General Aucton Mechansm for Search Advertsng Gagan Aggarwal S. Muthukrshnan Dávd Pál Martn Pál Keywords game theory, onlne auctons, stable matchngs ABSTRACT Internet search advertsng s often sold by an onlne automated aucton. Typcally a fxed number of slots k s avalable, and have to be allocated among n advertsers each of whom desres to dsplay an ad. Several mechansms to prce the slots and allocate them to advertsers have been studed, ncludng varants of the Generalzed Second Prce (GSP) mechansm, as well as mechansms from the Vckrey-Clarke-Groves (VCG) famly that are desgned to be truthful for proft maxmzng bdders. Extensons of these mechansms to account for thngs lke poston constrants and reserve prces have also been proposed. Many any of these aucton mechansms can be vewed as computng a bdder-optmal stable matchng wth sutably defned preferences of the aucton partcpants. Ths allows us to apply the theory of stable matchngs poneered by Gale and Shapley [13] to search auctons. In ths paper, we defne a general stable matchng model wth money n whch many of the exstng and new auctons can be expressed. We show that n ths model, a bdder-optmal stable matchng always exsts (under a mld non-degeneracy assumpton), and that a mechansm based on computng such matchng s truthful. Importantly, we gve an algorthm to compute a bdder-optmal matchng n polynomal tme of O(nk 3 ). As a result, we obtan the frst known, truthful mechansm for a varety of bdders. 1. INTRODUCTION Search engne companes lke Yahoo!, Google or MSN dsplay advertsements on web pages wth search results or varous knd of Google, Inc., Mountan Vew, CA. gagana@google.com Google, Inc., New York, NY. muthu@google.com Davd R. Cherton School of Computer Scence, Unversty of Waterloo, Waterloo, ON, Canada. dpal@cs.uwaterloo.ca. Work done durng summer 2007 nternshp at Google New York. Google, Inc., New York, NY. mpal@google.com Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, to republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. ACM Conference on Electronc Commerce 2008 Chcago, Illnos USA Copyrght 2008 ACM X-XXXXX-XX-X/XX/XX...$5.00. content. Typcally, the web page has a number of separately marked slots reserved for the ads, and an aucton s run to determne the set of ads to be dsplayed out of a pool of elgble ads that match the search query or content of the page. Typcally, each advertser must submt an ad together wth a bd ahead of tme. The purpose of the aucton s to determne an assgnment of ads to slots, as well as determne how much each wnnng advertser should pay for dsplayng her ad. The payment scheme may charge the advertser each tme the ad s shown, or only n the event that a user clcks on the ad. More nvolved schemes charge the advertser only when the user performs a pre-specfed acton n response to the ad, such as makng a purchase from the advertser s onlne store. GSP s a popular aucton mechansm used by maor search engnes. It assumes that there s a natural orderng on the ad slots (such as top to bottom or left to rght). It asks each advertser to submt a bd (ths can be a per-mpresson, per-clck or per-acton bd), ranks the advertsers n decreasng order of ther bds, and assgns the top k advertsers to the k avalable slots. From the pont of vew of a bdder, GSP enoys smple semantcs: each slot has a prce (that depends on the bds of other bdders), and the bdder s smply assgned the hghest slot whose prce does not exceed her bd. If the bdder s a maxmum prce bdder, n that her goal s smply to get the hghest possble slot whle makng sure that her cost does not exceed a certan threshold, t s her domnant strategy to submt the threshold as her bd. Ths s true for any chargng scheme, per mpresson, per clck, or any generally defned acton. The basc GSP mechansm can be extended n varous ways, for example by scalng each bd by a bdder-specfc multpler or by ntroducng a mnmum prce. These extensons are useful tools for the search engne: for example, gvng a hgher multpler to better ads ncreases the overall ad qualty, whle gvng a hgher multpler to ads lkely to be clcked on n a per-clck aucton may mprove revenue. The class of VCG mechansms [22, 7, 15] has been desgned as a truthful mechansm for an mportant class of proft maxmzng bdders. In the context of advertsng auctons, we assume that a proft maxmzng bdder derves a certan (expected) value v from her ad beng placed n poston. The advertser s proft s then equal to the (expected) proft from placng her ad, mnus the (expected) payment she s charged for the slot. In a smpler model where the clck probabltes of dfferent slots are known to the auctoneer, the advertser may smply submt her value of a clck. The auctoneer then fnds a maxmum-weght assgnment (whch maxmzes the overall value to all bdders), and determnes payments accordng to a formula that ncents the bdders to bd truthfully. Our paper s motvated by the observaton that the above mechansms really compute a bdder-optmal stable matchng between a
set of slots and a set of advertsers. We wll defne a model of bdder preferences and stable matchngs wth sde payments n whch ths observaton can be formalzed. Indeed, provng that the outcomes of these mechansms are bdder-optmal stable matchngs n our model s not dffcult. Wth ths observaton n mnd, we set out to study a general scenaro that accomodates the dfferent set of bdders as well as the ncorporates varous features that search engnes employ; ths scenaro not only ncludes mechansms descrbed above, but also other extensons. In our model, each bdder specfes a maxmum prce she s wllng to pay for an mpresson n a gven slot, as well as a value of that slot to her (whch may be greater than her wllngness to pay for the slot). In addton, the search engne may specfy a reserve (mnmum) prce for each bdder and each slot. We show that n ths model, a bdder-optmal stable matchng exsts and can be computed n O(nk 3 ) tme, where n s the number of bdders and k the number of slots. Further, we show that the mechansm that elcts bdder s preferences and computes a bdder-optmal stable matchng based on them s truthful n that for every bdder whose preferences can be expressed wthn our model, t s a domnant strategy to express her true preferences. In partcular, ths shows that there s a truthful aucton mechansm for any combnaton of maxmum prce and proft maxmzng bdders who can pay per mpersson, per clck or per acton, and can have constrants on the set of postons they may to appear n. Thus, what results s the frst-known general aucton mechansm that s truthful for a dverse set of bdders. 