Connected Price Dynamics with Revealed Preferences and Auctioneer s Discretion in VCG Combinatorial Auction

Transcription

1 CIRJE-F-960 Conneced Prce Dynamcs wh Revealed Preferences and Auconeer s Dscreon n VCG Combnaoral Aucon Hosh Masushma The Unversy of Tokyo February 2015 CIRJE Dscusson Papers can be downloaded whou charge from: hp:// Dscusson Papers are a seres of manuscrps n her draf form. They are no nended for crculaon or dsrbuon excep as ndcaed by he auhor. For ha reason Dscusson Papers may no be reproduced or dsrbued whou he wren consen of he auhor.

2 1 Conneced Prce Dynamcs wh Revealed Preferences and Auconeer s Dscreon n VCG Combnaoral Aucon * Hosh Masushma ** Deparmen of Economcs Unversy of Tokyo November Ths Verson: February Absrac We nvesgae a general class of dynamcal open-bd combnaoral aucon proocols ermed prce-demand procedures where he auconeer asks buyer-dependen prce vecors and buyers reveal demand ses. Such revelaons are easer o pracce han he revelaons of enre valuaons a once. Wh revealed preference acvy rule and connecedness we characerze he class of all procedures ha acheve he VCG oucome correcly. Snce buyers can deec wheher he auconeer succeeds o acheve he VCG oucome jus by observng he hsory hey can leave he selecon of procedure he auconeer s dscreon. The auconeer can save rrelevan nformaon leakage by selecng a sgnal-conngen shorcu. Keywords: Combnaoral Aucons Prce-Demand Procedures VCG Mechansms Revealed Preference Acvy Rule Connecedness Represenave Valuaon Funcons. JEL Classfcaon Numbers: D44 D47 D61 D82 D86 * Ths paper s a subsanal revson of my earler work Masushma 2011)). I repors he fndngs of a sudy ha was suppored by a gran-n-ad for scenfc research KAKENHI ) from he Japan Socey for he Promoon of Scence JSPS) and he Mnsry of Educaon Culure Spors Scence and Technology MEXT) of he Japanese governmen. I am graeful o he co-edor and an anonymous referee for her useful commens. All errors are mne. ** Deparmen of Economcs Unversy of Tokyo Hongo Bunkyo-ku Tokyo Japan. E-mal: hosh[a]e.u-okyo.ac.jp

3 2 1. Inroducon Ths paper nvesgaes a general class of dynamcal open-bd combnaoral aucon proocols ermed prce-demand procedures for allocang heerogeneous commodes n he effcen manner. In he connuous me horzon he auconeer seller) connues o ask a prce vecor for all packages of commodes o each buyer parcpans) and hs buyer reveals a se of packages as hs or her) demand se. We perm he auconeer o ask dfferen prce vecors across buyers. The auconeer evenually sop askng prce vecors whn a me lm and based on he resulan hsory of asked prce vecors and revealed demand ses he auconeer deermnes he allocaon of commodes and he sde paymens from he buyers o he auconeer. Hence we defne a prce-demand procedure as a combnaon of a prcng rule and an endng rule ha are hsory-conngen. The purpose of hs paper s o clarfy he possbly ha a prce-demand procedure acheves he VCG Vckery-Clarke-Groves 1 ) oucome allocaon and sde paymens).e. acheve effcency under he consrans of sraegy-proofness where each buyer s wllng o reveal all he bes-response packages assocaed wh hs rue valuaon funcon a all mes as a domnan sraegy. We make he followng wo assumpons on prce-demand procedures.e. he revealed preference acvy rule and he connecedness. The revealed preference acvy rule mples ha each buyer s requred o reveal demand ses n he manner ha here exss a sngle valuaon funcon assocaed wh whch hs revealed demand se s always he same as he se of all he bes responses hrough me. Wh hs rule we can replace sraegy-proofness wh he requremen for each buyer o conform o hs rue valuaon funcon hrough me as a domnan sraegy. The connecedness mples ha he auconeer never makes hs or her) asked prce vecor jump o any prce vecor ha he has never asked before. Wh hs connecedness from he observaon of he hsory he auconeer can always calculae he dfference n valuaon for any buyer beween any par of packages ha hs buyer revealed n hs hsory. 1 See Vckery 1961) Clarke 1971) and Groves 1973).

4 3 Wh he revealed preference acvy rule and he connecedness we can demonsrae a characerzaon resul for he class of all prce-demand procedures ha always acheves he correc VCG oucome n he followng manner. A any me we can unquely specfy a valuaon funcon ermed represenave valuaon funcon for each buyer n he hsory-conngen manner; assgns o any revealed package he mnmal relave valuaons and assgns o any unrevealed package he maxmal absolue value. In hs case any revealed package mus be a bes response for he represenave valuaon funcon. Based on hs specfcaon we show as hs paper s man heorem ha a prce-demand procedure acheves he correc VCG oucome f and only f he effcen allocaon and he effcen allocaons whou a sngle buyer assocaed wh he profle of represenave valuaon funcons ha we specfy a he endng me are all revealed n he resulan hsory. Hence all he auconeer has o do for he achevemen of he correc VCG oucome s o connue o ask prce vecors unl he recognze ha he effcen allocaon and he effcen allocaons whou a sngle buyer assocaed wh he resulan profle of represenave valuaon funcons are all revealed n he hsory. I s no a dffcul ask for he auconeer o fnd ou such a profle of represenave valuaon funcons. In fac he auconeer can do hs manner jus by ascendng he prce for any revealed packages and descendng he prce for any unrevealed packages. Imporanly we can show ha he effcen allocaon and he VCG paymens assocaed wh he profle of represenave valuaon funcons ha we specfy a he endng me s jus he same as he effcen allocaon and he VCG paymens assocaed wh he rue profle of valuaon funcons. Hence any buyer can easly deec wheher he auconeer succeeds o acheve he correc VCG oucome jus by observng he resulan hsory. Based on he above argumens s suffcen for he achevemen of he correc VCG oucome ha he auconeer makes he followng deal-free conracual agreemen wh he buyers: ) The auconeer connues o ask a prce vecor o each buyer under he consran of connecedness and requres each buyer o reveal a demand se under he consran of revealed preference acvy rule.

