Dynamic Efcient Auctions

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1 Dynamc fcent Auctons Drk Bergemann y and Juuso Valmak z ABSTRACT (xtended Abstract) We consder the truthful mplementaton of the socally e cent allocaton n a dynamc prvate value envronment n whch agents receve prvate nformaton over tme. We show that a sutable generalzaton of the Vckrey-Clark-Groves mechansm, based on the margnal contrbuton of each agent, leads to truthtellng n every perod. A leadng example of a dynamc allocaton model s the sequental aucton of a sngle good n whche current wnner of the object receves addtonal nformaton about her valuaton. We show that a mod ed sequental second prce aucton n whch only the current wnner makes a postve payment leads to truthtellng. In general allocatons problems, the margnal contrbuton mechansm contnues to nduce truthtellng n every perod but may now nclude postve transfers for many agents. Keywords ACM proceedngs, L A TX, text taggng Categores and Subject Descrptors H.4.m [Informaton Systems] Aucton Theory General Terms Aucton theory (Produces the WWW2007-spec c release, locaton and copyrght nformaton). For use wth www2007- submsson.cls V1.4. Supported by ACM. y Department of conomcs, Yale Unversty, New Haven, CT , U.S.A., drk.bergemann@yale.edu. z Department of conomcs, Helsnk School of conomcs and Unversty of Southampton, Helsnk, Fnland, juuso.valmak@hse.. Copyrght s held by the author/owner(s). WWW2007, May 8 12, 2007, Banff, Canada.. 1. INTRODUCTION The semnal analyss of second prce auctons by Vckrey (1961) establshed that sngle or multple unt dscrmnatory auctons can be used to mplement the socally e - cent allocaton n prvate value models n (weakly) domnant strateges. The subsequent contrbutons by Clarke (1971) and Groves (1973) showed that the nsght of Vckrey extends to more general allocaton problems n prvate value envronments. By requrng that the transfer payment of agent match her externalty cost on the remanng agents, agent nternalzes the socal objectve and s led to report her type truthfully. The resultng net utlty for agent corresponds to her margnal contrbuton to the socal value. In ths paper, we generalze the dea of a margnal contrbuton mechansm to dynamc envronments wth prvate nformaton. We desgn an ntertemporal sequence of transfer payments whch allow each agent to receve her ow margnal contrbuton n every perod. In other words, each agent wll pay her externalty cost n a tme consstent manner. In consequence, each agent s wllng to truthfully report her nformaton n every perod. The basc dea of the dynamc mechansm s rst explored n the context of the sequental allocaton of an ndvsble object wth ntally uncertan value to the bdders. We assume that the ntal estmate of the value s prvate nformaton to the bdder. In subsequent perods, a bdder receves addtonal nformaton only n those perods n whche object s allocated to her. The structure of the payo s n the model, and n partcular the resoluton of uncertanty, therefore resembles the mult-armed bandt problem. The rst result reports the constructon of a dynamcally e cent aucton that allocates the object n each perod accordng to the utltaran welfare crteron under symmetrc but mperfect nformaton. We show that a dynamc second prce aucton truthfully mplements the socally e cent allocaton perod by perod subject to Bayesan (and n fact even subject to ex post) ncentve constrants. The bandt framework consttutes a natural settng to analyze the repeated allocaton of an object or a lcense over tme. The key assumpton n the mult-armed bandt settng s that only the current user gans more nformaton about her valuaton of the object. If we thnk about the object as a lcense to use a faclty or to explore a resource for a lmted tme, t s natural to assume that the current nsder gans nformaton relatve to the outsders. A conceptual advantage of the sequental allocaton problem s that the structure of the socally e cent program s well understood. As the monetary transfers allow each agent to capture her margnal

2 contrbuton, the propertes of the socal program translate nto propertes of the margnal program. In the case of the dynamc aucton, we therefore obtan surprsngly explct and nformatve expressons for the ntertemporal transfer prces. The second result s the descrpton of a dynamc Vckrey- Clark-Groves mechansm n whch each agent receves n every perod her ow margnal contrbuton to the socal value. We obtan the second result for a general spec caton of the utlty of each agent and the arrval of prvate nformaton over tme. Throughout the paper we mantan the assumptons of quas-lnear utlty and of a prvate value envronment. The objectve of the dynamc mechansm s to mplement the socally e cent polcy. Wtransferable utltes, the socal objectve s smply to maxmze the expected dscounted sum