The Core and Incentive Compatibility of Ascending Proxy Package Auctions

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1 The Core and Incentve Compatblty of Ascendng Proxy Package Auctons Hroyuk Adach Ths Verson: January 3, 2011 Abstract Ausubel and Mlgrom (2002, Secton 8) have ntroduced a generalzed ascendng proxy aucton to analyze package auctons and exchanges wth general contracts under nontransferable utlty. They have shown that ther algorthm fnds an NTU-core allocaton wth respect to reported preferences. Instead of consderng general contracts, we ncorporate nearly contnuous transfers to the problem and ntroduce a mechansm whch corresponds to ther algorthm n ths envronment. Ths mechansm allows non-quaslnear envronments, s effcent and weakly strategy-proof for buyers, and gves the auctoneer a weakly hgher payoff than her lowest core payoff even wthout the substtutes condton. JEL Classfcaton: D44 Keywords: Gale and Shapley s marrage problem; Combnatoral Auctons; Package Exchanges; Complementarty; Substtutablty Emal: hkky.adach@gmal.com. Any errors or omssons are unntentonal and the author s responsblty. Ths work has been fnancally supported by Grant-n Ad for Young Scentsts and by the Shkshma Foundaton. 1

2 In combnatoral auctons or package auctons, a sngle auctoneer has multple ndvsble tems and each buyer can obtan a package of tems under some terms of trade. Package auctons can amelorate undesrable stuatons where buyers purchase some tems but lose other ndspensable tems, especally when tems to be sold are not necessarly substtutes. To analyze package auctons, Ausubel and Mlgrom (2002, Secton 8) have ntroduced a generalzed ascendng proxy aucton for nontransferable utlty (NTU) settngs or approxmatons of transferable utlty (TU) settngs. They have stated that ther algorthm fnds an NTU-core allocaton (wth respect to reported preferences) n package auctons even when goods are not substtutes or there are severe feasblty constrants. More recently, Hatfeld and Mlgrom (2005, Secton V) have reformulated a generalzed ascendng proxy aucton as the cumulatve offer process, whch s closely related to Gale and Shapley s deferred acceptance algorthm. In a sense, they have extended the Gale- Shapley marrage problems to package auctons. Ther model subsumes package exchanges, more general than package auctons, where each agent can sell and/or buys tems through a central auctoneer. In ths paper, we nvestgate the property of the core and domnant-strategy ncentve compatblty of a generalzed ascendng proxy aucton or the cumulatve offer process wth nearly contnuous transfers. In Sectons 1 and 2, we descrbe a generalzed ascendng proxy aucton wth general contracts under NTU ntroduced by Ausubel and Mlgrom. In Secton 3, nstead of consderng general contracts, we ncorporate nearly contnuous transfers to the problem and show that wth a te-breakng rule there s a sngle core allocaton best for all buyers among all core allocatons. Then we ntroduce a drect mechansm, called the COP mechansm wth transfers, whch corresponds to Ausubel and Mlgrom s algorthm n ths envronment. Our man result (Theorem 10) shows that under mld condtons ths mechansm mplements a Pareto effcent allocaton n weakly domnant strateges. In contrast to the VCG mechansm (appled to package auctons/exchanges), the COP mechansm wth transfers allows non-quaslnear preferences and gves the auctoneer a weakly hgher payoff than the lowest core payoff even wth complementary goods. The man restrcton s that the auctoneer s preference relaton over allocatons s fxed. Below we brefly overvew related studes. When utlty functons are quaslnear, the VCG mechansm developed by Vckrey (1961), Clarke(1971) and Croves (1973) can be appled to package auctons/exchanges wthout the substtutes condton. If the VCG mechansm s used to package auctons, tellng true preferences s a domnant strategy for each buyer and the outcome s effcent. But t has been known that the 2

