Robust Mechanisms for Risk-Averse Sellers

Transcription

1 Robust Mechaisms for Risk-Averse Sellers Mukud Sudararaja Google Ic., Moutai View, CA, USA Qiqi Ya Departmet of Computer Sciece, Staford Uiversity, Staford, CA, USA ABSTRACT The existig literature o optimal auctios focuses o optimizig the expected reveue of the seller, ad is appropriate for risk-eutral sellers. I this paper, we idetify good mechaisms for risk-averse sellers. As is stadard i the ecoomics literature, we model the risk-aversio of a seller by edowig the seller with a mootoe, cocave utility fuctio. We the seek robust mechaisms that are approximately optimal for all sellers, o matter what their levels of risk-aversio are. We have two mai results for multi-uit auctios with uit-demad bidders whose valuatios are draw i.i.d. from a regular distributio. First, we idetify a posted-price mechaism called the Hedge mechaism, which gives a uiversal costat factor approximatio; we also show for the ulimited supply case that this mechaism is i a sese the best possible. Secod, we show that the VCG mechaism gives a uiversal costat factor approximatio whe the umber of bidders is eve a small multiple of the umber of items. Alog the way we poit out that Myerso s characterizatio [11] fails to exted to utility-maximizatio for risk-averse sellers, ad establish iterestig properties of regular distributios ad mootoe hazard rate distributios. Categories ad Subject Descriptors J.4 [Computer Applicatios]: Social ad Behavioral Scieces Ecoomics Geeral Terms Ecoomics, Theory, Algorithms Keywords risk-aversio, optimal auctios, reveue maximizatio 1. INTRODUCTION Supported by a Staford Graduate Fellowship 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. Copyright 200X ACM X-XXXXX-XX-X/XX/XX...$ Auctio theory (cf. [11, 1]) typically seeks to optimize the seller s expected reveue, which presumes that the seller is risk-eutral. The focus of this work is to idetify good auctio mechaisms for sellers who care about the riskiess of the reveue i additio to the magitude of the reveue 1. There is a iheret trade-off betwee the magitude ad riskiess of reveue. Cosider the auctio of a sigle-item to a bidder whose valuatio is draw from the uiform distributio over the iterval [0, 1]. Recall that every truthful sigle-bidder mechaism offers the bidder a take-it-or-leaveit price. If the seller is risk-eutral ad cares about mea reveue, we must select a price p that maximizes the product of the price p times the probability of sale 1 p. The price p = 1/2 is optimal here, achievig a mea reveue of 1/4, but it yields zero reveue with probability 1/2. Prices lower tha 1/2 reduce the expected reveue, but icrease the certaity with which positive reveue is obtaied. A systematic ad stadard (cf. Stiglitz ad Rothschild [14]) way to express a bidder s trade-off betwee the magitude ad riskiess of reveue is to edow the seller with a cocave utility fuctio u : [0, ) [0, ) ad seek to maximize the seller s expected utility. We will assume throughout that this utility fuctio is mootoe ad ormalized i the sese that u(0) = 0. Let Rev(M,v) deote the reveue of mechaism M for the iput bid-profile v, the the expected utility of M w.r.t. a utility fuctio u is E v[u(rev(m,v))]. The cocavity of the utility fuctio models risk-aversio For istace, the optimal sigle-bidder mechaism for the utility fuctio u(x) = x sets a price p = 1/3 ad maximizes the expected utility p (1 p). Icreasig the cocavity of the utility fuctio icreases the emphasis o riskaversio the optimal price for the cube-root utility fuctio is p = 1/4. The liear utility fuctio u(x) = x models a risk-eutral seller. The goal of this paper is to idetify truthful mechaisms that are simultaeously good for the class of all risk-averse agets, i.e. we look for mechaisms that yield ear-optimal expected utility for all possible cocave utility fuctios. A useful side-effect of such a guaratee is that we do ot eed to kow the seller s utility fuctio i order to deploy the mechaism this is useful whe the auctioeer is coductig the auctio o behalf of a seller (thik ebay), whe the seller does ot kow its utility fuctio precisely, or whe 1 We seek ex-post icetive compatible mechaisms.this is i cotrast to the stadard Bayesia auctio theory literature (cf. [11, 1]) that studies Bayesia icetive compatible mechaisms. Bidders will therefore maximize utility by truth-tellig, ad do ot have to deal with ucertaity or risk; our model of risk applies oly to sellers.