of the ndvdual utltes. The soluton to ths dynamc optmzaton problem s by necessty tme consstent. In consequence, the dynamc Vckrey-Clark-Groves mechansm s tme consstent and the socal choce functon can be mplemented by a sequental mechansm wthout any ex ante commtment by the desgner. In contrast, n revenue maxmzng problems, the ratchet e ect leads to very dstnct solutons for mechansms wth and wthout ntertemporal commtment ablty (see Frexas, Guesnere, and Trole (1985)). In contrast to the statc envronment, the thruthtellng strategy n the dynamc settng forms an ex-post equlbrum rather than an equlbrum n weakly domnant strateges. The weakenng of the equlbrum noton s due to the dynamc nature of the game. If the connecton between other agents current announcements and ther mplcatons on the future contnuaton payo s s broken, then truthtellng s not necessarly ndvdually optmal. In recent years, a number of papers have been wrtten wthe am to explore varous ssues arsng n dynamc allocaton problems. Athey and Segal (2006) consder a nte tme horzon model wtransferable utltes and prvate values. Ther man result s the constructon of a balanced budget mechansm n the nte horzon allocaton model. Ther constructon of a rebalancng mechansm s based on a team mechansm n whche monetary transfers are pad only at the termnal perod and are equal to the sum of the other agents termnal utltes. In contrast, we desgn a sequence of transfers whch support the ow margnal contrbuton as the net utlty of each agent n every perod. In consequence we do not need a nte termnal tme to establsh the transfers. Bapna and Weber (2005) consder a sequental allocaton problem for a sngle, ndvsble object by a dynamc aucton. The basc optmzaton problem s a mult-armed bandt problem as n the aucton we dscuss here. Ther analyss attempts to use the Gttns ndex of each alternatve allocaton as a su cent statstc for the determnaton of the transfer prce. Whle the Gttns ndex s su cent to determne the e cent allocaton n each perod, the ndces, n partcular the second hghest ndex s typcally not a su cent statstc for the ncentve compatble transfer prce. Bapna and Weber (2005) present necessary and su - cent condtons when an a ne but report-contngent combnaton of ndces can represent the externalty cost. In contrast, we consder a drect mechansm and determne the transfers from general prncples of the ncentve problem. In partcular we do not requre any assumptons beyond the prvate value envronment and transferable utlty. In symmetrc nformaton envronments, Bergemann and Välmäk (2003), (2006) use the noton of margnal contrbuton to construct e cent equlbra n dynamc rst prce auctons. In ths paper, we emphasze the role of a tme-consstent utlty ow, namely the ow margnal contrbuton, to encompass envronments wth prvate nformaton. Ths paper s organzed as follows. Secton?? sets up the basc aucton model. Secton?? contans the constructon of the e cent dynamc aucton. Secton?? extends the constructon to general prvate value envronments. Secton?? concludes. 2. MODL Settng. We consder a dynamc aucton model n dscrete tme wth an n nte horzon. In every perod t; a sngle ndvsble object can be allocated to a bdder 2 f1; ; Ng. The true valuaton of bdder s gven by! 2 = [0; 1]. The pror dstrbuton about the valuaton! s gven by F (! ) and the dstrbutons are ndependent across bdders. In perod 0, bdder does not know the realzaton of!, nstead she receves an nformatve sgnal s 0 2 S = [0; 1] about her true value of the object. The sgnal s s generated by a condtonal dstrbuton functon G (s j! ). In each subsequent perod t, only the wnnng bdder n perod t 1 receves addtonal nformaton about her valuaton! n the form of an addtonal and condtonally ndependent sgnal s t 2 S from the condtonal dstrbuton G (s j! ). ach sgnal s t s prvate nformaton to bdder and s not observed by any other agent. 1 We denote the prvate hstory of bdder by = s 0 ; ; s t 1 The posteror belef of agent about! can be calculated by Bayes rule usng The expected value of the object for bdder gven hs prvate hstory s denoted by v =!. ach agent has quas-lnear utlty and the net value of gettng the object n perod t s v h t p t ; where p t s the transfer prce pad n perod t. ach agent dscounts the future wth a common dscount factor ; 0 < < Mechansm. A dynamc drect mechansm asks the bdders to report ther sgnals n every perod t. The report bs t may or may not be truthful. We de ne the ntal reports by b h 0 = bs 0 1; ; bs 0 N ; and nductvely the hstory of reports by b = b 1 ; bs t 1; ; bs t N. 1 We descrbe the arrval of new nformaton as a Bayesan samplng process. The equlbrum characterzaton n Theorem 1 would contnue to hold for any stochastc process, possbly non-markovan, provded that the sgnal realzatons are ndependent across agents and that sgnals only arrve for wnnng bdders.

