly more tha half of the optimal reveue. Usig our radomized post-the-sample algorithm i place of the origial post-the-sample algorithm i the auctio of Dhagwatotai et al. (4), we have the followig corollary. Corollary. I a sigle-item multi-bidder auctio, where bidders values are draw idepedetly from regular distributios, ad where for each bidder there is at least aother bidder whose value is draw from the same distributio, there is a prior-idepedet auctio whose reveue is strictly better tha 8 of the reveue of the optimal auctio which kows all distributios. 9
I Theorem 7, the improvemet over the oe half approximatio will be o the order of 8, ad this is admittedly maily of theoretical iterest (we also made o effort i optimizig the parameters). The cocrete values of ζ ad ɛ are give i the proof ear the ed of the sectio. We first aalyze the performace of the three igrediet mechaism i the radomized post-the-sample algorithm, give i the ext three lemmas. Lemma 8. For ay [, ], the approximatio ratio of the -iflated post-the-sample algorithm is at least + [ + + + + log ( q + ) ]. (3) Proof. The reveue from a sigle bidder by postig twice the sample is exactly half of the reveue from a two bidder -iflated secod price auctio where the two bidders values are draw i.i.d. from the same regular distributio. The lemma the follows from Lemma 5 ad Remark. Lemma 9. For ay regular distributio, if postig a price of ρ obtais a reveue that is β fractio of the optimal reveue R( ), the the reveue of the ρ-shaded post-the-sample algorithm is at least [ ( ) ] q + ( ρ) β + βρ R( ). (4) This is the most techically ivolved proof of this sectio. We relegate it to Appedix C. Lemma. If for a regular distributio, postig a price of v ρ v obtais a reveue that is β fractio of the optimal reveue R( ), the the reveue of the post-the-sample algorithm is at least ( + ρβ)r( ). Proof. The reveue of the post-the-sample algorithm is equal to the total area uder the reveue curve (Dhagwatotai et al., 4). We will therefore give a lower boud o this area. I geeral, subject to cocavity ad fixig R( ), the area uder the reveue curve is miimized at R() by the triagle. Give β, the fractio of R( ) extracted by postig a price v /( ρ), the quatile of v /( ρ) is give by βr( )( ρ). At this quatile, the triagle reveue curve would have a reveue of R( )β( ρ), but the curret reveue is R( )β. The two differ by ρβr( ). The extra area over the triagle is therefore at least ρβr( ). Proof of Theorem 7. We will show that settig δ =, ρ =., ad sufficietly small ɛ ad ζ, with ɛ = 4ζ, will guaratee a reveue better tha of the optimal reveue. Note that (4) is a decreasig fuctio i β for ρ <.5. For. ad ay value of β,.8 (4)+. (3) is at least.55r( ), by takig the worst value of β =. For β.5 ad ay value of,.8 (4)+. (3) is at least.58r( ), by takig the worst value of β =.5, ad the miimum is take at =. For the oly remaiig case, i.e., >. ad β >.5, Lemma gives that the reveue of the post-the-sample algorithm is at least.55r( ). Therefore, ruig post-the-sample with probability 6 guaratees a reveue of better tha.5+ 6 fractio of R( ) i this case. With the remaiig probability of 6, ruig the δ-iflated post-the-sample with probability 7 ad the ρ-shaded post-the-sample with probability 8 7 guaratees a reveue of.5 + 5 9 fractio of the optimal reveue i the cases aalyzed above. Overall,