5 4 ) Ths procedure connues unl he hsory occurs ha reveals he effcen allocaon and he effcen allocaons whou any sngle buyer assocaed wh he hsory-conngen profle of represenave valuaon funcons. ) The auconeer evenually ends he procedure whn a me lm and hen acheves he VCG oucome assocaed wh he resulan profle of represenave valuaon funcons. Noe ha he buyers do no need o know he deal of prce-demand procedure ha he auconeer follows; provded he auconeer s suffcenly penalzed whenever he fals o sasfy hese requremens he above-menoned conracual agreemen can suffcenly ncenvze he auconeer o selec a prce-demand procedure ha acheves he correc VCG oucome. Because of he deal-free naure he buyers can leave he selecon of prce-demand procedure he auconeer s dscreon. In hs case he auconeer can make he selecon of prce-demand procedure even conngen on hs prvae sgnal. Ths sgnal could nclude non-rval nformaon regardng he buyers valuaon funcons bu does no need o be verfable o he cour. I s ofen ha a buyer s afrad ha any nformaon abou hs valuaon funcon whch s no necessarly relevan o he allocaon problem s leaked o he publc. The auconeer can preserve such a buyer s prvacy concern by selecng a shorcu procedure. The exen o whch he nformaon regardng a buyer s valuaon funcon s leaked o he publc s well descrbed by he resulan represenave valuaon funcon as well as he se of all revealed packages. The auconeer can save such an rrelevan nformaon leakage by ulzng hs prvae nformaon and soppng he procedure as soon as he recognze ha he effcen allocaon and he effcen allocaons whou any sngle buyer are all revealed n he hsory. As he prevous works such as Rohkopf Tesberg and Kahn 1990) have poned ou 2 he sandard pracce of he VCG mechansm wheren buyers drecly announce he enre valuaons for he enormous number of packages has flaws n pracce. In fac s generally oo complcaed for hem o assess and repor he enre valuaons and hen compue he VCG oucome a one me. 2 See also Mlgrom 2004) Ausubel and Mlgrom 2006) Parkes 2006) and Rohkopf 2007).

6 5 Hence s mporan o search for he possbly of replacng such a sandard pracce wh a dynamcal prce adjusmen such as a prce-demand procedure because he sequenal revelaon of all he bes responses o asked prce vecors hrough me would be nherenly much easer o make han he revelaon of he enre valuaons a once. For example Ausubel ) demonsraed an open-bd proocol ermed Ausubel mechansm or s varans as he replacemen of he sandard pracce. Chen and Takeuch 2010) conduced laboraory expermens and showed he beer performance of an open-bd proocol ermed BEA aucon han he sandard pracce. The presen paper demonsraes no only such easer revelaon pracces bu also a concree and racable mehod for dervng he VCG oucome from he resulan hsory. There exs many oher relevan works such as Gul and Sacche 2000) whch examned her respecve open-bd verson of he VCG mechansm 3. Among hem Parkes and Ungar 2002) Lahae Consann and Parkes 2005) Lahae and Parkes 2004) and Mshra and Parkes 2007) nroduced an nvolved marke equlbrum noon ermed unversal compeve equlbrum for undersandng such praccal ssues. The unversal compeve equlbrum s defned as a buyer-dependen prce vecor ha s no only a compeve equlbrum bu also a compeve equlbrum whou any sngle buyer. We can derve he effcen allocaon and he VCG paymens from a unversal compeve equlbrum. Because of s mahemacal form we can regard a profle of represenave valuaon funcons as a buyer-dependen prce vecor. We can show ha he profle of represenave valuaon funcons ha we specfy a he endng me s a unversal compeve equlbrum for he rue profle of valuaon funcons. Rohkopf 2007) argued abou buyers prvacy concern and poned ou he dffculy of hdng he revealed npus by usng crypography. Ths paper assumes ha all such npus are auomacally leaked o he publc. Wh hs assumpon he auconeer s careful selecon of a shorcu procedure becomes que subsanal. The remander of hs paper s organzed as follows: Secon 2 models he combnaoral aucon problem. Secon 3 nroduces prce-demand procedure and he 3 See also Kelso and Crawford 1982) Bkhchandan and Osroy 2002) Ausubel Cramon and Mlgrom 2006) Parkes and Ungar 2002) Lahae and Parkes 2004) Lahae Consann and Parkes 2005) Mshra and Parkes 2007) and ohers.

7 6 assumpon of revealed preference acvy rule. Secon 4 nroduces he assumpon of connecedness. Secon 5 nroduces represenave valuaon funcon and shows he characerzaon resul. Secon 6 clarfes he relaonshp beween represenave valuaon funcon and unversal compeve equlbrum. Secon 7 explans he deal-free conracual agreemen and he auconeer s dscreon. Secon 8 explans he auconeer s sgnal-conngen selecon of prce-demand procedure. Secon 9 argues he ssue on nformaon leakage. Secon 10 concludes.