2. RELATED WORK The GSP aucton s the maor vehcle for sellng ads on the nternet. It has been observed that although t s not truthful for proft maxmzng bdders, t does have a Nash equlbrum that s effcent and ts resultng prces are equal to VCG prces, see e.g. [11, 2]. A varant of GSP n whch the bdder can specfy the lowest (maxmum) acceptable poston has been proposed n [3], whch also has a Nash equlbrum equvalent to a sutably defned VCG aucton. The recent manuscrpt [12] explores the effect of addng mnmum prces to GSP. The GSP mechansm assumes a fxed orderng on avalable slots. Wth ncreasngly complex web page layouts, ths assumpton s no longer unversally vald; f there are several regons on the page where ads can be placed, t s not obvous that one slot or regon should be unversally preferred to another by all advertsers. Also, allowng the advertser to choose the acton she s charged for (mpresson vs clck) destroys ncentve propertes of the aucton, even f as s natural, one consders multplyng each per-clck bd by the probablty of a clck). As long as dfferent postons have dfferent clck probabltes, t s not dffcult to come up wth examples where swtchng from a per-clck bd to per-mpresson bd or vce versa lowers the overall cost to the bdder whle gvng her the same or better slot. The general class of VCG mechansms follows from works [22, 7, 15]. For an overvew of the VCG mechansm appled to sponsored search, see e.g. [1, 2]. VCG s a natural mechansm, but bdders may fnd t unntutve to nterpret the prces they are charged. Also, t does not drectly support maxmum-prce bdders who naturally ft nto GSP. The stable matchng model has been ntroduced by Gale and Shapley [13] n 1962 and has been studed extensvely snce then. The monograph [19] gves a great overvew of mportant results n the area; we only menton themes that are drectly relevant to our work. In the basc model ntroduced n [13], a set I of men s to be matched to a set J of women n a one to one fashon. Each man has a preference orderng on the set of women, and each woman has a preference orderng on the set of men. The goal s to fnd a matchng that s stable n that there s no man and a woman n whch the man would prefer the woman to hs partner n the matchng and vce versa. Gale and Shapley [13] gve a deferred acceptance algorthm to compute a man-optmal stable matchng. The stable matchng has been generalzed to allow sde payments between members of a matched par. In such models, each partcpant has a preference relaton on the set of possble potental (partner, payment) pars. In models where the sde payments are allowed to be arbtrary real numbers, the preference relaton s often gven by a set of utlty functons, one for each man-woman par expressng the man s preferences, and one woman s. A model wth utlty functons that are lnear n money has been studed by [20, 18, 9]. It has long been known that n the lnear utlty model, a bdder-optmal stable matchng s equvalent to VCG allocaton and prces, [16, 6]. Arbtrary (non-lnear) ncreasng contnuous utlty functons were consdered n [8, 9, 4, 5]. These models crucally depend on the utlty functons beng contnuous and defned on the whole set R, an assumpton we have to drop n our paper. The paper [8] shows that even n such a general model, there exsts a bdder-optmal stable matchng, but no algorthm was gven to fnd t. Moreover they show that n a mechansm based on the man-optmal matchng, t s weakly domnant for each man to reveal hs true utlty functon. Our paper bulds heavly on ths body of work. A feature that dstngushes our work s that n order to model maxmum prce bdders and reserve prces we need to ntroduce preferences that can not be expressed as contnuous, strctly monotone utlty functons. Ths seemngly nnocent change ntroduces techncal dffcultes and makes the model harder to work wth. Stll, we are able to transfer the man structural results to our model. Under the assumpton that the bdder preferences are n a general poston, we can stll prove the exstence of a bdder-optmal matchng, and we gve a very effcent algorthm to fnd t. The general poston assumpton can be lfted by adoptng a sutable te-breakng rule, whch allows us to show that the stable matchng mechansm s truthful. Ths gves us the frst known truthful mechansms for a varety of bdders. 3. THE MAX-VALUE MODEL Our model conssts of the set I = {1, 2,..., n} of bdders and the set J = {1, 2,..., k} of slots. We use letter to denote a bdder and letter to denote a slot. Each bdder has a value v, for each slot how much s that slot worth to her, and a maxmum prce m, she s able and wllng to pay for the slot. To motvate why v, and m, mght be dfferent, consder buyng a house whose value you estmate sgnfcantly hgher than your bank. Whle your value for the house s hgh, the amount of money your bank s wllng to lend you s lower. Allowng the bdder to specfy both a value and a maxmum s also needed to model the GSP aucton. In addton to bdder preferences, the seller specfes for each bdder and each slot a reserve prce r,. For smplcty we assume that the reserve prces are known to the bdders n advance. For each and each we assume that r, 0, v, 0, m, v,. If bdder s nterested n the slot he specfes m, r,. Otherwse, f bdder has no nterest n slot he specfes negatve m,. We denote by v, m, r the n k matrces wth entres v,, m,, r, respectvely. We refer to the trple (v, m, r) as an aucton nstance or smply aucton. We wsh to fnd an assgnment of slots to bdders, and compute how much each wnnng bdder should pay for her slot. To study strategc behavor of the bdders, we need to specfy ther relatve
preference for possble outcomes. Bdder Preferences. We assume that each bdder s ndfferent among varous outcomes as long as her assgned slot (f any) and payment s the same. Let us defne the utlty of a bdder who s offered a slot at prce p as follows. If p m,, we set u = v p. If p > m,, we set u = 1. Ths utlty, nterpreted as a functon of the prce, s not contnuous at p = m,. If the bdder s not matched (at zero prce), her utlty s 0. Gven a choce between slot 1 at prce q 1 m,1 and slot 2 at prce p 2 m,2, the bdder prefers the offer wth hgher utlty, and s ndfferent among offers that have the same utlty. In partcular, the bdder prefers to be not matched to beng matched to a slot at prce that exceeds her maxmum prce m. The bdder s ndfferent between beng matched wth utlty 0 and not beng matched. 3.1 Stable Matchng We formalze the noton of a matchng n the followng defntons. DEFINITION 1 (MATCHING). A matchng s a trple (u, p, µ), where u = (u 1, u 2,..., u n) s a non-negatve utlty vector, p = (p 1, p 2,..., p k ) s a non-negatve prce vector, and µ I J s a set of bdder-slot pars such that no slot and no bdder occurs n more than one par. If a par (, ) µ, we say that bdder s matched to slot. We use µ() to denote the slot matched to a bdder, and µ() to denote to denote the bdder matched to a slot. Bdders and slots that do not belong to any par n µ are sad to be unmatched. DEFINITION 2 (FEASIBLE MATCHING). A matchng (u, p, µ) s sad to be feasble for an aucton (v, m, r), whenever for every (, ) µ, p [r,, m,], (1) u + p = v,, (2) and for each unmatched bdder s u = 0 and for each unmatched slot s p = 0. DEFINITION 3 (STABLE MATCHING). A matchng (u, p, µ) s stable for an aucton (v, m, r) whenever for each (, ) I J at least one of the followng nequaltes holds: u + p v,, (3) p m,, (4) u + r, v,. (5) A par (, ) I J whch does not satsfy any of the three nequaltes s called blockng. Geometrc nterpretaton of nequaltes (3), (4), (5) s explaned n Fgure 1. Note that f a bdder s not nterested n a slot, then (4) s trvally satsfed. A feasble matchng does not have to be stable, and a stable matchng does not have to be feasble. However, we wll be nterested n matchngs that are both stable and feasble. More specfcally, we wll be nterested n a partcular matchng (u, p, µ ) that s stable, feasble, and s, wth respect to each bdder s preferences, superor to any other feasble stable matchng (u, p, µ). It s surprsng that such a matchng exsts. Its exstence for other smpler models, e.g., wth contnuous utlty vs. prce curves or other preference relatons, s a core result of the theory of stable matchngs. u utlty of bdder v, v, r, v, m, r, (p, u ) m, v, p prce of slot Fgure 1: Matchng s stable whenever for each bdder I and each slot J the pont wth coordnates (p, u ) les outsde the gray regon. 