the (.8 6,.,. 6, )-radomized post-the-sample algorithm gives a approximatio ratio of at least.5 + 5 9. Refereces Azar, P., Daskalakis, C., Micali, S., ad Weiberg, S. M. (3). Optimal ad efficiet parametric auctios. I Proceedigs of the Twety-Fourth Aual ACM-SIAM Symposium o Discrete Algorithms, pages 596 64. SIAM. Azar, P. ad Micali, S. (). Optimal parametric auctios. Mit-csail-tr--. Bergema, D. ad Schlag, K. (). Should i stay or should i go? search without priors. mimeo, available at: http://www.gtceter.org/archive//cof/schlag34. pdf (last accessed 8 April 4). Bergema, D. ad Schlag, K. H. (8). Pricig without priors. Joural of the Europea Ecoomic Associatio, 6(-3):56 569. Bulow, J. ad Klemperer, P. (996). Auctios versus egotiatios. The America Ecoomic Review, pages 8 94. Bulow, J. ad Roberts, J. (989). The simple ecoomics of optimal auctios. Joural of Political Ecoomy, 97(5):6 9. Cole, R. ad Roughgarde, T. (4). The sample complexity of reveue maximizatio. I Symposium o Theory of Computig, STOC 4, New York, NY, USA, May 3 - Jue 3, 4, pages 43 5. Devaur, N. R., Hartlie, J. D., Karli, A. R., ad Nguye, C. T. (). multi-parameter mechaism desig. I WINE, pages 33. Prior-idepedet Dhagwatotai, P., Roughgarde, T., ad Ya, Q. (4). Reveue maximizatio with a sigle sample. Games ad Ecoomic Behavior. Fu, H., Haghpaah, N., Hartlie, J. D., ad Kleiberg, R. (4). Optimal auctios for correlated buyers with samplig. I ACM Coferece o Ecoomics ad Computatio, EC 4, Staford, CA, USA, Jue 8-, 4, pages 3 36. Fu, H., Hartlie, J. D., ad Hoy, D. (3). Prior-idepedet auctios for risk-averse agets. I ACM Coferece o Electroic Commerce, pages 47 488. Huag, Z., Masour, Y., ad Roughgarde, T. (4). Makig the most of your samples. arxiv preprit arxiv:47.479. Myerso, R. (98). Optimal auctio desig. Mathematics of Operatios Research, 6():pp. 58 73. Roughgarde, T., Talgam-Cohe, I., ad Ya, Q. (). Supply-limitig mechaisms. I ACM Coferece o Electroic Commerce, pages 844 86.
Wilso, R. B.,. (989). Game-theoretic approaches to tradig processes. I Advaces i ecoomic theory. Cambridge Uiversity Press. A Missig Argumet from Sectio Throughout the paper we have made the assumptio R() = R() =. This is stadard i the literature, ad we briefly show that this is without loss of geerality. A value distributio whose support icludes satisfies R() = ; otherwise, oe ca mix ito the distributio with probability ɛ a uiform distributio o [, v], where v is the ifimum of the support. As ɛ approaches, this mixture coverges to the origial distributio, ad for sufficietly small ɛ, the resultig reveue curve remais cocave if the origial reveue curve is. As for the assumptio R() =, we first justify it for bouded support distributios. Whe there is o poit mass o the supremum of the support, its reveue will be at quatile. If this is ot the case, we ca mix ito the distributio with probability ɛ a uiform distributio o [ v, v + δ], where v is the supremum of the support, ad δ > is a small positive real umber. For sufficietly small δ ad ɛ, the resultig reveue curve will be cocave if the origial oe is; as ɛ approaches, the mixture also coverges to the origial distributio. Ay ubouded distributio s reveue ca be approached arbitrarily well by takig its trucatio at higher ad higer values. The trucated distributio is bouded. All our aalysis would ot be affected by all such asymptotic approximatios, ad we will simply assume R() = R() =. B Missig Proofs from Sectio 3 Proof of Lemma 6. We estimate the itegral i (). For δ =, ( q) ( + δq) dq = = ( ) ( q) ( + q) dq = ( q) ( q ) dq + ( q) dq + ( q) ( q ) dq ( q) ( q ) dq [ ( = ( ) ( + ) + ) ( q) ( ) dq ( q ) ] ( ). Substitutig this ito (), ad otig that the optimal reveue is at most R( ), we obtai the lemma. C Missig Proofs from Sectio 4 Proof of Lemma 9. We first itroduce some otatios. Let ϕ : [, ] [, ] be a fuctio that maps a quatile q to the quatile whose value is /( ρ) times the value correspodig to q.