3 VCG mechansm has some drawbacks when tems may not be substtutes. For nstance, ts payoff vector may le outsde the core when goods are not substtutes, as shown by Bkhchandan and Ostroy (2002, Theorem 6.1) and Ausubel and Mlgrom (2002, Theorem 12). When such cases happen, buyers payoffs are too hgh and the seller s payoff s too low for the payoff vector to be wthn the TU-core. In addton, the VCG mechansm has the so-called monotoncty problem and s vulnerable to shll bddng. Yokoo et al. (2004) have showed that under TU shll bdng s a general problem n all effcent combnatoral aucton mechansms, ncludng the VCG mechansm. In ths paper, we exclude the possblty that buyers use shll bddng. See Cramton et al. (2006) for recent surveys on package auctons by researchers n computer scence, economcs and operatons research. The followng papers have also examned package auctons under transferable utlty wthout the substtutes condton. Bernhem and Whnston (1986) and Bkhchandan and Ostroy(2002) have studed package auctons and exchanges. Bernhem and Whnston have proposed frst prce auctons for package auctons and exchanges and studed Nash equlbra n truthful strateges. Bkhchandan and Ostroy, usng nteger and lnear programmng, have analyzed package exchanges between many sellers and many buyers where package auctons are treated as a specal case. See also Mlgrom (2007). Wurman and Wellman (2000) and Parkes and Ungar (2000) both have ntroduced ascendng package auctons based on the classcal work on the assgnment problem by Kuhn (1955), Koopmans and Beckmann (1957), and Shapley and Shubk (1972). Wurman and Wellman s Ascendng k-bundle Aucton (AkBA) acheves effcent allocatons wth anonymous nonlnear prces. Parkes and Ungar s Bundle(2) and Bundle(3) attan effcent allocatons wth anonymous nonlnear prces and non-anonymous nonlnear prces, respectvely. Parkes and Ungar s algorthms are an extenson of Bertsekas (1981) for the assgnment problem, whch s related to the Hungaran method by Kuhn (1955). Note that Crawford and Knoer (1981), who have ncorporated transfers nto the marrage model, have mentoned a connecton between the Hungaran method and ther generalzaton of the Gale-Shapley algorthm. See Demange et al. (1986) for a follow-up study. Ausubel and Mlgrom (2002, Sectons 3-7) have ntroduced an ascendng proxy aucton under TU, ntmately lnked to Bundle(3) by Parkes and Ungar, and have analyzed Nash equlbra n sem-sncere strateges. In ther mechansm the payoff vectors n Nash equlbra supported by sem-sncere strateges are the bdder-pareto-optmal ponts n the TU-core. Under the assumpton that goods are gross substtutes, Kelso and Crawford (1982) have explored many-to-one matchng problems and have showed the exstence of compettve equlbrum 3

4 and mnmum-prce equlbrum. Ther model can be consdered as a mult-object smultaneous ascendng aucton where goods are substtutes. 1 Gul and Stacchett (1999) have consdered a smlar problem (under quaslnear preferences) by ntroducng the sngle mprovement property, closely related to the gross-substtutes condton, and show that the the gross substtutes condton s necessary for the exstence of compettve equlbrum and that the set of compettve equlbra s a lattce. In a sequel, Gul and Stacchett (2000) have analyzed a mult-object Englsh aucton and show that under some condton t converges to mnmum-prce compettve equlbrum and ts outcome s dentcal to that of the VCG mechansm. Mlgrom (2000) has also researched a model closely related to Kelso and Crawford (1982). 1 Package auctons and exchanges In package auctons, there s a sngle auctoneer, called h, who has several tems to sell. There are n buyers. Denote by I the set of all buyers and by a typcal buyer. Each buyer can purchase a package of tems under some terms of condton. We call such terms of trade a contract. A contract can nclude any terms of trade between auctoneer h and the correspondng buyer. A typcal contract between h and s denoted x. Because a contract can specfy any terms of trade, each agent can buy and/or sell tems through auctoneer h. Thus, the Ausubel and Mlgrom (2002, Secton 8) model subsumes package exchanges wth a central auctoneer under NTU. Keepng ths n mnd, we contnue to call each agent a buyer and h an auctoneer for clarty. Let X denote the set of all contracts assocated wth buyer. For each buyer, the set X contans only a fnte number of elements and t ncludes an opton, denoted φ, for buyer not to take part n any trade. Buyer can always choose null contract φ f he prefers. Let X = I X be the entre contract set among the auctoneer and buyers. Each buyer has weak preferences R over hs set of contracts X. We denote the correspondng strct preferences (the asymmetrc part of R ) by P. Alternatvely we can express s preferences by hs choce functon C ( ) : 2 X X. That s, for each subset X of X, C (X ) s s favorte contracts n X. Because X contans only fnte contracts, set X s parttoned n the followng 1 The Kelso-Crawford salary-adjustment process works even when preferences are not quaslnear. Ther defnton of the gross substtutes condton s order-theoretc. Ausubel and Mlgrom (2002, Thoerem 10) provde a convenent characterzaton of the substtutes condton under quaslnear preferences. Ther characterzaton usng submodularty s cardnal. 4