8 7 2. The Model Le us nvesgae an combnaoral aucon problem wheren l 1 mulple heerogeneous ems are raded alogeher. Le L {1... l} denoe he se of all ems. The se of all buyers s denoed by N {1... n} where n 2. An allocaon s defned as a a1... a n ) where a L mples he package of ems ha s assgned o buyer and a a j for j. Le A denoe he se of all allocaons. An allocaon whou buyer N s defned as a aj ) jn\{ } where aj L and a j a h for h j. Le u A denoe he se of all allocaons whou buyer N. We assume quas-lneary. A valuaon funcon for buyer N s denoed by :2 L where u ) 0 and R 1) u a) u a ) f a a and a a. Le U denoe he se of all valuaon funcons for buyer. Le U U and N U U. jn\{ } j An allocaon a A s sad o be effcen for a profle of valuaon funcons u U f N u a ) u a ) for all a A. N Le A * u) A denoe he se of all effcen allocaons for u U. An allocaon a A whou buyer s sad o be effcen for a profle of valuaon funcons whou buyer N u uj) jn\{ } U f Le u f : U u a ) u a ) for all a A. j j jn\{ } jn\{ } * A u ) A denoe he se of all effcen allocaons whou buyer for U. A drec mechansm herenafer a mechansm s defned as G f x) where n A denoes an allocaon rule and x : U R denoes a paymen rule. Le

9 8 f u) f u)) A x x ) x : U N N n R and x u) x u)) R. A N mechansm G s sad o be effcen f f u for all u U. * ) A u) A mechansm G f x) s sad o be VCG Vckery-Clarke-Groves) f s effcen and for all N x u) max u a ) u f u)) j j j j a A j N \{ } j N \{ } and u U. Noe ha a VCG mechansm G f x) s sraegy-proof n ha ruh-ellng s a domnan sraegy.e. for every N u U and u U u f u)) x u) u f u u )) x u u ).

10 9 3. Prce-Demand Procedures and Revealed Preference Acvy Rule A prce vecor for buyer N s denoed by L 2 p p a )) R where a L p ) 0 and 2) p a) p a ) f a a and a a. Le P denoe he se of all prce vecors for buyer. Le P P and N p p ) P. N In he connuous me horzon [0 ) we consder an open-bd proocol ermed a prce-demand procedure as follows 4. A he begnnng of he nal me 0 he auconeer observes a prvae sgnal. The se of possble prvae sgnals s denoed by. Conngen on he observed sgnal he auconeer selecs a prce-demand procedure denoed by T ) T) wheren ) N denoes a prcng rule and T : U [0 ) denoes an endng rule whch wll be defned laer on. Unl Secon 8 we wll no explcly ake no accoun he sgnal-conngence of prce-demand procedure. Hence from hs secon o Secon 7 we wll smply denoe by T) a prce-demand procedure nsead of T ). A any me [0 ) he auconeer asks a prce vecor p p ) P o each buyer N and requres hs buyer o reveal a se of packages m m ) 2 L as hs 4 Ths paper depends on he assumpon of connuous me horzon. We may expec o replace hs assumpon wh he dscree me horzon whou volang he subsance of hs paper. In hs case however we need o assume he presence of a posve prce grd weaken he ncenve requremens and even abandon he exac achevemen of he VCG oucome o a ceran exen.

11 10 demand se 5. We perm m o nclude he null package whle we do no perm m o be empy.e. m L 2 2 \{ }. The auconeer s permed o ask dfferen prce vecors across buyers. A combnaon of he asked prce vecor and he revealed demand se L 2 p m) P 2 \{ } s sad o be conssen wh a valuaon funcon u U for buyer f m s he se of all bes response packages o m arg max{ u a ) p a )}. L a2 p for u.e. A hsory for buyer N up o me 0 ) s denoed by L 2 :[0 ) 2 \{ } h P where h ) p ) m )). A hsory h s sad o be conssen wh u U f h ) s conssen wh u for all [0 ). Le h 0 denoe he null hsory. Le H u ) denoe he se of all hsores for buyer up o me ha s conssen wh u. Le H H u ) uu H H N H and H h h) N H [0 ) H { h }. 0 0 For every h H le us defne he se of all valuaon funcons for buyer wh whch h s conssen by U h ) { u U h H u )}. For every h H we defne he se of all packages ha buyer reveals n he hsory h by L A h) { a2 am ) for some [0 )}. Le Uh ) U h) Ah ) A h) and A h ) Aj hj). N N A prcng rule ) N s defned by jn\{ } 5 We wll denoe p a ) p ) a ) where p p ).

12 11 : H P for all N. 0 A any me [0 ) where he hsory h H has occurred he auconeer asks he prce vecor h ) P o each buyer N. Le h h u ) denoe he hsory up o me ha occurs when he buyers connue o reveal her demand ses n he conssen manner wh u : and h u ) H u) p ) h ) for all N and [0 ) where we denoe h u ) h u )) N and h u ) ) h ) p ) m )) for all N and [0 ). Imporanly hs paper wll assume he revealed preference acvy rule whch requres each buyer o reveal he demand ses n he conssen manner wh a sngle valuaon funcon; for any buyer a any me he occurred hsory h mus sasfy ha U h ) s non-empy.e. U h ). Wh hs rule we can descrbe any observaon of a buyer s behavor as he consequence of hs selecon of a valuaon funcon as a proxy assocaed wh whch he connues o reveal he se of all bes response packages for hs seleced valuaon funcon hrough me. Provded he buyers confrm o a profle of valuaon funcons u U n he above-menoned manner he auconeer sops askng prce vecors a he endng me gven by T u) [0 ). Imporanly we wll assume ha for every u U h ) and u U h ) ) [ Tu ) Tu )] [ u U h T u u ))]. Ths s a necessary assumpon because he auconeer canno drecly make he selecon of endng me conngen on he rue profle of valuaon funcons; whenever ) T u )) he resulan hsory mus be he same beween u and u : u U h u ) ) and Tu ) Tu ). We defne he se of all possble hsores as T u) T u) h u h u