3.2 Our Results One of our techncal contrbutons s a proof of the exstence of stable, feasble matchng n our model. THEOREM 4 (EXISTENCE OF BIDDER-OPTIMAL MATCHING). If the aucton (v, m, r) s n a general poston, t has a unque bdder-optmal stable matchng. We defer the precse defnton of general poston to Defnton 12. In essence, any aucton (v, m, r) can be brought nto general poston by arbtrarly small (symbolc) perturbatons. In practce ths assumpton s easly removed by usng a consstent te-breakng rule. We propose an aucton mechansm that, for any reserve prces r specfed by the auctoneer, and any valutons v and maxmum prces m specfed by the bdders, computes the bdder-optmal stable matchng (u, p, µ ), assgns the slots to the bdders accordng to µ and charges the matched bdders prces p correspondngly. We call ths mechansm the stable matchng mechansm. We study ths mechansm from the game-theoretc perspectve and prove that the mechansm s truthful. THEOREM 5 (TRUTHFULNESS). The stable matchng mechansm s a truthful mechansm for bdders n the max-value model. That s, submttng her true vectors v and m s a domnant strategy for each bdder. Our fnal contrbuton s an algorthm that computes the bdderoptmal stable matchng. THEOREM 6. There s an algorthm that fnds the bdder-optmal stable matchng n the max-value model n tme O(nk 3 ). Thus, there s a truthful mechansm for max-value bdders that can be mplemented n ths runnng tme. Taken together, these results yeld the frst known truthful mechansm that s effcent to mplement for all bdders who can be represented n our max-value model. Ths not only ncludes the wellknown GSP or VCG and ther varants by search engnes, but much more. 4. MODELING ADVERTISING AUCTIONS In ths secton, we wll present examples of aucton mechansms commonly used n sponsored search. We wll show how to model
these mechansms n our max-value model. In the next secton we gve examples of novel combned mechansms that can be mplemented n our model. 4.1 Exstng Mechansms GSP pay-per-mpresson. In a Generalzed Second Prce aucton, each advertser submts a sngle number b as her bd, whch s the maxmum amount she s wllng to pay for dsplayng her ad. The auctoneer orders bdders n decreasng order of ther bds, and assgns the frst k advertsers to the k avalable slots n ths order. The -th allocated advertser pays amount equal to the (+1)-st bd for each mpresson. GSP pay-per-clck. An alternatve s to charge the advertser only n the event of a clck on her ad. The bd b s nterpreted as a maxmum the advertser s wllng to pay for a clck. Agan, the advertsers are ordered by ther per-clck bd, and each allocated advertser pays the next hghest bd n the event of a clck. In a qualty-weghted varant, the ads are ordered by the product of ther q qualty score q and bd b ; the -th advertser pays b +1 +1 q n the event of a clck. Note that the expected cost per mpresson q b +1 +1 q ctr, depends not only on the next hghest bd but also on the poston, as long as the probablty ctr, of clckng on the ad n poston depends on the poston. Thus, there s no drect way to translate a per-clck bd to a per-mpresson bd, wthout lookng at the compettor s bds. The VCG mechansm for proft-maxmzng bdders. In a varant of the VCG mechansm consdered e.g. n [2], each bdder states her value V for a clck. The auctoneer derves the expected value of each slot v, = V ctr, for that bdder by usng an estmate ctr, of the probablty that the ad would be clcked on f placed n poston. The auctoneer computes a maxmum-weght matchng n the bpartte graph on bdders and postons wth v, as edge weghts. The maxmum weght matchng µ gves the fnal allocaton. For prcng, the VCG formula sets the prce per mpresson of slot = µ () to be p = P k I\{} v k,µ (k) v k,µ (k) where µ s a maxmum-weght matchng wth the set of bdders I \ {}. Note that the per-mpresson prce p can be translated to a per-clck prce by chargng bdder prce p /ctr, for each clck. (Smlar translaton can be done for a generally defned user acton other than a clck, as long as the probablty of the acton can be estmated.) For each of the above mechansms, we defne a correspondng type of bdder n the max-value model. Max-per-mpresson bdder has a target cost per mpresson b. She prefers payng b or less per mpresson to any outcome where she pays more than b. Gven that her cost per mpresson s at most b, she prefers hgher (wth lower ndex) poston to lower poston. Gven a fxed poston, she prefers payng lower prce to hgher prce. A max-per-mpresson bdder can be translated nto the maxvalue model by settng her m, = b for all postons J, and settng her value v, = M(k + 1 ) where M s a suffcently large number (M > b s enough). Max-per-clck bdder dffers from a max-per-mpresson bdder n that she s not wllng to pay more than b per clck. We translate her per-clck bd nto our framework usng predcted clck probabltes: set m, = b ctr, for I and v, = M(k + 1 ) where M > b max ctr,. Proft-maxmzng bdder seeks the poston and payment that maxmzes her expected proft (value from clcks mnus payment). If we assume that her value per clck s V, such bdder s modeled by settng v, = m, = V ctr,. We formalze the correspondence between the mechansms and correspondng bdder types n the followng theorem. THEOREM 7. The outcome (allocaton and payments) of a (1) per-mpresson GSP, (2) per-clck GSP, (3) VCG aucton, respectvely s a bdder-optmal stable matchng for a set of (1) max-permpresson bdders, (2) max-per-clck bdders, (3) proft-maxmzng bdders, respectvely. PROOF. Part (3) of the theorem has been observed by multple authors ncludng [16]. Chapter 7 of [19] as well as [6] dscuss the relatonshp of the VCG mechansm for assgnments and stable matchngs. We gve a proof for part (1), per-mpresson GSP. The proof of part (2) for per-clck GSP s smlar. For smplcty, we assume that n > k and all reserve prces are zero. Let b 1 > b 2 > > b n be the per-mpresson bds of the bdders. Wthout loss of generalty, the bdders are ordered by decreasng order of ther bds. (By the general poston assumpton, assume bds are dstnct.) Recall that we encode a max-per-mpresson bdder by settng v, = M(k + 1) and m, = b. The matchng produced by the GSP aucton s as follows: the matched pars are µ = {(1, 1), (2, 2),..., (k, k)}, bdder s utltes u = M(k + 1) b +1 for 1 k, u = 0 for > k, and prces p = b +1 for = 1, 2,..., k. It s easy to verfy that the matchng s feasble and stable accordng to Defntons 2 and 3. Frst we show that any feasble matchng n whch the assgnment s dfferent from µ s not stable. Indeed, such a matchng (u, p, µ ) must have a bdder k such that was not allocated a slot among the frst slots, and a slot that s ether unmatched or matched to some bdder >. From feasblty we have that p = 0 f slot s unmatched and p b n case t s matched. In ether case, p < b. Also, snce bdder s matched to some slot > (or unmatched), we know that u v, = M(k + 1). We now clam that (, ) s a blockng par. Snce v, u M[(k +1) (k +1)] M, nequaltes (3) and (5) are volated, and snce p < b, nequalty (4) s volated as well. Now consder any matchng wth the assgnment µ = {(1, 1),..., (k, k)}. It s easy to verfy that n order to be stable, t must be that p b +1, otherwse the par (+1, ) would be a blockng par. Hence the matchng wth prces p = b +1 has the lowest possble prces and hence s bdder-optmal. Mnmum prces. Some search engnes mpose a mnmum prce r for each ad (for example, based on perceved qualty of the ad). In GSP, only bdders whose bd s above the reserve prce can partcpate. The allocaton s n decreasng order of bds, and each bdder pays the maxmum of her reserve prce and the next bd. Mnmum GSP prces are easly translated to the max-value model by settng r = r (f payng per mpresson) or r = r ctr, (f payng per clck). Our model allows for separate reserve prces for dfferent slots (e.g. hgher reserve prce for certan premum slots) that are not easly mplemented n the GSP world. 