2 the seller s risk attitude chages with time. The followig example illustrates the challege i the cotext of a sigle-item sigle-bidder auctio. Cosider two sellers with utility fuctios u risk-eutral (x) = x, which expresses risk-eutrality, ad u risk-averse (x) = mi(x,ǫ) for some very small ǫ > 0, which expresses strog risk-aversio. Suppose, as before, that there is a sigle bidder whose valuatio is draw from the uiform distributio with support [0, 1]. The uique optimal mechaism for the first utility fuctio makes a take-it-or-leave-it offer of 1/2. This gives the first seller a utility of 1/4, ad gives the secod seller a utility of ǫ (1 F(1/2)) = ǫ/2. Lowerig the price to ǫ improves the secod seller s utility to (1 ǫ) ǫ, but reduces the first seller s utility from 1/4 to (1 ǫ) ǫ. Our challege i geeral is to idetify mechaisms that simultaeously appease sellers with differet levels risk-aversio ragig from risk-eutral sellers who care about expected reveue to very risk-averse oes who oly care about the certaity with which a positive reveue is obtaied. 1.1 Orgaizatio Sectio 2 describes our auctio model, our distributioal assumptios ad formalizes our auctio objective. Sectio 3 describes the difficulty i characterizig our bechmark ad defies a stroger, simpler bechmark. Sectio 4 idetifies uiversally approximate posted-price mechaisms for ulimited ad limited supply. Sectio 5 bouds the uiversal approximatio of the VCG mechaisms for multi-uit auctios as a fuctio of the ratio of the umber of bidders to the umber of items. Sectio 6 cocludes with ope directios. 2. PRELIMINARIES 2.1 Auctio Model Our ivestigatio focuses o multi-uit auctios. We adopt the followig stadard auctio model. There are uitdemad bidders 1,2,...,, ad k idetical idivisible items for sale. A bidder i has a private valuatio v i for wiig a item, ad 0 for losig. A mechaism M = (x,p) first collects a bid b i from each bidder i, the determies the wiers by the allocatio rule x : b {0, 1}, i.e., bidder i wis a item if ad oly if x i(b) = 1, ad fially uses the paymet rule p : b [0, ) to charge each bidder i a price p i(b). We will focus our attetio o ex post icetive compatible, a.k.a., truthful, 2 ad ex post idividual-ratioal 3 mechaisms. Hece we will use the terms bid ad valuatio iterchageably. We make the stadard assumptio that valuatios are draw i.i.d. from a distributio F. The distributio F is kow to the seller, but the valuatios ca be private to buyers. 2.2 Auctio Objective Let Rev(M,v) deote the reveue of mechaism M for the iput bid-profile v. The the expected reveue of M is E v[u(rev(m,v))] otice that the expectatio is over the bids (or valuatios) which is the stadard auctio objective i Bayesia reveue maximizatio. We model the riskattitude of a specific seller by edowig the seller with a 2 For ay possible bids b i of the other bidders, bidder i always maximizes her utility v i x i(b) p i(b), by settig her bid b i to be her true valuatio v i. 3 A bidder is ever charged more tha her bid, ad is oly charged whe she wis. cocave utility fuctio u : [0, ) [0, ). We will assume throughout that this utility fuctio is mootoe ad ormalized i the sese that u(0) = 0. The the expected utility of M w.r.t. a utility fuctio u is E v[u(rev(m,v))]. As discussed i the itroductio, the cocavity of the utility fuctio models risk-aversio. Recall that the goal of this paper is to idetify truthful mechaisms that are simultaeously good for the class of all risk-averse agets, i.e. we look for mechaisms that yield ear-optimal expected utility for all possible cocave ormalized utility fuctios. More precisely, for each riskaverse seller, the truthful mechaism M u that maximizes the seller s expected utility is a bechmark agaist which we measure our proposed mechaism (say M) we quatify the goodess of this mechaism for this seller by the approximatio ratio U(M)/U(M u), where U(X) deotes the expected utility of mechaism X. The goodess of the mechaism is the the worst-case approximatio ratio over all cocave utility fuctios, i.e. ρ = mi uu(m)/u(m u); i this case, we will say that the mechaism is a uiversal ρ- approximatio. For each of the auctio settigs we cosider, we will try to fid a mechaism M that maximizes ρ. 2.3 Distributioal Assumptios For techical coveiece, we will assume that the distributio F has a smooth positive desity fuctio, ad has o-egative support. We will i additio assume that the distributio F from which the valuatio is draw satisfies a stadard regularity coditio (cf. [11], [1]). Every distributio fuctio F correspods to a reveue fuctio R from domai [0, 1] (or (0,1] if the support of F is ifiite) to the o-egative reals defied as follows: for all q, R F(q) = q F 1 (1 q). (we will drop the subscript whe it is clear from the cotext) Note that R(0) = 0 ad R(1) = 0, ad we ca ofte defie a distributio F by specifyig the correspodig R F( ) fuctio. We say a distributio F is regular if the reveue fuctio R F( ) w.r.t. F is strictly cocave. This is also equivalet to the more commoly used defiitio that virtual valuatio φ F(v) = v 1/h(v) is odecreasig i v, where h(v) = f(v) is the hazard rate fuctio w.r.t. F. We say F satisfies the mootoe hazard rate 1 F(v) coditio (or simply F is m.h.r.), if h(v) is odecreasig i v. May importat distributios are regular ad m.h.r, icludig uiform, expoetial, ormal, while other distributios such as some power-law distributios are regular but ot m.h.r. [6]. To justify our use of the regularity assumptio, the followig example shows that o uiversal costat factor approximatio is possible without assumptios o the distributio F. Example 1. Recall the utility fuctios u risk-eutral ad u risk-averse defied i the itroductio. Defie R as R(0) = R(1) = 0, R(ǫ) = 1, R(2ǫ) = ǫ, R(1 ǫ) = ǫ, ad let R be liear i all four itervals betwee these five poits; here ǫ refers to the quatity i the defiitio of u risk-averse (see itroductio). Smoothe R by a egligible amout such that the correspodig F fuctio satisfies our smoothess assumptio o distributios. Cosider a sigle bidder whose valuatio fuctio is draw from F, which is clearly a irregular distributio. Thus to achieve a costat fractio of optimal utility for u risk-eutral meas that we have to sell at probability at price