3 The set of possble hstores of reports n perod t s denoted by b H t. The allocaton rule for a dynamc drect revelaton mechansm s x t b H t! [0; 1] N The allocaton n perod t s a vector x t = x t 1; ; x t N ; where x t denotes the probablty of assgnng the object to n t wth NX x t = 1. =1 The transfer (or prcng) rule s gven by p t H b t! R N D A dynamc mechansm M = x; p; H b s a trple where x = x t 1 ; p = p t 1 and b H = n bh t o 1 are the sequences of publc decsons and publc reports (hstores) qulbrum. The bdders evaluate payo s accordng to the dscounted expected payo crteron. A reportng strategy for agent s a mappng m t S! S. For a gven mechansm M, the expected payo for bdder from reportng a sequence bs = fbs t g of sgnals gven that the others are reportng bs = fbs t g s gven by 1X h t x t b 1 ; bs t ; bs t v p t b 1 ; bs t ; bs t Gven the mechansm M and the reportng strateges bs, the optmal reportng strategy of bdder solves a sequental optmzaton problem whch can phrased recursvely n terms of value functons, or max bs t 2S 8 < V ( b 1 ; ) = x t b 1 ; bs t ; bs t p t b 1 ; bs t ; bs t v + V b ; +1 9 = ; We say that the dynamc drect mechansm M s Bayesan ncentve compatble, f for every agent, n every perod t, truthtellng s a best response gven that all other agents report truthfully. In terms of the value functon, t means that for all and all t, the soluton to the dynamc programmng equaton max bs t 2S V ( 1 ) = x t 1 ; bs t ; s t v p t 1 ; bs t ; s t + V 1 ; bs t ; s t s to report truthfully,.e. to choose bs t = s t. Fnally, we say that the mechansm M s ex post ncentve compatble f truthtellng s a best response for agent regardless of the dstrbuton of sgnals of the other agents, or x t 1 ; bs t s 2 arg max ; s t v bs t 2S p t 1 ; bs t ; s t + V 1 ; bs t ; s t ; ; for all s t 2 S. In the dynamc context, ex post ncentve compatblty has to be qual ed n the sense that s ex post wth respect to all sgnals receved n perod t, but not ex post wth respect to sgnals arrvng after perod t. Consequently, the value functon V 1 ; bs t ; s t s stll the future expected value condtonal on 1 ; bs t ; s t. 3. DYNAMIC AUCTION We start wthe sngle good allocaton problem and show that t s possble to mplement the socally e cent allocaton n ex post equlbrum (and hence n Bayesan Nash equlbrum). The constructon resembles to some extent a second prce aucton n each perod. The transfer prce of the wnnng bdder s calculated n each perod by comparson to the optmal allocaton polcy wthn the set of bdders where the current wnner s excluded. As a result, the wnnng bdder nternalzes her e ect on the welfare of other bdders. The transfer prce of the loosng bdders wll be equal to zero provded that only the wnnng bdder receves addtonal nformaton. The exact constructon of the transfer prces follows the sprt of the Vckrey prcng, but the ntertemporal trade-o s are fully taken nto account Socal fcency. The socally e cent assgnment polcy s obtaned by maxmzng the utltaran welfare crteron, namely the expected dscounted sum of utltes. Gven a hstory of sgnals h s n perod s, the socally optmal program can be wrtten smply as W (h s ) = max fx t ( )g 1 t=s 1X t=s =1 NX t s x t v Alternatvely, we can represent the socal program n ts recursve form ( N ) W (h s ) = max X x s x s (h s (h s ) v (h s ) + W (h s ; x s ) ) =1 The expected value W (h s ; x s ) represents the optmal contnuaton value condtonal upon the state h s and the allocaton x s today. The socally optmal assgnment problem s a standard mult armed bandt problem and the optmal polcy s characterzed by an ndex polcy (see Gttns (1989) and Whttle (1982) for a textbook ntroducton). In partcular, we compute for every bdder the Gttns ndex based exclusvely on the nformaton about bdder. The ndex of bdder n state s the soluton to the followng optmal stoppng problem ( P ) = max s=0 s v +s P s=0 s The socally e cent allocaton polcy x = x t 1 s to choose n every perod a bdder wthe maxmal ndex x t > 0 f j j for all j Margnal Contrbuton. In the statc Vckrey aucton, the prce of the wnnng bdder s equal to the hghest valuaton among the loosng bdders. The hghest value among the remanng bdders

