Games and Economic Behavior

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1 Games and Economc Behavor 73 ( Conens lss avalable a ScenceDrec Games and Economc Behavor Sequenal aucons wh randomly arrvng buyers Maher Sad Oln Busness School, Washngon Unversy n S. Lous, S. Lous, MO 6330, Uned Saes arcle nfo absrac Arcle hsory: Receved 3 Augus 00 Avalable onlne 9 January 0 JEL classfcaon: C73 D44 D83 Keywords: Dynamc markes Sequenal aucons Endogenous opons Random arrvals Sochasc equvalence Symmerc Markov equlbrum We analyze a dynamc marke n whch buyers compee n a sequence of prvae-value aucons for dfferenaed goods. New buyers and new objecs may arrve a random mes. Snce objecs are mperfec subsues, buyers values are no perssen. Insead, each buyer s prvae value for a new objec s a new ndependen draw from he same dsrbuon. We consder he use of second-prce aucons for sellng hese objecs, and show ha here exss a unque symmerc Markov equlbrum n hs marke. In equlbrum, buyers shade her bds down by her connuaon value, whch s he (endogenous opon value of parcpang n fuure aucons. We characerze hs opon value and show ha depends no only on he number of buyers currenly presen on he marke and he dsrbuon of her values, bu also on ancpaed marke dynamcs. 0 Elsever Inc. All rghs reserved.. Inroducon In hs paper, we consder a model of dynamc markes n whch buyers and sellers arrve randomly o he marke and examne he equlbrum behavor of marke parcpans n response o hese arrvals. In parcular, we nvesgae he role of curren and ancpaed fuure marke condons n deermnng endogenous ousde opons n hs seng. We characerze he nfluence of hese ousde opons on equlbrum prce deermnaon, and sudy he manner n whch equlbrum payoffs and behavor are affeced by boh curren condons and fuure dynamcs. In our model, buyers face an nfne sequence of aucons for heerogeneous bu sochascally equvalen objecs each buyer s prvae value for a new objec s a new ndependen draw from a fxed dsrbuon ha arrve a random mes. Moreover, new buyers probablscally arrve on he marke n each perod. Thus, n any gven aucon, buyers are presened wh he ousde opon of parcpang n a fuure aucon for an mperfecly subsued equvalen objec, bu wh a poenally dfferen number of compeors. We descrbe he unque symmerc Markov equlbrum of hs model by dervng a precse characerzaon of hs opon value. Essenally, losng n an aucon oday yelds he opporuny o parcpae n anoher aucon n he fuure; however, he poenal for enry by addonal buyers and he random arrval mes of aucons mples, n conras o much of he Ths paper s based on a chaper of my dsseraon a ale Unversy, and was subsanally revsed durng a year-long say a Mcrosof Research New England. I am especally graeful o Drk Bergemann, Johannes Hörner, Ben Polak, and Larry Samuelson for her generous advce and suppor. I would also lke o hank Marnx Amand, Lug Ballea, Alessandro Bona, Rahul Deb, Eduardo Fangold, and Coln Sewar for her many helpful commens and suggesons. Parcpans n he ale Mcro Theory Lunch, he 008 World Congress of he Game Theory Socey, and he 008 European Meeng of he Economerc Socey provded valuable feedback, as dd an anonymous advsory edor and referee. Ths work was generously suppored by a Naonal Scence Foundaon Graduae Research Fellowshp and a Cowles Foundaon Anderson Fellowshp. E-mal address: sad@wusl.edu /$ see fron maer 0 Elsever Inc. All rghs reserved. do:0.06/j.geb

2 M. Sad / Games and Economc Behavor 73 ( leraure on sequenal aucons, ha he compeve envronmen n he fuure may dffer sgnfcanly from he presen one. Thus, he expeced payoff of a buyer currenly presen on he marke s drecly lnked o her expeced payoffs wh a dfferen number of compeors presen n fuure perods. Ths leads o a dfference equaon characerzng he (endogenously deermned n equlbrum ousde opon avalable o each buyer. Whle hs dfference equaon adms, n general, an nfne number of soluons, we show ha here s a unque bounded soluon correspondng o he unque symmerc Markov equlbrum of he game. Ths mples ha each buyer s equlbrum ousde opon s, n fac, an appropraely dscouned sum of expeced payoffs from parcpang n each of he nfne sequence of aucons wh dfferng numbers of parcpans, where he wegh on each aucon s a combnaon of pure me dscounng and he lkelhood of marke dynamcs leadng o he correspondng sae. Our model of sequenal aucons capures several salen feaures of many real-world markes. Consder, for nsance, a buyer n he housng marke. When decdng on a bd o place on a parcular home, he buyer mus consder he marke dynamcs offers wll