5 way. Gven buyer s preferences, let [x (1) ] = C (X ) be hs best contract set. Any contract n set [x (1) ] s utlty-equvalent to each other and x (1) s a typcal element of [x (1) ]. Smlarly, let [x (2) ] = C (X \[x (1) ]) be hs second best contract set,..., [x (k) ] = C (X \ m=1 k 1 [x(m) ]) hs k-th best contract set, and so forth. Gven R, we wrte R : [x (1) ],[x (2) ],,[φ ],,[z ]. As buyer s always free to opt out, under R null contract φ s a least acceptable contract n X. Off course, ths preference rankng depends on R but we omt ts dependency untl we dscuss ncentve compatblty of ascendng proxy auctons n Secton 3. Each buyer eventually makes one and only one contract x (ncludng φ ) wth the auctoneer. Denote by x = {x } I a profle of contracts where each buyer gets x. Not every profle x of contracts s feasble. For nstance, n a package aucton wth several tems to bo sold, one tem cannot be sold to more than one buyer. In problems lke menu auctons studed by Bernhem and Whnston (1986), there can be very complex feasblty constrants. A profle x of contracts s called an allocaton f t s feasble. We ncorporate possbly complcated feasblty constrants nto the model by functon ρ( ). For each set of contracts X X, let ρ(x) denote the set of all feasble allocatons n X. The auctoneer can always choose φ for any I, that s, she can exclude any buyer from any trade. Denote by φ I = {φ } I null allocaton where every buyer obtans nothng φ, whch s always avalable to her. We treat φ I and each φ as an empty set. We place Assumpton 1 Auctoneer h has strct preferences, denoted by P h, over ρ(x ). Null allocaton φ I s a least acceptable allocaton for her. We fx the preferences of the auctoneer throughout the paper. Followng Hatfeld and Mlgrom (2005, Secton I.A), h chosen set C h (X) s h s favorte allocaton n X for each X X. We regard that any X X contans null allocaton φ I as an empty set, snce φ I s always avalable to her and a least acceptable allocaton for her. Because h has strct preferences, C h (X) pcks a sngle allocaton for each X X. In addton, h s rejected set s defned by r h (X) X\C h (X) for each X X. We do not mpose the assumpton that functon r h s ncreasng n the usual set ncluson order. See Hatfeld and Mlgrom who characterzed the substtutes condton by the monotoncty of r h. Now defne a core allocaton for the problem. An allocaton x = {x } I s called ndvdually ratonal f xr h φ I and ( I) x R φ, where R h s weak preferences assocated wth P h. Consder 5