4.2 New Aucton mechansms Let us gve a few examples of new aucton mechansms that are specal cases of the max-value model. GSP wth arbtrary poston preferences. Consder an advertser who wshes for her ad to appear only n certan slots. For example, [3] propose a GSP varant n whch each bdder has the opton to specfy a prefx of postons {1, 2,..., β } for some β she s nterested n and exclude the remanng slots. Also, tools lke Google s Poston Preference allow the advertser to specfy arbtrary poston ntervals [α, β ]. We are however not aware of any publshed
work that dscusses more sophstcated poston preferences. One would magne that n the world of content advertsng where there may be multple areas desgned for ads on a sngle page, havng a rcher language n whch to express the preferences over slots would be benefcal to the advertser. Such preferences are readly expressble n the max-value model. Combnng clck and mpresson bdders n GSP. Snce both pay per clck and pay per mpresson models are wdely used n practce, t s useful to have a way of combnng these two bddng modes. Ths can be easly done by computng a stable matchng for a mxed pool of bdders. The followng smpler approach s not approprate, as t does not have the proper ncentve structure. Suppose we allow each bdder to specfy both a maxmum prce b, as well as a payment type τ {I, C}. A nave combned aucton orders bdders by decreasng b. Each advertser wth τ = I s charged the next hghest bd b +1 for showng the ad. Each advertser wth τ = C s charged b +1 n the event that the user clcks on the ad. Note, ths scheme gves advertsers a strong ncentve to report τ = C regardless of ther true type (as long as the probablty of user clckng s less than 1). To offset ths ncentve, the auctoneer may ntroduce multplers 0 < q C < 1 and q I = 1 and set the effectve bd of each bdder to be b eff = b q τ. In the modfed GSP aucton, bdders are be sorted by ther effectve bd. Each bdder who reports type τ = I s charged b eff +1 for each mpresson, whle each bdder reportng τ = C s charged b eff +1/q C n the event of a clck. For any value of 0 < q C < 1, there s a smple nstance n whch some bdder can gan by msreportng her type. Let ctr 1 and ctr 2 be the probablty that an user wll clck on an ad n poston 1 and 2 respectvely. Assume ths probablty s the same for all ads, and that ctr 1 > ctr 2. Suppose that the frst slot s won by a bdder of type I, the second slot s won by a bdder of type C, and that there s at least one more bdder wth postve bd. If q C > ctr 2, the bdder n the second poston can lower her overall cost whle keepng the same poston by reportng type C and keepng the same effectve bd. On the other hand, f q C < ctr 1, bdder n the frst poston can lower her cost by reportng type I, and adustng her bd so that her effectve bd stays the same. Dverse bdders. There are many types of bdders wth dfferent goals. Some lke to thnk n terms of a maxmum prce per clck or mpresson. Some prefer to target only certan postons (e.g. top of the page) for consstency or brandng reasons. Others try to maxmze ther proft and are able to estmate the value of a specfc user acton. Each bdder may specfy her goal n a language famlar to her. We are not aware of any pror research on aucton mechansms for such dverse set of bdders. detal. To do so, we ntroduce the concept of an update graph. DEFINITION 8 (UPDATE GRAPH). Gven an aucton (v, m, r), the update graph for a matchng (u, p, µ) s a drected weghted bpartte multgraph wth partte sets I and J { 0}, where 0 s the dummy slot. The update graph conssts of fve types of edges. For each bdder and each slot J there s a forward edge from to wth weght u + p v,, f p [r,, m,); a backward edge from to wth weght v, u p, f (, ) µ, a reserve-prce edge from to wth weght u + r, v,, f u + r, > v, and m, > r,, a maxmum-prce edge from to wth weght u + m, v,, f u + m, > v, and m, > r,, a termnal edge from to 0 wth weght u f u > 0. An alternatng path n the update graph starts wth an unmatched bdder vertex 0 wth u 0 > 0, follows a sequence of forward and backward edges, and ends wth a reserve-prce, maxmum-prce or termnal edge. We place the restrcton that all vertces of the alternatng path must be dstnct, wth the possble excepton that the last vertex s allowed to appear once agan along the path. The weght w(p ) of an alternatng path P s the sum of weghts of ts edges. Let (u (t), p (t), µ (t) ) be a matchng and G (t) be the correspondng update graph. A sngle teraton of the algorthm conssts of the followng steps. 1. If there s no alternatng path, stop and output the current matchng. Otherwse, let P be an alternatng path n G (t) of mnmum weght. Let w (t) (P ) denote ts weght, and let P = ( 0, 1, 1, 2, 2,..., l, l, l+1 ) for some l 0. 2. Let d (t) ( 0, y) be the length of the shortest path n G (t) from 0 to any vertex y, usng only forward and backward edges. If a vertex y s not reachable from 0, d (t) ( 0, y) =. 3. Compute utlty updates for each bdder I. The vector u (t+1) gves the fnal utltes for the teraton. u (t+1) = u (t) max w (t) (P ) d (t) ( 0, ), 0 (6) 5. ALGORITHM FOR COMPUTING THE BIDDER-OPTIMAL MATCHING In ths secton we present an algorthm that for gven aucton (v, m, r) (n general poston) computes the bdder-optmal stable matchng. The algorthm starts wth an empty matchng (u (0), p (0), µ (0) ) whch s defned as follows. Utlty of each bdder s u (0) = B, where B s a large enough number, such that B > max{v, (, ) I J}. Prce of each slot s p (0) = 0. There are no matched pars,.e. µ (0) =. In each teraton, the algorthm fnds an augmentng path, and updates the current matchng (u (t), p (t), µ (t) ) to the next matchng (u (t+1), p (t+1), µ (t+1) ). The algorthm stops when no more updates can be made, and outputs the current matchng (u (T ), p (T ), µ (T ) ) at the end of the last teraton. We now descrbe an teraton n more 4. Compute prce updates for each slot J. p (t+) = p (t) + max The fnal prces p (t+1) w (t) (P ) d (t) ( 0, ), 0 (7) are equal to p (t+) wth one excepton. In case the last edge of P s a reserve-prce edge, we set the prce of slot l+1, the last vertex of P to be p (t+1) = max(p (t+), r l, l+1 ). 5. Update the assgnment µ (t) along the alternatng path P to obtan the new assgnment µ (t+1). We have not specfed how should the set of assgnment edges be updated. Before we do that, let us state two nvarants mantaned by the algorthm. (A1) The matchng (u (t), p (t), µ (t) ) s stable for the aucton (v, m, r).