4 represents the socal opportunty cost of assgnng the object to the wnnng bdder. In a dynamc framework, the socal opportunty cost s determned by the optmal contnuaton plan n the absence of the current wnner. It s therefore useful to de ne the value of the socal program after removng bdder from the set of agents 1X X W (h s ) = max t s x t j v j j fx t (ht )g 1 t=s t=s j6= The margnal contrbuton M of bdder at hstory s then naturally de ned by M = W W (1) The margnal contrbuton s the change n socal value due to the addton of agent and hence the possblty of assgnng the object to. The margnal contrbuton of agent may be thought of as the nformaton rent that agent may be able to secure for herself n the drect mechansm. If bdder can secure her margnal contrbuton n a tme consstent manner, she should be able to receve the ow margnal contrbuton m n every perod. The ow margnal contrbuton accrues ncrementally over each perod M = m + M ; x t As n the notatons of the value functons above, M ; x t represents the margnal contrbuton of agent n the contnuaton problem condtonal on the hstory and the allocaton x t today. The ow margnal contrbuton can be expressed more drectly usng the de nton of the margnal contrbuton (1) as m = W W (2) W ; x t W ; x t Dynamc Second Prce Aucton. The ow margnal contrbuton s a natural canddate for the net utlty that each bdder should receve n each perod t. We now construct a transfer prce suchat under the e cent allocaton, each bdder s net payo concdes wth her ow margnal contrbuton. We then show that ths prcng rule makes truthtellng ncentve compatble n the dynamc mechansm. The wnnng bdder receves the object n perod t. To match her net payo to her ow margnal contrbuton, we must have m = v p (3) The remanng bdders, j 6=, do not receve the object n perod t and ther transfer prce must o set the ow margnal contrbuton m j = p j Consder rst the e cent bdder n perod t. We expand the ow margnal contrbuton n (2) by notng that s the e cent assgnment and that another bdder, say k, would consttute the e cent assgnment n the absence of bdder m = v v k k W ; W ; k (4) The optmal assgnment polcy s wthout loss of generalty a determnstc polcy as a functon of the hstory. We therefore replace the vector x t by the assgnment decson whch determnes the dentty of the wnnng bdder. Thus, n (4), W ; and W ; k represent the contnuaton value of the socal program wthout, condtonal on the hstory and the current assgnment beng or k respectvely. We notce that wth prvate values, the contnuaton value of the socal program wthout and condtonal on and gvng the object to agent n perod t s smply equal to the value of the program condtonal on alone, or W ; = W The addtonal nformaton generated by the assgnment to agent only pertans to agent and hence has no value for the allocaton problem once s removed. We can therefore rewrte the ow margnal contrbuton of the wnnng agent as m = v (1 ) W The ow margnal contrbuton of s therefore her expected ow value mnus the delay n the accrual of the socal bene t arsng from the optmal assgnment among all agents excludng agent. It follows that the transfer prce should smply be gven by p = (1 ) W, (5) whch s the ow socal opportunty cost of assgnng the object today to agent. A smlar analyss, based on the ow margnal contrbuton (4) leads to the determnaton of the transfer prce for the losng bdders. Consder a bdder j who should not get the object n perod t. Her ow utlty s clearly zero n perod t. Moreover, by the optmalty of the ndex polcy, the removal of alternatve j from the set of possble allocatons does not change the optmal assgnment today. In consequence, the dentty of the wnnng bdder does not depend on the presence of alternatve j. In other words the e cent assgnment to wll reman e cent after we remove j. As a result the ow margnal contrbuton of the loosng bdder s zero, and we have p j = m j = 0. Theorem 1 (Dynamc Second Prce Aucton). The socally e cent allocaton rule x s ex post ncentve compatble n the dynamc drect mechansm wthe payment rule p where p j (1 ) W = j f x t j = 1; 0 f x t j = 0 Proof. By the unmprovablty prncple, t s su cent to prove that f an agent receves n all future perods her margnal contrbuton as her contnuaton value, then truthtellng s ncentve compatble for an agent n perod t. Suppose then that at, t s socally e cent to assgn the object to agent and suppose that all agents except report truthful. The ncentve constrant for agent s then gven by p + M ; M ; j (6) v for some j 6=. By the determnaton of the transfer prce p, t follows that (6) can be wrtten as follows M M ; j (7)

5 and by de nton of the margnal contrbuton, we can rewrte (7) n terms of the socal value functons W W W ; j W ; j ; and expandng by v, we have W W v + W ; j v but then the result s W ; j ; W W ; j W W ; j (8) The nequalty (8) follows from the fact that the sze of the loss due to a suboptmal choce j (weakly) ncreases n the number of alternatves present. For the case of an ne cent agent j n perod t, we have M j v j j p j + M j ; j. (9) As the transfer prce s ndependent of the report of agent j, and gven by (5), we can rewrte (9) as follows M j v j j (1 ) W j + M j ; j. After replacng the margnal contrbutons by the socal value functons, we have W W j v j j (1 ) W j + W ; j W j ; j. But as W j ; j = W j, the terms nvolvng the value functons of j all drop out and we are left wth W v j j + W ; j, (10) whch s a vald nequalty snce j s by hypothess not the e cent choce n perod t. The ncentve compatble prcng rule has a few nterestng mplcatons. Frst, we observe that n the case of two bdders, the formula for the dynamc second prce reduces to the statc soluton. If we remove one bdder, the socal program has no other choce but to always assgn t to the remanng bdder. But then, the expected value of that assgnment polcy s smply equal to the expected value of the object for bdder j n perod t by the martngale probablty of the Bayesan posteror. In other words, the transfer s equal to the current expected value of the next best compettor. Wth more than two bdders, the socal program wthout bdder wll contan an opton value due to the possblty of assgnng the object to the more favorable bdder. In consequence the socal opportunty cost s hgher than the hghest expected valuaton among the remanng bdders. Second, we observe that the transfer prce of the wnnng bdder s ndependent of her own nformaton about the object. Ths means, that for any number of perods n whch the ownershp of the object does not change, the transfer prce wll stay constant as well, even thoughe valuaton of the object by the wnnng bdder may undergo substantal change. The desgn of the transfer prce pursued the objectve to matche ow margnal contrbuton of every agent n every perod. The determnaton of the transfer prce s based exclusvely on the reported sgnals of the other agents, rather than ther true sgnals. For ths reason, truthtellng s not only Bayesan ncentve compatble, but ex post ncentve compatble, f we qualfy ex post to mean condtonal on all sgnals receved up to and ncludng perod t. An mportant nsght from the statc analyss of the prvate value envronment s the fact that ncentve compatblty can be guaranteed n weakly domnant strateges. Ths strong result does not carry over nto the dynamc settng due to the nteracton of the strateges. In a dynamc settng, each agent can condton her strategy on the past reports of the other agents. In partcular, the strategy of truthtellng after all hstores fals to be a weakly domnant strategy as t removes the ablty to respond to past announcements. Yet our argument shows that the weaker condton of ex post ncentve compatblty can be sats ed. The vtal assumpton n the dynamc aucton model pertaned to the ow of nformaton ach bdder receves addtonal prvate nformaton n perod t + 1 f and only f she receved the object n perod t. Ths s the essental nformatonal hypothess n mult-armed bandt framework. Yet we mght be nterested n a settng n whch each bdder may learn more about the value of the object even n perods n whch she does not control the object. The ncentve analyss s agan based on the ow margnal contrbuton. But once we leave the bandt framework, then some loosng bdders may have to pay a postve prce even n perods n whchey do not receve the object. Consder a loosng bdder j and suppose that the removal of bdder j would change the e cent assgnment polcy from agent to agent k. The ow contrbuton of the loosng bdder j would now be equal to m j = v vk k + W j ; W j ; k < 0 In other words, f the presence of j changes the e cent assgnment polcy, then ths leads to an externalty cost created by agent j and hence strctly postve transfer prces even n perods n whch agent j does not receve the object. 4. GNRAL PRIVAT VALU NVIRON MNT In ths secton we extend the prvate value envronment from a sngle unt aucton to a general allocaton model. In addton, we substantally generalze the statstcal model of nformaton. The net expected ow utlty of agent n perod t s now determned by the ( ow) allocaton a t 2 A, the prvate hstory and the transfer prce p t v a t ; p t. The utlty functon v represents the expected utlty to agent from an allocaton a t gven the prvate nformaton. The set of avalable allocatons s gven by a compact and tme nvarant set A. The prvate sgnal of agent n perod t + 1 s generated accordng to a condtonal dstrbuton functon s t+1 G s t+1 a t ; We generalze the nformaton ow by allowng the sgnal s t+1 of agent n perod t+1 to be dependent on the current allocatve decson a t and the entre past hstory of prvate sgnals receved by agent. The allocaton rule for the drect

6 mechansm s now gven by x t b H t! (A) ; and the transfer rules are gven by p t b H t! R N As before, we denote the socally e cent polcy by x = D x t 1. The drect dynamc mechansm M = x ; p ; b H extends the Vckrey-Clark-Groves mechansm to general ntertemporal envronments by the margnal contrbuton argument as developed earler n the context of the sngle unt allocaton problem. Theorem 2 (Dynamc VCG Mechansm). The socally e cent allocaton rule fx g s ex post ncentve ncentve compatble wthe payment rule p p t x ; = m v x ; (11) Proof. The basc dea of the proof generalzes the margnal contrbuton argument n Theorem 1. By the unmprovablty prncple, t su ces to prove that f agent wll receve as her contnuaton value her margnal contrbuton, then truthtellng s ncentve compatble for agent n perod t, or v x ; v a; p t x ; + M x ;(12) p t a; + M ; a ; for all ; t and a 2 A. By constructon of the transfer prce, the lhs of (12) represents the margnal contrbuton of agent. Smlarly, we can express the contnuaton margnal contrbuton M ; a n terms of the values of the d erent socal programs W W (13) v a; p t a; + W ; a W ; a By constructon of the transfer prce, we can represent the prce that agent would have to pay f allocaton a were to be chosen n terms of the margnal contrbuton f the reported hstory were the true sgnal receved by agent. By constructon, we have as n (11) p t x ; = m v x ; The ow margnal contrbuton of agent s gven by m ; = IX j=1 v j so that the prce s gven by a; ; X j6= v j x ; (14) + W ; a W ; x p t = X v j x ; X v j a; ; (15) j6= j6= + W ; x W ; a We can now nsert the prces nto (13) to obtan W W v a; X v j x ; X v j a; ; + W ; x j6= + W ; a j6= But now we can reconsttute the entre expresson n terms of the socal value of the program wth and wthout agent and we are lead to the nal nequalty W W W ; a W ; where the later s true by the optmalty of x at. We observe that the prcng rule (11) for agent depends on the report of agent only throughe determnaton of the socal allocaton whch already appeared as a promnent feature n the statc envronment. Theorem 2 gves a general characterzaton of the transfer prces. In spec c envronment (such as a publc good provson model), we can then gan addtonal nsghts nto the structure of the e cent transfer prces by analyzng how the polces would change wthe addton or removal of an arbtrary agent. 5. CONCLUSION Ths paper suggest the constructon of a drect dynamc mechansm n prvate value envronments wtransferable utlty. The desgn of the monetary transfers reles on the notons of margnal contrbuton and ow margnal contrbuton. These notons allow us to transfer the nsghts of the Vckrey-Clark-Groves mechansm from a statc envronment to general dynamc settngs. In the case of the sequental allocaton of a sngle ndvsble object, we show that the noton of margnal contrbuton and ts relatonshp to the socal program allow us to gve explct solutons of the monetary transfers n each perod. Many nterestng questons are left open. The dynamc mechansm consdered here sats es the ncentve compatblty and ndvdual partcpaton constrants n every perod. In partcular, we do not requre that the monetary transfer satsfy a balanced budget condton n every perod. The recent analyss of Athey and Segal (2006) suggests that a sequental verson of AGV mechansm mght be able to acheve budget balancng n every perod as well. Ths paper s slent on the ssue of revenue maxmzng mechansms. In order to make progress n that drecton, a characterzaton of mplementable allocatons n dynamc settng wll rst be necessary. Fnally, we restrcted our attenton to prvate value envronments. A recent lterature, begnnng wth Maskn (1992) and Dasgupta and Maskn (2000) showed how to extend the VCG mechansm to nterdependent value envronments. In dynamc settngs, the sngle crossng condton wll then typcally nvolve a dynamc element whch wll ntroduce some complcatons. These tasks are left for future research. 6. ADDITIONAL AUTHORS Addtonal authors Juuso Valmak (Department of conomcs, Helsnk School of Busness and Unversty of Southampton) emal juuso.valmak@hse.f)

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