13 12 T u) H T) { h H h h u ) for someu U}. A mechansm G f x) s sad o be conssen wh a prce-demand procedure T) f for every h H T) u U h ) and u U h ) f u) x u)) f u) x u)). Whenever he mechansm G s conssen wh he prce-demand procedure T) hen he auconeer can gaher he suffcen nformaon regardng he profle of valuaon funcons for achevng he oucome.e. he allocaon and he paymen vecor nduced by G. The followng lemma shows ha whenever a mechansm s effcen and conssen wh a prce-demand procedure he allocaon nduced by hs mechansm mus be revealed n he resulan hsory. Lemma 1: If a mechansm G f x) s effcen and conssen wh a prce-demand procedure T) hen for every u U f u ) s revealed n he resulan hsory.e. T u) f u) A h u )). Proof: See Appendx A. In order o undersand hs lemma suppose ha f u) and was no revealed. Then whou loss of generaly we can assume ha he valuaon for he null package.e. u f u)) u ) s very small compared wh any non-null package. Ths however mples ha any effcen allocaon never assgns buyer he null package. Ths s a conradcon. Nex suppose ha f u) and was no revealed. Then whou loss of generaly we can assume ha here s a proper subse a f u) such ha u f u )) s greaer han bu very close o u a ). In hs case we can mprove he welfare by assgnng buyer a nsead of f u ) and assgnng some buyer j aj { f u)\ a} nsead of a j. Ths s a conradcon. Hence f u ) mus be revealed.

14 13 4. Connecedness A hsory h for buyer up o me s sad o be conneced f for every 0 ) eher p ) lm p ) or p ) p ) for some 0 ). The connecedness mples ha he auconeer never makes hs asked prce vecor jump o any prce vecor ha he has never asked before. Because of hs connecedness from he observaon of he hsory he auconeer can calculae he dfference n valuaon for any buyer beween any par of packages whenever hs buyer revealed hese packages n he hsory. Lemma 2: For every conneced hsory h H and { a a} A h ) here unquely exss x a a h ) R such ha x a a h ) u a ) u a) for all u U h ). Proof: See Appendx B. A prce-demand procedure T ) s sad o be conneced f for every 0 ) u U and N h u ) H s conneced. The followng proposon shows a necessary and suffcen condon for he exsence of a VCG mechansm ha s conssen wh a conneced prce-demand procedure; s necessary and suffcen ha assocaed wh any profle of valuaon funcons he effcen allocaons wh and whou any sngle buyer are all revealed n he hsory. Proposon 3: There exss a VCG mechansm G ha s conssen wh a conneced prce-demand procedure T ) f and only f for every h H T) here exs * a h ) A h ) and * a h ) A h ) for each N such ha for every u U h ) 3) * * a h ) A u) and

15 14 4) a h ) A u ) for all N. * * Proof: See Appendx C. If he effcen allocaons wh and whou any buyer are all revealed n he hsory follows from Lemma 2 ha he dfference n valuaon beween he effcen allocaon and he effcen allocaon whou any sngle buyer s unquely deermned from he observaon of hs hsory. Ths guaranees ha here exss a VCG mechansm ha s conssen wh he prce-demand procedure because he VCG paymen for each buyer s equvalen o he sum of he oher buyers dfferences n valuaon beween he effcen allocaon and he effcen allocaon whou buyer. On he oher hand f eher he effcen allocaons or he effcen allocaon whou a buyer s unrevealed he hsory fals o deermne he VCG paymens ha are common o all he profles of valuaon funcons ha are conssen wh hs hsory.

16 15 5. Represenave Valuaon Funcons For every N [0 ) and conneced hsory h H we defne he represenave valuaon funcon u [ h ] U as follows. Assume [ u h ] ) 0. Fx an arbrary package ha belongs o A h ) denoed by a A h ). For every a A h )\{ a } le us specfy and for every a A h ) u a ) u a ) x a a h ) [ h ] [ h ] u a ) nf { u a) p ) a) p ) a )}. [ h ] [ h ] [0 ) a m ) The represenave valuaon funcon assgns he maxmal absolue value o any unrevealed non-null package n he conssen manner wh he hsory. I s clear by defnon ha he represenave valuaon funcon [ h ] u exss unquely. Le [ h ] [ h ] ) N u u denoe he profle of represenave valuaon funcons assocaed wh he conneced hsory h H. The followng proposon shows ha he represenave valuaon funcon [ h u ] assgns any revealed package a A h ) wh he mnmal possble valuaon n relave erms. Proposon 4: For every [0 ) conneced hsory h H and u U holds ha u U h ) f and only f for every a A h ) and u a u a u a u a for all a A h ) [ ] [ ] ) ) h h ) ) u a u a u a u a for all a A h ). [ ] [ ] ) ) h h ) ) In hs case for every a A h ) and [ h ] ) ) u a u a u a u a f and only f A h ). [ h ] ) )

17 16 Proof: See Appendx D. The followng heorem shows ha he necessary and suffcen condon n Proposon 3 can be replaced wh anoher condon mplyng ha assocaed wh he profle of represenave valuaon funcons he effcen allocaons wh and whou any sngle buyer are all revealed n he hsory. Hence all we have o do for evaluang he suffcency s o examne jus he represenave valuaon funcons. Theorem 5: There exss a VCG mechansm G ha s conssen wh a conneced prce-demand procedure T ) f and only f for every h H T) 5) * [ h ] ) u ) Ah A and 6) A h A u for all N. * [ h ] ) ) Proof: See Appendx E. By followng he conneced prce-demand procedure T ) sasfyng 5) and 6) he auconeer can acheve a VCG mechansm f x ) whch s specfed as follows; for every u U and T u) * [ h u )] T u) f u) A u ) A h u )) for all N T u) T u) [ h u )] [ h u )] u u j j uj j a A j N \{} jn \{} x ) max a ) f u)) Snce he profle of represenave valuaon funcons. [ h ] u mnmzes he dfferences n valuaon beween he effcen allocaons and oher allocaons he requremens of effcency for [ h ] u would be he severes among all he profles of valuaon funcons ncluded n Uh ). Hence s suffcen o examne jus he profle of represenave valuaon funcons [ ] h u.