3 at least 1/2, which implies that we get at most 2ǫ 2 utility for u risk-averse, compared to ǫ(1 ǫ) at the price ǫ/(1 ǫ). 2.4 Results ad Techiques We first show that the virtual-value based approach employed by Myerso [11] for the risk-eutral case exteds to risk-averse sigle-item auctios, but ot (to the best of our kowledge) to auctios of two or more items (see Sectio 3). We the preset three results. First, whe the supply is ulimited (or equivaletly, the umber of items k is equal to the umber of bidders ), we idetify a mechaism called the Hedge mechaism that is a uiversal 1/2-approximatio (see Theorem 6). The ratio improves to early 0.7 with the assumptio that the distributio satisfies a stadard hazard rate coditio. The Hedge mechaism is a posted pricig mechaism, which offers every bidder a take-it-or-leave-it offer p. We choose the price p to be less tha the optimal price for a risk-eutral seller so as to guaratee a good probability of sale to ay bidder at a good reveue level. Moreover, this mechaism is the best possible i the sese that o mechaism ca be a uiversal ρ-approximatio for ρ > 1/2 (see Theorem 8) this impossibility result idetifies a certai heavy-tailed regular distributio, called the lefttriagle distributio that exhibits the worst-case trade-off betwee riskiess ad magitude of reveue over all regular distributios. Secod, whe there is limited supply (umber of items k is less tha the umber of bidders ), we idetify a sequetial posted pricig mechaism that gives a uiversal 1/8-approximatio by modifyig the Hedge mechaism to hadle the supply costrait (see Theorem 12). The key to this modificatio is to use a certai limited supply auctio to guide the choice of the posted price. Third, we will show that the VCG mechaism [16, 3, 8] yields a uiversal approximatio ratio close to 1/4 uder moderate competitio, i.e., whe is a reasoable multiple of k (see Theorem 15). Recall that for a k-item auctio the VCG mechaism is a k + 1-st price auctio, i which the top k bidders wi ad get charged the k + 1-st highest bid. We prove our result by establishig a probability boud for the k + 1-st order statistic of i.i.d. draws from a regular distributio. 2.5 Related Work Myerso [11] idetifies the optimal sigle-item mechaism for a risk-eutral seller ad has ispired much work (cf. Chapter 13 from [12]). There is some work that tackles risk i the cotext of auctios. Eso [5] idetifies a optimal mechaism for a risk-averse seller, which always provides the same reveue at every bid vector by modifyig Myerso s optimal mechaism; ufortuately, this mechaism does ot satisfy expost (or eve ex-iterim) idividual ratioality, ad charges bidders eve whe they lose. Maski ad Riley [10] idetify the optimal Bayesia-icetive compatible mechaism for a risk-eutral seller whe the bidders are risk-averse. I our model, we idetify mechaisms that are ex-post icetive compatible. So the buyers optimize their utility biddig truthfully for every realizatio of the valuatios, ad thus have o ucertaity or risk to deal with. Hu et al. [9] study risk-aversio i sigle-item auctios. Specifically, they show for both the first ad secod price mechaisms that the optimal reserve price reduces as the level of risk-aversio of the seller icreases. I cotrast, we idetify the optimal truthful mechaism for a risk-averse seller i Sectio 3 (it happes to be a secod price mechaism with a reserve), study auctios of two or more items ad idetify mechaisms that are simultaeously approximate for all risk-averse sellers. A alterative, simpler a model of auctio-risk compared to the oe we adopt is to optimize for a trade-off betwee the mea ad the variace of the auctio reveue, i.e., E[R] t V ar[r]. However, as Sectio 2A i Stiglitz ad Rothschild [15] shows, this approach does ot capture all the types of behavior ituitively cosistet with risk-aversio, because this approach restricts the form of seller utility fuctios. Our model of risk-aversio is ispired i part by Stiglitz ad Rothschild [14]. There is sigificat literature o prior-free optimal auctios (see Chapter 13 from [12]). I this framework, the bechmark (i the ulimited supply case, the reveue from the optimal price for that bid vector costraied to serve at least 2 bidders) is defied idepedetly for each bid vector, ad the performace of the mechaism is measured worstcase over all bid-vectors. I cotrast, i our framework, as i all Bayesia auctio theory, the mechaism s performace is measured i expectatio over the distributio of the bids. However, we believe that it is worth ivestigatig the risk properties of the mechaisms proposed i this literature, which ought to yield uiversal costat factor approximatios i several auctio settigs. Fially, we metio papers that ispire our proof techiques. Chawla et al. [2] idetifies posted price mechaisms they use a auctio mechaism to guide the selectio of the prices. We use a similar idea i Sectio 4.2. Bulow ad Klemperer [1] shows that the VCG mechaism with k extra bidders yields better expected reveue tha the optimal mechaism so log as the bidder valuatios are draw i.i.d. from a regular distributio. Dughmi et al. [4] exteds the result of Bulow ad Klemperer [1] to matroid settigs, ad itroduces the problem of desigig markets with good reveue properties. We use ideas from these papers to boud the performace of the VCG mechaism i Sectio ON UTILITY-OPTIMAL MECHANISMS Recall from the itroductio that we would like to desig mechaisms that yield a good approximatio of the optimal expected utility for each utility fuctio. Our bechmark for a specific utility fuctio u is the truthful, idividually ratioal mechaism that maximizes the expected