(A2) For every matched par (, ) µ (t), u (t) and p (t) satsfy (1) and (2). An mportant consequence of nvarant (A1) s that forward edges have non-negatve weght. Indeed, t can be easly checked that a forward edge wth a negatve weght would be blockng par. Invarant (A2) guarantees that backward edges have zero weght. Smlarly, nvarant (A2) mples that the weght of every backward edge must be zero. Fnally, each reserve-prce, maxmum-prce and termnal edges has non-negatve weght by defnton. LEMMA 9. All edge weghts n each update graph G (t) are non-negatve. Wth non-negatve edge weghts, sngle-source shortest paths can be computed usng Dkstra s algorthm n tme proportonal to the square of the number of vertces reachable from the source. Snce no unmatched vertex s reachable from any other vertex, there are at most 2k reachable vertces at any tme, thus the shortest alternatng path P and dstances d (t) ( 0, y) can be computed n tme O(k 2 ). Fnally, let us deal wth updatng the assgnment µ. Snce the alternatng path alternates between usng forward (.e. non-matchng) and backward (.e. matchng) edges, a natural move s to remove all the matchng edges of P and replace them by non-matchng edges of P. Care must be taken however to take nto account the specal nature of the last edge of P as well as the fact that the last vertex of P may be vsted twce. We consder three cases: Case 1: P ends wth a termnal edge,.e. l+1 s the dummy slot. Flp matchng and non-matchng edges along the whole length of P. Bdder l ends up beng unmatched, and for x = 0, 1,..., l 1, bdder x wll be matched to slot x+1. Case 2: P ends wth a maxmum-prce edge. Consder two subcases: (a) l+1 = l. Ths means that the prce bdder l was matched to reached hs maxmum prce. Flp matchng an non-matchng edges along P. Ths leaves bdder l unmatched, and for x = 0, 1,..., l 1 bdder x s matched wth slot x+1. (b) Otherwse, the maxmum prce was reached on a non-matchng edge. Keep the matchng unchanged. That s, µ (t+1) = µ (t). Case 3: P ends wth a reserve-prce edge. Ths s the most complex case. Consder three subcases: (a) Item l+1 s unmatched n µ (t). Ths case ncreases the sze of the matchng. For x = 0, 1,..., l, match bdder x wth slot x+1. (b) Item l+1 s matched n µ (t) and the reserve prce r l, l+1 offered by bdder l does not exceed the current prce p (t+) l+1 of the slots. Keep the matchng unchanged, that s, µ (t+1) = µ (t). (c) Item l+1 s matched n µ (t) to some bdder l+1 and r l, l+1 > p (t+) l+1. If P s a path, that s, f P does not vst slots l twce, we smply unmatch bdder l+1, and flp matchng and non-matchng edges of P. (Ths keeps the sze of the matchng the same, as bdder 0 gets matched and bdder l+1 unmatched.) If P vsts l+1 twce, t must be that l+1 = d for some d. Note that t s not the case that d = l, snce ths would mean that l was matched to l+1. Ths s mpossble because the reserve prce on ths edge has been reached ust now. Ths way, the end of P forms a cycle wth at least 2 bdders and 2 slots. We flp the matchng and non-matchng edges along the cycle, but leave the rest of P untouched. Ths leaves bdder x matched to slot x+1, for x = d, d + 1,..., l. 6. ANALYSIS OF OUR ALGORITHM In ths secton we show that the algorthm from Secton 5 ndeed computes a bdder-optmal stable matchng. It s not obvous that a stable matchng even exsts for any aucton nstance (v, m, r). Our algorthm provdes a constructve proof of ths fact. An alternate proof of exstence can be done usng lmt arguments and the deferred acceptance algorthm on a sequence of dscretzatons of bdder s preference lsts. The detals are deferred to the full verson of ths paper. LEMMA 10. The matchng (u (T ), p (T ), µ (T ) ) computed by the matchng algorthm s feasble and stable. PROOF. Stablty follows drectly from nvarant (A1). Feasblty follows from nvarant (A2) and the fact that snce there are no alternatng paths, t must be that u (T ) = 0 for every unmatched bdder. We shall prove the nvarants later n ths secton. Whle a feasble stable matchng always exsts, there may not always be a bdderoptmal matchng, as the followng example shows. Consder the case of a sngle slot and two bdders wth dentcal maxmum bds. There are two stable matchngs. In each matchng, the slot s allocated to one of the bdders at maxmum prce. Each matchng s preferred by one bdder over the other, hence there s no matchng preferred by both of them. Fortunately t turns out that the example above s degenerate, and that a bdder-optmal matchng exsts for every non-degenerate, or "general poston" aucton. To make ths precse, we need the followng two defntons. DEFINITION 11 (AUCTION GRAPH). The aucton graph of an aucton (v, m, r) s a drected weghted bpartte multgraph wth partte sets I and J { 0}, where 0 s the dummy slot. The aucton graph contans fve types of edges. For each bdder and each slot J there exst a forward edge from to wth weght v,, a backward edge from to wth weght v,, a reserve-prce edge from to wth weght r, v,, a maxmum-prce edge from to wth weght m, v,, a termnal edge from to 0 wth weght 0. DEFINITION 12 (GENERAL POSITION). An aucton (v, m, r) s n general poston f for every bdder, no two alternatng walks n the aucton graph that start at bdder, follow alternatng forward and backward edges and end wth a dstnct edge that s ether a reserve-prce, maxmum-prce or termnal edge, have the same weght. Any aucton (v, m, r) can be brought nto general poston by a symbolc perturbaton. In the algorthm mplementaton, ths can be also acheved by breakng tes lexcographcally by the dentty of the fnal edge of the walk. The next secton s pretty techncal. It establshes nvarants needed to prove Theorem 6.
6.1 Invarants Besdes nvarants (A1) and (A2) ntroduced n Secton 5, we clam three more nvarants. (A3) Each unmatched slot has zero prce. (B1) f a bdder s nterested n slot and u (t) + m, = v,, then (, ) µ (t). (B2) If a bdder s nterested n a slot and u (t) then (, ) µ (t) or p (t) r,. + r, = v,, All the fve nvarants are proved by nducton on t. Invarants (B1) and (B2) are techncal and we omt ther proofs n ths verson of the paper. However, we use them n the nducton step to prove the frst three nvarants. Both (B1) and (B2) rely on the general poston assumpton. PROOF OF THE INVARIANTS. The base case, t = 0, s readly verfed. Invarant (A1) follows from that u (0) = B for all I, p (0) = 0 for all J, and hence (3) s satsfed. Invarants (A2) and (A3) hold trvally. Let us prove that (u (t+1), p (t+1), µ (t+1) ) satsfes (A3). Note that p (t+1) p (t). The slots matched n µ (t) reman matched n µ (t+1), at most one addtonal slot s matched n µ (t+1). The remanng slots are not reachable from 0 n G (t), snce for any such slot, p (t) = 0 and for any I, r, > 0 by the general poston assumpton, thus