18 17 6. Unversal Compeve Equlbrum A profle of prce vecors p p ) P s sad o be a compeve equlbrum N CE for u U f here exss an allocaon denoed by a u) A ha maxmzes he payoffs for he seller and he buyers.e. CE p a ) p a) for all a A N N and for every N and a A CE CE u a ) p a ) u a ) p a ). A profle of prce vecors p P s sad o be a compeve equlbrum whou buyer for u U f here exss an allocaon whou buyer denoed by CE a u) A ha maxmzes he payoffs for he sellers and he buyers excep for buyer sasfyng ha for all a A CE p j aj ) pj aj) jn\{ } jn\{ } and for every j N \{ } and a j A j u a p a u a p a. CE CE j j ) j j ) j j) j j) Accordng o he same manner as he prevous works such as Parkes and Ungar 2002) Lahae and Parkes 2004) Lahae Consann and Parkes 2005) and Mshra and Parkes 2007) we defne a unversal compeve equlbrum for u U as a profle of prce vecors p P such ha s a compeve equlbrum for u and also for every N s a compeve equlbrum whou buyer for u. We mus noe ha whenever p s a unversal compeve equlbrum for u hen he allocaons CE CE a u ) and a u) could sasfy he effcency n he manner ha a u A u CE * ) ) and ) ) for all N. CE j* a u A u Because of 1) and 2) we can express he represenave valuaon funcon by a A -dmensonal vecor [ h ] [ h ] A )) aa whch could be regarded as a prce u u a R vecor for buyer.e. u [ h ] P.

19 18 The followng proposon saes ha he profle of represenave valuaon funcons assocaed wh he hsory a he endng me s a unversal compeve equlbrum. Proposon 6: For every h H f properes 5) and 6) are sasfed hen he profle of represenave valuaon funcons [ h ] u s a unversal compeve equlbrum for all u U h ). Proof: See Appendx F. Clearly here exss a VCG mechansm ha s conssen wh he followng conneced prce-demand procedure; he auconeer sars wh declnng he prce vecor for each buyer o a prce vecor ha s suffcenly close o he zero vecor. The auconeer hen adjuss he prce vecor for each buyer n he ascendng manner ha whenever he buyer reveals any package a he curren me hen he auconeer ncreases he prce for hs package. In hs manner any buyer evenually comes o reveal all packages ncludng he null package a one me mplyng ha we can denfy all relave valuaons a one me achevng he correc VCG oucome. In hs case he profle of prce vecors a he endng me corresponds o a unversal compeve equlbrum assocaed wh he rue profle of valuaon funcons. The auconeer can replace he above procedure wh a shorcu procedure; he auconeer can sop askng prce vecors as soon as he recognzes ha he effcen allocaons wh and whou any sngle buyer assocaed wh he profle of represenave valuaon funcons were all revealed n he resulan hsory. Ths s relevan o he ssue abou he exen o whch he nformaon abou he rue profle of valuaon funcons s leaked o he publc hrough he procedure he deal of whch wll be dscussed n Secon 8.

20 19 7. Deal-Free Conracual Agreemens Based on he argumens n he prevous secons we can consder he suaon n whch he auconeer he seller) makes a conracual agreemen wh he parcpans he buyers) as he bundle of he followng requremens. ) The auconeer connues o ask a prce vecor o each buyer under he consran of connecedness and requre each buyer o reveal a demand se under he consran of he revealed preference acvy rule. ) Ths procedure connues unl he hsory h occurs ha reveals he effcen allocaons wh and whou any sngle buyer assocaed wh he profle of represenave valuaon funcons [ h ] u.e. * [ h ] ) u ) Ah A and A h A u for all N * [ h ] ) ) where denoes he endng me. ) The auconeer evenually ends he procedure whn a me lm and hen acheves he VCG oucome assocaed wh he profle of represenave valuaon funcons a he endng me [ h ] n u.e. deermnes as ) A R sasfyng ha * [ h ] ) ) a A h A u and s u a u a for all N [ ] [ ] max h h ) ) j j j j a A j N \{ } j N \{ } where denoes he endng me h denoes he resulan hsory and ) N s s. Wh he revealed preference acvy rule we can descrbe any observaon of a buyer s behavor as he consequence of hs selecon of a proxy valuaon funcon assocaed wh whch he connues o reveal he se of all bes response packages. Ths propery jon wh he achevemen of he VCG oucomes n he manner of ) can ncenvze he buyers o selec her rue valuaon funcons as domnan sraeges revealng he se of all bes response packages assocaed wh her rue valuaon funcons.

21 20 I s no a dffcul ask for he auconeer o follow hese enre requremens. For example as shown a he boom of Secon 6 he auconeer can easly desgn a conneced prce-demand procedure wh whch a VCG mechansm s conssen. Under he above-menoned conracual agreemen he buyers can leave he selecon of prce-demand procedure he auconeer s dscreon. In hs case noe ha he buyers and he hrd pares such as he cours can easly deec wheher he auconeer acually follows he enre requremens jus by observng he occurred hsory of asked prce vecors and revealed demand ses; he buyers do no need o know he deal of he prce-demand procedure ha he auconeer follows. Hence we can say ha her conracual agreemen could be deal-free. Provded he auconeer s suffcenly penalzed whenever he does no follow hese requremens he conracual agreemen n he above manner can suffcenly ncenvze he auconeer o selec a conneced prce-demand procedure wh whch a VCG mechansm s conssen.