utility w.r.t. u i this sectio we focus o gettig a hadle o such a mechaism for a fixed utility fuctio u. We show that the result of Myerso [11] ca be exteded to idetify the optimal mechaism for the sigle item case, but ot for auctios of two or more items. For the rest of the paper, we use the stroger, simpler bechmark from Fact 3. Myerso s characterizatio says that the expected reveue of ay truthful mechaism equals the expected total virtual valuatio served by the mechaism. It geerates a prescriptio for the allocatio ad paymets of the optimal riskeutral truthful mechaism o a specific iput bid vector. I the sigle-item case, to geeralize Myerso s characterizatio to auctios with risk-averse sellers, we geeralize the otio of virtual valuatio to take risk-aversio ito accout: give a distributio F ad a cocave utility fuctio u, we defie the virtual utility fuctio as φ u F(v) = u(v) u (v)/h(v). As i the case of virtual valuatios, the virtual utility φ u d F(v) is the derivative u(v) (1 F(v)) of the d(1 F(v)) expected utility from a bid-idepedet take-it-or-leave-it

4 offer v to a sigle bidder. We the have the followig: Lemma 2. I a sigle-item auctio, for ay mechaism M = (x,p) ad cocave utility fuctio u, the expected utility of the mechaism, E v[u(rev(m,v))], is equal to the expected virtual valuatio served E v[ P i φu F(v i) x i(v)]. Proof. The expected utility of the mechaism is: E v[u(rev(m,v))] = E v[u( X i = X i = X i = E v[ X i p i(v))] E v i [E vi [u(p i(v))]] E v i [E vi [φ u F(v i) x i(v)]] φ u F(v i) x i(v)] Here the secod equality holds because we sell to at most 1 bidder. The third equality holds because whe v i is fixed, the mechaism iduces a fixed offer price, say p, for bidder Ri so E vi [u(p i(v))] = u(p )(1 F(p )), which is equal to (u(v) u (v)/h(v))f(v)dv, which is E p vi [φ u F(v i) x i(v)], the expected virtual utility we get from bidder i. We ca ow use the lemma to show that the optimal mechaism for a seller with utility fuctio u is a secod price auctio with a reserve price a mechaism that is well-kow to be truthful. Cosider the secod price mmechaism with a reserve r u, where r u solves that φ u F(r u) = 0. Whe the distributio is regular, the virtual utility fuctio is odecreasig i the valuatio (see Lemma 19 i the appedix). So the above mechaism allocates the item to the bidder with the highest virtual utility, so log as there is at least oe bidder with o-egative virtual utility. (Whe the distributio is ot regular, ad i particular whe the virtual utility fuctio is ot mootoe, oe ca apply the iroig procedure of Myerso to idetify the optimal mechaism as the oe that maximizes the total iroed virtual utility served.) I Sectio 5 we will preset aother applicatio of the above characterizatio that shows that the sigle-item Vickrey auctio has good reveue properties. However, this characterizatio does ot exted to auctios where more tha oe items are for sale. The first step of the proof of Lemma 2, which sums the cotributios of the bidders idepedetly, oly works because a sigle-item auctio sells to ad charges at most oe bidder. Whe there are more tha oe items for sale, that step is still soud if the utility fuctio is liear (the risk-eutral case), eablig Myerso s theory to be very geeral, but it does ot work for strictly cocave utility fuctios. We ow idetify a upper boud o the expected utility of utility-optimal mechaism that applies to auctio settigs beyod sigle-item auctios. We will use this upper boud as a bechmark for aalysis. For ay mechaism M ad cocave utility fuctio u, the expected utility of the mechaism E v[u(rev(m,v))] is upper-bouded by the utility fuctio applied to the expected reveue u(e v[rev(m,v)]) by Jese s iequality, which is the upper-bouded by the utility fuctio applied to the expected reveue of Myerso s reveue-optimal mechaism Mye, u(e v[rev(mye,v)]), because a utility fuctio is mootoe. So we have the followig: Fact 3. For ay mechaism M 4, ad ay cocave utility fuctio u, the expected utility of M is upper-bouded by the utility fuctio applied to the expected reveue of Myerso s mechaism, i.e., E v[u(rev(m,v))] u(e v[rev(mye,v)]). 4. UNIVERSALLY APPROXIMATE SEQUEN- TIAL POSTED PRICING MECHANISMS I this sectio we propose sequetial posted pricig mechaisms (or SPM i short) for multi-uit auctios. I a SPM, a take-it-or-leave-it price is offered to each bidder oe by oe i arbitrary order, as log as supply lasts. A obvious advatage of such mechaisms is that they ca be applied to both offlie ad olie settigs ad are collusio-resistat i the sese of Goldberg ad Hartlie [7]. 4.1 The Ulimited Supply Case Fix a regular distributio F from which the valuatios are draw i.i.d. We ow idetify a SPM that offers every bidder the same take-it-or-leave-it offer p, ad show that this mechaism is 1/2-approximate for all regular distributios, ad 0.69-approximate for all m.h.r. distributios. Let p is the optimal price that maximizes p(1 F(p)), ad q = 1 F(p). Settig the offer price p to be p yields the optimal expected reveue, but the probability of sale for each bidder ca be very low. Itriguigly, we fid that reducig the offer price to p q is optimal, i.e., the discout factor is precisely the probability of sale at the optimal price for a risk-eutral seller i a sigle item-sigle bidder auctio. We call this the Hedge Mechaism. Theorem 6 shows that this achieves a 1/2 approximatio for regular distributios ( 0.7-approximatio for m.h.r. distributios), ad Theorem 8 shows that we caot do better. To aalyze the performace of the Hedge mechaism, the followig property of regular distributios is crucial. Lemma 4. For all regular distributio F, we have 1 F(p q ) 1/2. Proof. Let q = 1 F(p q ). Note that q q because p q p. Let R( ) be F s reveue fuctio, which is cocave by regularity. The fact that q 1/2 follows from the followig iequalities: 4 We shall oly work with determiistic mechaisms, but i fact we ca allow the mechaism here to be radomized.

