Efficient dynamic mechanisms in environments with interdependent valuations: The role of contingent transfers

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1 Theorecal Economcs , / Effcen dynamc mechansms n envronmens wh nerdependen valuaons: The role of conngen ransfers Heng Lu Deparmen of Economcs, Unversy of Mchgan Ths paper addresses he problem of mplemenng socally effcen allocaons n dynamc envronmens wh nerdependen valuaons and evolvng prvae nformaon. In he case where he agens nformaon s correlaed across me, we consruc effcen and ncenve compable drec dynamc mechansms. Unlke he mechansms wh hsory-ndependen ransfers n he exsng leraure, hese mechansms feaure hsory-dependen ransfers. Moreover, hey are remnscen of he classcal Vckrey Clarke Grove VCG mechansm, even hough he laer s no ncenve compable wh nerdependen valuaons. We furher show ha he VCG aspec of he drec mechansms suggess naural ways for mplemenaon n some repeaed aucons. Keywords. Dynamc mechansm, nerdependen valuaon, neremporal correlaon. JEL classfcaon. C73, D61, D Inroducon Ths paper sudes effcen mechansm desgn n dynamc allocaon problems wh nerdependen valuaons. A canoncal real-world example of such problems s he followng. Perodcally, he U.S. governmen uses aucons o sell lcenses for he rgh o drll for ol n adjacen offshore areas. Bdders n hese aucons are ol frms. Presumably, hese frms conduc geologcal surveys o esmae he amoun of ol n each area before bddng n each aucon, so ha he nformaon obaned by one frm s also valuable for he oher frms. The effcen allocaon of lcenses depends on he evolvng prvae nformaon of he frms, so he governmen should carefully desgn he aucons o nduce ruhful revelaon by he frms n every perod. More absracly, n he problems of neres, a sequence of decsons needs o be made over me: n each perod an allocaon s o be made among a group of agens, who have me-varyng, payoff-relevan Heng Lu: henglu29@gmal.com I am graeful o Paulo Barell and Har Govndan for her gudance and encouragemen. I hank he edor and hree referees for many consrucve suggesons ha mproved he exposon and precson of he paper. I also hank Guy Are, Drk Bergemann, Tlman Börgers, John Duggan, Albn Erlanson, Ben Golub, Asen Kochov, Claudo Mezze, Konrad Merendorff, Davd Mller, Romans Pancs, Alessandro Pavan, Davd Rahman, Phl Reny, Karol Szwagrzak, Wllam Thomson, Bob Wlson, and semnar parcpans a Rocheser, Mchgan, Unversy of Norh Carolna a Chapel Hll, Washngon Unversy n S. Lous, he 213 Cowles Summer Conference n Economc Theory, he 213 Osaka Rocheser Theory Conference, he 24h Sony Brook Game Theory Fesval, and he 214 Canadan Economc Theory Conference for helpful commens. All remanng errors are my own. Copyrgh 218 The Auhor. Ths s an open access arcle under he erms of he Creave Commons Arbuon-Non Commercal Lcense, avalable a hp://econheory.org. hps://do.org/1.3982/te2234

2 796 Heng Lu Theorecal Economcs prvae nformaon. Effcen mechansm desgn s he queson of how o ruhfully mplemen socally effcen allocaons,.e., how o handle he ncenve compably consrans mpled by he evolvng prvae nformaon. Followng he leraure, we resrc ourselves o he case of quas-lnear preferences and prvae nformaon ha follows a general Markov decson process whose evoluon depends on allocaons. In hs envronmen, and under he assumpon ha valuaons are prvae,.e., no nerdependen, Bergemann and Välmäk 21 and Ahey and Segal 213 have successfully addressed hs queson by means of dynamc exensons of he classc VCG Vckrey 1961, Clarke 1971, andgroves 1973 andagvd Aspremon and Gérard-Vare 1979 and Arrow 1979 mechansms. 1 However, wh nerdependence, s well known ha he VCG mechansm and s dynamc exensons are no ncenve compable whou addonal srong assumpons. The key nsgh of he VCG mechansm makng each agen a resdual claman s no applcable when an agen s nformaon affecs ohers ules. In fac, n generc envronmens wh muldmensonal and sascally ndependen prvae nformaon, Dasgupa and Maskn 2 and Jehel and Moldovanu 21 have shown ha no effcen mechansm, VCG or no, s Bayesan ncenve compable. 2 Alernavely, wh correlaed prvae nformaon, he loery mechansm of Crémer and McLean 1988 s effcen and Bayesan ncenve compable. Ye n dynamc envronmens, a perod-by-perod exenson of Crémer and McLean s mechansm may no be ncenve compable, because agens have more opporunes o devae. 3 Bu noce ha long-erm neracons offer a rcher famly of ransfer schemes compared o he sac case; n parcular, ransfers can be made hsory-dependen. Wh such ransfers, an agen s curren repor affecs no only her curren payoff bu also he enre sream of fuure ransfers. Therefore, one mgh be able o resore ncenve compably wh a careful choce of neremporal rade-offs. We show ha hs s ndeed he case. For he above-menoned dynamc allocaon problems, we consruc effcen and ncenve compable dynamc mechansms, provded ha nformaon s correlaed over me, as we explan below. In addon, he mechansms ensure ha each agen becomes a resdual claman, as n he VCG mechansm. Tha s, n each perod and regardless of he hsory, an agen s expeced connuaon payoff equals he connuaon socal surplus when all agens ruhfully repor her prvae nformaon. In oher words, no only do we provde a soluon o he dynamc ncenve compably ssue wh nerdependence, bu also he soluon shares some of he man feaures of he VCG mechansm. 4 Furhermore, as n he prvae-valuaon case, he consruced dynamc mechansms sasfy a srong ncenve compably requremen perodc ex pos ncenve compably, whch requres ruh-ellng o be a bes response of an agen a every sage, rrespecve of he pas messages and allocaons and oher agens curren 1 Also see Parkes and Sngh Jehel e al. 26 furher prove ha only consan allocaon rules are ex pos ncenve compable n generc models wh muldmensonal sgnals. 3 See he example n Secon From a praccal vewpon, he consruced hsory-dependen ransfers also pon oward a new way o lnk nformaon ha has been largely gnored n he desgn of varous economc mechansms.

3 Theorecal Economcs Envronmens wh ndependen valuaons 797 prvae nformaon. 5 Consrucng a perodc ex pos ncenve compable dynamc mechansm s no only of heorecal neres; also suggess naural ways o mplemen he drec mechansm wh dynamc aucons. In Secon 4, n a class of repeaed allocaon problems where no sac aucon forma s effcen, we defne a dynamc forma wh conngen ransfers ha has an effcen symmerc equlbrum n monoone sraeges. The neremporal correlaon ha s requred for our resuls resembles he correlaon condons n Crémer and McLean 1988 when he sae space of he Markov decson process s fne. Tha s, we requre convex or lnear ndependence condons on he assocaed ranson marces. 6 In Secon 5, we exend he resuls o he nfnesgnal case. Generalzng he convex and lnear ndependence condons, we consruc effcen dynamc mechansms ha are approxmaely ncenve compable. 7 Moreover, under sronger correlaon condons, here are mechansms wh conngen ransfers ha are perodc ex pos ncenve compable. Therefore, he resuls n he dynamc mechansms conras sharply wh hose n he sac counerpars, where one can only acheve approxmae ncenve compably or approxmae surplus exracon. 8 Fnally, n he Supplemenal Maeral avalable n a supplemenary fle on he journal webse, hp://econheory.org/sup/2234/supplemen.pdf, we address he ssues of budge balance and surplus exracon. Specfcally, by modfyng he ransfers, we consruc an average exernaly mechansm ha balances he budge, 9 and a loery-augmened mechansm à la Crémer and McLean 1988 and McAfee and Reny 1992 ha exracs all he surplus of he agens n he fne case and vrually all he surplus n he nfne case. Whle he man resuls requre neremporal correlaon, n he Supplemenal Maeral, we also sudy he case where each agen s prvae nformaon evolves ndependenly. We focus on sengs wh one-dmensonal prvae nformaon and consruc ransfers ha are he dynamc counerpars of he generalzed VCG mechansm cf. Crémerand McLean1985, Jehel and Moldovanu 21, Bergemann and Välmäk 22. In he prvae-valuaon specal case, hese ransfers reduce o he dynamc pvo mechansm consruced by Bergemann and Välmäk 21. Inhegeneral nerdependence case, we denfy dynamc sngle-crossng condons ha ensure ncenve compably. 5 Ahey and Mller 27, Bergemann and Välmäk 21, and Ahey and Segal 213 nroduce he noon of ex pos ncenve compably n every perod n he sudy of dynamc mechansms wh prvae valuaon. In hs paper, we follow Bergemann and Välmäk 21 and call perodc ex pos ncenve compably. 6 These condons are relaed o, bu dfferen from, hose n Crémer and McLean 1988 for sac mechansms wh correlaed sgnals. Specfcally, we do no mpose any resrcon on he nformaon srucure whn a perod. 7 The convex ndependence condon s smlar o McAfee and Reny s exenson cf. McAfee and Reny 1992orCrémer and McLean See McAfee and Reny 1992 and Mller e al The mechansm s relaed o he balanced eam mechansm consruced n Ahey and Segal 213, whch generalzes he AGV mechansm nroduced by Arrow 1979 and d Aspremon and Gérard-Vare 1979 o dynamc envronmens wh ndependen prvae valuaons.

4 798 Heng Lu Theorecal Economcs Relaed leraure Effcen mechansms wh nerdependen valuaons In addon o he papers menoned above, our dynamc mechansms are also relaed o he wo-sage VCG mechansm n Mezze 24, Mezze provdes one way o bypass he above mpossbly resuls, under he assumpons ha agens can observe her realzed ules and ha ransfers can be made based on he repored ules. From an appled perspecve, hese are srong assumpons. More mporanly, n Mezze s mechansm, agens are ndfferen among all messages when hey repor her ules. If s cosly o repor ules, hen agens would raher walk away from he mechansm a hs sage. In comparson, we consder drec mechansms ha ask agens o repor her prvae sgnals n each perod, n whch ruh-ellng consues a perfec equlbrum. Furhermore, for each agen and each sgnal profle, here are messages ha yeld dfferen expeced payoffs n every perod. Dynamc mechansm desgn Mos of he recen leraure on dynamc mechansms assumes ndependen prvae valuaons e.g., Bergemann and Välmäk 21, Ahey and Segal 213, Sad 212, andpavan e al. 214, wh an excepon of Gershkov and Moldovanu 29. Gershkov and Moldovanu consder a problem of sequenal allocaons of objecs o myopc agens who arrve over me. 11 In her model, he me horzon s fne, valuaons are prvae, and sgnals are one-dmensonal. They show ha f he dsrbuon of sgnals s unknown, hen nerdependence arses endogenously as a resul of learnng, whch may preven effcen mplemenaon wh onlne mechansms. 12,13 Snce agens are mpaen n Gershkov and Moldovanu s model, he ncenve problems are sac. They denfy sngle-crossng condons on he underlyng uncerany ha ensure he exsence of effcen mechansms. Relaed o he hsorydependen mechansms n hs paper, hey also pon ou ha effcen mechansms n her model exs f all ransfers can be delayed o he las perod. Two closely relaed papers are Hörner e al. 215 and Noda 218. Independen o hs paper, Hörner e al. 215 also sudy he role of neremporal correlaon n dynamc Bayesan games wh communcaon. They consder he case n whch sgnal spaces are fne and he evoluon of sgnals s saonary. Addonally, hey sudy ruhful Bayes Nash equlbra of he nfnely repeaed game wh prvae nformaon. 14 In he case wh correlaed sgnals and nerdependen values, hey exend he nsgh of Crémer and McLean 1988 and also he sac budge-balanced mechansm n Kosenok and Severnov 28 o dynamc games and denfy an neremporal full-rank condon ha s suffcen o oban a folk heorem n ruhful equlbra. They show ha even 1 See Deb and Mshra 214 for a relaed recen sudy. 11 See also Gershkov and Moldovanu Gershkov and Moldovanu 21, 212 for sudes of relaed quesons. 12 Segal 23 also emphaszes hs feaure n a sac model. 13 The erm onlne mechansm s mosly used n he algorhmc game heory leraure o sudy allocaon problems wh arrvals and deparures; requres ha allocaons and ransfers of an agen are made when she s presen. 14 Truhful Bayes Nash equlbra, defned by Hörner e al. 215, generalze perfec publc equlbra n repeaed games wh mperfec publc monorng.

5 Theorecal Economcs Envronmens wh ndependen valuaons 799 n repeaed games where ransfers are no allowed, hey use connuaon payoffs as effecve ransfers, hereby brdgng he gap beween dynamc games and mechansm desgn. By conras, we consder a dynamc mechansm desgn seng wh ransferable ules and nerdependen valuaons, where he evoluon of prvae nformaon can vary over me; our resuls cover boh he fne and nfne sgnal space cases and emphasze he VCG feaure of hsory-dependen ransfers, whch s absen from her game-heorec analyss. Moreover, snce he soluon concep adoped n hs paper perodc ex pos ncenve compably s sronger han her ruhful equlbra n he case wh nerdependen values, we denfy sronger neremporal full-rank condons. Fnally, n he case wh ndependen sgnals, Hörner e al. 215resrc aenon o he prvae-valuaon sengs, whereas we consder he general seng wh nerdependen valuaons and exend he exsng posve resuls n he sac envronmens o dynamc envronmens. Noda 218 also sudes a queson smlar o ours, assumng sgnal spaces are fne. Noda 218 generalzes he convex ndependence condon n Crémer and McLean 1988 o dynamc sengs ha guaranees mplemenably and surplus exracon. Dfferen from Noda s work, hs paper consders boh fne- and nfne-sgnal spaces, gves suffcen condons for he exsence of perodc ex pos ncenve compable mechansms, and consrucs he correspondng conngen ransfers. For he case where sgnal spaces are fne, he neremporal convex ndependence condon n Noda 218s weaker han ha denfed n hs paper, alhough boh condons are genercally sasfed n he fne-horzon case. Moreover, we also generalze he spannng condons n Crémerand McLean 1988o dynamc envronmens, whereas Noda 218 only sudes convex ndependence. 2. Model 2.1 The envronmen We consder a dynamc nerdependen valuaon envronmen wh N N 2 agens. Tme s dscree, ndexed by {1 2 T},whereT. 15 In each perod,eachagen {1 2 N} prvaely observes a payoff-relevan sgnal,where s a fne se. The exenson o he nfne sgnal space case s suded n he Supplemenal Maeral. The sgnal space n perod s = N =1 wh a generc elemen = 1 N. For each and, denoe he prvae nformaon held by agens oher han n perod by = N j j. In each perod, he flow uly u of agen s deermned by he curren sgnal profle, he curren allocaon a A, and he curren moneary ransfer p R,where A s he fne se of socal alernaves n perod. The flow uly of each agen s assumed o be quas-lnear n moneary ransfers, and agens have a common dscoun facor δ 1. Gven sequences of sgnals { } T =1, allocaons {a } T =1, and moneary 15 We sudy boh he cases of fne and nfne horzon.

6 8 Heng Lu Theorecal Economcs ransfers {p 1 pn } T =1, he oal payoff of each agen s T δ 1[ u a p ] =1 The agen s prvae sgnals evolve over me followng a Markov decson process. Specfcally, n he nal perod, he sgnal profle 1 s drawn from a pror probably 1 1. In each perod >1, he dsrbuon of curren sgnal profle s deermned by he realzed sgnal profle 1 and he allocaon decson a 1 n he prevous perod, represened by a ranson probably : A 1 1. The uly funcons u,hepror 1, and he ranson probables are assumed o be common knowledge. In conras o prevous works ha ofen assume ndependen pror and ransons across agens, here we specfy a general Markov decson process for he evoluon of sgnals, whch allows correlaon of prvae nformaon. Whle n prvae-valuaon envronmens he exsence of effcen mechansms does no depend on wheher correlaon s allowed or no, as shown by Ahey and Segal 213, wll be clear n Secon 3 how correlaon makes a dfference n dynamc sengs wh nerdependen valuaons. 2.2 Effcency and mechansms A socally effcen allocaon rule s a sequence of funcons {a : A } T =1 ha solves he socal program [ T N ] max E δ 1 u a {a } T =1 =1 where he expecaon s aken wh respec o he processes { } and {a }. Snce he flow uly depends only on he curren sgnal profle, whch s assumed o be Markov, he socal program can be wren n he recursve form: for each {1 2 T}, W = max a A =1 =1 N u a + δe [ ] W a where W s he socal surplus sarng from perod gven he realzed sgnal profle,andw T +1. By he prncple of opmaly, a solves he socal program f and only f s a soluon o hs recursve problem. We focus on ruhful equlbra of drec publc mechansms ha mplemen he socally effcen allocaons {a }T =1.InSecon 4, we sudy ndrec mechansms ha mplemen he drec mechansms. In a drec publc mechansm, n each perod, each agen s asked o make a publc repor r of her curren prvae sgnal. Then a publc allocaon decson a and a ransfer p for each agen are made as funcons of he curren repor profle r = r N =1 and he perod- publc hsory h.theperod-

7 Theorecal Economcs Envronmens wh ndependen valuaons 81 publc hsory conans all repors and allocaons up o perod 1,.e., h = r 1 a 1 r 2 a 2 r 1 a 1 16 Le H denoe he se of possble perod- publc hsores. Formally, an effcen drec revelaon mechansm Ɣ ={ a p } T =1 consss of as he message space n each perod, a sequence of allocaon rules a : A, and a sequence of moneary ransfers p : H R N. The perod- prvae hsory h of each agen conans he perod- publc hsory and he sequence of her realzed prvae sgnals unl perod,.e., h = r 1 a 1 1 r 2 a 2 2 r 1 a 1 1 Le H denoe he se of agen s possble perod- prvae hsores. Wh a slgh abuse of noaon, a sraegy for agen s a sequence of mappngs r ={r }T =1,where r : H,haassgnareporoeachofherperod- prvae hsores. A sraegy for agen s ruhful f always repors agen s prvae sgnal ruhfully n each perod, regardless of her prvae hsory. Gven a mechansm Ɣ ={ a p } T =1 and a sraegy profle r ={r } N =1,agen s expeced dscouned payoff s E T δ 1[ u a r p h r ] 17 =1 The equlbrum concep we adop s perodc ex pos equlbrum defned by Bergemann and Välmäk 21 and Ahey and Segal 213. We say hahe mechansm s perodc ex pos ncenve compable or, equvalenly, he ruhful sraegy profle s a perodc ex pos equlbrum f for each agen and n each perod, ruh-ellng s always a bes response regardless of he prvae hsory and he curren sgnals of oher agens, gven ha oher agens adop ruhful sraeges. Formally, le V h be agen s connuaon payoff gven perod- prvae hsory, gven ha oher agens repor ruhfully. Tha s, V h = max E [ u a r r p h r ] + δv h 18 The effcen mechansm s perodc ex pos ncenve compable f for each,,andh, arg max u a r r p h r [ + δe V h a r ] 16 We assume ha agens do no observe he realzed per-perod payoffs. Also noe ha snce he mechansm s publc, an agen can also nfer he ransfers for all oher agens. 17 In he nfne-horzon T = case, we requre ha all agens expeced dscouned payoffs are well defned under he mechansm Ɣ, ha s, he expecaon and he nfne sum n agens payoffs are nerchangeable. 18 In he fne horzon case, we se VT +1.

8 82 Heng Lu Theorecal Economcs for each. Defne he perod- ex pos connuaon payoff o be V h ; h = u a p h [ + δe V h a ] As suggesed by Bergemann and Välmäk 21, ex pos ncenve compably noons need o be qualfed whn each perod n a dynamc envronmen, snce an agen may wsh o change her repor n some prevous round based on he new nformaon she has receved n laer perods. Gven he fac ha nerdependen valuaons render domnan sraegy ncenve compably mpossble, perodc ex pos ncenve compably s he bes we can hope for n he curren seup. Fnally, he Vckrey Clarke Groves VCG mechansm s an effcen mechansm Ɣ ={ a p } T =1 under whch each agen s connuaon payoff s equal o he connuaon socal surplus ne of a erm ha s ndependen of her curren and fuure repors,.e., for each and, here s a funcon W such ha V h ; h = W W 1 for all h, h,and. 3. Effcen mechansm desgn 3.1 An example Before presenng he general resuls, we presen a wo-perod repeaed aucon example o explan he man deas. 19 Two frms, A and B, compee for lcenses o drll for ol on wo adjacen offshore areas. The wo lcenses are sold sequenally n wo aucons {1 2} and he allocaon n aucon s a {A B}, wherea = means ha frm {A B} obans he lcense for he correspondng area. Each frm s payoff from obanng a lcense depends on s drllng cos and he amoun of ol s n ha area: u A s = 2s 1 u B s = 3s 6 Suppose ha here s no dscounng and each frm cares abou s oal prof from boh aucons. Each frm {A B} observes a prvae sgnal n aucon. Suppose ha pror o he aucons each frm can perform a es n one of he areas. In parcular, frm A s prvae sgnal 1 A {4 6} ndcaes he amoun of ol n area 1, denoed A 1 = s 1,and frm B learns prvaely from 2 B {4 6} he expeced amoun of ol n area 2, denoed 2 B = s 2. In addon, we assume ha he jon dsrbuon of 1 A and B 2, denoed by 1 A B 2,s [ ] [ ] /8 1/8 = /8 3/8 so ha he condonal dsrbuon of 2 B gven A 1, denoed by B 2 A 1,s [ ] [ ] /4 1/4 = /4 3/4 19 The example s adaped and exended from Dasgupa and Maskn 2.

9 Theorecal Economcs Envronmens wh ndependen valuaons 83 Fnally, we assume ha frm B does no learn any relevan nformaon n he frs aucon, and neher does frm A n he second aucon. Tha s, 1 A and B 1 are ndependenly dsrbued, and so are 2 A and B 2. We frs noce ha effcency and ncenve compably are ncompable f only he frs aucon s conduced. To see hs, noe ha effcency n he frs aucon requres frm A o gve up he lcense when s more profable,.e., { A a 1 = B f A 1 = 4 f A 1 = 6 Ths mples ha frm A needs o be compensaed for reporng r1 A = 4. Specfcally, we have he ncenve compably condons r A 1 = 6 raher han p A 1 4 pa 1 6 p A pa 1 4 Summng up he wo nequales gves Thus, no ncenve compable ransfer exss. Alernavely, when only he second lcense s beng auconed, frm B s ncenve consran maers and s sraghforward o verfy ha he ransfer for frm B, { f r p B B 2 = 2 = 4 11 f r2 B = 6 ruhfully mplemens he effcen allocaon a 2 n he second aucon, where a 2 s gven by { A a 2 = B f B 2 = 4 f B 2 = 6 Now we show ha by lnkng he wo aucons, dynamc effcency s mplemenable, despe he mpossbly for sac effcency. The dea s o use he correlaon beween 1 A and B 2 and consruc a hsory-dependen ransfer for frm A n he second aucon so ha frm A s wllng o repor s rue sgnal n he frs aucon. For nsance, consder he ransfer schedule p A 2 a 1 r2 B gven by 4 5 f a 1 = B r2 B = 4 p A 2 = 14 5 f a 1 = B r2 B = 6 oherwse We clam ha he dynamc mechansm Ɣ lnk {a 1 a 2 pa 2 pb 2 } s ex pos ncenve compable. Recall ha ruh-ellng s opmal for frm B gven p B 2. Snce he ransfer pa 2 has no effec on frm B s ncenve consrans, under {p A 2 pb 2 },frmb s sll wllng o repor s rue sgnal n he second aucon. Now consder frm A s ncenve consrans. Frm A, when reporng s sgnal, akes no accoun he fac ha s fuure ransfer depends on he curren allocaon a 1 and he opponen s repor r2 B n he nex aucon. As

10 84 Heng Lu Theorecal Economcs a consequence, he ncenve compably consrans are sasfed gven he specfed condonal dsrbuon of sgnals: The nuon for hs mechansm s as follows. Noe ha by consrucon, he lefhand sdes of he above wo nequales are equal o he socal surplus gven frm s A prvae sgnal. By explong he neremporal correlaon beween 1 A and B 2,heransfer p A 2 makes frm A a claman of he socal surplus n he frs aucon whou affecng any frm s ncenve consrans n he second aucon. Gven ha frm B adheres o ruhful sraeges, s opmal for frm A o be ruhful so as o maxmze he socal surplus and hence s own prof. Now le us modfy he example o llusrae he role of neremporal correlaon and s dfference from whn-perod correlaon Crémer and McLean 1988 ndynamc mechansms. We remove he assumpon ha 1 A and B 1 are ndependen and suppose ha before frms learn her payoff relevan sgnals, frm A has access o some prvae sgnal A { 1} ha deermnes he jon dsrbuon A 1 B 1 A of A 1 and B 1 : [ ] [ ] /8 3/8 = /8 1/8 [ ] [ ] /8 1/8 = /8 3/8 Tha s, 1 A and B 1 are negavely correlaed f A = and are posvely correlaed f A = 1. Fnally, he jon dsrbuon of A 1 and B 2 remans he same and s assumed o be ndependen of A. Suppose ha he auconeer wans o explo he correlaon beween 1 A and B 1 o ncenvze frm A. Ths amouns o consrucng loery ransfers for frm A based on frm B s frs perod repor r1 B. However, for such loeres o work, he auconeer needs o know he jon dsrbuon of 1 A and B 1,whchsfrmA s prvae nformaon. Gven a loery scheme n he frs aucon, frm A may have an ncenve o msrepor s sgnal A. To see hs, suppose ha he auconeer beleves ha frm A s nal repor r A { 1} s ruhful, and hus uses he followng ransfers pa 1 ra 1 rb 1 ; ra for frm A: [ ] [ ] p A 1 4 4; pa 1 4 6; 13 5 p A 1 6 4; = pa 1 6 6; [ ] [ ] p A 1 4 4; 1 pa 1 4 6; p A 1 6 4; 1 = pa 1 6 6; 1 Gven he jon dsrbuons, s sraghforward o check ha under p A 1 ra 1 rb 1 ; ra,f frm B reporsssgnalsruhfully,hensopmalforfrma o reveal 1 A and oban

11 Theorecal Economcs Envronmens wh ndependen valuaons 85 zero surplus n he frs aucon, had repored s nal prvae sgnal A ruhfully. However, gven p A 1 ra 1 rb 1 ; ra,frmacould benef from msreporng A.Forexample, when A =, he followng conngen devaon of frm A s profable: frs repors r A = 1 so ha he ransfer n he frs aucon s pa 1 ra 1 rb 1 ; 1; hen afer learnng A 1, always repors he oppose r1 A A 1.WhenA 1 = 4, frma repors ra 1 = 6 and loses he frs aucon wh no surplus, when A 1 = 6,frmA wns by reporng ra pa 1 6 4; pa 1 6 6; 1 = ; = 4 and receves a posve surplus, pa 1 4 4; pa 1 4 6; 1 = 4 Smlar conngen devaons of frm A exs when A = 1. Fnally, we noe ha snce he neremporal correlaon canno be manpulaed by eher frm, he dynamc mechansm Ɣ lnk consruced before remans ex pos ncenve compable Man resuls In hs secon, we consruc perodc ex pos ncenve compable effcen dynamc mechansms under general ranson dynamcs. Theorem 3.1 shows ha under a generc neremporal correlaon condon and some resrcons on uly funcons and sgnal spaces n he las perod, such a dynamc mechansm always exss. 21 In parcular, we show ha n each perod he correlaon beween and can be used o consruc hsory-dependen ransfers such ha agen s ncenve s algned wh he socal ncenve. Moreover, he resulng ransfers are remnscen of boh he VCG ransfers and he loery ransfers n Crémer and McLean In Theorem 3.2, weshow ha a slghly sronger neremporal correlaon condon ensures dynamc effcency wh a sequence of VCG-ype ransfers. We make he followng assumpons on he uly funcons and he evoluon of prvae nformaon. Assumpon 1 Bounded payoffs. For each agen, max a 1 T δ 1 u a < =1 2 In hs example, he whn-perod rank condon of Crémer and McLean 1988 fals, whch mples ha mplemenng he effcen allocaons wh sac mechansms s mpossble. If we also assume ha frm B receves a prvae sgnal n perod ha s correlaed wh frm A s perod- sgnal, hen a perodby-perod Crémer McLean mechansm would mplemen he effcen allocaons, bu s only Bayesan ncenve compable. 21 For he nfne-horzon case, no such resrcons are mposed.

12 86 Heng Lu Theorecal Economcs Assumpon 2 Convex ndependence. For each 1 T, N, a A,and, no column of he marx M a [ a ] s a convex combnaon of oher columns,.e., for each, a where Conv{ a { a nf a / Conv { a } \{ } } \{ } s he convex hull generaed by he se of vecors } \{ }. Moreover, he ranson probables sasfy ds 2 a Conv { a } \{ } > 22 Assumpon 3 Spannng condon. For each 1 T, N, a A,and he column vecors of he marx M a [ a ], are lnearly ndependen,.e., here does no exs a collecon of real numbers {η }, whch are no all equal o zero, such ha η a = for all any T,any, a and sasfes.moreover,ft =, hen here exs D R + and T N + such ha for, he norm of he pseudo-nverse of he marx M a M a + D 23 Assumpon 1 says ha he payoff funcon of each agen s well defned. Ths assumpon s vacuous n he case where allocaon and sgnal spaces are mendependen. Assumpons 2 and 3 requre ha ranson probables exhb neremporal correlaon among dfferen agens sgnals and he neremporal correlaon does no vansh n he nfne-horzon case. 24,25 In parcular, for each agen and 22 The funcon ds 2 C s he Eucldean dsance beween a pon and a se C. Noda 218 mposes a smlar condon n he nvesgaon of surplus exracon mechansms n he nfne-horzon case. 23 The pseudo-nverse A + of a full column-rank marx A s defned as A + = A A 1 A,whereA s he ranspose of A. The norm of a marx A s defned as A =sup{ Ax : x = 1}. 24 Crémer and McLean 1988 consder smlar condons n he sudy of sac mechansm desgn wh correlaed nformaon. 25 I hank an anonymous referee for ponng ou an error n he prevous verson and suggesng srenghenng of he assumpons for he nfne-horzon case. The nonvanshng neremporal correlaon condon n Assumpon 2 s based on he analyss n Noda 218. The correspondng condon n Assumpon 3 s new.

13 Theorecal Economcs Envronmens wh ndependen valuaons 87 n each perod, condonal on any a and,agen s curren prvae sgnal s correlaed wh oher agens sgnals n he nex perod. Independen evoluon of prvae nformaon across agens s ruled ou by hese assumpons. The assumpons of nonvanshng neremporal correlaon wll guaranee ha agens dscouned payoffs are well defned n he nfne-horzon case under our dynamc mechansms. 26 One example ha he neremporal correlaon does no vansh s when he ranson probables are saonary,.e., for each, =, A = A,and a = a. To movae he nformaon correlaon assumpons, suppose ha here s an underlyng sae of naure ω wh possble values n a se n each perod. In addon, ω follows a hdden Markov process ha evolves over me and s no observed by any agen. In each perod, he relaonshp beween he sae of naure ω and he agens prvae nformaon s descrbed by a jon dsrbuon ξ over. If each agen s prvae sgnal provdes useful nformaon abou ω,.e., he condonal ξ ω vares wh,henaslongasω s no ndependenly dsrbued, s correlaed wh,even condonal on and a. Inhefne-horzoncaseT<, we also mpose he followng ex pos ncenve compably assumpon on he allocaon rule a T. Assumpon 4 Ex pos ncenve compably n perod T. If T <, henheeffcen allocaon n perod T, a T, s ex pos ncenve compable. In our seup, he allocaon problem n perod T s essenally a sac one. Thus, we can adop a se of suffcen condons from he exsng leraure Bergemann and Välmäk 22 n parcular on sac mechansm desgn. The suffcen condons for ex pos ncenve compably n sac models are resrcve gven he mpossbly resuls n Dasgupa and Maskn 2, Jehel and Moldovanu 21, and Jehel e al. 26. In parcular, perod-t sgnals have o be one-dmensonal and he uly funcons have o sasfy a sngle-crossng condon. We also emphasze ha no such assumpons are mposed on he prvae sgnals and uly funcons from perod 1 o T 1. We can hnk of a suaon where agens rade a new asse wh each oher n mulple perods. Inally, each agen s prvae nformaon may be muldmensonal snce here s much uncerany abou many aspecs of he asse. As agens rade over me, hey gradually learn more nformaon abou he asse. In he las perod, each agen s sgnal s smply a real number ha represens her esmaon of he asse value. Now we sae he man resuls ha generalze he dea of he example n Secon 3.1. All he proofs of he resuls n Secon 3 are relegaed o Appendx A. Theorem 3.1. Under Assumpons 1, 2, and4, hereexss asequenceofransfers p : A R < T such ha he effcen dynamc mechansm {a p } s perodc ex pos ncenve compable. 26 The unform lower bound ɛ n Assumpon 2 and he unform upper bound D n Assumpon 3 can be furher relaxed o allow for me-dependen bounds as long as agens payoffs under he consruced mechansms are well defned.

14 88 Heng Lu Theorecal Economcs Here we gve a heursc argumen. Recall ha n he prvae-valuaon case, he hsory-ndependen ransfers n he VCG mechansm or eam mechansm n Ahey and Segal 213, p = u j a = u j a j 1 j j are ncenve compable. However, wh nerdependen valuaons, ransfers n 1 depend drecly on agen s repor, whch creaes an ncenve for msreporng. To fx hs problem, we consder general hsory-dependen ransfers p h. I urns ou ha under Assumpons 1, 2, and4, s enough o use ransfers ha depend on he hsory n he prevous round. Specfcally, we show ha f T =, here exs ransfers p ; a under whch a ruhful sraegy profle s perodc ex pos equlbrum. These hsory-dependen ransfers work as follows. In each perod, heransfer p for agen does no depend on her curren repor r,soagen s ncenve n perod s unaffeced by p. Insead, her ransfer n he nex perod p depends on r and a, whch means ha a ruh-ellng ncenve n perod s provded hrough p.under he ruh-ellng sraegy profle, n perod + 1 agen receves he sum of perod- flow payoffs of all oher agens, so agen s connuaon payoff n perod s equal o he socal surplus from perod onward. Furhermore, he ransfer for agen n perod + 1 s such ha here wll be no expeced gan from lyng n perod. Therefore, agen has no ncenve o devae from ruh-ellng n perod. The above argumen also suggess he necessy of a boundary condon for he ncenve problem n he las perod when T s fne. Snce he allocaon problem n perod T s sac and here s no avalable nformaon aferward, Assumpon 4 s needed. 27 The nex resul shows ha under a slghly sronger condon on he ranson probables, he dynamc effcen allocaons are ncenve compable wh a sequence of VCG-ype ransfers for each agen n he sense ha each agen s repor n each perod affecs her payoff only hrough he deermnaon of allocaon. Theorem 3.2. Under Assumpons 1, 3, and 4, here exss a sequence of ransfers p : A R < T such ha he effcen dynamc mechansm {a p } s perodc ex pos ncenve compable. The effcen mechansm n Theorem 3.2 shares anoher dsncve feaure of he VCG mechansm: each agen s repor affecs her own ransfers only hrough he mpac on allocaons. The nuon n hs case s even smpler. The ransfer p for agen does no depend on or 1. Insead, an ncenve for ruh-ellng n perod s agan guaraneed hrough p :under p,agen s connuaon payoff n perod s equal o he socal surplus from perod onward. 27 Bayesan ncenve compably of a T s no enough for our resul o hold, as agens have he opporuny o manpulae he desgner s perod-t belef by msreporng n perod T 1.

15 Theorecal Economcs Envronmens wh ndependen valuaons 89 In he above wo heorems, here seems o be a gap beween he nfne- and he fne-horzon cases, as he posve resul n he laer requres more condons han ha n he former. However, he nex corollary bulds a connecon beween hese cases: n he fne horzon case, by replacng he effcen allocaon n he las perod wh a consan allocaon whch s ex pos ncenve compable bu neffcen, effcency can be acheved n all bu he las perod; moreover, as he me horzon grows o nfny, he neffcency n he las perod vanshes n he lm. The proof follows drecly from ha of Theorem 3.1. Corollary 3.3. In he fne-horzon case T <, under Assumpons 1 and 2, here exss a sequence of ransfers p : A R < T such ha he almos effcen dynamc mechansm {a p <T ā T }, where, for all T, ā T T ā for some ā A T, s perodc ex pos ncenve compable. Remark 3.4. If for each and, hen Assumpons 2 and 3 are genercally sasfed n he fne-horzon case even f n each perod sgnals are ndependenly dsrbued condonal on all he avalable nformaon. 28 Accordngly, effcen dynamc mechansms exs n a large class of dynamc envronmens provded ha ex pos ncenve compably s achevable n he las perod Assumpon 4. Moreover, f he me horzon s nfne, hen Assumpon 4 has no be. Therefore, nsead of creang dffcules for effcen mechansms as one would magne, repeaed neracons, n fac, faclae he consrucon of ncenve compable ransfers. We also noe ha boh Assumpons 2 and 3 rule ou ceran nformaon envronmens ha are relevan n applcaons. For nsance, n he drllng example n Secon 3.1, boh assumpons fal f a frm s sgnal consss of a common value componen abou he amoun of ol ha s correlaed across aucons and a frm-specfc prvae cos componen ha s ndependenly dsrbued. Neverheless, f hese wo componens are addvely separable n a frm s valuaon, hen we can have an effcen mechansm ha merges he dynamc mechansms consruced above and he dynamc VCG mechansm for he prvae valuaons. 29 Remark 3.5. We have consdered suffcen condons for he exsence of hsorydependen ransfers ha mplemen he effcen allocaon. There exs oher weaker condons on he ranson probables. For example, each agen s perod- sgnal s > of oher agens. Formally, for each,, and,hereexsss>such ha for any sequence a a a s 1 s 1 τ= A τ, here does no exs a and a collecon of real numbers {ξ } \{ } such ha could be correlaed wh all fuure sgnals s ξ for all \{ } and 28 These wo assumpons are also generc n he nfne-horzon case f he ranson probables are saonary. 29 I hank an anonymous referee for suggesng hs dscusson.

16 81 Heng Lu Theorecal Economcs s s all s a a a s 1 = ξ s s, s a a a s 1 for where s s a a a s 1 s he condonal probably dsrbuon of s, gven and a a a s 1,.e., s s a a a s 1 = s s 1 a s 2 s 1 a s 2 s 2 s s s a s 1 s 1 If so, agen s ruh-ellng ncenve n each perod could be provded hrough all fuure repors of oher agens. An alernave suffcen condon, whch shares some smlares of Mezze s wo-sage VCG mechansm and guaranees he consrucon of our VCG-ype dynamc mechansm wh hsory-dependen ransfers, s ha each agen s perod- + 1 sgnal generaes an unbased predcon of hs realzed uly n perod,.e., for each and, here exss a funcon b : A R such ha u a = a b a A common feaure n he above suffcen condons s ha he ranson probables nvolve condonng on all agens prvae nformaon n perod ; hs s he crcal condon for perodc ex pos ncenve compably. Remark 3.6. By replacng he sequence of effcen allocaons wh an arbrary sequence of allocaon funcons, can be shown sraghforwardly ha n he nfnehorzon case, under he neremporal correlaon assumpon, any dynamc allocaon s perodc ex pos ncenve compable. Thus our possbly resuls for effcen desgn should be aken under he same cavea as he Crémer McLean mechansm: he resuls are somewha unrealsc and may sugges some lmaons of he mechansm desgn heory. In hs regard, our resuls could also be nerpreed as sronger negave resuls n dynamc mechansm desgn: enough neremporal correlaon of dfferen agens nformaon solves agens ncenve problems n a robus way. 3 Smlar o he Crémer McLean mechansm, our mechansms rely on he assumpons ha he ranson probables are common knowledge, here s no compeon on he desgner s sde, and agens are rsk-neural, have unlmed lably, and canno collude or defaul a he ex pos sage n each perod. Wheher hese assumpons are reasonable n dynamc envronmens depends on he parcular applcaons. Noneheless, our resuls pon oward an mporan channel, namely neremporal correlaon of prvae nformaon, hrough whch he desgner can fully explo he benefs from long-erm neracons among agens. 3 Noe ha perodc ex pos ncenve compably s weaker han ex pos ncenve compably. Thus our resuls do no conradc he negave resul n Jehel e al. 26.

17 Theorecal Economcs Envronmens wh ndependen valuaons Indrec mplemenaon wh aucons: An example In he prevous secon, we focused on drec dynamc mechansms o address feasbly ssues: he exsence of effcen dynamc mechansms ha are perodc ex pos ncenve compable. A naural queson s wheher here are ndrec mechansms, such as aucons, ha mplemen he drec mechansms. One dffculy of hs s ha he hsory-dependen ransfers n our drec mechansms are complex n general. Neverheless, he VCG aspec of he drec mechansms suggess a naural way for ndrec mplemenaon: sac aucons combned wh conngen ransfers. Here we presen a repeaed allocaon problem n whch no sac aucon forma s effcen, bu hsory-dependen ransfers faclae mplemenng our effcen drec mechansms wh famlar aucon formas. In every perod = 1 2, an ndvsble objec s o be allocaed o a bdder {1 2 N}. The allocaon a {1 2 N} deermneswhchbddergesheobjecn perod. Weassume habdder s valuaon of he objec n perod s symmerc and gven by v = + γ j j where γ> s a measure of nerdependence n valuaons. We also assume ha he allocaon does no affec he evoluon of agens prvae nformaon. Ths mples ha s effcen o allocae each objec o he agen wh an arbrary e-breakng rule whose valuaon of he objec s he hghes. Fnally, we assume ha for each,,and, here exss a map η : R such ha 1 N j j = η 2 Condon 2, whch s sronger han Assumpon 3, saes ha he average of all bdders prvae sgnals oday s an unbased esmaon of an ndex ha aggregaes all bu one bdder s sgnal omorrow. For nsance, hs condon holds when here s an unobserved sae of he world ω ha s a marngale process and agens sgnals are dencally dsrbued wh margnal dsrbuon ω such ha ω = ω and /N = ω. In hs case, we have η 1 = N 1 j Frs noe ha when γ 1], he sandard sngle-crossng condon on valuaons s sasfed; hs ensures ha he symmerc equlbrum of a repeaed sealed-bd secondprce aucon s effcen. 31 Alernavely, when γ>1, s well known ha no sandard aucon forma s effcen. 32 Applyng he nsgh from he drec mechansms 31 The generalzed VCG mechansm s also perodc ex pos ncenve compable when γ 1]. 32 Smlarly, here s no effcen and perodc ex pos ncenve compable sac mechansm. j

18 812 Heng Lu Theorecal Economcs wh hsory-dependen ransfers, we consder he followng dynamc wnner-pay aucon forma: Sep 1. Bdder subms a sealed bd b R n perod. Sep2.Theobjecs hen allocaed o he bdder who submed he lowes bd wh an arbrary e-breakng rule, a b 1 b N = mn { {1 N}:b b j j } Sep 3. The wnner n perod pays he second lowes bd n hs perod; oher bdders does no pay. Sep 4. The wnner also pays a conngen ransfer n perod + 1 ha depends on all oher bdders bds n boh perod and + 1. Formally, f bdder wns n perod,hepaysb j = mn{bk : k } n perod and r n perod + 1,whchsgven by r b b Nγ = [η δ b 1 + γn 1 b j ] 1 + γn 1 I s sraghforward o verfy ha a symmerc and monoone equlbrum n he consruced aucon s, for all and, b = 1 + γn 1. Moreover, hs symmerc sraegy profle remans an equlbrum of he dynamc aucon rrespecve of he bds or wnners ha he auconeer may choose o dsclose o some bdders. Remark 4.1. In he above mplemenaon resul, we have assumed symmery n bdder s valuaons so as o oban a symmerc equlbrum. The logc exends o he asymmerc valuaon case, alhough here s no symmerc equlbrum. For nsance, n he example n Secon 3.1, he sngle-crossng condon s volaed n he frs aucon; consequenly, o have an effcen equlbrum, frm A pays an amoun ha s ndependen of s bd n he frs aucon n hs case, s zero snce only frm A needs o subm a nonrval bd, and he ncenve o follow he equlbrum sraegy s provded from he conngen bonuses based on frm B s bd n he second perod. 5. Infne sgnal spaces In hs secon, we sudy he case where agens sgnal spaces are nfne and focus on he nfne-horzon seng T =. We frs denfy condons on he ranson probables under whch here exs mechansms ha are approxmaely perodc ex pos ncenve compable, hereby esablshng nfne-sgnal versons of Theorems 3.1 and 3.2 under a weaker soluon concep. We hen show ha under sronger condons here are mechansms ha are perodc ex pos ncenve compable. Suppose for each and, s he un nerval [ 1] endowed wh he Borel sgma algebra, A = A,whereA s a fne se, and u a s connuous n for each a A. 33 In addon, we assume ha he ranson probably a s saonary ndependen of and has a connuous densy represenaon f a. The margnal densy on s denoed by f a. 33 The resuls n hs secon hold when each s a compac and convex subse of an Eucldean space.

19 Theorecal Economcs Envronmens wh ndependen valuaons Approxmae perodc ex pos ncenve compably Frs consder a weakenng of perodc ex pos equlbrum, whch requres ha afer any hsory, ruh-ellng s almos a bes response f all oher agens repor ruhfully. Formally, for any ε>, we say ha he mechansm {a p } 1 s ε-perodc ex pos ncenve compable f for each,, h,and, u a p h [ + δe V h a ] u a r p h r [ + δe V h a r ] ε for any r,wherev h s he connuaon payoff of agen f all agen repor ruhfully from perod + 1 onward. The condon mples ha afer any hsory, any one-sho devaon from ruh-ellng would yeld an agen a mos ε mprovemen n hs connuaon payoff. Noe ha because of dscounng, f a mechansm s ε-perodc ex pos ncenve compable, hen ruh-ellng consss of a conemporaneous ε1 δ 1 -perfec ex pos equlbrum. In he followng wo lemmas, we denfy condons on he ranson denses f a such ha for every ε>, here exs ransfer schedules p ha are ε- perodc ex pos ncenve compable. Lemma 5.1. Fx any,, a,and. If for every and every, f a = f a d { } = 1 3 hen for any ε>, here exs ransfers ha are p ; a measurable n and connuous n and such ha max u j a δ p ; a f a d ε 4 j and for any r. p ; a f a d p r ; a f a d 5 Lemma 5.2. Fx any,, a,and. If here does no exs a nonzero sgned measure η on he Borel subses of such ha f a η d = 6

20 814 Heng Lu Theorecal Economcs hen for any ε>, here exs connuous ransfers p ; a such ha max u j a δ p ; a f a d ε 7 j The proofs of he resuls n hs secon are relegaed o Appendx B. Condon 3 n Lemma 5.1 s a drec exenson of McAfee and Reny 1992 o he dynamc case. Followng he proof of Theorem 3.1, mples ha here are ε-perodc ex pos ncenve compable ransfers of he form p : A R. The spannng condon n Lemma 5.2 s new. I guaranees he exsence of ε-perodc ex pos ncenve compable ransfers of he form p : A R. Smlar o he mechansms presened n Secon 3, each agen s almos a resdual claman and, hence, never gans by more han ε from msreporng n any perod. Fnally, we noe ha whou furher resrcons on he uly funcons and ranson probables, soluons o eher he nequaly sysem 4and5or7 may no exs for ε =. 34 In oher words, n general s unlkely o acheve -perodc ex pos ncenve compably wh conngen ransfers consdered n Lemmas 5.1 and 5.2. Inuvely, here may no be enough varaon of wh respec o n he densy f a o accoun for he varaon of n j hj a. However, our resuls show ha under eher condon 3or6, he ses of expeced values of all hese conngen ransfers are dense n he se of possble uly funcons, whch delvers ε-perodc ex pos ncenve compably. 5.2 Perodc ex pos ncenve compably The lemmasn Secon 5.1 generalze he man resuls n Secon 3. However, hey are no very sasfacory, especally n he dynamc envronmens. Tha s, agens may well devae from ruh-ellng under ε-perodc ex pos ncenve compably, ye hey evaluae her connuaon payoffs assumng ohers are always ruhful. In hs secon, we srenghen he resuls o full perodc ex pos ncenve compably under sronger correlaon condons. Noe ha he conngen ransfers ha delver ε-perodc ex pos ncenve compably n Secon 5.1 depend on he repors one perod ahead, whereas n prncple hey could depend on repors n he more dsan fuure see Remark 3.5. Therefore, we consder he conngen ransfers p : A τ A τ R τ> Inuvely, f agen s curren prvae sgnal s correlaed wh oher agens fuure sgnals {τ } τ>, hen provded ha oher agens always repor ruhfully, s possble o use he enre sequence, {τ } τ>, o provde ncenve for agen o repor ruhfully. To pu dfferenly, we mgh fll he gap n ε-ncenve compably wh an nfne sequence of correlaed sgnals. We formalze hs nuon n he nex wo proposons. 34 For nsance, when ε =, 7 reduces o a Fredholm negral equaon of he frs knd, whch may no have soluons.

21 Theorecal Economcs Envronmens wh ndependen valuaons 815 For each,, andτ>,lefτ τ a a τ 1 denoe he margnal densy on τ gven any a a τ and. Proposon 5.3. Fx any,, and. If for every τ>, a a τ A A τ,,and τ, f τ a a τ 1 = τ{ } = 1 f a a τ 1 d hen here exss a sequence of ransfers p τ τ ; a a τ 1 τ> measurable n and connuous n τ and such ha j u j a = τ= δ τ τ p τ τ ; a a τ 1 f τ τ a a τ 1 d τ and τ p τ τ τ ; a a τ 1 p τ τ r ; a a τ 1 f τ τ a a τ 1 d τ f τ τ a a τ 1 d τ for any r and τ>. Proposon 5.4. Fx any,, and. If for every τ>, a a τ A A τ, here does no exs a nonzero sgned measure η τ on he Borel subses of such ha fτ a a τ 1 η τ d = hen here exss a sequence of ransfers p τ τ ; a a τ 1 τ> measurable n and connuous n τ and such ha j u j a = τ= δ τ τ p τ τ ; a a τ 1 f τ τ a a τ 1 d τ Proposons 5.3 and 5.4 mply ha here are conngen ransfers under whch an agen becomes a resdual claman as n he VCG mechansm when her curren sgnal s correlaed wh ohers sgnals n he enre fuure. To provde an nuon of he resuls, frs noe ha he convex ndependence condon n Lemma 5.1 mples ha he closure of he se of funcons generaed by all one-perod-ahead conngen ransfers equals he se of all connuous funcons on he un nerval. Therefore, for any g C[ 1] and

22 816 Heng Lu Theorecal Economcs ε>, here s an nfne sequence of connuous funcons {h n } n=1 such ha for each n, here exss a measurable funcon p n s wh h n s = T p ns f sd, and sup gs s [ 1] n h n s ε 2 n m=1 Snce g s bounded, he nfne sum n=1 h n s well defned and equals g. Hence,forany fxed sequence of allocaons and n any perod, we can fnd a sequence of conngen ransfers, whch are used o provde ncenves for agens o repor ruhfully n ha perod. One suble dfference beween our consrucon and Crémer and McLean s mechansm s ha we use he assumpon ha for any gven allocaon a, he uly funcons are connuous n agens sgnals, whereas n Crémer and McLean s mechansm, agens ge zero payoff f beng ruhful and ge negave payoff f lyng. 6. Concludng remarks Dynamc mechansm desgn feaures a rcher famly of hsory-dependen ransfers compared wh he sac counerpar. Ths paper has aken a frs sep oward undersandng he mplcaons of such rchness on effcen mplemenaons n general envronmens wh nerdependen valuaons. In parcular, we have shown how neremporal correlaon of prvae nformaon leads o conngen ransfers ha resemble dynamc VCG mechansms. We also emphasze ha whle he heorecal possbly resuls n hs paper serve as a benchmark for he desgn of effcen mechansms, he praccaly of conngen ransfers may vary wh specfc economc problems. We conclude by nong ha he model can be exended o accommodae he possbly of arrval and deparure of poenal agens. In parcular, he neremporal correlaon condon can be generalzed sraghforwardly o hs case. Several new ssues need o be addressed. Frs, wh nerdependen valuaons, agens arrval and deparure would change boh he nformaon srucure and he uly funcons, snce each acve agen holds nformaon ha drecly affecs oher agens payoffs. Second, agens are requred o make conngen ransfers n he dynamc mechansms. Thus, ransfers o an agen may occur even f she s no longer acve. Ths may be problemac n some suaons where moneary ransfers have o be made along wh he physcal allocaons. Thrd, he arrval or deparure mes may also be agens prvae nformaon. 35 Moreover, here may be uncerany n arrval or deparure raes, whch furher complcaes he ncenve compably consrans. Appendx A: Proofs of he resuls n Secon 3 Theorems 3.1 and 3.2 consder boh he nfne-horzon and he fne-horzon cases. We prove Theorem 3.1 for he nfne-horzon case, usng he one-sho devaon prncple. Then we prove Theorem 3.2 for he fne horzon case, usng backward nducon. The proofs of he oher wo cases he fne-horzon case n Theorem 3.1 and he 35 See Gershkov e al. 215 and Merendorff 216 for examples.

23 Theorecal Economcs Envronmens wh ndependen valuaons 817 nfne-horzon case n Theorem3.2 followsmlarlnes and, herefore, arerelegaedo he Supplemenal Maeral. Proof of Theorem 3.1. Here we prove he nfne-horzon case; he proof for fne horzon case s gven n Secon S5.1 of he Supplemenal Maeral. The proof consss of hree lemmas. Lemma A.1. Suppose ha Assumpons 1 and 2 hold. For each and, here exss a ransfer funcon p r ; a such ha, for each a and, he followng wo condons are sasfed: For each, j u j a = δ p ; a a For each and r, p ; a a p r ; a a where a shemargnalof a on. Proof. Frs noe ha he frs par of Assumpon 2 s equvalen o he followng condon: for each,, a,and, and for each π :, π a = a π = 1 8 To see hs, suppose Assumpon 2 holds. If π : sasfes π a = a hen π = 1 and π = for all and. Therefore, π = 1. Conversely, suppose condon 9 holds bu he frs par of Assumpon 2 s volaed. Tha s, here exss a and a collecon of nonnegave numbers {ξ } such ha a = ξ a