there s no forward edge to. Hence the prce of any such slot remans zero. Let us prove that (u (t+1), p (t+1), µ (t+1) ) satsfes (A1). We consder three cases for any par (, ) I J: Case 1: p (t) [r,, m,). (u (t), p (t), µ (t) ) s stable by the nducton hypothess and hence u (t) w (t) (P ), then u (t+1) = u (t) + p (t) and p (t+1) v,. If d (t) ( 0, ) p (t) p (t+1) satsfy (3). On the other hand, f d (t) ( 0, ) < w (t) (P ), then, thus u(t+1) and u (t+1) = u (t) (w (t) (P ) d (t) ( 0, )), (8) p (t+1) p (t+) p (t) + (w (t) (P ) d (t) ( 0, )). (9) Snce from to there s a forward edge n G (t), d (t) ( 0, ) d (t) ( 0, ) + (u (t) + p (t) v,). (10) We add (8) to (9), subtract (10), and we get that u (t+1) satsfy (3). Case 2: p (t) m,. Snce p (t+1) p (t) (Ths case apples also f s not nterested n.) Case 3: p (t) and p (t+1), (4) holds for p(t+1). < r, and s nterested n. (u (t), p (t), µ (t) ) s stable by the nducton hypothess and hence u (t) d (t) ( 0, ) w (t) (P ), then u (t+1) = u (t) satsfes (5). On the other hand, f d (t) ( 0, ) < w (t) (P ), then u (t+1) satsfes (5). If also and hence u (t+1) = u (t) (w (t) (P ) d (t) ( 0, )). (11) We clam that n G (t) there s reserve-prce edge from to and thus w (t) (P ) d (t) ( 0, ) + (u (t) + r, v,). (12) To prove the exstence of the reserve-prce edge we show that u (t) + r, > v,. The non-strct nequalty holds snce u (t) satsfes (5). The strctness follows snce, by the nducton hypothess, (u (t), p (t), µ (t) ) satsfes (A2) and (B2). By subtractng (12) from (11) we get that u (t+1) satsfes (5). Frst, let us prove that (u (t+1), p (t+), µ (t) ) satsfes (A2). Consder any par (, ) µ (t). In G (t) there s a backward edge from to. By nducton hypothess, (u (t), p (t), µ (t) ) satsfes (A2) and hence the backward edge has zero weght. Hence d (t) ( 0, ) = d (t) ( 0, ). (13) Therefore, from the updates (6), (7) follows u (t+1) u (t) + p (t) and hence (1) remans to hold. If w (t) (P ) d (t) ( 0, ), then p (t+) satsfed by p (t+) by the update (7) for prces p (t+) + p (t+) = = p (t) and thus (2) remans. On the other hand, f w (t) (P ) > d (t) ( 0, ), then = p (t) + (w (t) (P ) d (t) ( 0, )). (14) We also clam that there exsts maxmum-prce edge from to and thus w (t) (P ) d (t) ( 0, ) + (u (t) + m, v,). (15) To prove the exstence of the maxmum-prce edge we show that u (t) + m, > v,. The non-strct nequalty holds snce p (t) m, and thus u (t) + m, u (t) + p (t) = v, snce by the nducton hypothess (u (t), p (t), µ (t) ) satsfes (A2). Strctness follows snce, by the nducton hypothess, (u (t), p (t), µ (t) ) satsfes (B1). Summng (13), (15), (14) and cancelng common terms gves p (t+) (u (t) + p (t) v,) + m, = m,, where u (t) + p (t) v, = 0 follows from the nducton hypothess. Hence, snce p (t+) p (t) r,, (2) remans to hold for p (t+). Fnally, let us prove that (u (t+1), p (t+1), µ (t+1) ) satsfes (A2). For any par (, ) µ (t) µ (t+1) we have already done t, snce p (t+1) = p (t+). It remans to consder pars n µ (t+1) \ µ (t). Let P = ( 0, 1, 1,..., l, l, l+1 ) be the alternatng path used to obtan µ (t+1) from µ (t). Any par (, ) µ (t+1) \ µ (t) s an edge lyng P and has the form (, ) = ( x, x+1). We consder two cases. Case 1: x < l. In ths case (, ) = ( x, x+1) s a forward edge and has weght u (t) + p (t) v,, and snce t les on a mnmumweght path, d (t) ( 0, ) = d (t) ( 0, ) + (u (t) + p (t) v,). (16) Snce w (t) (P ) d (t) ( 0, ) and w (t) (P ) d (t) ( 0, ), the updated quanttes are u (t+1) p (t+1) The equalty (1) for u (t+1) = u (t) (w (t) (P ) d (t) ( 0, )), (17) = p (t) + (w (t) (P ) d (t) ( 0, )). (18) and p (t+1) follows by summng (17), (18) and subtractng (16). Let us verfy that p (t+1) satsfes (2). Snce (, ) s a forward edge, p (t) [r,, m,). By the nducton hypothess (u (t), p (t), µ (t) ) s stable, thus u (t) + p (t) v,, hence u (t) + m, > v, and consequently n G (t) there s a maxmum-prce edge from to of weght u (t) + m, v,. Therefore w (t) (P ) d (t) ( 0, ) + u (t) + m, v,. (19)
We add (18) to (19) and from that we subtract (16), we cancel common terms and we have p (t+1) m,. The verfcaton of (2) for p (t+1) s fnshed by observng that p (t+1) p (t) r,. Case 2: x = l. Snce we assume that (, ) = ( l, l+1 ) belongs to µ (t+1) \ µ (t), t can be nether a termnal edge nor a maxmumprce edge, and thus t must be a reserve-prce edge and has weght u (t) + r, v,. By the same argument p (t+) r,, hence p (t+1) = r, and clearly satsfes (2). Observe that u (t+1) = u (t) (w (t) (P ) d (t) ( 0, )), w (t) (P ) = d (t) ( 0, ) + (u (t) + r, v,). Subtractng the two equatons shows that u (t+1) and p (t+1) satsfy (1). 6.2 Runnng Tme We bound the number of teratons by O(nk) n the clam below. Snce each teraton can be mplemented n tme O(k 2 ), ths gves us overall runnng tme O(nk 3 ). LEMMA 13. The matchng algorthm fnshes after at most n(2k+ 1) teratons. PROOF. Consder the number of edges n the update graph. Intally, the graph G (0) has at most nk reserve-prce, nk maxmumprce and n termnal edges. We clam that n each teraton, the number of edges n the update graph s reduced by one. Snce the algorthm must stop when there are no more edges left, ths bounds the total number of teratons. Consder an teraton t of the algorthm. We clam that n the alternatng path P = ( 0, 1, 1,..., l, l, l+1 ), the last edge (, ) = ( l, l+1 ) wll not appear n the update graph G (t+1). Ths s easly verfed by consderng three cases: Case 1: If (, ) s a termnal edge, then w (t) (P ) = d (t) ( 0, ) + u (t) and hence u (t+1) = u (t) (w (t) (P ) d (t) ( 0, )) = 0. Case 2: If (, ) s a maxmum-prce edge, then w (t) (P ) = d (t) ( 0, )+ (u (t) + m, v,) and hence u (t+1) + m, = u (t) (w (t) (P ) d (t) ( 0, )) + m, = v,. Case 3: If (, ) s a reserve-prce edge, then w (t) (P ) = d (t) ( 0, ) + (u (t) +r, v,) and hence u (t+1) +r, = u (t) (w (t) (P ) d (t) ( 0, )) + r, = v,. The utltes never ncrease and the prces never decrease throughout the algorthm, thus the edge ( l, l+1 ) does not appear n any update graph G (t ) for any t > t. 6.3 Bdder Optmalty LEMMA 14. Let (v, m, r) be an aucton n general poston, and let (u, p, µ ) be any feasble stable matchng. Then n any teraton t of the matchng algorthm, we have that u u (t) for all I and p p (t) for all J. Wthout loss of generalty assume that (u, p, µ) s such that there does not exst a par (, ) µ such that p = m,. If there was such a par, then we can decrease prces of some of the tems and ncrease utltes of some of the bdders such that p < m,. Ths s possble because of the general poston assumpton. See full verson of the paper. We prove Lemma 14 by nducton on t. The base case, t = 0, trvally holds true, snce by feasblty of (u, p, µ ), p 0 for all J and u B for all I. In the nductve case, assume that u (t) u