22 21 8. Sgnal-Conngen Prce-Demand Procedure Accordng o he success of he deal-free conracual agreemen explaned n Secon 7 he auconeer can make he selecon of a prce-demand procedure even conngen on hs prvae sgnal. Ths sgnal could nclude non-rval nformaon regardng he buyers valuaon funcons. Imporanly hs sgnal does no need o be verfable o he buyers and he cour. Even f he sgnal s nformave he auconeer he seller) canno ulze for hs self-neresed purpose such as hs revenue maxmzaon. The buyer can deec ha he auconeer seleced an neffcen allocaon jus by observng he hsory and hen can suffcenly penalze he auconeer. We defne a sgnal-conngen prce-demand procedure by T ). A mechansm G f x) s sad o be conssen wh a sgnal-conngen prce-demand procedure f s conssen wh T ) for all. Le us consder a mappng :U ; we assume ha he auconeer observes he prvae sgnal gven by u) when u s he rue profle of valuaon funcons. By observng hs sgnal he auconeer recognzes ha he profle of valuaon funcons belongs o he se prce-demand procedure T ). 1 ) U and hen selecs he correspondng As he exreme case le us consder he suaon wheren he auconeer observes he complee nformaon regardng he profle of valuaon funcons: U and u) u for all u U. The auconeer can selec he followng conneced prce-demand procedure ) T ) ha s conssen wh a VCG mechansm. A he nal me 0 he auconeer asks he profle of prce vecors ha corresponds o he unversal compeve equlbrum assocaed wh u 1 ). The resulan endng me Tu ) could be he nal me 0; he auconeer can mmedaely verfy he correc effcen allocaons wh and whou any sngle buyer achevng a VCG oucome a he nal me 0.

23 22 9. Informaon Leakage We can regard he se of all valuaon funcons for each buyer N wh whch he conneced hsory h up o me s conssen.e. U h ) exen o whch he nformaon regardng buyer as descrbng he s ' valuaon funcon s leaked o he publc. Proposon 4 mples ha U h ).e. he exen of nformaon leakage for buyer s ' valuaon funcons can be unquely denfed from he represenave valuaon funcon [ h ] u for buyer and he se of all packages ha buyer reveals A h ); he combnaon of u and A h ) s he suffcen sascs for he exen o [ h ] whch he nformaon abou buyer s ' valuaon funcon was leaked o he publc. The auconeer can preserve he buyers prvacy concern by selecng a shorcu prce-demand procedure ha decreases he exen of nformaon leakage as much as possble. For example le us consder a case of wo buyers and wo ems.e. buyer 1 buyer 2 em A and em B n whch he profle of valuaon funcons u s gven by and u { A}) u { B}) W u { B}) u { A}) u { A B}) u { A B}) 100 W. 1 2 We assume ha he auconeer knows he above-menoned form of valuaon funcons bu does no know he value of W 1 ). Noe ha he ses of all effcen allocaons wh and whou any sngle buyer are as follows. If W 100 hen If 1 W 100 hen A * u) {{ A}{ B})} ) {{ }} and 2* A u1 A B A * u) {{ A B} ) { A B})} 1* A u2 A B ) {{ }}. 2* A u1 A B ) {{ }} and 1* A u2 A B ) {{ }}. If W 100 hen A * u) {{ A}{ B}){ AB } ) { AB })} 2* A u1 A B ) {{ }} and 1* A u2 A B ) {{ }}.

24 23 We nvesgae he followng conneced prce-demand procedure. A he nal me 0 he auconeer asks he profle of prce vecors p p1 p2) gven by and p { A}} p { B}) p { A}} p { B}) p { A B}} p { A B}) Noe ha hs profle p corresponds o he profle of represenave valuaon funcons a he nal me 0. Case 1: W 100. A he nal me 0 each buyer s wllng o reveal he se of all bes response packages m gven by m1 {{ A}{ A B}} and m 2 {{ B}{ A B}}. The effcen allocaon { A}{ B }) he effcen allocaon whou buyer 2 { AB } and he effcen allocaon whou buyer 1 { A B } whch are assocaed wh he profle of represenave valuaon funcons p a he nal me 0 were all revealed a he nal me 0. Hence he auconeer can acheve he VCG oucome a he nal me 0: f u) { A}{ B}) x u) p { A B}) p { B}) 100 and x u) p { A B}) p { A}) The prce-demand procedure ends a he nal me 0 leakng o he publc jus he followng nformaon: 1) Each buyer s valuaon for { A B } s greaer han or equals ) Buyer 1 s valuaon for { A } s greaer han ha { B }. Buyer 2 s valuaon for { B } s greaer han ha { A }. 3) For buyer 1 he dfference n valuaon beween { A B } and { A } s equal o 100. For buyer 2 he dfference n valuaon beween { A B } and { B } s equal o 100. Case 2: W 100. gven by A he nal me 0 each buyer reveals he se of all bes response packages m

25 24 m1 {} and m 2 {}. The effcen allocaons { A}{ B }) { AB } ) and { A B}) he effcen allocaon whou buyer 2 { A B } and he effcen allocaon whou buyer 1 { A B } whch are assocaed wh he profle of represenave valuaon funcons p a he nal me 0 were no revealed a he nal me 0. Hence afer he nal me 0 he auconeer adjuss he profle of asked prce vecors p p ) n he followng descendng manner: a each me 0 p { A}} p { B}) p { A}} p { B}) and p { AB }} p{ AB }) Up o he me 100 W he buyers connue o reveal he same demand ses as a he nal me 0. A me 100 W where he auconeer asks he profle of prce vecors p p100 W) gven by and W p1{ A}} p1{ B}) p2{ A}} p2{ B}) 50 2 p { AB }} p{ AB }) 100 W 1 2 each buyer s wllng o reveal he se of all bes response packages m1 {{ A B} } and m 2 {{ A B} }. m gven by Noe ha he effcen allocaons { AB } ) and { A B}) he effcen allocaon whou buyer 2 { A B } and he effcen allocaon whou buyer 1 { A B } whch are assocaed wh he profle of represenave valuaon funcons p a he nal me 0 were all revealed a he me 100 W. Hence he auconeer can acheve he correc VCG oucome: f u) { A B} ) x u) p { A B}) p ) 100 W and x u) p { A B}) p { A B})

26 25 The prce-demand procedure ends a he me 100 W leakng o he publc jus he followng nformaon: 1) Each buyer s valuaon for { AB } s equal o 100 W. 2) Buyer 1 s valuaons for { A } and { B } and buyer 2 s valuaons for { A } and W { B } are all less han Compared wh he prce-demand procedure a he boom of Secon 6 he above-menoned prce-demand procedure can dramacally save he exen of nformaon leakage. In parcular he auconeer can perfecly hde he rrelevan nformaon abou buyer 1 s valuaon for em B and he buyer 2 valuaon for em A.