6 two allocatons x = {x } I and y = {y } I. We wrte xr y f x R y. Defnton 2 An ndvdually ratonal allocaton x s a core allocaton f there does not exst another allocaton y such that yp h x and yr j x j { I : y φ }. If such an allocaton y exsts, we say that allocaton y blocks allocaton x. Let R I = (R ) I denote a profle of buyers weak preference relatons. The set of all core allocatons s called the core and denoted by Core(R I,P h ). 2 A profle R I of buyers weak preferences s a partal order on the set of all allocatons when we defne that, for any x,y ρ(x ), xr I y f ( I) xr y. Therefore, Core(R I,P h ),R I s a partally ordered set. Defnton 3 A core allocaton x s called a buyer-maxmal core allocaton f there exsts no other core allocaton y wth ( I) y R x and ( I) y P x. That s, a buyer-maxmal core allocaton s a maxmal element n Core(R I,P h ) wth respect to partal order R I. 2 Ausubel and Mlgrom s cumulatve offer process We descrbe Ausubel and Mlgrom s generalzed ascendng proxy aucton or the Cumulatve Offer Process (the COP) n Hatfeld and Mlgrom (2005, Secton V). Suppose a preference profle of buyers looks as follows: R 1 : [x (1) 1 ],[x(2) 1 ],,[φ 1],.. R : [x (1) ],[x (2) ],,[φ ],.. R n : [x (1) n ],[x (2) n ],,[φ n ],. 2 We have that the strct core the core by Defnton 2 the weak core. Note that under transferable utlty the strct core concdes wth the weak core and so wth the core by Defnton 2. In ths paper we wll use the above defnton. 6

7 The COP proceeds n the followng steps. Step 1: Set X (0) := φ for each I and X h (0) := I X (0). Step 2: Each buyer offers hs best contract set [x (1) ] =: X (1) and the set of contracts auctoneer h has receved from buyers s X h (1) := X h (0) ( I X (1)). The auctoneer selects her favorte allocaton C h (X h (1)) =: {x } I and rejects the other set of contracts, X h (1)\C h (X h (1)) = r h (X h (1)). (The auctoneer can always choose null allocaton φ I f she prefers.) Buyer s called a provsonal wnner f x φ. He s called a provsonal loser f x = φ. Step t: Each provsonal wnner keeps offerng the same set X (t 1) of contracts as n the prevous step. Each provsonal loser offers the set X (t 1) of contracts he has made n the prevous step (whch were rejected by h) plus, f [φ ] X (t 1), hs next preferred contract set C (X \X (t 1)). That s, hs offer s X (t) := X (t 1) C (X \X (t 1)) f [φ ] X (t 1) and X (t) := X (t 1) otherwse. The set of contracts h has receved s X h (t) := I X (t). The auctoneer selects her favorte allocaton C h (X h (t)) and rejects the other set of contracts X h (t)\c h (X h (t)) = r h (X h (t)). The process contnues untl t termnates; X h (t + 1) = X h (t). If x φ, buyer s a wnner. If x = φ, he s a loser. Ausubel and Mlgrom (2002, Theorem 14), Mlgrom (2004, Theorem 8.12) and Hatfeld and Mlgrom (2005, Theorem 16) showed that the allocaton produced by the COP algorthm s a core allocaton. It s apparently a buyer-maxmal core allocaton from the constructon of the COP algorthm. Theorem 4 (Ausbel and Mlgrom) The COP algorthm produces a buyer-maxmal core allocaton n Core(R I,P h ),R I. Wth general contracts, there are multple buyer-maxmal core allocatons. 3 Contnuous transfers and strategy-proofness In ths paper, we ncorporate contnuous transfers nto the problem. Instead of examnng package auctons/exchanges wth contnuous transfers drectly, we consder the problems wth dscrete transfers where the mnmum unt of measurement s small enough. Usng some fxed te-breakng rule, we show that every core allocaton nvolves the same assgnment of packages. Ths mples that the set of wnnng buyers s dentcal among all core allocatons. Moreover, wth a te-breakng rule there s a sngle core allocaton best for all buyer. We then devse the COP mechansm wth 7