5 q = R(q)/(p q ) R(q ) 1 q «q + R(1)q /(p q ) 1 q 1 q (p q ) 1 q «/(p q ) 1 q = 1 q 1 q 1 q The first step is by the defiitio of q. The secod step is by the cocavity of R. (I the above figure, ote that (q, R(q)) is above the lie segmet coectig (q, R(q )) ad (1, R(1))). The third step is because R(q ) = p q ad R(1) is o-egative. Whe the distributio F is further assumed to be m.h.r., we ca improve the costat to e 1/e. Lemma 5. For a m.h.r. distributio F, we have 1 F(p q ) e 1/e Proof. W.l.o.g., we ca let p = 1 by scalig the valuatio R space. Let cumulative hazard rate fuctio H(x) be x h(t)dt, ad ote that the mootoe hazard rate coditio implies that H(x) is mootoe, covex ad ormalized 0 (H(0) = 0). Note that at the price p = 1, the virtual valuatio is 0, i.e., 1 1/h(1) = 0. So h(1) = 1. Further, the fuctio h is odecreasig. So H(1) = R 1 h(t)dt 1 h(1) = 1. 0 Our claim follows from the followig iequalities: q = 1 F(p q ) = 1 F(q ) = e H(q ) = e H(1 F(p )) = e H(e H(p ) ) = e H(e H(1) ) e H(1)e H(1) e 1/e The secod ad sixth steps are because p = 1. The third ad fifth steps are because the distributio fuctio ca be writte i terms of the cumulative hazard rate fuctio: F(x) = 1 e H(x). The seveth step is because H(e H(1) ) e H(1) H(1) by the covexity of H ad that H(1) 1. The last step holds because e x x is at most 1/e for x [0, 1]. We ow use the bouds i the previous two lemmas to complete the proof of the theorem. Theorem 6. I a multi-uit auctio with ulimited supply, where bidders valuatios are draw i.i.d. from a regular (or m.h.r) distributio F, the Hedge mechaism is a uiversal 0.5 (or e 1/e )-approximatio. Proof. We prove for the regular case; for the proof of the m.h.r. case we simply use the boud from Lemma 5 istead of the boud from Lemma 4. Fix a cocave utility fuctio u. For each bidder i, let 0-1 radom variable X i idicate whether bidder i s bid is at least p q. Expected Utility of Hedge = E[u( X X i p q )] i P i E[ Xi ] u(p q ) 0.5 u(p q ) 0.5 Optimal Expected Utility The first step is because the sale price is p q. The secod step is by mootoicity ad cocavity of u ad because 0 P i Xi p q p q. The third step is by Lemma 4, ad hece E[ P i Xi] /2. Applyig Fact 3 completes the proof. Remark 7. If bidders valuatios are draw from oidetical but idepedet regular distributios, we ca idetify a distict offer prices for each bidder i, p i q i, (here p i is the price that maximizes the expected reveue i a sigle bidder-sigle item auctio with bidder i; ad q i is the sale probability at that price), such that the guaratee i Theorem 6 holds. The followig lemma shows that the ratios i Theorem 6 caot be improved. The proof idetifies a certai lefttriagle distributio that exhibits worst-case behavior over regular distributios, ad shows that the expoetial distributio exhibits worst-case behavior over all m.h.r. distributios. The proof elucidates why the price p q is critical for the sigle-bidder case ad justifies its use i the Hedge mechaism. Theorem 8. There exists a regular (or m.h.r) distributio such that o mechaism yields a uiversal approximatio with ratio larger tha tha 1/2 (or e 1/e ) for a sigle-bidder sigle-item auctio, respectively. Proof. Cosider a sigle-item sigle-bidder auctio. Cosider two possible seller utility fuctios, u risk-eutral, ad u risk-averse as defied i the itroductio. The optimal utility w.r.t. u risk-eutral is p q, achieved at price p, ad the optimal utility w.r.t. u risk-averse is roughly ǫ (as ǫ 0), achieved at price ǫ. We argue that the sale probability q = 1 F(p q ) at the price p q is a upper-boud o the best uiversal approximatio possible. The expected reveue at price p q is qp q. So, the approximatio ratio for the risk-eutral seller is precisely q. The expected utility for the risk-averse seller at price p q is approximately ǫq. So, the approximatio ratio for this seller is also q. Now suppose a price lower tha p q is offered. The the expected reveue deteriorates, ad the approximatio ratio for the risk-eutral seller drops below q. O the other had, suppose a price higher tha p q is offered. The the sale probability drops below q, ad so does the approximatio ratio for the risk-averse seller. The it suffices to show that there is regular distributio with sale probability 1/2 at price p q, ad there is a m.h.r. distributio with sale probability e 1/e at price p q. First we defie the left-triagle distributio via its reveue fuctio R L( ) as follows. Let R L(0) = R L(1) = 0, R L(ǫ) = 1 for some small ǫ > 0, ad let R L(q) be liear betwee these poits, ad smoothe by a egligible amout to make sure that the correspodig F is a valid distributio. (It is essetially a shifted Pareto distributio.) So p q is 1, ad clearly the sale probability at price 1 is roughly 1/2.

6 Secod, cosider the expoetial distributio F(p) = 1 e p, which satisfies the mootoe hazard rate coditio. p = 1, q = 1/e, ad it follows that 1 F(p q ) = e 1/e. Remark 9. Our bouds i Theorem 8 ad Theorem 6 are worst-case over the umber of bidders, ad the mechaism we propose does ot require kowledge of. I geeral, the kowledge of is useful: As icreases it makes sese to icrease the price from the heavily discouted price p q towards the optimal risk-eutral price p, because for large, the resultig reveue as a radom variable is well cocetrated. 4.2 The Limited Supply Case I this sectio we idetify a SPM that yields a uiversal 1/8-approximatio for limited supply auctios. I this case, we have k items to sell, where k ca be less tha the umber of bidders, ad this allocatio costrait imposes a additioal challege: usig the posted price idetified i the previous sectio will cause us to hit the supply costrait without havig collected eough reveue. To defie the price to use i our posted pricig mechaism i this cotext, we apply a trick itroduced i [2] as follows. Give a mechaism that hoors the supply costrait, for a fixed bidder, defie the allocatio probability q to be the probability that she wis i ruig this mechaism, where the radomizatio is over all valuatio profiles. As the valuatios are idetically distributed, q is idetical for all bidders. The posted price to use is the p = F 1 (1 q). This takes care of the supply costrait. The key for us is the to fid the right mechaism to draw the allocatio probability from. Recall that the optimal risk-eutral mechaism is the VCG mechaism with reserve p. I order to have better cotrol over the distributio of the reveue of the mechaism, we derive the allocatio probability from the VCG mechaism with a discouted reserve r = p q. By Lemma 4, at least half of the bidders meet the reserve i expectatio, ad as we will show it follows that the allocatio probability q is bouded betwee k ad k. Moreover, the loss i expected reveue 2 due to this sub-optimal reserve is bouded. We formalize these i the followig two claims. Lemma 10. Rev(V CG r=p q ) 0.5 Rev(V CGp ). Proof. For otatioal coveiece, let ˆR(p) = p(1 F(p)). Fix a bidder i, fix the bids b i of the other bidders, ad let t be the threshold iduced by the V CG mechaism (with o reserve) for bidder i. The the threshold bids of bidder i i V CG p ad V CG r are max{t, p } ad max{t, r} respectively. It suffices to show that the expected reveue of bidder i i V CG r, which is ˆR(max{t, r}), is at least half of that i V CG p, which is ˆR(max{t, p }), ad our claim follows by summig over i ad b i. There are two cases. If t p, the t p q = r, ad so the offered prices ad the expected reveues from the two auctios are idetical. Otherwise, t < p, so bidder i is offered p (with reveue p q ) by V CG p, ad a price i the iterval [p q, p ] by V CG r. As reveue is mootoically decreasig as price goes dow from p to 0, the reveue of V CG r is miimized whe the offer price is p q. By Lemma 4 the resultig reveue p q (1 F(p q )) is at least, completig the proof. p q 2 Lemma 11. Let q be the allocatio probability of ay fixed bidder. The q lies i the iterval [ k 2, k ]. Proof. Let X be the umber of bidders with bids at least r. The expected umber of wiers of V CG r is mi(k, X). By defiitio of q, q is the expected umber of wiers i V CG r. So, q = E[mi(k, X)] ad hece, q k/. By defiitio of r, each bidder s bid is at least r with probability at least 0.5, ad so, E[X] 0.5. Therefore q = E[mi(k, X)] E[ k X] = k 0.5 = 0.5k. Now we ca defie our Hedge mechaism (for the limitedsupply case). The hedge mechaism is a SPM which makes a take-it-or-leave-it offer at price p = F 1 (1 q) to bidders oe by oe, as log as the supply lasts. Theorem 12. I a multi-uit auctio with k items ad bidders, where bidders valuatios are draw i.i.d. from a regular distributio F, the Hedge mechaism is a uiversal 1/8-approximatio to optimal expected utility. Notice that the reveue of Hedge is p mi(y, k), where Y is the umber of bidders who bid at least p, which is a biomial variable with parameter (, q). Hece E[Y ] = q 0.5k. Crucial to our aalysis is the followig property about capped biomial variables: Lemma 13. Let Y be a biomial radom variable with parameter (, q) where q 0.5k for some positive iteger k, the E[mi(Y,q)] 0.25 q. Proof. Clearly E[Y ] = q 0.5k. First let k = 1, ad hece 0.5 q 1. Note that E[mi(Y, q)]/q = Pr[Y > 0] = 1 (1 q), which is at least 1 (1 0.5/) 1 e 0.5 = 1 e 0.5 > Next let k > 1, ad hece q 0.5k 1. By [13], oe of q, q is the media of Y, ad hece Pr[Y q ] 0.5. It follows that E[mi(Y, q)] Pr[Y q ] q 0.5 q. q 0.5q, ad our claim follows. We ow complete the proof of Theorem 12. Proof. (of Theorem 12) The expected utility of Hedge: E v[u(p mi(y,k))] E v[u(p mi(y, q))] mi(y, q) E v[u(pq) ] q 1/4 u(pq) 1/4 u(rev(v CG r)) 1/8 u(rev(v CG p )) The secod step is by cocavity of u, the fourth step is by mootoicity of the utility fuctio with the followig additioal justificatio for ay bidder i, she wis with probability q i V CG r. O the other had, the optimal way to maximize expected reveue subject to the costrait that she wis with probability q is to set a sigle price p ad get expected reveue qp. The last step is by Lemma 10. Applyig Fact 3 completes the proof. We do ot have a aalog of Theorem 8 for the limited supply case we do ot kow if our aalysis is tight or if it possible to idetify a better posted price mechaism. 5. THE VCG MECHANISM I this sectio, we quatify the uiversal approximatio ratio of the VCG mechaism i multi-uit auctios. This is useful because the VCG mechaism (k + 1-st price auctio) or a slight variatio of it is ofte used i practice.

7 5.1 The Sigle-Item Case We first restrict our attetio to the case whe the umber of items is 1. The mai result of this sectio is that the Vickrey mechaism (the secod-price mechaism) is a uiversal (1 1/)-approximatio whe there are bidders. Theorem 14. For a sigle item auctio with bidders, whe valuatios are draw i.i.d. from a regular distributio F, the Vickrey mechaism is a uiversal (1 1/)- approximatio to optimal expected utility. This theorem is a geeralizatio of a result of Dughmi et al. [4], which was for the risk-eutral case; most of the proof steps are similar, ad so we oly metio the proof structure, which is also used i the ext sectio. Let OPT be the mechaism which first rus the utility-optimal mechaism OPT o the 1 bidders, ad the allocates the item for free to the other bidder i case it is still available. Our theorem follows from three statemets. First, the reveue (ad hece utility) of OPT o bidders is equal to that of OP T o 1 bidders. Secod, amog all mechaisms that always sell the item, icludig Vickrey ad OPT, Vickrey maximizes the wier s valuatio ad hece virtual utility, ad hece by the characterizatio Lemma 2, Vickrey o bidders has a higher expected utility tha that of OPT o 1 bidders. Third, as we will show more more geerally i Lemma 18, the optimal expected utility from 1 bidders is at least 1 1/ fractio of that from bidders. These three statemets altogether imply our theorem. 5.2 The Multi-Uit Case I this sectio we prove a result aalogous to Theorem 14 for multi-uit auctios. Theorem 15. I a multi-uit auctio with k items ad bidders, where bidders valuatios are draw i.i.d. from a regular distributio F, the VCG mechaism is a uiversal ( k)/4-approximatio to optimal expected utility. The result implies that as log as the umber of bidders is a small multiple of the umber of items, the uiversal approximatio ratio of VCG mechaism is close to 1/4. The proof structure is similar to that of Theorem 14, but the details are differet because Lemma 2 does ot exted to the multi-uit case (as discussed i Sectio 3). Recall that the reveue of the VCG mechaism is exactly k times the k + 1-st highest bid (let the + 1-th highest bid be 0). The followig probability boud o the k + 1-st highest bid is crucial to our aalysis. Lemma 16. For ay regular distributio F, ad 1 < t, let Y be the t-th largest of i.i.d. radom draws from F, ad the Pr[Y E[Y ]] 1/4. Proof. Our proof cosists of two steps. First, give a regular distributio F, we costruct a slightly o-regular distributio F such that Pr[Y E[Y ]] Pr[Ỹ E[Ỹ ]], where Ỹ is the t-th largest valuatio of i.i.d. draws from F. This ew distributio F has correspodig reveue fuctio such that R(q) = a q + b for q (0,1] for some b > 0 ad a + b 0, ad it the suffices to show that Pr[Ỹ E[Ỹ ]] 1/4 for such distributios. Give ay regular distributio F, let z = 1 F(E[Y ]), ad cosider the distributio F correspodig to the reveue fuctio R such that R(z) = R(z) ad R (q) = R (z) for all q (0,1]. I other words, R is the lie segmet that is taget with R at z. By cocavity of R, we have R(q) R(q) for all q (0,1], To aid the aalysis, let Q t, be the t-th order statistics (i.e., the t-th smallest valuatio) of i.i.d. draws from the uiform distributio over [0, 1]. Therefore for all y, Pr[Y y] = Pr[Q t, 1 F(y)] ad similarly for Ỹ ad F. Let z = 1 F(E[Ỹ ]). The to show that Pr[Y E[Y ]] Pr[Ỹ E[Ỹ ]], it suffices to show that Pr[Qt, z] Pr[Qt, z], or simply that z z. Recall that R(q) R(q) for all q. Therefore F(v) F(v) for all v, ad hece E[Ỹ ] E[Y ]. Also recall that R(z) = R(z). Therefore F 1 (1 z) = F 1 (1 z) = E[Y ] E[Ỹ ] = F 1 (1 z). So z z. Now we prove that Pr[Ỹ E[Ỹ ]] 1/4. Let distributio F be such that the correspodig reveue fuctio is R(q) = a q + b for some b 0 ad a + b 0. Let f t,(q) =! (t 1)!( t)! qt 1 (1 q) t be the desity fuctio of Q t,. The E[Ỹ ] = Z 1 = q=0 Z 1 q=0 = a + b f t,(q) R(q) dq q f t,(q) (a + b q )dq t 1, where we use the facts that 1 ft,(q) = ft 1, 1(q) ad q t 1 that f t, ad f t 1, 1 as desity fuctios both itegrate to 1. Note that whe q = t 1, R(q)/q = a + b/q = E[ Ỹ ]. t 1 Therefore 1 F(E[Ỹ ]) =, ad hece Pr[Ỹ E[Ỹ ]] = Pr[Q t, t 1 ]. Note that for i.i.d. draws from the uiform distributio over [0, 1], the t-th order statistic is at most t 1 if ad oly if the umber of draws that are at most t 1 is at least t. 1 Let B be this umber, which is a biomial variable with parameter ad t 1 t 1. The Pr[Qt, ] = Pr[B t], ad by properties of biomial distributio, Pr[B t] is at least 1/4, where 1/4 is achieved whe t = = 2. Based o Lemma 16, we ca prove the followig approximate versio of the classical result of Bulow ad Klemperer [1]. (We suspect that a exact versio holds without the approximatio factor 1/4; to recover the statemet origial result, replace utility by reveue ad remove the 1/4.) Lemma 17. Suppose valuatios of bidders are draw i.i.d. from a regular distributio. The optimal expected utility whe sellig k items to bidders is at most 1/4 times the expected utility of the VCG mechaism whe sellig k items to + k bidders. Proof. We will let superscripts i V CG k, or Mye k, deote that we are sellig k items to bidders. By Fact 3, the optimal expected utility of sellig k items to bidders is at most u(e v[rev(mye k,,v)]), which by the classic Bulow-Klemperer result [1] ad the mootoicity of u is at most u(e v[rev(v CG k,+k,v)]). Note that the reveue of V CG k,+k is k times Y = F 1 (1 Q k+1,+k ),