27 Concluson Ths paper nvesgaed a general class of open-bd combnaoral aucon proocols n he connuous me horzon ermed prce-demand procedures; he auconeer asks buyer-dependen prce vecors and each buyer reveals demand ses as he se of all bes-response packages hrough me. We assumed he revealed preference acvy rule and he connecedness. We demonsraed he characerzaon resul for he class of all prce-demand procedures ha acheve he VCG oucome. Tha s we showed ha a prce-demand procedure acheves he VCG oucome f and only f he effcen allocaon and he effcen allocaons whou any sngle buyer assocaed wh he resulan profle of represenave valuaon funcons are all revealed n he hsory. Ths class has he advanage over he sandard pracce of he VCG mechansm because he revelaon of bes responses hrough me n he procedure s much easer han he revelaon of he enre valuaons a one me n he sandard pracce. We argued ha s possble for he auconeer o acheve he VCG oucome n hs manner jus by makng he deal-free conracual agreemen wh he buyers. In hs case he buyers can easly deec wheher he auconeer succeeds o acheve he correc VCG oucome jus by observng he resulan hsory. Hence hey can leave he selecon of prce-demand procedure he auconeer s dscreon whou volang he auconeer s ncenve. The auconeer can make hs selecon even conngen on hs unverfable prvae nformaon. He can save rrelevan nformaon leakage o a ceran exen by selecng a shorcu procedure.

28 27 References Ausubel L. 2004): An Effcen Ascendng-Bd Aucon for Mulple Objecs Amercan Economc Revew Ausubel L. 2006): An Effcen Dynamc Aucon for Heerogeneous Commodes Amercan Economc Revew Ausubel L. P. Cramon and P. Mlgrom 2006): The Clock-Proxy Aucon: A Praccal Combnaoral Aucon Desgn n Combnaoral Aucons ed. by P. Cramon Y. Shoham and R. Senberg. MIT Press. Ausubel L. and P. Mlgrom 2006): The Lovely bu Lonely Vckrey Aucon n Combnaoral Aucons ed. by P. Cramon Y. Shoham and R. Senberg. MIT Press. Bkhchandan S. and J. Osroy 2002): The Package Assgnmen Model Journal of Economc Theory Chen Y. and K. Takeuch 2010): Mul-un Aucons wh Package Bddngs: An Expermenal Comparson of BEA and Vckrey Games and Economc Behavor 68 pp Clarke E. 1971): Mulpar Prcng of Publc Goods Publc Choce Groves T. 1973): Incenves n Teams Economerca Gul F. and E. Sacche 2000): The Englsh Aucon wh Dfferenaed Commodes Journal of Economc Theory Kelso A. and V. Crawford 1982): Job Machng Coalon Formaon and Gross Subsues Economerca Lahae S. F. Consann and D. Parkes 2005): More on he Power of Demand Queres n Combnaoral Aucons: Learnng Aomc Languages and Handlng Incenves Proc. 19 h Inern. Jon Conf. Arf. Inell. Lahae S. and D. Parkes 2004): Applyng Learnng Algorhms o Preference Elcaon n he Generalzed Vckrey Aucon mmeo Harvard Unversy. Masushma H. 2011): Prce-Based Combnaoral Aucon: Connecedness and Represenave Valuaons Dscusson Paper CIRJE-F-806 Unversy of Tokyo.

29 28 Mlgrom P. 2004): Pung Aucon Theory o Work Cambrdge Unversy Press: Cambrdge. Mshra D. and D. Parkes 2007): Ascendng Prce Vckrey Aucons for General Valuaons Journal of Economc Theory Parkes D. 2006): Ierave Combnaoral Aucons n Combnaoral Aucons ed. by P. Cramon Y. Shoham and R. Senberg. MIT Press: Cambrdge. Parkes D. and L. Ungar 2002): An Ascendng-Prce Generalzed Vckrey Aucon mmeo Harvard Unversy. Rohkopf M. T. Tesberg and E. Kahn 1990): Why Are Vckrey Aucons Rare? Journal of Polcal Economy Segal I. 2006): Communcaon Requremens of Combnaoral Allocaon Problems n Combnaoral Aucons ed. by P. Cramon Y. Shoham and R. Senberg MIT Press. Vckrey W. 1961): Counerspeculaon Aucons and Compeve Sealed Tenders Journal of Fnance

30 29 Appendx A: Proof of Lemma 1 For every 0 and every u U we defne u U by L u a) u a) for all a 2 \ { }. Assume ha here exss u U and N such ha Suppose f u A h u. T u) ) )) f u A h u. T u) ) )) Then h T u) u ) mus be conssen wh u for all 0.e. u U h T for all 0 T u) )) whch along wh conssency and effcency mples ha f u u ) f u) A u u ) for all 0. * Ths s a conradcon because any effcen allocaon * a A u u ) for u u ) never sasfes Suppose a f u) provded s suffcenly large.. Then we can selec 2 L a such ha a f u) and a f u) and ha for every a 2 L sasfyng ha a { a f u)} and a f u) u f u)) u a ) u f u)) u a). 6 From 1) we can selec u u wh whch h ) T) s conssen.e. T u u U h T. T u) )) Here we can selec u and j N \{} n a manner ha u f u)) u a) s close enough o zero and A-1) u f u)) u a ) u f u) f u)\ a ) u f u)). j j j j 6 Noe ha f f u ) s a sngleon here exss no such a. Hence a sasfes he requremens. auomacally

31 30 Le us specfy â A by aˆ a aˆ f u) f u)\ a and j j From A-1) aˆ f u) for all h N \{ j}. h h u f u)) u f u)) u aˆ ) u aˆ ) h h h h hn\{ } hn\{ } T mplyng ha f u ) s no effcen for u u ). However snce u h ) u ) s conssen wh u mus hold ha f u u ) f u). Ths s a conradcon.