(Formally, ϕ( q) = q(f ( q)/( ρ)). 3 ) We cosider the differece of ρ-shaded post-the-sample algorithm as compared with the plai v post-the-sample algorithm. The two algorithms differ oly whe s falls i the iterval [v, ρ ]: the post-the-sample algorithm does ot serve, whereas the ρ-shaded post-the-sample algorithm does. Coditioig o v, this evet happes with probability q(v) ϕ(q(v)). The reveue of the post-thesample algorithm is R (q)( q) dq, ad therefore the reveue of the ρ-shaded post-the-sample algorithm is R (q)( q) dq + R (q)(q ϕ(q) dq = R (q) dq R (q)ϕ(q) dq = R (q)ϕ(q) dq. We eed to lower boud this term. Whe the buyer s value v is smaller tha v, its virtual value is egative. Here we would like to lower boud ϕ(q). Let γ be the ratio of the reveue of postig v ( ρ) to the optimal reveue R( ). Let q γ be the quatile of the value v ( ρ), the because v ( ρ)q γ = γv, we have q γ = γ /( ρ). For v v ( ρ), we kow that the reveue of postig the price v is smaller tha that of postig v/( ρ). I other words, vq(v) ϕ(q(v)), ad therefore we get the lower boud ϕ(q) ( ρ)q, q [q γ, ]. Let the reveue of postig v ρ be βr(). The the quatile of the value v ρ is β( ρ). We kow that, for all q [, q γ ], ϕ β ( ρ). Now we have that qγ v ρ R (q)ϕ(q) dq R (q)ϕ(q) dq R (q)β ( ρ) dq ( ρ) = R q γ (q)ϕ(q) dq β ( ρ)r(q) ( ρ) R(q)q R(q) dq q γ q γ q γ R (q)q dq Usig R(q γ ) = γr( ), this is equal to [ q R (q)ϕ(q) dq ( ρ) β (γ )R( ) γr( ) = = γq ρ q γ R(q) dq R (q)ϕ(q) dq + ( ρ)β ( γ)r( ) + γ R( ) + ( ρ) R(q) dq q γ R (q)ϕ(q) dq + ( ρ)β ( γ)r( ) + γ R( ) + ( ρ)γr( ) R (q)ϕ(q) dq + ( ρ)β ( γ)r( ) + γ R( ) + γr() ( ρ γ ), ] ( γ ρ where i the iequality we used the cocavity of R ad the fact R(q γ ) = γr( ). Note that the first term is a quatity uaffected by the value of γ. Therefore we ca take the partial derivative of this lower boud with respect to γ ad get R( ) [ ( ρ)β + γ + ρ 3 If v/( ρ) is beyod the support of the distributio, just let ϕ be. ]. ) 3
Sice q γ, v ( ρ)q γ ( ρ)v, ad hece γ ρ. 4 Therefore the partial derivative is lower bouded by ( R( )( ρ) ( β) + ). Therefore, to miimize the reveue, the adversary should miimize γ. Hece, substitutig γ by ρ, we get the followig lower boud o the reveue: [ ] q R ( ρ)( + ) (q)ϕ(q) dq + + βρ R( ). (5) Now we cosider boudig the first term. For q, its virtual value R (q) is positive, therefore we eed to upper boud ϕ(q). Recall that β is the umber such that the reveue of postig v ρ is βr(). Let q deote the quatile of the value v /( ρ), ad we kew that q = β ( ρ). Notice that, for every q [, ], ϕ(q) q. We have R (q)ϕ(q) dq q R (q) dq = β ( ρ)r( ). Substitutig this to (5), we see that the reveue is lower bouded by [ ( ) ] q + ( ρ) β + βρ R( ). 4 A tighter lower boud for γ would be ρ ρ+ρ, but the loose boud ρ will suffice for our purpose. 4