and p (t) p. We frst prove that PROPOSITION 15. u (t+1) u and p (t+) p. We look contnuously at updates (6) and (7). For that purpose we defne for each I a contnuous non-ncreasng functon u (x), u (x) = u (t) max x d (t) ( 0, ), 0, and for each J a contnuous non-decreasng functon p (x), p (x) = p (t) + max x d (t) ( 0, ), 0. Clearly, u (t+1) = u(w (t) (P )) and p (t+) = p(w (t) (P )). To prove that u (t+1) u and p (t+) p, suppose by contracton that there exsts y [0, w (t) (P )] such that ether u (y) < u for some I or p (y) > p for some J. We choose nfmal such y. Clearly, u(y) u, p(y) p and y < w (t) (P ). Consder the sets I = { I u (y) = u and d (t) ( 0, ) y}, J = { J p (y) = p and d (t) ( 0, ) y}. CLAIM 16. Each slot J s matched n µ (t) to some I. PROOF OF THE CLAIM. Let J. If was unmatched, then ether d (t) ( 0, ) = w (t) (P ) or d (t) ( 0, ) = ; however both optons contradct the choce of y and that J. Thus s matched to some I, hence n G (t) there s a backward edge from to and thus d (t) ( 0, ) = d (t) ( 0, ) and therefore u (y) + p (y) = v,. Further, nvarants (A2) and (B1) mply that p (t) [r,, m,). Consequently, there s a maxmum-prce edge from to, w (t) (P ) d (t) ( 0, ) + (u (t) + m, v,), and hence p = p (y) < p (t+) = p (t) +(w (t) (P ) d (t) ( 0, )) m,. Therefore p [r,, m,), and snce (u, p, µ ) s stable, u + p v, and hence u (y) = v, p (y) = v, p u. On the other hand, by nfmalty of y, u (y) u. Thus I. CLAIM 17. Each bdder I s matched n µ to some J. PROOF OF THE CLAIM. Snce n G (t) there s a termnal edge from to the dummy slot, w (t) (P ) d (t) ( 0, ) + u (t). Hence u = u (y) = u (t) (y d (t) ( 0, )) > u (t) (w (t) (P ) d (t) ( 0, )) 0, and thus bdder s matched n µ to some slot J. By feasblty of (u, p, µ ), p [r,, m,]. By the assumpton made at the begnnng p m,. Therefore n G (t) there s a forward edge from to and thus d (t) ( 0, ) d (t) ( 0, ) + (u (t) Clearly, snce I, By the prce update rule + p (t) v,). (20) u (y) = u (t) (y d (t) ( 0, )). (21) p (y) p (t) + (y d (t) ( 0, )). (22) We add (21) to (22) and subtract from that (20) and we obtan p (y) v, u (y). Hence, snce by feasblty of (u, p, µ ), u + p = v,, we have p (y) v, u (y) = v, u = p.
Recallng that p(y) p we see that p (y) = p. Subtractng (21) from (20) and cancellng common terms we have d (t) ( 0, ) y + (u (y) + p (t) v,). We upper-bound the rght sde of the nequalty usng that u (y) = u, p (t) p (y) and u + p = v, and we have Thus J. d (t) ( 0, ) y + (u + p v,) = y. From the two clams t follows that I = J and that µ (t) bectvely matches I wth J. In partcular 0 I. Choose J wth smallest d (t) ( 0, ). Consder the mnmum-weght path n G (t) from 0 to whch uses only forward and backward edges. The vertex on the path ust before s a bdder I. Clearly, y d (t) ( 0, ) > d (t) ( 0, ) and hence u (y) < u. There s a forward edge from to, thus p (t) [r,, m,) and also u (y) + p (y) = v,, and hence (*) u + p < v,. Snce n G (t) there s a maxmum-prce edge from to, p = p (y) < m,, whch together wth (*) contradcts stablty of (u, p, µ ). Ths proves Proposton 15. To prove Lemma 14 t remans to show that p (t+1) p. Ths amounts to show that f (u (t+1), p (t+1), µ (t+1) ) was obtaned from (u (t), p (t), µ (t) ) by updatng along an alternatng path P of whch the last edge, (, ) = ( l, l+1 ), was a reserve-prce edge and p (t+) < r,, then r, p. (23) Snce (u, p, µ ) s stable, ether u + p v, or p m,. In former case, (23) follows from that u (t+1) = v, r,, Proposton 15 and that (u, p, µ ) s stable. In latter case, (23) follows snce the presence of the reserve-prce edge from to guarantees that m, > r,. The dscusson thus far completes the proof of Theorem 4. 7. INCENTIVE COMPATIBILITY In ths secton we wll prove Theorem 5. A mechansm based on computng men-optmal stable matchng has been shown to be truth-revealng n several contexts. For the basc stable matchng problem wthout payments, a concse proof can be found n [17]. For the case of contnuous utltes, a proof was gven n [8]. Our proof for the max-value model mmcs the overall structure of ts predecessors. Frst, we show that there s no feasble matchng n whch every sngle bdder would be better off than n the bdderoptmal matchng. (Note that f an agent or set of agents were to successfully le about ther preferences, the mechansm would stll output a matchng that s feasble wth respect to the true preferences.) Ths property s known as weak Pareto optmalty of the bdder-optmal matchng. LEMMA 18 (PARETO OPTIMALITY). Let (v, m, r) be an aucton n general poston and let (u, p, µ ) be the bdder-optmal matchng. Then for any matchng (u, p, µ) that s feasble for (v, m, r), there s at least one bdder I such that u u. Second, we show that every feasble matchng s ether stable, or has a blockng bdder-slot par that nvolves a bdder who s not better off n ths matchng than n the bdder-optmal matchng. Versons of the followng lemma appear n [14, 10, 19]. The orgnal statement n a model wthout money s attrbuted to J. S. Hwang. LEMMA 19 (HWANG S LEMMA). Let (u, p, µ) be a matchng that s feasble for an aucton (v, m, r) n general poston and let (u, p, µ ) be the bdder-optmal matchng for that aucton. Let I + = { I u > u }. If I + s non-empty, then there exsts a blockng par (, ) (I I + ) J. Theorem 5 drectly follows from Lemma 19. In fact, the lemma mples the followng stronger statement. THEOREM 20. There s no way for a bdder or a coalton of bdders to manpulate ther bds n a way such that every bdder n the coalton would strctly beneft from the manpulaton. PROOF. Suppose there s a coalton I + of bdders that can beneft from submttng false bds. Let (v, m, r) be an aucton that reflects the true preferences of all bdders, and let (v, m, r) be an aucton that reflects the falsfed bds. Note that v = v and m = m except for bdders I +. Let (u, p, µ) be the bdder-optmal stable matchng for the aucton (v, m, r). Frst observe that the matchng (u, p, µ) must be feasble for the true aucton (v, m, r). Ths s because for each bdder I I +, the feasblty constrants are the same n both auctons. For bdders I +, we need to verfy that p m, whenever (, ) µ. Ths follows because the true bdder-optmal matchng (u, p, µ ) respects maxmum prces, and any outcome that respects maxmum prces s preferred over an outcome that doesn t. Snce (u, p, µ) s feasble, we can apply Lemma 19 and conclude that there s a par (, ) wth I I + that s blockng for the aucton (v, m, r). The rest of ths secton s devoted to the proofs of Lemmas 18 and 19. PROOF OF LEMMA 18. For the sake of contradcton, suppose that there s a feasble matchng (u, p, µ) such that u > u for all I. Note that every bdder must be matched n µ, snce u > u 0. For each bdder I, consder the slot = µ() matched to bdder n the matchng