be affeced f a large number of homes are expeced o come ono he marke n he near fuure, or, alernaely, f fuure demand s ancpaed o be hgh due o (for nsance rapd job creaon n he regon. Moreover, houses are heerogeneous: alhough houses whn a parcular neghborhood are smlar, hey are ypcally suffcenly dfferenaed ha dfferen buyers wll rank hem (and value hem dfferenly. Therefore, homebuyers offers wll depend no only on he currenly avalable housng sock, bu also expecaons abou curren and fuure marke ghness. Smlarly, he model apples o sengs n whch frms parcpae repeaedly n pure common-values aucons; for nsance, frms ha bd n ol rghs aucons. Such frms have prvae esmaes of he value of a parcular rac, upon whch bds wll be based. Such bds wll no only ake no accoun he wnner s curse, bu also he frequency of fuure aucons for smlar racs and expecaons abou he compeveness of such aucons. In he presence of capacy consrans on he number of new projecs ha can be underaken smulaneously, accounng for he opon value of fuure aucons s a crucal par of he bd deermnaon process. The presen work follows n a long leraure of models examnng dynamc markes wh randomly arrvng buyers and sellers. Wolnsky (988 consders he seady sae of a dynamc marke where buyers are randomly mached o sellers. As n our model, buyers values are redrawn from he same dsrbuon whenever hey face a new seller; sellers, however, are long-lved and sraegc. De Fraja and Sákovcs (00 examne a smlar dynamc marke, wh he crucal dfference ha objecs are homogeneous, and hence buyers values do no vary over me. More recen conrbuons o he leraure nclude Saerhwae and Shneyerov (007, 008, who show ha, n he presence of perssen frcons, he seady-sae equlbrum arsng from forward-lookng bddng n a dynamc marke converges o he sac Walrasan equlbrum. These models are all focused on he analyss of he seady sae of dynamc markes, when he rae of enry o he marke s perfecly balanced wh he rae of ex. Our paper, on he oher hand, s concerned wh he behavor of agens n response o he changng marke condons of a dynamc envronmen. Ths s n lne wh work such as Taylor (995, whch examnes barganng power and prce formaon as relaes o he number of raders on each sde of a dynamc marke. He assumes ha agens are homogeneous all buyers have he same commonly known value for purchasng an objec and ha rade s conduced va posed prces. As n Coles and Muhoo (998, our model enrches hs seng by allowng for heerogeneous buyers and objecs. Insead of a combnaon of ulmaum barganng and Berrand compeon, however, we model he surplus-dvson process by explcly employng an aucon mechansm for prce deermnaon. In addon, we decouple he arrval of buyers and sellers. The presen work s also closely lnked o elemens of he heorecal leraure on sequenal aucons. Engelbrech- Wggans (994 sudes sequenal aucons n whch a fxed number of perfecly paen buyers wh sngle-un demand compee n a fne sequence of second-prce aucons for sochascally equvalen objecs. Hs model, however, does no allow for several feaures of he presen work; n parcular, does no allow for he enry of new buyers or consder he role of marke dynamcs n prce deermnaon. Jofre-Bone and Pesendorfer (003 assume sochascally equvalen objecs and allow for he enry of new poenal buyers, bu hese agens are shor-lved bdders who do no ake he fuure no accoun. Boh Budsh (008 and Zehammer (00 consder a smlar envronmen, resrced o wo perods (and wo objecs, and sudy he effcency properes of sequenal aucons under dfferen nformaon revelaon regmes. Sad (0 looks a a seng wh enry dynamcs smlar o he presen work, bu makes he complemenary (and oppose assumpon of ndependen prvae values ha are fxed across me, focusng on he desgn of opmal mechansms. As n he presen work, ousde opons are endogenously deermned; however, he presence of perssen prvae values nroduces an elemen of learnng ha s no presen heren.. The model We consder he connuous-me lm of an nfne-horzon dscree-me marke model; perods of lengh are ndexed by N. There s a fne number n of rsk-neural buyers wh sngle-un demand on he marke n any gven Inders (008 and Fuchs and Skrzypacz (00 also examne he non-seady-sae mpac of new buyer arrvals on he payoffs of buyers and sellers. However, hey do hs n a framework of blaeral (one-o-one barganng whch does no capure he dynamc naure of drec compeon beween several curren and fuure poenal marke parcpans. There s also a growng emprcal leraure examnng bddng n sequenal aucons on ebay. See, among ohers, Saler (008, Zehammer (006, Ingser (009, and Backus and Lews (00.