32 31 Appendx B: Proof of Lemma 2 Snce h s conneced here exss a fne sequence ) l ) l k a ) l 1 such ha k 2 a 1) a a k ) a and ) l [0 ) and a l1) l) a m ) for all l {1... k} ) l ) l m ) for all l {2... k}. For every u U h ) and every l {2... k} snce ) l l1) ) l { } ) a a m follows u a p a u a p a.e. ) l ) l ) l l1) ) l l1) ) ) ) ) ) ) Hence B-1) u a u a p a p a l) l1) l) l) l) l1) ) ) ) ) ) ) k ) l l 1) ) ) { ) )} l2 u a u a u a u a k ) l ) l ) l l1) { p ) a ) p ) a )} l2. Le us specfy x a a h ) R as k ) l ) l ) l l1) ) { ) ) ) )} l2 x a a h p a p a. Snce hs specfcaon does no depend on he selecon of u U h ) follows from B-1) ha for every u U h ) and every { a a} A h ) u a ) u a) x a a h ).

33 32 Appendx C: Proof of Proposon 3 We prove he f par as follows. Suppose ha for every h H T) here exs * a h ) A h ) and * a h ) A h ) for each N ha sasfy 3) and 4) for all u U h ). Then we can specfy f : U A n a manner ha for every h H T) and every u U h ) * f u) a h ). We can also specfy x : U h H T) and every u U h ) R for each N n a manner ha for every * ) j j ) j ) ) jn\{ } x u x a h f u h. From Lemma 2 and 4) follows ha x u) max u a ) u f u)) j j j j a A j N \{ } j N \{ } whch along wh 3) mples ha he specfed mechansm G f x) s VCG. We prove he only f par as follows. Assume ha G f x) s VCG and conssen wh T ). Noe from 1) and 2) ha for every N every h H and every { a a } A h ) here exss { u u } U h ) such ha u a ) u a ) u a ) u a ). Hence for every u U f eher T u) f u) A h u )) or A h u A u for some j N j T u) j* )) j ) hen here exs j N and u j U j such ha T u) uj u j) U h u )) and for every N \{ j} x u u ) max{ u a ) u a )} { u f u)) u f u))} j j j j h h j j h h a A hn\{ j} hn\{ j} ) max u a ) u f u)) x u. h h h h a A h N \{ } h N \{ }

34 33 Ths conradcs he supposon ha G s conssen wh T). Hence we have proved ha for every u U and T u) f u) A h u )) A h u A u for all j N. j T u) j* )) j ) Suppose ha here exs {} uu U j N and a j j A such ha T u) u U h u )) a A h u j j T u) j )) j a A j* u ) j and j a A j* u ). j Whou loss of generaly we can selec u sasfyng ha Snce x u) max u a ) u f u)) j j j a A N \{ j } N \{ j } u a ) u f u)). N\{ j} N\{ j} T u) f u) A h u )) and a A h u j j T u) j )) follows ha u a ) u f u)) u a ) u f u)) x u) N\{ j} N\{ j} N\{ j} N\{ j} whch mples ha x u ) x u). Ths conradcs he supposon ha G s conssen wh T). Hence we have proved ha for every u U and every j N A h u )) A u )) j T u) j* j T u) u ju j h j u )) j whch mples ha here exss * a h ) A h ) ha sasfes 4). Moreover Lemma 1 mples ha here exss * a h ) A h ) for each N sasfyng 3). From he above observaons we have proved he only f par.

35 34 Appendx D: Proof of Proposon 4 The proof of he f par s sraghforward from he defnon of u [ h ]. From he defnon of [ h u ] and a A h ) follows ha f u U h ) a A u a u a u a u a and a A h ) f and only f [ ] [ ] ) ) h h ) ) u a u a u a u a x a a h [ ] [ ] ) ) h h ) ) ) hen for every where we have used he assumpon of revealed preference acvy rule and Lemma 2. By leng a from u [ h ] ) u ) 0 and [ h ] ) ) u a u a follows ha u a u a f and only f A h ). [ h ] ) )

36 35 Appendx E: Proof of Theorem 5 From Proposon 4 and he specfcaon of every a A h) and every a A [ h ] u follows ha for every N u a ) u a ) u a ) u a ) for all u U h ). Hence for every a A h ) [ h ] [ h ] a A * u) for all u U h ) f * [ h ] a A u ). From he specfcaon of [ h ] u follows ha for every a A h ) * [ h ] a A u ) f a A * u) for all u U h ). Hence we have proved ha for every a A h ) * [ h ] a A u ) f and only f a A * u) for all u U h ). Ths mples ha 5) s equvalen o 3). In he same manner for every j N and j j every a A h ) j j j* [ h ] j a A u ) f and only f j j* a A u ) for all u U h ). j j j j Ths mples ha 6) s equvalen o 4).

37 36 Appendx F: Proof of Theorem 6 From 5) and 6) we can selec * a h ) A h ) and * a h ) A h ) for each N such ha * * [ h ] a h ) A h ) A u ) and Hence a h A h A u for all N. * * [ h ] ) ) ) u a h )) u a ) for all a A N [ h ] * [ h ] N and for every N )) ) for all a A. [ hj] j* [ hj] uj a h uj aj jn\{ } jn\{ } From Proposon 4 * a h ) A h ) and * a h ) A h ) follows ha for every u U h ) and every N and for every j N \{ } * [ ] * [ ] )) h h )) ) ) u a h u a h u a u a for all a A * [ h j] * [ hj] j j j j j j j j j j u a h )) u a h )) u a ) u a ) for all a j A j. These observaons mply ha [ h ] u P s a unversal compeve equlbrum.