8 transfers. Our man result (Theorem 10) s that the COP mechansm wth transfers mplements a Pareto effcent allocaton n the orgnal envronment n weakly domnant strateges and gves the auctoneer a weakly hgher payoff than her smallest core payoff. Suppose each contract conssts of a par, x = (a,t ), where t T R s a transfer buyer pays to auctoneer h and a, called a package, s part of the contract except transfer t. A package a can contan pecunary elements not captured by transfer t. Let A be a fnte set of all packages for buyer. Null package A s an opton that buyer buys/sells nothng or excluded from any trade. Let φ = (,0) denote a contract where buyer buys/sells nothng and pay 0. The set of all package profle s A = A. A profle of packages a = (a ) I s called a package assgnment f t s feasble. We denote by I = ( ) I null package assgnment, whch s always feasble. For each set of package profles A A, denote by ρ(a) the set of all feasble package assgnments n A. Any package profle a n A \ρ(a ) s nfeasble. An allocaton x = (a,t) s a par of a package assgnment a and a profle t = (t ) I of transfers. Null allocaton s denoted by φ I = {φ } I = ( I,(0) I ). The auctoneer can always choose null allocaton φ I = ( I,(0) I ), whch s a least acceptable allocaton for her. For each buyer, consder a set of von Neumann-Morgenstern utlty functons {u (a,t ;θ ) : θ Θ } ndexed by θ Θ, where Θ s a compact subset of a metrc space. We sometmes dentfy θ as a bnary relaton representng s weak preferences by regardng that, for any two (a,t ) and (b,s ) n A R, (a,t )θ (b,s ) f and only f u (a,t ;θ ) u (b,s ;θ ). The set Θ s buyer s type space or hs preference relatons. Buyer s typcal type s denoted by θ. Let Θ = I Θ be the set of all type profles and θ = (θ ) Θ Θ a typcal type profle. We assume that, for any buyer of any type θ, null contract φ = (,0) s a least acceptable outcome, n whch case hs utlty s u (,0;θ ) = 0. We put the assumptons on each buyer s preference relatons n terms of hs utlty functons. Assumpton 5 (a) A set T of contnuous transfers s a closed nterval n R whch ncludes 0. The type space Θ of each buyer s a compact subset of a metrc space. (b) For each θ Θ and each a A, u (a,t ;θ ) s strctly decreasng n t T and bounded from above. (c) For any θ Θ and a A, u (a, n 1 maxt ;θ ) < 0. (d) For each a A, u (a,t ;θ ) s (unformly) contnuous n (t,θ ) on T Θ. 3 The auctoneer s utlty functon s fxed and denoted by u h (a,t h ) where a A and t h = n =1 t. 3 On a compact subset of a metrc space, a functon s contnuous f and only of t s unformly contnuous. 8

9 We denote R h the auctoneer s preference relatons on all allocatons whch s consstent wth u h ( ). We normalze that u h ( I, 0) = 0. For any nfeasble package profle a A \ρ(a ), u h (a,t h ) s assumed to take a negatve value. Her utlty functon s assumed to satsfy Assumpton 6 (a) For each a ρ(a ), u h (a,t h ) s contnuous and strctly ncreasng n t h T and bounded from above. (b) For any a ρ(a ), u h (a,mnt )) < 0. Now we want to defne a core allocaton for a fxed preference profle (θ,r h ). If we wrte the asymmetrc part of auctoneer s weak preferences R h as P h and take A T n as the set of all contracts, then a core allocaton s defned n the same way as Defnton 2. We denote by Core(θ,R h ;A T n ) the set of all core allocatons n A T n at a preference profle (θ,r h ). Because there s only one auctoneer, there s a smallest auctoneer s utlty n the core, whch we wrte as u h = mn{u h (a, I t ) : (a,t) Core(θ,R h ;A T n )}. (1) Core allocatons n whch the auctoneer attans her lowest core payoff u h are buyer-maxmal core allocatons. Fx a buyers preference profle θ Θ and consder a maxmzaton problem: w(θ) =max t T n max u h(a, n =1 t ) (2) a ρ(a ) s.t. u (a,t ;θ ) 0 for all I, u h (a, n =1 t ) 0. For each θ, there s at least one soluton (a(θ),t(θ)) to the maxmzaton problem (2) because the constrant set s compact and the object functon s contnuous by Assumptons 5 and 6. Let us defne a subset of package assgnments, A(θ) = {a ρ(a ) : (a,t) solves (2) for some t T n }. Any soluton (a,t) to (2) s an auctoneer-best core allocaton under (θ,p h ). On the other hand, f (b,s) s a core allocaton, by ncreasng the buyers payments (s ) n some way we can obtan an auctoneer-best core allocaton whch nvolves the same package assgnment b. Therefore, we call a package assgnment a A(θ) a core package assgnment n A T n under (θ,p h ). 9

10 When we thnk of (2) as a famly of maxmzaton problems ndexed by θ Θ, by Berge s theorem of the maxmum, w(θ) s contnuous n θ. For each θ and each t, let f (t,θ) denote the maxmzed value of the nner maxmzaton problem n (2). Then f (t, θ) s (unformly) contnuous n (t,θ) on T n Θ. Ths mples n partcular that ( ε > 0)( δ > 0)( θ Θ)( t,t T n wth d(t,t ) < δ), f (t,θ) f (t,θ) < ε, (3) where d( ) s the Eucldean dstance. Now we want to dscretze the set T of contnuous transfers. For any δ > 0, defne a set T δ R of the form T δ = {m δ : m s are ntegers.} (4) wth T δ beng a largest such subset of T. Ths s a set of dscrete transfers for agents where δ s the mnmum unt of measurement. By takng X = A Tδ n as a fnte set of all contracts, we want to apply the method n Sectons 1 to ths dscretzed envronment. For each θ Θ, let us defne the followng preference relaton R θ over A T δ whch s a restrcton of θ over A T to A T δ. For any (a,t ) and (b,s ) n A T δ, (a,t )R θ (b,s ) f u (a,t ;θ ) u (b,s ;θ ) (5) Snce A T δ s fnte, so s the set R {R θ over A T δ. The asymmetrc part of R θ : θ Θ } of buyer s all (weak) preference relatons s denoted by P θ. Let R θ I = (Rθ ) I. Consder the followng maxmzaton problem where T n s replaced wth T n δ w δ (θ) =max t T n δ n (2): max u h(a, n =1 t ) (6) a ρ(a ) s.t. u (a,t ;θ ) 0 for all I, u h (a, n =1 t ) 0. Because the nner maxmzaton problem s the same as n (2), w δ (θ) = max t T n δ f (t,θ). As before, w δ (θ) s contnuous n θ. Snce Tδ n T n, w(θ) w δ (θ). We want to show that, for a suffcently small δ > 0, the two values are equal at any θ. Lemma 7 ( δ > 0)( θ Θ), w(θ) w δ (θ) = 0. 10

11 Proof. Suppose not. Then ( δ > 0)( θ Θ) w(θ) w δ (θ) > 0. Because both w(θ) and w δ (θ) are contnuous n θ and Θ s compact, we have max θ Θ w(θ) w δ (θ) ε > 0. But ths contradcts the unform contnuty of f (t,θ) on T n Θ or (3) n partcular. From now on, we choose a δ > 0 as n the above Lemma and take a set of dscrete but nearly contnuous transfers T δ n the form of (4). Defne A δ (θ) = {a ρ(a ) : (a,t) solves (6) for some t T n δ }. Any soluton (a,t) to (6) s an auctoneer-best core allocaton n A T n δ under (Rθ I,P h). On the other hand, f (b,s) s a core allocaton, then by ncreasng the buyers payments (s ) n some way we can obtan an auctoneer-best core allocaton whch nvolves the same package assgnment b. So, we call a package assgnment a A δ (θ) a core package assgnment n A T n δ under (Rθ I,P h). The above Lemma mples that f a A δ (θ) then a A(θ). That s, A δ (θ) A(θ) for each θ. In general, A δ (θ) s not sngle-valued,.e., there may be multple core package assgnments. To select a sngle element from A δ (θ), we use a pre-determned lnear order > a over ρ(a ). We also use a lexcographc order > l over T n δ. For any two t = {t } I and s = {s } I n T n δ, t > l s f t = s for the frst k buyers and t k+1 > s k+1. Ths lnear order > l s used to select a sngle element from the set of buyer-maxmal core allocatons. Defne h s strct preferences P br h allocatons (a,t) and (b,s) n ρ(a ) T n δ, (a,t)p br h (b,s) f [u h(a, t ) > u h (b, s )] or f [u h (a, t ) = u h (b, s ) and a > a b] or f [u h (a, t ) = u h (b, s ), a = b and t > l s]. as follows. For any two When the auctoneer s strct preference relaton Ph br s fxed, for each R θ I R I there s a unque core package assgnment, denoted a (θ), n A δ (θ) where a core allocaton n A Tδ n s defned wth respect to (R θ I,Pbr h ). Let I(a) = { I : a } be the set of buyers who take part n the package assgnment a. Lemma 8 (a) All core allocatons n A Tδ n under (Rθ I,Pbr h ) nvolve the same core package assgnment a (θ) A δ (θ). (b) In partcular, the set of wnnng buyers I(a (θ)) = { I : a (θ) } s dentcal among all core allocatons n A T n δ under (Rθ I,Pbr h ). 11