3 38 M. Sad / Games and Economc Behavor 73 ( perod. Snce he objecs under consderaon are dfferenaed and hence mperfecly subsuable, buyers have dfferen values for dfferen goods. Moreover, snce buyers are heerogeneous, her evaluaon of hese dsnc goods dffer from one anoher n he ermnology of Engelbrech-Wggans (994, objecs are sochascally equvalen. Ths also arses f objecs are common-value goods and each buyer receve a new unbased sgnal abou each objec. Thus, each buyer {,...,n } has a prvae valuaon v for he objec avalable a me, where v s ndependenly (across buyers and perods drawn from he dsrbuon F on R +. We assume ha F has fne varance and a connuous densy funcon f. Fnally, buyers dscoun he fuure exponenally wh dscoun facor δ = e r, where r > 0 s he dscoun rae. In each perod, here s a mos one seller presen. The arrval of sellers s sochasc; n parcular, here s some exogenously fxed probably p = λ, where λ>0, ha a new seller arrves on he marke n each perod. Smlarly, addonal buyers may arrve on he marke n each perod. For smplcy, we wll assume ha a mos one buyer arrves a a me, and ha hs arrval occurs wh some exogenously gven probably q = ρ, where ρ > 0. Noe, however, ha we assume ha hs arrval occurs mmedaely and wh probably f here s only a sngle buyer remanng on he marke from prevous perods here are always a leas wo buyers presen. In he even ha a seller has arrved on he marke, each buyer observes boh v and he number of compeng buyers presen. The seller hen conducs a second-prce aucon o allocae her objec. We assume ha sellers are nonsraegc hey are unable o se a reserve prce and canno reman on he marke for more han one perod. Conversely, buyers mus parcpae n each aucon ha akes place when hey are presen on he marke, bu may subm bds less han her rue values. 3 There are wo man advanages o assumng ha reserve prces are no used. Frs, we are able o avod he ssues of poolng (paral or oherwse ha can occur a he reserve prce when here are allocaonal exernales see, for nsance, Hale (000 and Jehel and Moldovanu (000. Second, equlbrum n he presence of reserve prces s characerzed by a nonlnear second-order dfference equaon. Solvng such a dfference equaon numercally, le alone analycally, s ypcally an exercse n fuly. And, compuaonal ssues asde, he lack of a closed-form soluon precludes many avenues of addonal nvesgaon, ncludng (for nsance sandard comparave sacs exercses. Ths s n sharp conras o, for nsance, Coles and Muhoo (998, who arrve a a lnear second-order (homogeneous dfference equaon n her model, n par due o her assumpons of bnary values and prce-seng va Berrand compeon. As s sandard, (n denoes he k-h hghes of n ndependen draws from F,whG (n and g (n denong he correspondng dsrbuon and densy funcons, respecvely. In addon, we wll defne, for all n k k k N, Ŷ (n := E [ (n ] [ (n ] E. Ths s he expeced dfference beween he hghes of n and n ndependen draws from F, where, by convenon, we le E[ (0 ]=0. I s useful o noe ha Ŷ (n s decreasng n n. 3. Symmerc Markov equlbrum Le V (v,n denoe he expeced payoff o a bdder when her valuaon s v and she s one of n bdders presen on he marke. Recall ha a seller mus be currenly presen on he marke for buyers o be aware of her valuaons. Furhermore, le W (n denoe he expeced value o a buyer when she s one of n buyers presen on he marke a he begnnng of a perod, before he realzaon of he buyer and seller arrval processes. A he begnnng of a perod when here are n buyers presen, here are four possble oucomes: wh probably pq, boh a buyer and a seller arrve, leadng o an aucon wh n + parcpans; wh probably p( q only a seller arrves, yeldng an aucon wh n parcpans; wh probably ( pq only a buyer arrves, leadng o he nex perod sarng wh n + parcpans; or, wh he remanng probably ( p( q, neher a buyer nor a seller arrve, leadng o he nex perod beng dencal o he curren one. Thus, for all n, W (n := pqe [ V ( v,n + ] + p( qe [ V ( v,n] + ( pqδw (n + + ( p( qδw (n. ( Le us now consder he problem facng buyer when here are n buyers on he marke and an objec s currenly avalable (and, hence, an aucon s abou o occur. Ths buyer, wh valuaon v, mus choose her bd b. If she wns he aucon, she receves a payoff of v less he second-hghes bd. On he oher hand, f she loses, she remans on he marke as one of n buyers omorrow, yeldng her a payoff of δw (n. Therefore, ( { ( V v,n = max Pr b b > { max b j} [ E v { } max b j b > { max b j} ] ( + Pr b < { max b j} } δw (n. We may use hs expresson n order o deermne equlbrum bd funcons, as demonsraed n he followng resul. Noe ha we focus on he unque symmerc Markov equlbrum of hs dynamc game. Oher equlbra ceranly exs; however, 3 I s easy o see ha, n equlbrum (gven a parcular dscoun facor, bds wll always be posve f he suppor of he value dsrbuon s suffcenly bounded away from zero.

4 M. Sad / Games and Economc Behavor 73 ( as all oher equlbra are ruled ou by nerm domnance-ype argumens, s naural o focus on he symmerc Markov equlbrum. