Robust Auctions for Revenue via Enhanced Competition

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1 Submitted to Opertions Reserch mnuscript 0 Authors re encourged to submit new ppers to INFORMS journls by mens of style file templte, which includes the journl title. However, use of templte does not certify tht the pper hs been ccepted for publiction in the nmed journl. INFORMS journl templtes re for the exclusive purpose of submitting to n INFORMS journl nd should not be used to distribute the ppers in print or online or to submit the ppers to nother publiction. Robust Auctions for Revenue vi Enhnced Competition Tim Roughgrden, Inbl Tlgm-Cohen Deprtment of Computer Science, Stnford University, Stnford, CA 94305, tim@cs.stnford.edu, itlgm@stnford.edu Qiqi Yn Google Inc., Mountin View, CA 94043, contct@qiqiyn.com Most results in revenue-mximizing mechnism design hinge on getting the price right offering to sell good to bidders t price low enough to encourge sle, but high enough to grner non-trivil revenue. Getting the price right cn be hrd work, especilly when the seller hs little or no priori informtion bout bidders vlutions. Moreover, this pproch becomes prohibitively chllenging when there re multiple indivisible goods on the mrket, in which cse getting the prices right is long-stnding open problem, even for mtching mrkets with symmetric bidders (ech of whom seeks single good). In this pper we pply robust pproch to designing uctions for revenue. Insted of relying on prior knowledge regrding bidder vlutions, we let the mrket do the work nd let prices emerge from competition for scrce goods. We nlyze the revenue gurntees of one of the simplest imginble implementtions of this ide: first, enhnce competition in the mrket, whether by incresing demnd or by limiting supply; second, run stndrd second-price (Vickrey) uction. Enhncing competition is nturl wy to bypss lck of knowledge seller who does not know how to set prices cn insted set quntities (of bidders nd/or goods on the mrket). We prove tht simultneously for mny vlution distributions, this chieves expected revenue t lest s good s the optiml revenue in the originl mrket or gurntees constnt pproximtion to it. Our robust nd simple pproch thus provides hndle on the elusive optiml revenue in multi-item mtching mrkets, nd shows when the use of welfre-mximizing Vickrey uctions is justified even if revenue is priority. By estblishing quntittive trde-offs, our work provides guidelines for seller in choosing mong lterntive revenue-extrcting strtegies: sophisticted pricing bsed on mrket reserch, dvertising to drw dditionl bidders, nd limiting supply to crete scrcity on the mrket. Key words : Bidding nd uctions, pricing, mtchings 1

2 2 Article submitted to Opertions Reserch; mnuscript no Introduction 1.1. The Revenue-Mximiztion Problem Consider set of m indivisible goods for sle, nd the problem of mtching them to n buyers with privte vlues, ech of whom wnts no more thn single good. This problem hs been studied extensively with respect to the gol of mximizing economic efficiency; e.g., this is the topic of the clssic pper on Multi-Item Auctions of Demnge et l. (1986). In this pper we focus on n lterntive importnt gol mximizing the seller s revenue. To demonstrte our setting, consider for-profit trvel website selling overnight ccommodtion fced with the tsk of ssigning m vilble rooms to n interested buyers. Ech buyer needs single room for the night, nd hs different privte vlues for different rooms bsed on their loction, size etc. Uncertinty of the seller regrding buyer vlues is cptured by probbilistic (Byesin) model, in which the vlues for every good j [m] re ssumed to be independent drws from distribution F j (where F j stisfies stndrd regulrity condition). The fct tht F j is common to ll buyers mkes our model symmetric with respect to buyers (but not with respect to goods). The seller wishes to mximize its expected revenue by designing deterministic uction in which no buyer cn do better thn to prticipte nd revel his true vlues (i.e., dominnt strtegy truthful). 1 When m = 1, tht is when there is single good on the mrket, Myerson (1981) chrcterizes the revenue-optiml truthful uction under the ssumption tht the distribution F 1 from which vlues for the good re drwn is fully known to the seller. The optiml uction in this cse turns out to be the well-known second-price uction (Vickrey 1961), with n dditionl reserve price r tilored to the distribution F 1. The resulting uction is very simple: the bidders report their vlues to the seller, the bidder with the highest bid bove r wins, nd the winner pys the second-highest bid bove r if there is one or r otherwise. Myerson s chrcteriztion of optiml mechnisms lso pplies to mrkets with multiple copies (units) of the single good, where ech bidder seeks t most one copy. More generlly, it pplies to ll single-prmeter mrkets, in which every bidder cn either win or lose nd hs single privte vlue for winning. 2 Since Myerson s seminl work there hve been efforts to extend it in severl directions. A direction tht hs ttrcted much ttention is to generlize the optiml uction chrcteriztion beyond the m = 1 cse, to multi-prmeter mrkets (e.g., Vincent nd Mnelli 2007). In prticulr, there is no known chrcteriztion for the mtching mrkets described bove, in which there re multiple goods nd ech buyer seeks t most one good. Another importnt direction tht hs become known s Wilson s doctrine is to design lterntive, robust uctions for revenue, in the sense tht they do not depend on the seller s full knowledge of the vlue distributions (Wilson 1987). A third direction is inspired by the simplicity of Myerson s uction second-price uction with reserve

3 Article submitted to Opertions Reserch; mnuscript no. 0 3 nd ims to design similrly simple uctions for revenue in more generl settings (e.g., Hrtline nd Roughgrden 2009). In this pper we contribute to ll three gols bove by pplying robust pproch to revenue mximiztion. We develop frmework for designing mechnisms tht re robust, simple, nd gurnteed to work well for vriety of mrket environments including mtching mrkets. Our mechnisms re bsed on the nturl ide of enhncing bidder competition 3 for the goods, either by dding competing bidders in the mnner of Bulow nd Klemperer (1996) or by rtificilly limiting the supply, nd then running vrint of the Vickrey uction. Despite voiding ny reference to the vlue distributions, the expected revenue chieved by these mechnisms exceeds or pproximtes the expected revenue of the optiml mechnisms tilored to the distributions. Besides leding to good mechnisms, our pproch sheds light on trde-offs mong possible seller strtegies, including how mny more buyers re needed, or how mny units of good to produce reltive to the mrket size, in order to replce the need to rigorously lern the preferences of existing buyers. We now demonstrte our pproch vi simple motivting exmple. Our tretment of robustness in the reminder of this introduction is intuitive; for forml discussion of how we define robustness nd comprison to other robustness notions see Section Motivting Exmple: Multi-Unit Mrkets As simple motivting exmple we consider symmetric multi-unit mrkets (Exmple 1), nd in prticulr specil cse in Exmple 2 nd generliztion in Exmple 3. Exmple 1 (Multi-unit). There re k identicl copies, or units, of single good for sle, nd n k bidders who ech wnt t most one unit. The bidders vlues for unit re i.i.d. smples from the vlue distribution. For survey on multi-unit uctions see Nisn (2014). 4 For exmple, the units cn be identicl rooms t lrge hotel. Another exmple is copies of digitl good such s n e-book, in which cse there is no limit on the number of copies tht cn be mde. Exmple 2 (Digitl goods). A multi-unit mrket with k = n units, where n is the number of bidders. We lso consider generliztion in which there re fesibility constrints, i.e., not ll sets of bidders cn fesibly be llocted units, even if they include less bidders thn the number of vilble units. The next exmple demonstrtes the kind of fesibility constrints we consider (dditionl such mtroid constrints pper in Bikhchndni et l. 2011):

4 4 Article submitted to Opertions Reserch; mnuscript no. 0 Exmple 3 (Job Scheduling). A multi-unit mrket where the units re slots for running jobs on mchine, nd subset of bidders is fesible if ech bidder s job cn be mtched to slot between its rrivl time nd dedline. Techniclly, multi-unit mrket with no constrints cn be thought of s finding mtching of bidders to units in complete biprtite grph, while in the job scheduling exmple the corresponding biprtite grph hs some suitble structure. Our Approch We present two pproches to robust revenue mximiztion in the bove exmples: ugmenting demnd nd limiting supply. Both pproches re inspired by common business prctices ugmenting demnd corresponds to dvertising the uction nd drwing more prticipnts, nd limiting supply corresponds to prctices like limited editions, limited runs of rtwork, or rtificil scrcity (for n exmple of this phenomenon in the dimond mrket see McEchern 2012, other exmples include scrcity of newly-lunched technology products etc.). On theoreticl level, both pproches rethink the stndrd definition of n uction environment, in which the demnd nd supply re considered exogenous, treting these insted s n endogenous prt of the mechnism design problem. Augmenting Demnd in Multi-Unit Mrkets A well-known result sttes the following: Theorem 1 (Bulow nd Klemperer (1996)). When selling single good to bidders whose vlues re i.i.d. drws from distribution stisfying regulrity, the expected revenue of the revenueoptiml mechnism with n bidders is t most tht of the Vickrey uction with n + 1 bidders. In other words, when the demnd is ugmented by dding single dditionl bidder competing for the good, the simple Vickrey uction chieves t lest the mximum revenue possible with the originl demnd. This is despite being oblivious to the vlue distribution, wheres the optiml Myerson mechnism for n bidders depends on this knowledge to set the reserve price. We remrk tht the regulrity constrint on the distribution is stndrd; for bidder whose vlue is drwn from the distribution, regulrity mens tht the revenue curve describing the trde-off between selling to the bidder often t low price nd selling less often t higher price is concve. This is stisfied by ll common distributions (uniform, norml, power-lw, etc.), nd without it no result long the lines of Theorem 1 is possible see Section 3 for detils. The Bulow nd Klemperer theorem generlizes to the multi-unit setting in Exmple 1: When there re k units of the good, the expected revenue of the revenue-optiml mechnism with n bidders is t most tht of the Vickrey uction with n + k bidders (Bulow nd Klemperer 1996). It lso pplies to constrined settings such s the job scheduling exmple (Exmple 3): When the best schedule is ble to mtch ρ k bidders to the k slots without violting n rrivl time/dedline,

5 Article submitted to Opertions Reserch; mnuscript no. 0 5 the expected revenue of the revenue-optiml mechnism with n bidders is t most tht of the Vickrey uction with n + ρ bidders, where the ρ dditionl bidders cn be scheduled simultneously (Dughmi et l. 2012). In generl, Bulow-Klemperer-type theorem sttes tht insted of running the optiml mechnism on the originl uction environment, we cn get s much revenue in expecttion by running vrint of the Vickrey uction on n environment suitbly ugmented with dditionl bidders. This cn be seen s theoreticl justifiction to tret bidder prticiption in uctions s first-order concern when iming for revenue, perhps even t the expense of sophisticted pricing. Limiting Supply in Multi-Unit Mrkets supply. The flip side of incresing demnd is limiting Mechnism 1 Supply-Limiting Mechnism 1. Set supply limit l = n/2 equl to hlf the number of bidders. 2. Run the Vickrey uction subject to supply limit l. Mechnism 1 is supply-limiting mechnism for digitl goods (Exmple 2). In the second step of the lgorithm, the Vickrey uction subject to supply limit l is simple vrition on the stndrd second-price uction: it ssigns copies to the l buyers with highest bids (even though there re enough copies for ll buyers), nd chrges them ech the (l + 1)th highest bid. The resulting mechnism is simple nd nturl, nd does not rely on knowledge of vlue distributions. Intuitively, enhncing competition by limiting the supply hs similr effect on revenue s enhncing competition by dding bidders in Bulow-Klemperer-type theorems. The difference between the two pproches is tht the former requires ugmenting the resources in this cse bidders vilble to the uction, while the ltter requires the bility to withhold supply (mny sellers, e.g. compnies like Apple Inc., hve the bility to do both). This difference trnsltes into different revenue gurntees: while in Bulow-Klemperer-type theorems the expected revenue of the ugmented Vickrey uction usully exceeds tht of the optiml mechnism, the expected revenue of the supply-limiting mechnism pproximtes tht of the optiml mechnism. In prticulr, the supply-limiting mechnism chieves t lest hlf of the optiml revenue in expecttion, despite remining oblivious to the vlue distribution on which the optiml mechnism depends. The performnce gurntee for Mechnism 1 cn be seen s justifiction to the following rule of thumb for sellers: ssuming no production costs nd buyers whose vlues re distributed similrly, produce number of units equl to constnt frction of the mrket size. I.e., when sellers lck the necessry informtion to set prices, they cn set quntities insted, nd this works well simultneously for mny vlue distributions.

6 6 Article submitted to Opertions Reserch; mnuscript no. 0 The Connection between Augmenting Demnd nd Limiting Supply Our technicl pproch to estblishing pproximtion gurntees of supply-limiting mechnisms utilizes the intuition bove, by which limiting supply hs similr effect s incresing demnd. This intuition is formulted by bsing the proofs of the pproximtion fctors on reduction mong mrkets, which enbles the ppliction of n pproprite Bulow-Klemperer-type theorem. Reduction 2 Digitl Goods 0. Strt with originl mrket with n buyers nd n units Denote the optiml expected revenue by OPT. 1. Restrict to mrket with n/2 buyers nd supply limit n/2 The optiml expected revenue is OPT /2, by subdditivity of revenue in the buyers (Lemm 6 below) nd since the supply limit hs no effect here. 2. Augment to get mrket with n buyers nd supply limit n/2 The expected revenue of the Vickrey uction is OPT /2, by the Bulow-Klemperer-type theorem for multi-unit mrkets pplied to the restricted mrket. In prticulr, Reduction 2 shows tht Mechnism 1 gurntees hlf the optiml revenue in expecttion s follows: Strting with the originl mrket, define new mrket by dropping hlf of the bidders nd setting supply limit of l = n/2. Consider the resulting restricted mrket with hlf of the originl bidders nd corresponding supply limit. One cn show tht if we were to restrict the optiml mechnism to run on this mrket insted of the originl one, its expected revenue would hve been t lest hlf of its originl expected revenue. Now conceptully dd bck the n/2 removed bidders but without chnging the supply to get the ugmented mrket, nd run the Vickrey uction. It follows from the Bulow-Klemperer theorem for multi-unit mrkets tht the expected revenue is t lest s high s the optiml expected revenue for the restricted mrket. Therefore the supply-limiting mechnism gurntees t lest hlf of the optiml expected revenue in the originl mrket Our Contribution As demonstrted in Section 1.2, our min contribution is in formulting nd proving robust revenue gurntees of competition enhncement in uctions, through incresed demnd or limited supply, for vriety of mrkets, including types of mrkets where the optiml mechnism remins unknown (nd is presumbly very complex). We show tht under miniml regulrity ssumptions, the simple nd robust mechnisms bove nd their revenue gurntees Vickrey with dditionl bidders nd Vickrey with supply limit generlize to significntly more complex settings. In other words, we

7 Article submitted to Opertions Reserch; mnuscript no. 0 7 identify mrkets in which such mechnisms re gurnteed to chieve optiml or pproximtelyoptiml expected revenue. We remrk tht by using these mechnisms for revenue, the seller lso gurntees tht the welfre is pproximtely optiml, nd in fct one cn chieve other trde-offs between revenue nd welfre by setting suitble supply limits. Our technicl contribution is in proving novel Bulow-Klemperer-type theorems for different mrkets, nd designing supply-limiting mechnisms whose pproximtion gurntees follow from the Bulow-Klemperer-type theorems. While Bulow-Klemperer-type theorems hve been studied before, they hve never been ttempted beyond single-prmeter buyers, i.e., for multiple different goods. To our knowledge, supply-limiting mechnisms hve lso not been studied before, nor hs the connection between incresing demnd nd limiting supply been explicitly formulted s in our reductions. Proving Bulow-Klemperer-type theorems for mtching mrkets is the most techniclly chllenging component of this work. The nlysis for multi-prmeter settings is chllenging due to dependency issues the competition for item j tht drives its price depends on the buyers vlues for the other items. We overcome dependency chllenges vi technique from the nlysis of rndomized lgorithms clled the principle of deferred decision, combined with the combintoril properties of optiml mtchings. Results: Augmented Demnd We prove the first generliztion of Bulow nd Klemperer s theorem (Theorem 1) to multi-prmeter mrkets. Theorem 2 (Bulow-Klemperer-Type Theorem for Mtching Mrkets (Informl)). For every mtching mrket with n bidders nd m goods, ssuming symmetry nd regulrity, the expected revenue of the Vickrey uction with m dditionl bidders is t lest the optiml expected revenue in the originl mrket. The forml sttement ppers in Theorem 6. We emphsize tht the symmetry ssumption in this theorem is cross bidders, not goods. Tht is, vlues of different bidders for the sme good re i.i.d. smples from the sme distribution, but different goods cn hve different vlue distributions. This kind of symmetry mkes sense in prcticl pplictions, where the seller knows it is selling very different kinds of goods, but sees the bidders whose identities nd chrcteristics re unknown s homogeneous (Chung nd Ely 2007). In ddition to Theorem 2, we prove Bulow-Klemperer-type theorems tht chieve better gurntees for mtching mrkets with more supply thn demnd (n m), nd tht pply to symmetric mrkets where bidders vlues for good my belong to different distributions (see Section 7).

8 8 Article submitted to Opertions Reserch; mnuscript no. 0 Results: Limited Supply We design supply-limiting mechnisms for both single-prmeter nd multi-item mrkets. The former include digitl good mrkets (recll Mechnism 1), s well s more generl multi-unit mrkets, possibly with constrints or symmetric bidders (see Section 4). For multi-item mtching mrkets, we first define notion of setting limit on supply where the supply is heterogeneous rther thn homogeneous. A multi-item uction subject to supply limit l mens tht no more thn l goods my be ssigned, with no limittion on which l goods these shll be. Intuitively, this lets the mrket do the work of choosing which prt of the supply to limit. This is in line with our robust pproch, s seller with no knowledge of how the vlues for the different goods re distributed cnnot mke this decision without risking big loss in revenue. Notice tht the simple supply-limiting mechnism we designed for multi-unit mrkets (Mechnism 1) is now well-defined for multi-item mrkets s well, nd we cn prove the following theorem: Theorem 3 (Supply-Limiting Mechnism for Mtching Mrkets (Informl)). For every mtching mrket with n 2 bidders nd m goods, the expected revenue of Mechnism 1 is t lest constnt frction of the optiml expected revenue. Qulittively, Theorem 3 is interesting since it shows tht simple robust mechnism cn chieve frction of the optiml expected revenue tht is independent of the size of the mrket, s mesured by prmeters n nd m. Moreover, the constnt frctions we chieve re quite good in mny cses, e.g., we chieve frction of 1/4 when the number of bidders equls the number of goods (Theorem 8). An interesting open problem is whether this is the best possible by ny robust mechnism. The nlysis of the pproximtion gurntees re vi generl reduction (Reduction 3) long the lines of Reduction 2, instntited with pproprite Bulow-Klemperer-type theorems. Reduction 3 Approximtion Gurntees vi Bulow-Klemperer-Type Theorems 0. Strt with originl mrket with n buyers Denote the optiml expected revenue by OPT. 1. Restrict to mrket with < n buyers nd supply limit l The optiml expected revenue is constnt frction of OPT, by subdditivity. 2. Augment to get mrket with n buyers nd supply limit l The expected revenue of the Vickrey uction is constnt frction of OPT, by suitble Bulow- Klemperer-type theorem.

9 Article submitted to Opertions Reserch; mnuscript no Orgniztion In Section 2 we discuss our pproch to robustness nd survey relted literture. In Section 3 we formlly present our model nd preliminries. Section 4 includes our nlysis of competition enhncement for multi-unit mrkets. Sections 5 nd 6 nlyze multi-item mtching mrkets where n is proportionl to m nd contin our min technicl results for incresing demnd nd limiting supply, respectively. Extensions nd generliztions cn be found in Section 7. Section 8 concludes. 2. Prior-Independent Robustness nd Relted Work In this section we discuss our pproch to robust revenue gurntees, nd present relted work Definition Robustness hs been long-time gol of mechnism nd mrket design. Intuitively, robust mechnisms re mechnisms tht perform well for lrge rnge of economic environments. Their performnce is insensitive to the environment s precise detils nd for this reson robustness is lso referred to s detil-freeness. To formulte robustness one must specify wht it mens to perform well nd for which rnge of environments should the performnce gurntee hold. There re severl lterntive formultions in the literture, including robust optimiztion nd others, nd we discuss these in Section 2.3. To define the prior-independent notion of robustness, we focus for simplicity on the single good cse. Consider first prticulr distribution F from which the buyers i.i.d. vlues for the good re drwn. Let OPT F be the optiml expected revenue tht truthful deterministic mechnism with full knowledge of F cn chieve in this mrket. Let α (0, 1] be n pproximtion fctor. A mechnism is α-optiml with respect to F if its expected revenue is t lest α OPT F. This is similr to verge-cse pproximtion in combintoril optimiztion, where n lgorithm s pproximtion gurntee holds for inputs drwn from known distribution; the difference is tht the benchmrk OPT F is with respect to truthful mechnism insted of n lgorithm. Now let F be set of vlue distributions, clled the priors. A mechnism is robustly α-optiml with respect to F if for every distribution F in this set, the mechnism is α-optiml with respect to F. In this cse we lso sy tht it gives n α-pproximtion to the optiml expected revenue. We hve thus defined wht it mens for robust mechnism to perform well : it must chieve expected revenue tht either exceeds or pproximtes the optiml expected revenue simultneously for every distribution in clss of distributions F. In this pper we set the lrge rnge of distributions F to be ll regulr distributions. Roughly these re distributions whose til is no hevier thn tht of power lw distribution. Regulrity is stndrd ssumption in uction theory, nlogous to tht of downwrd-sloping mrginl revenue

10 10 Article submitted to Opertions Reserch; mnuscript no. 0 in monopoly theory, nd without it no good robust revenue gurntees re possible (see Bulow nd Klemperer 1996, nd Section 3 for precise definition nd negtive exmples). The bove definition of robustness is n interesting mixture of verge- nd worst-cse gurntees. On one hnd, performnce is mesured in expecttion over the rndom input; on the other it is mesured in the worst cse over ll distributions tht belong to F. Such robustness is referred to s prior-independence, since there is n underlying ssumption tht vlues re smpled from priors, nd yet the robust mechnism must be independent of the priors s it must work well for ll of them Rtionles Wht re the rtionles behind prior-independent robustness? In prticulr, why mesure whether robust uction is performing well by compring it to the optiml mechnism with ccess to the prior distribution? And why choose lrge rnge of distributions with miniml ssumptions (rther thn incorporte prtil informtion tht the seller my hve bout the distributions in order to nrrow it down)? First, our min results show tht for lrge rnge of distributions on which little is ssumed, we cn get constnt pproximtion to the mbitious benchmrk of OPT F, even in chllenging environments like multi-item mrkets for which the optiml mechnism remins elusive: Theorem 2 cn be rephrsed s stting tht Vickrey with m dditionl bidders is robustly 1-optiml, nd Theorem 3 cn be rephrsed s stting tht Vickrey with supply limit n/2 is robustly α-optiml for some constnt 1/α. The two choices bove thus serve to strengthen our results. An lterntive pproch to robustness could be, rther thn to pproximte the optiml mechnism for every distribution in F, to design mechnism tht mximizes the minimum expected revenue where the minimum is tken over ll distributions in the set F. Such n pproch would run into the open problem of finding the optiml mechnism for multi-item mrkets, even in the degenerte cse where F contins only one distribution. Seeking pproximtion rther thn mximiztion is wht enbles us to circumvent the open problem. In ddition, choosing OPT F s benchmrk llows the seller to mke informed decisions regrding how much to invest in obtining informtion on F. The choice of setting F to be the clss of ll regulr distributions hs the dvntge of cpturing situtions in which the seller hs no informtion bout the vlue distribution, s is the cse for new seller or new good on the mrket nd for goods whose distribution is constntly shifting, s well s situtions in which the seller s informtion is prohibitively expensive or highly noisy nd thus too risky to rely upon. In other words, the sme reson for voiding dependence on prior distributions in mechnism design lck of relible, ccessible informtion lso justifies

11 Article submitted to Opertions Reserch; mnuscript no voiding dependence on prior informtion bout the distributions. Moreover, ssuming no prtil informtion leds to simple nd nturl mechnisms, thus reinforcing our chosen robustness notion Robustness in the Literture Robustness in mechnism design hs been studied from three different perspectives economics, opertions reserch nd computer science. The Wilson (1987) doctrine in economics clls for the development of mechnisms independent of the detils of the economic environment, s fr s these re not relly common knowledge mong the buyers nd seller. Wilson writes tht the importnce of repeted wekening of the common knowledge ssumption is tht only in this wy will the theory pproximte relity. In opertions reserch, Scrf (1958) observed tht we my hve reson to suspect tht the future demnd will come from distribution tht differs from tht governing pst history in n unpredictble wy. In this context, Bertsims nd Thiele (2014) note in their essy on Modern Decision-Mking Under Uncertinty tht the need for non-probbilistic theory hs become pressing. In computer science, the dominnt prdigm of worst-cse nlysis hs been extended to mechnism design, reflecting the expectncy tht mechnisms (like lgorithms) should work well cross rnge of settings, s well s generl mistrust in designers bility to ccurtely cpture rel-world distributions (Nisn 2014). Given its importnce, it is not surprising tht there is rich literture on robustness in mechnism design. Our pproch contributes to this literture by simultneously chieving robustness nd simplicity while being pplicble to multi-item environments. Prior-Independence for Single-Prmeter Mrkets For single-item nd other singleprmeter mrkets, Bulow nd Klemperer (1996) were the first to study the effect of ugmented demnd. A simplified proof of their min result ws given by Kirkegrd (2006). Dughmi et l. (2012) generlize this result to mrkets with mtroid-bsed constrints, nd use the generlized version to investigte conditions under which the Vickrey uction indvertently yields pproximtelyoptiml revenue. Hrtline nd Roughgrden (2009) develop similr result for symmetric buyers who do not shre the sme vlue distributions. Fu et l. (2015) study the tightness of the Bulow nd Klemperer result. Using techniques relted to n erly version of this work (Roughgrden et l. 2012), Sivn nd Syrgknis (2013) develop version of Bulow nd Klemperer s result for convex combintions of distributions stisfying regulrity. A nturl pproch to prior-independent robustness is to instntite the optiml mechnism developed by Myerson (1981) with n empiricl rther thn known distribution, where the smples come from the buyers bids. This pproch is explored by Segl (2003) nd Blig nd Vohr (2003), nd is symptoticlly optiml s the size of the mrket goes to infinity. It lso llows

12 12 Article submitted to Opertions Reserch; mnuscript no. 0 the seller to incorporte into the mechnism prior informtion bout the clss of possible vlue distributions ( higher level of prior informtion not bout the possible vlues but rther bout their possible distributions). Dhngwtnoti et l. (2010) tke the smpling pproch further by designing simple mechnisms using only single smple, which re nevertheless robustly α-optiml for smll constnt 1/α prmeters. Other Robustness Notions for Single-Prmeter Mrkets Vlue distributions re not used in the design of prior-independent uctions, but they re used in their nlysis, nmely in the definition of robustly α-optiml which is bsed on comprison to the optiml expected revenue. In prior-free uction design, distributions re not even used to evlute the performnce of n uction. This rises the question of which benchmrk to use. Neemn (2003) studies the revenue performnce of the English uction in our setting n scending-uction version of the Vickrey uction compring it to the benchmrk of welfre, which is clerly n upper bound on revenue in uctions in which buyers hve no better choice thn to prticipte. A different pproch is initited by Goldberg et l. (2006), who define notion of resonble uctions to compete ginst. The reltions between the different notions of robustness hs lso been studied (see, e.g., Hrtline nd Roughgrden 2014). Multi-Prmeter Mrkets In contemporneous work with n erly version of this pper, Devnur et l. independently consider similr set of problems s us, but using different mechnisms nd nlyses (2011). Their mechnisms re rgubly more complex nd less nturl since they re bsed on crefully-constructed price menus (rther thn on enhnced competition). Following our erly version, Azr et l. (2014) studied mtching mrkets in which prtil informtion bout the vlue distributions is vilble to the seller in the form of limited number of smples. Closely relted to our work, Bndi nd Bertsims (2014) pply the robust optimiztion pproch to optiml mechnism design in multi-item mrkets. Their model differs from our model in severl importnt spects, including considertion of dditive vlutions rther thn mtching mrkets, nd divisible rther thn indivisible goods. A min gol of their pper is orthogonl to ours to study the importnt issues of budgets nd correlted vlues in mechnism design. They lso ddress uctions without budget constrints but in their setting these reduce to single-item uctions, which is fr from the cse in our model. It is lso interesting to compre our robustness notion to theirs, where the ltter is inspired by the robust optimiztion prdigm. Bndi nd Bertsims model the seller s knowledge bout the vlues by n uncertinty set, thus ccommodting for prtil knowledge bsed on historicl bidding dt, nd then optimlly solve the relted robust optimiztion problem. They use simultions to show tht their robust optimiztion pproch improves upon the revenue performnce of Myerson s

13 Article submitted to Opertions Reserch; mnuscript no mechnism for single items, when the seller s knowledge of the prior distribution is inccurte. We use different notion of robustness inspired by pproximtion lgorithms for combintoril optimiztion, nd our gol for single items is to surpss or pproximte the performnce of Myerson s mechnism tilored to the ccurte distribution (which our mechnism is oblivious to). Recently there hve been significnt dvnces on the problem of prior-dependent optiml mechnism chrcteriztion for multi-item mrkets. Ci et l. (2013) give chrcteriztion for optiml mechnisms given ccess to the prior distributions, nd with the relxed requirement of Byesin, rther thn dominnt strtegy, truthfulness. The relxed truthfulness notion requires tht no buyer cn do better in expecttion over the other buyers vlutions thn to prticipte nd bid truthfully in the uction. It thus relies on common knowledge of the prior distributions mong the buyers s well s for the seller (cf. Chung nd Ely 2007). Chwl et l. (2010) give n upper bound on the optiml expected revenue for mtching mrkets, nd our techniques utilize one of their reductions. They chieve prior-dependent 1/6.75- pproximtion for mtching mrkets with multiple units nd symmetric buyers, nd lso 3/32- pproximtion for n even more generl environment (nmely grphicl mtroid with unitdemnd buyers). Simplicity in Mechnism Design Another mechnism design considertion tht hs drwn ttention in recent yers is simplicity. Chwl et l. (2007, 2010) study (prior-dependent) postedprice mechnisms, where buyers simply choose from menu of priced lloctions. Hrtline nd Roughgrden (2009) seek conditions on single-prmeter mrkets such tht the simple Vickrey uction with (prior-dependent) reserves chieves ner-optiml revenue. This simple uction formt or generlized version of it re common in online dvertising nd sponsored serch, the min source of revenue for compnies like Google Inc. or Yhoo! Inc. (Lhie et l. 2007, Celis et l. 2014). The extension of the Vickrey uction to multi-item mrkets, clled the VCG mechnism (Vickrey 1961, Clrke 1971, Groves 1973), is rgubly not s simple (Ausubel nd Milgrom 2006, Rothkopf 2007). Yet in the mtching mrkets we consider, mny of the complictions of VCG do not occur, nmely, communicting the bids nd running the uction re both computtionlly trctble, nd our competition enhncement methods ensure tht the revenue does not collpse. Chwl et l. (2013) nlyze the VCG mechnism s performnce in job scheduling context, nd some of our techniques re inspired by their nlysis. Another simple mechnism formt tht hs been proposed recently (Bbioff et l. 2014) is lottery between running Myerson s mechnism for the grnd bundle of ll goods, nd between seprte runs of Myerson s mechnism for every good.

14 14 Article submitted to Opertions Reserch; mnuscript no Preliminries In Section 3.1 we describe our model including multi-unit nd mtching environments, in Section 3.2 we review the bsics of optiml mechnism design nd in Section 3.3 we discuss two technicl tools (regulrity nd representtive environments) Model An uction environment (or mrket) hs s set {1,..., m} of m goods (or items) for sle to set {1,..., n} of n bidders (or buyers). As convention we use the index i for bidders nd j for items. Throughout we mke the distinction between items nd units, where the ltter re different copies of the sme item so bidders hve the sme vlue for them. We describe two min environments of interest; two extensions pper in Section 7. Multi-Unit nd Other Single-Prmeter Environments Consider generl model of single-prmeter environments: An environment is defined by non-empty collection I 2 [n] of bidder sets, ech contining bidders who cn win simultneously. The sets in I re clled fesible lloctions. Every subset of fesible lloction is lso fesible (i.e., the set system ([n], I) is downwrd-closed). We ssume tht every bidder belongs to t lest one fesible lloction. Every bidder i hs privte vlue v i [0, ) for winning, which is drwn independently t rndom from distribution F i with density function f i positive over nonzero intervl support. The density function is smooth with one exception constnt mount of probbility mss cn concentrte on the highest point in the support. This exception is useful for Proposition 1. The described environment is clled single-prmeter since the vlue for winning is fully described by v i. Throughout we ssume risk-neutrl qusi-liner utility model, in which bidder s utility for winning is his vlue minus the pyment he is chrged, nd bidders im to mximize their expected utilities. We sy tht single-prmeter bidders re i.i.d. (or symmetric) if their vlue distributions re identicl. The environment is i.i.d. if the bidders re i.i.d. We cn now define multi-unit (or k-unit) environment: It is single-prmeter environment in which subset of bidders is fesible lloction if nd only if its size is t most k. This models k units for sle to n k unit-demnd bidders who re interested in t most one unit. We will sometimes impose n dditionl supply limit of l k, restricting fesible lloctions to size t most l. A mtroid environment is single-prmeter environment in which the set system ([n], I) of bidders nd fesible lloctions forms mtroid (Oxley 1992). Job scheduling mrkets (Exmple 3) re n exmple of mtroid environments, nd k-unit environments re specil cse (corresponding to the k-uniform mtroid). See Section 7.1 for further detils.

15 Article submitted to Opertions Reserch; mnuscript no Multi-Item Mtching Environments In mtching environment there re m different items for sle with one unit vilble of ech item. Fesible lloctions re ll mtchings of items to bidders (ech bidder wins t most one item nd ech item is llocted to t most one bidder). This models unit-demnd bidders. We will sometimes impose n dditionl supply limit of l m, restricting the mtchings to size t most l. Every bidder i hs privte vlue v i,j [0, ) for winning item j, which is drwn independently t rndom from distribution F i,j with smooth density function f i,j positive over nonzero intervl support. A mtching environment is thus multi-prmeter. We gin ssume risk-neutrl qusi-liner utility model. We sy the bidders re i.i.d. (or symmetric) if F i,j does not depend on the identity of bidder i, i.e., ech item j hs n ssocited distribution F j nd F i,j = F j. In other words, for every item j the vlues {v i,j } i [n] re i.i.d. smples from F j. Note tht different items j, j hve different distributions F j, F j, s necessry for pplictions such s the trvel website in Section 1. Independence of the vlues is mintined cross both bidders nd items. For tretment of symmetric bidders see Sections 7.2 nd Optiml Mechnism Design Mechnisms By the reveltion principle, without loss of generlity we my restrict ttention to direct mechnisms, which receive vector of bids b. In the single-prmeter cse b R n 0 where b i is bidder i s bid for winning, nd in the mtching cse b R nm 0 winning item j. We focus on deterministic mechnisms, comprised of: where b i,j is bidder i s bid for 1. An lloction rule x = x(b), which mps bid vector b to fesible lloction; in the singleprmeter cse x {0, 1} n, where x i = x i (b) indictes whether bidder i wins, nd in the mtching cse x {0, 1} nm, where x i,j = x i,j (b) indictes whether bidder i wins item j. 2. A pyment rule p = p(b), which mps bid vector b to pyment vector. The pyment vector p belongs to R n 0, where p i = p i (b) is the pyment chrged to bidder i. Fixing bid vector b, the mechnism s welfre in the single-prmeter cse is i x iv i, nd in the mtching cse i,j x i,jv i,j. The mechnism s revenue is i p i. Bidder i s utility in the singleprmeter cse is x i v i p i, nd in the mtching cse j x i,jv i,j p i. A mechnism is (dominnt strtegy) truthful if for every bidder i nd bid profile b i of the other bidders, i mximizes his utility by prticipting nd bidding truthfully, i.e., bidding b i = v i in the single-prmeter cse nd b i,j = v i,j for ll j in the mtching cse. All the mechnisms we study re truthful, so from now on we no longer distinguish between bids nd vlues nd use v i or v i,j to denote both. We will minly be interested in mechnism s expected revenue E v [ i p i], where p = p(v) nd the expecttion is tken over i.i.d. vlues drwn from the vlue distributions.

16 16 Article submitted to Opertions Reserch; mnuscript no. 0 The revenue benchmrk ginst which we mesure the performnce of our mechnisms is the following: By optiml expected revenue we men the mximum expected revenue over ll (dominnt strtegy) truthful, deterministic mechnisms. Chwl et l. (2010b) show tht for mtching environments, the expected revenue from the optiml deterministic mechnism is within constnt fctor of the expected revenue from the optiml rndomized mechnism. Thus our results for deterministic mechnisms pply to rndomized mechnisms up to constnt fctor. Mximizing Welfre nd the Vickrey Auction The generl form of the Vickrey uction is clled the VCG mechnism, nd it works for ny mrket whether single-prmeter or multi-item. VCG is remrkble in being both truthful nd welfre-mximizing for every vlue profile v. Its lloction rule chooses fesible lloction tht mximizes welfre; its pyment rule chrges every bidder i pyment equl to i s externlity the difference in the mximum welfre of the other bidders when i does not prticipte in the uction nd when i does prticipte in it. In the context of mtching environments, the VCG lloction rule cn be implemented s mximum weighted mtching over biprtite grph, where vertices on one side re the bidders, vertices on the other side re the items, nd the weight of every edge (i, j) is v i,j (Bertseks 1991). The pyment rule lso solves biprtite mtching problems to compute the pyments. For singleprmeter k-unit environments, Vickrey s lloction rule finds k bidders with highest vlues, nd for mtroid environments it uses simple greedy lgorithm to find fesible lloction with highest welfre (nd similrly for the Vickrey pyment rules). For our supply-limiting mechnisms, we dd to the VCG or Vickrey mechnisms supply limit l nd denote them by VCG l nd Vic l, respectively. Mximizing Revenue nd Myerson s Mechnism For single-prmeter environments, Myerson (1981) chrcterized the optiml mechnism tht mximizes expected revenue. (In fct, Myerson showed n even stronger result his mechnism mximizes the expected revenue over ll Byesin truthful, rndomized mechnisms!) Let F be regulr distribution with density f (see Section 3.3 for discussion of regulrity). Define its virtul vlue function φ F : R 0 R to be φ F (v) = v 1 F (v). Myerson showed the following. f(v) Lemm 1 (Myerson). Given single-prmeter environment nd truthful mechnism (x, p), for every bidder i nd vlue profile v i of the other bidders, E vi F i [p i (v)] = E vi F i [x i (v)φ Fi (v i )]. Myerson s lemm sys tht in expecttion over bidder i s vlue, his pyment is equl to his virtul vlue when he is llocted. By summing over ll bidders, this lemm implies tht in expecttion over the vlue profile, mximizing the revenue is equivlent to mximizing the totl virtul vlue of llocted bidders, quntity known s the virtul surplus. Myerson s mechnism mximizes expected revenue by finding the fesible lloction with mximum virtul surplus. For exmple, in k-unit environment this will be the set of k bidders with the highest positive virtul vlues.

17 Article submitted to Opertions Reserch; mnuscript no Technicl Tools Regulrity We sy tht bidders re regulr if their vlues re drwn from regulr distributions. Definition 1 (Regulr Distribution). A distribution F is regulr if its virtul vlue function is monotone non-decresing. Most commonly-studied distributions re regulr, including the uniform, exponentil nd norml distributions, nd distributions with log-concve densities. The ssumption tht bidders re regulr is stndrd in optiml mechnism design nd is necessry for designing good prior-independent mechnisms, s demonstrted by Dhngwtnoti et l. (2010): Fix vlue z nd number n of bidders, nd define n irregulr, long-tiled vlue distribution F z such tht the probbility for z is 1/n 2 nd otherwise the vlue is zero. Consider single-item environment with n bidders whose vlues re drwn from F z. The optiml uction hs expected revenue t lest z/n. But ny priorindependent truthful uction essentilly hs to guess the vlue of z, since the probbility tht the non-winning bids provide informtion bout z is smll. Thus its expected revenue cnnot be within constnt fctor of z/n for every F z. Representtive Environments A representtive environment is the single-prmeter counterprt of mtching environment. Consider mtching environment with m items, n symmetric bidders nd vlue distributions {F j } m j=1. The corresponding representtive environment hs the sme m items, but nm single-prmeter bidders every bidder in the mtching environment hs m representtives in the representtive environment. The jth representtive of bidder i is only interested in item j nd hs vlue v i,j F j for winning it. Every lloction in the representtive environment cn be trnslted to n lloction in the mtching environment if the jth representtive of i wins, then item j is llocted to bidder i in the mtching environment nd vice vers. An lloction in the representtive environment is fesible if the corresponding lloction in the mtching environment forms mtching, mening tht only one representtive per bidder wins. Intuitively, the representtive environment is more competitive thn the mtching one, since representtives of the sme bidder compete ginst ech other on who will be the winner. Thus the expected revenue chievble in the representtive environment should be t lest the optiml expected revenue in the mtching environment. Chwl et l. (2010) formlize this intuition by showing tht ny truthful mechnism M for the mtching environment trnsltes to truthful mechnism M rep for the representtive environment, such tht the expected revenue of M rep is only higher. Roughly this is by trnslting the lloction rule of M to n lloction rule in the representtive environment s bove, nd viewing the pyment rule of M s price menu, whose prices re exceeded in the representtive environment by chrging every representtive the minimum vlue it needs to bid in order to win.