12 Let Core(R θ I,Pbr h ;A T δ n ) denote the set of all core allocatons n A T n δ under (Rθ I,Pbr h ). Note that the set of an auctoneer-worst core allocatons s equal to the set of buyer-maxmal core allocatons. When the auctoneer has strct preferences Ph br, the set of auctoneer-worst core allocatons conssts of a sngle core allocaton, and so does the set of buyer-maxmal core allocatons. Snce as stated n Theorem 4 the COP algorthm produces a buyer-maxmal core allocaton wth general contracts, t produces a buyer-best core allocaton n ths envronment. Lemma 9 Core(R θ I,Pbr h ;A T δ n ),Rθ I has a greatest element, denoted by Core(R θ I,P br ). It s the allocaton produced by the COP algorthm n ths envronment. T n δ h ;A A greatest element Core(R θ I,Pbr h ;A T δ n) n Core(R θ I,Pbr h ;A T δ n ),Rθ I s a core allocaton best for all buyers among all core allocatons. It s called a buyer-best core allocaton n Core(R θ I,Pbr h ;A T n δ ). Now we can ntroduce the followng drect mechansm for a package aucton/exchange wth contnuous transfers, whch we call the COP mechansm wth transfers. The mechansm proceeds as follows. (1) Auctoneer pcks a δ whch satsfes Lemma 7. (2) Each buyer reports a type from Θ. (3) Auctoneer calculates and mplements an allocaton Core(R θ I,Pbr h ;A T n δ ) ρ(a ) T n. The man result of ths paper s the followng theorem. We wrte P θ for the asymmetrc part of buyer s weak preference relaton R θ and Core(R θ I,Pbr h ) for Core(Rθ I,Pbr h ;A T n δ ). Theorem 10 In the COP mechansm wth transfers, (a) tellng a true type s a weakly domnant strategy for every buyer ; (b) the correspondng outcome Core(R θ I,Pbr h ;A T δ n ) s a Pareto effcent allocaton n ρ(a ) T n wth respect to (θ,r h ); (c) the auctoneer s payoff s weakly hgher than her smallest payoff n Core(θ,R h ;A T n ). Proof. Part (a): Assume that the COP mechansm wth transfers s used and the auctoneer s preference relaton P br h over allocatons s fxed. Let θ be any profle of buyers types and pck any buyer. Impose Assumptons 5 and 6 so that Lemma 8 holds. Defne x = Core(R θ I,Pbr h ) and let x be s outcome. Suppose the theorem s wrong. Then there s s report τ θ Θ whch s a best reply to θ and obtans some contract x P θ x. If x = φ, then proof s done. So assume x φ. Denote x = Core(R τ,rθ,pbr h ), where Rτ and R θ are defned as n (5). Consder the followng preference relaton over A T δ. R : { x },{φ },. 12

13 Because R lsts x on top, we stll have x = Core( R,R θ,pbr h ). By Lemma 8 (b) and x = φ, any allocaton x whch nvolves φ s blocked by some allocaton y n A T n δ under ( R,R θ,pbr h ). Consder the followng preference relaton over A T δ. R θ ( x ) : [x (1) ],[x (2) ],,[ x ],{φ },, where R θ ( x ) s dentcal up to R θ. Denote z = Core(R θ ( x ),R θ,pbr h ). Under (Rθ ( x ),R θ,pbr h ), any allocaton x whch nvolves φ s blocked by the same allocaton y mentoned n the last paragraph. Therefore, z φ and so z R θ x. Because under (R θ ( x ),R θ,pbr h ) the set of contracts lsted below x can not be part of a blockng allocaton aganst z, we have z = Core(R θ,r θ,pbr h ). Recall that by defnton z = x. So z R θ x P θ z, contradctng that R θ s a ratonal preference relaton over A T δ. Ths completes the proof. Part (b): Ths holds because the buyers payments sum up to what the auctoneer receves and the resultng package assgnment s n A δ (θ) A(θ). Part (c): Suppose (a,t ) = Core(θ,Ph br ;A T n δ ) and let uδ h = u h(a, t ) be the auctoneer s payoff. The auctoneer s smallest core payoff wth contnuous transfers s u h mn{u h (a, t ) : (a,t) Core(θ,R h ;A T n )} mn{u h (a, s ) : (a,s) Core(θ,R h ;A T n )}. (7) The nequalty holds because a A δ (θ) A(θ) and there s some s such that (a,s) Core(θ,R h ;A T n ). Let (a,ŝ) attan the mnmum n (7). Part (c) clams that u δ h u h. If not, then I(a )t < I(a )ŝ. We can take γ = I(a )ŝ I(a )t > 0 and λ = ŝ γ/n for each I(a ). Then allocaton (a,λ) blocks (a,ŝ) n A T n under (θ,r h ). Ths contradcts that (a,ŝ) Core(θ,R h ;A T n ). The COP mechansm wth transfers s a drect mechansm where each buyer s message space s Θ and the outcome functon s Core(R θ I,Pbr h ;A T δ n ). Instead, we can thnk of an ndrect mechansm whch corresponds to t. Choose a δ as n Lemma 7. Let R be the set of all preference relatons over A T δ. The COP mechansm wth (dscrete) transfers s defned as follows. Each buyer submts a report from R and the auctoneer h mplements Core(R θ I,Pbr h ;A T n δ ). 13

14 Accordng to Theorem 10 (a), n the COP mechansm wth (dscrete) transfers t s a weakly domnant strategy for each buyer to report R θ R whch s a restrcton of hs true preference relaton θ over A T to A T δ as n (5). And Statements (b) and (c) of Theorem 10 also hold. The VCG mechansm (appled to package auctons/exchanges) and the COP mechansm wth transfers dffer n the followng way. The VCG mechansm s defned when preferences are quaslnear. It s effcent and strategy-proof and s the only mechansm wth that property f the profles of utlty functons are rch enough (e.g. nclude all contnuous functons). When goods are not substtutes, the auctoneer s revenues may be strctly lower than the smallest revenues n the core. The COP mechansm wth transfers accommodates envronments where preferences may not be quaslnear but the auctoneer s preference relaton over allocatons s fxed. It s effcent and weakly strategy-proof for buyers. The auctoneer s payoff s weakly hgher than her lowest core payoff even when goods are not substtutes. 4 Concluson Ths paper studed the property of the core and domnant-strategy ncentve compatblty of ascendng proxy auctons wthout the substtutes condton under NTU, proposed by Ausubel and Mlgrom (2002, Secton 8). In ths paper, we ntroduced (nearly) contnuous transfers nto the problem. Wth nearly contnuous transfers and a te-breakng rule, there s a buyer-best core allocaton. We then devsed the COP mechansm wth transfers whch corresponds to Ausubel and Mlgrom s algorthm n ths envronment. Ths mechansm mplements a Pareto effcent allocaton n weakly domnant strateges. Compared to the VCG mechansm (appled to package auctons/exchanges), the COP mechansm wth transfers accommodates non-quaslnear preferences and the auctoneer s payoff s weakly hgher than her lowest core payoff even when there are complementary goods. References [1] Ausubel, Lawrence M. and Paul R. Mlgrom Ascendng Auctons wth Package Bddng, Fronters of Theoretcal Economcs 1(1):

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