8 where Q k+1,+k is the k +1-th order statistics of +k i.i.d. draws from a uiform distributio over [0, 1]. By Lemma 16 Pr[Y E[Y ]] 1/4. Our lemma follows because the utility of V CG k,+k would be at least 1/4 u(k E[Y ]) = 1/4 u(e v[rev(v CG k,+k,v)]). The followig claim bouds the loss of optimal utility i droppig k bidders. Lemma 18. Suppose valuatios of bidders are draw i.i.d. from a regular distributio. The optimal expected utility whe sellig k items to k bidders is at least 1 k/ fractio of the optimal expected utility whe sellig k bidders to bidders. Proof. Let M be a utility-optimal mechaism for sellig k items to bidders N = {1, 2,..., }. For ay subset S of bidders, let radom variable R S be the reveue we collect from S i M. The the expected utility of ruig M o all bidders is E v[u(r N)]. Suppose we radomly select a set S of size k. The we have: E v,s[u(r S)] E v[e S[u(R N) RS R N ]] = E v[u(r N) E S[ RS R N ]] = E v[u(r N) (1 k )] = (1 k ) Ev[u(RN)] Here the iequality is by the cocavity of u ad that R S R N, ad the secod equality is due to the fact that every bidder s reveue is accouted i R S with probability 1 k/. By a averagig argumet, for some set S of k bidders, ad for some fixed bids v S of bidders outside of S, the mechaism M iduced o S has expected utility that is at least 1 k/ fractio of the expected utility of ruig M o all bidders. Our lemma follows because the utility-optimal mechaism o k bidders ca oly do better tha this iduced mechaism. Now Theorem 15 follows by chaiig the iequalities from Lemma 17 ad Claim CONCLUSIONS AND OPEN PROBLEMS I this paper, we idetify truthful mechaisms for multiuit auctios that offer simultaeous costat-factor approximatios for all risk-averse sellers, o matter what their levels of risk-aversio are. We hope that this paper spurs iterest i the desig ad aalysis of mechaisms for riskaverse sellers. We see several ope directios for istace, idetifyig better mechaisms for the auctio settigs studied i this paper, idetifyig mechaisms for more combiatorial auctio settigs, ad desigig olie mechaisms that adapt prices based o previous sales. We coclude by siglig out a specific challege: ca we characterize the utility-optimal mechaism for a seller with a fixed, kow utility fuctio? How about whe the seller s utility fuctio has additioal structure for istace, it satisfies costat (absolute or relative) risk aversio? (Sectio 3 discusses how the stadard approach from Myerso [11] does ot work for multi-item auctios.) 7. REFERENCES [1] J. Bulow ad P. Klemperer. Auctios versus egotiatios. America Ecoomic Review, 86(1): , [2] S. Chawla, J. Hartlie, D. Malec, ad B. Siva. Sequetial posted pricig ad multi-parameter mechaism desig. I Proc. 41st ACM Symp. o Theory of Computig (STOC), [3] E. H. Clarke. Multipart pricig of public goods. Public Choice, 11:17 33, [4] S. Dughmi, T. Roughgarde, ad M. Sudararaja. Reveue submodularity. I EC 09: Proceedigs of the teth ACM coferece o Electroic commerce, pages , New York, NY, USA, ACM. [5] P. Eso ad G. Futo. Auctio desig with a risk averse seller. Ecoomics Letters, 65(1):71 74, October [6] C. Ewerhart. Optimal desig ad ρ-cocavity. Workig Paper, [7] A. Goldberg ad J. Hartlie. Collusio-resistat mechaisms for sigle-parameter agets. I Proc. 16th ACM Symp. o Discrete Algorithms, [8] T. Groves. Icetives i teams. Ecoometrica, 41: , [9] A. Hu, S. A. Matthews, ad L. Zou. Risk aversio ad optimal reserve prices i first ad secod-price auctios. Workig Paper, [10] E. S. Maski ad J. G. Riley. Optimal auctios with risk averse buyers. Ecoometrica, 52(6): , November [11] R. Myerso. Optimal auctio desig. Mathematics of Operatios Research, 6(1):58 73, [12] N. Nisa, T. Roughgarde, E. Tardos, ad V. V. Vazirai. Algorithmic Game Theory. Cambridge Uiversity Press, New York, NY, USA, [13] J. B. R. Kaas. Mea, media ad mode i biomial distributios. Statistica Neerladica, 34:13 18, [14] M. Rothschild ad J. E. Stiglitz. Icreasig risk: I. a defiitio. Joural of Ecoomic Theory, 2(3): , September [15] M. Rothschild ad J. E. Stiglitz. Icreasig risk ii: Its ecoomic cosequeces. Joural of Ecoomic Theory, 3(1):66 84, March [16] W. Vickrey. Couterspeculatio, auctios, ad competitive sealed teders. J. of Fiace, 16:8 37, APPENDIX A. MISSING PROOFS A.1 Proof of Lemma 19 Lemma 19. Let F be a regular distributio. For ay cocave utility fuctio u, φu F(v) is odecreasig. Proof. Sice F is regular, φ F(v) = (v 1 h(v) ) = 1 +

9 h (v) 0. The: h 2 (v) dφ u F(v) dv = (u(v) u (v) h(v) ) = u (v) u (v)h(v) u (v)h (v) h 2 (v) = u (v) (1 + h (v) h 2 (v) ) u (v) h(v) (A.1) (A.2) (A.3) = u (v) φ F(v) u (v) (A.4) h(v) 0 (A.5)