18 18 Article submitted to Opertions Reserch; mnuscript no. 0 Lemm 2 (Chwl et l. (2010)). The expected revenue of M rep in the single-prmeter representtive environment is t lest the expected revenue of M in the mtching environment. 4. Multi-Unit Mrkets In this section we formlly prove the results presented in Section 1.2, demonstrting our generl frmework. The pproch of ugmenting demnd hs been studied for multi-unit uctions, nd slightly generlized version of result by Bulow nd Klemperer (1996) is the following: Theorem 4 (Bulow-Klemperer-Type Theorem for Multi-Unit Mrkets). For every k- unit environment with i.i.d. regulr bidders nd supply limit l, the expected revenue of the Vickrey uction with min{k, l} dditionl bidders is t lest the optiml expected revenue in the originl mrket. In other words, Vickrey with min{k, l} dditionl bidders is robustly 1-optiml. As for limiting supply, we instntite our generl reduction (Reduction 3) with the bove Bulow- Klemperer-type theorem to prove the following. For simplicity of presenttion ssume the number of bidders n is even. 5 Theorem 5 (Supply-Limiting Mechnism for Multi-Unit Mrkets). For every k-unit environment with n 2 i.i.d. regulr bidders, the expected revenue of the supply-limiting mechnism Vic n/2 is t lest mx{ 1, n k }-frction of the optiml expected revenue. In other words, 2 n Vickrey with supply limit n/2 is robustly α-optiml for α = mx{ 1, n k }. 2 n In the bove theorem, the supply limit of n/2 kicks in when the number of units k exceeds n/2, nd in this cse we get 1/2-pproximtion. If the supply k is limited to n/2 to begin with, the competition is inherently high nd Vickrey with no supply limit provides n n k-pproximtion. n Proof. We instntite Reduction 3 s follows. To go from the originl mrket to the restricted mrket, remove min{ n 2, k} bidders from the originl mrket, nd if k > n 2 set supply limit of l = n 2. Anlysis: We first clim tht the restriction of the originl mrket mintins t lest frction of mx{ 1, n k } of the optiml expected revenue in the originl mrket. This is becuse, s shown by 2 n Dughmi et l. (2012), the expected optiml revenue s function of the bidder set is submodulr. 6 Revenue submodulrity mens decresing mrginl returns to the expected revenue s more bidders re dded, so the first mx{ n, n k} bidders lredy cpture t lest mx{ 1, n k }-frction of 2 2 n the optiml expected revenue. Limiting the supply to n when k > n hs no effect since in this cse 2 2 the number of bidders remining in the restricted environment is n 2. We cn now pply the Bulow-Klemperer-type theorem for multi-unit mrkets (Theorem 4) to the restricted environment. In the first cse, k > n 2 nd the restricted environment hs n 2 bidders, k units nd supply limit l = n 2. In the second cse, k n 2 nd the restricted environment hs n k bidders, k units nd no supply limit (i.e., l = k). In both cses, by Theorem 4 the expected revenue of

19 Article submitted to Opertions Reserch; mnuscript no Vickrey with min{ n, k} dditionl bidders is t lest the optiml expected revenue in the restricted 2 environment. So running Vickrey with min{ n, k} dditionl bidders on the restricted environment 2 is mx{ 1, n k }-pproximtion to the optiml expected revenue in the originl environment. But 2 n this is equivlent to running the supply-limiting mechnism Vic n/2 on the originl environment, completing the proof. The pproximtion fctor in Theorem 5 is symptoticlly tight: Proposition 1. For every 0 < γ < 1, consider the supply-limiting mechnism Vic γn. There exists n n-unit environment with n i.i.d. regulr bidders such tht the expected revenue of Vic γn is t most ( 1 + o(1))-frction of the optiml expected revenue. 2 Proof. Consider first the cse tht 1/n γ 1/2, i.e., the supply limit is severe. Let the vlue distribution F be the uniform distribution over the support [1, 1 + ɛ] for sufficiently smll prmeter ɛ = ɛ(n). The optiml expected revenue is roughly n, while Vic γn cn extrct s revenue t most γn(1 + ɛ) n/2 + o(1). Now suppose 1/2 < γ n 1 z. For sufficiently lrge H, let the vlue distribution be F (z) = n 1+z 1 over the support [0, H] with point mss of t H. The optiml expected revenue is t lest the 1+H expected revenue chieved by offering posted price H to every one of the n bidders, which extrcts H(1 H ) = H 1 from every bidder in expecttion. In comprison, the expected revenue in 1+H 1+H Vic γn comes from the (γn+1)st highest bid. This bid is concentrted round z = 1 γ, the vlue of γ z such tht F (z) = 1 γ. So VCG chieves n expected revenue of roughly 1 γ γn = (1 γ)n < n. γ 2 5. Mtching Mrkets: Augmenting Demnd In this section we prove Bulow-Klemperer-type theorem for mtching environments the first generliztion of Bulow nd Klemperer (1996) to multi-item mrket. Recll wht we men by i.i.d. bidders in mtching environment: different items hve different distributions, but independence is both cross bidders nd cross items. Theorem 6 (Bulow-Klemperer-Type Theorem for Mtching Mrkets). For every mtching environment with i.i.d. regulr bidders nd m items, the expected revenue of the VCG mechnism with m dditionl bidders is t lest the optiml expected revenue in the originl mrket. In other words, VCG with m dditionl bidders is robustly optiml. Theorem 6 provides simple hndle on the unknown optiml expected revenue in mtching mrkets. For exmple, in mrket with two goods for sle, the best chievble revenue is t most wht VCG cn chive with two more bidders. Note tht in mrkets with plentiful supply, i.e. mrkets in which m n, the demnd ugmenttion tht is required is substntil. In Section