µ. Snce the par (, ) s not blockng for the bdder-optmal matchng (u, p, µ ), t must be that p > p. In partcular, the exstence of µ mples that there must be n slots wth postve prces n the bdder-optmal matchng µ, and that these slots are matched n µ as well. In the matchng algorthm of Secton 5, f a slot ever becomes matched to a bdder, t stays matched to some bdder throughout the algorthm. Thus before the last teraton, at most n 1 slots have postve prces. Suppose the last teraton, teraton T 1, ncreases the sze of the matchng to n, and let be the last slot to be matched. Let = µ() be the bdder matched to n the hypothetcal matchng µ. Let P be the shortest alternatng path found n Step 1 of the last teraton of the matchng algorthm. Recall that the frst vertex of the path s denoted by 0 and w (T 1) (P ) denotes ts length. If P ends wth the reserve-prce edge (, ), t must be that and are matched n both µ and µ at the same reserve prce, contradctng our assumpton that u > u. On the other hand, f P does not end wth the reserve-prce edge (, ), we show that there s a shorter alternatng path P that does nclude ths edge, whch agan leads to a contradcton. From Step (T 1) 3 of the last teraton we have u u = w (T 1) (P ) d (T 1) ( 0, ). Let s be the length of the reserve prce edge (, );
(T 1) recall from Defnton 8 that s = u + r, v,. Now consder the alternatng path P that conssts of the shortest path from 0 to followed by the reserve prce (, ) edge. We have w (T 1) (P ) w (T 1) (P (T 1) ) = u u s = v, r, u. Snce u < u v, r,, ths dfference s postve and hence P must be a shorter alternatng path than P. PROOF OF LEMMA 19. Wthout loss of generalty assume that (u, p, µ) s such that there does not exst a par (, ) µ such that u + r, = v,. If there was such a par, then we can decrease prces of some of the tems and ncrease utltes of some of the bdders such that u + r, > v,. (Ths s possble because of the general poston assumpton. See full verson of the paper.) The set I + would only grow by such operaton. Let us denote by µ(i + ), µ (I + ) the set of slots matched to bdders n I + n matchng respectvely µ, µ. We consder two cases: Case 1: µ(i + ) µ (I + ). For any I + we have u > u 0 and hence each bdder n I + s matched n µ to some slot. There exsts a slot µ(i + ), µ (I + ). Let = µ(). Snce I +, u > u. We argue that p < p : By the general poston assumpton p m,, and hence by feasblty of (u, p, µ), p [r,, m,) and u + p = v,. Hence u + p v,. Therefore p v, u > v, u = p. In partcular, s matched n µ to some, and by the choce of, I +. Thus u u. By feasblty of (u, p, µ ), p [r,, m,] and u + p = v,. By the assumpton on (u, p, µ) that we made at the begnnng of the proof, u v, r,. Now, t s not hard to see that (, ) s blockng par for µ. Ths s because p < p m,, u u = v, p v, r, u v, r,, u + p < u + p = v,. and Case 2: µ(i + ) = µ (I + ) = J +. Snce u > u for I +, by stablty of (u, p, µ ) t follows that p < p for J +. Consder a reduced aucton (v, m, r ) on the set of bdders I + and set of slots J +. We set the reserve prces to reflect the nfluence of bdders n I \I +. More specfcally, let I = { I \I + u v, r,}. For every I + and J +, we set r, = max `r,, max I mn(m,, v, u ). We also set v, = v, and m, = m, except that f m, r, we set m, = 1. It s not hard to show that f v, m, r s n general poston, then so s (v, m, r ), usng the fact that each utlty u was at some pont set to be equal to the length of some alternatng walk n the aucton graph. Now consder the matchngs µ and µ restrcted to the sets I +, J +. If the restrcted µ s not feasble for (v, m, r ), t must be because p < r, for some poston = µ(). Ths can only happen f r, > r, and hence r, = max(m,, v, u ) for some bdder I \ I +. On the other hand, t s easy to check that the restrcted matchng µ s feasble, stable and bdder-optmal for the aucton (v, m, r ). If the restrcted µ s feasble for ths aucton, by Lemma 18, there s a bdder I such that u u. Ths however contradcts the defnton of the set I +. 8. REFERENCES [1] Gagan Aggarwal. Prvacy Protecton and Advertsng n a Networked World. PhD thess, Stanford Unversty, 2005. [2] Gagan Aggarwal, Ashsh Goel, and Raeev Motwan. Truthful auctons for prcng search keywords. In ACM Conf on Electronc commerce, pages 1 7, 2006. [3] Gagan Aggarwal, S. Muthukrshnan, and Jon Feldman. Bddng to the top: VCG and equlbra of poston-based auctons. In WAOA, 2006. [4] Ahmet Alkan. Exstence and computaton of matchng equlbra. European Journal of Poltcal Economy, 5(2-3):285 296, 1989. [5] Ahmet Alkan and Davd Gale. The core of the matchng game. Games and Economc Behavor, 2(3):203 212, 1990. [6] Sushl Bkhchandan and Joseph M. Ostroy. From the assgnment model to combnatoral auctons. In Combnatoral Auctons. MIT Press, 2006. [7] Edward H. Clarke. Multpart prcng of publc goods. Publc Choce, 11(1):17 33, 1971. [8] G. Demange and D. Gale. The strategy structure of two-sded matchng markets. Econometrca, 53(4):873 888, 1985. [9] Gabrelle Demange, Davd Gale, and Marlda Sotomayor. Mult-tem auctons. Journal of Poltcal Economy, 94(4):863 872, 1986. [10] Gabrelle Demange, Davd Gale, and Marlda Sotomayor. A further note on the stable matchng problem. Dscrete Appled Mathematcs, 16:217 222, 1987. [11] Benamn Edelman, Mchael Ostrovsky, and Mchael Schwarz. Internet advertsng and the generalzed second prce aucton: Sellng bllons of dollars worth of keywords. Amercan Economc Revew, 97(1):242 259, March 2007. [12] Eyal Even-Dar, Jon Feldman, Yshay Mansour, and S. Muthukrshnan. On the effect of mnmum prces on poston auctons. Unpublshed manuscrpt, 2008. [13] Davd Gale and Lloyd S. Shapley. College admssons and the stablty of marrage. Am Math Monthly, 69(1):9 15, 1962. [14] Davd Gale and Marlda Sotomayor. Some remarks on the stable marrage problem. Dscrete Appled Mathematcs, 11:223 232, 1985. [15] Theodore Groves. Incentves n teams. Econometrca, 41(4):617 631, 1973. [16] Herman B. Leonard. Elctaton of honest preferences for the assgnment of ndvduals to postons. Journal of Poltcal Economy, 91:461 479, 1983. [17] Noam Nsan, Tm Roughgarden, Eva Tardos, and Vay V. Vazran, edtors. Algorthmc Game Theory. Cambrdge Unversty Press, Septermber 2007. [18] M. Qunz. Core and compettve equlbra wth ndvsbltes. Internatonal Journal of Game Theory, 13(1):41 60, 1984. [19] Alvn E. Roth and Marlda A. Olvera Sotomayor. Two-sded matchng: A study n game-theoretc modelng and analyss. Cambrdge Unversty Press, 1990. [20] Lloyd S. Shapley and Martn Shubk. The assgnment game : The core. Intl J. of Game Theory, 1(1):111 130, 1971. [21] Hal R. Varan. Poston auctons. Internatonal Journal of Industral Organzaton, 2006. [22] Wllam Vckrey. Counterspeculaton, auctons, and compettve sealed tenders. J.of Fnance, 16(1):8 37, 1961.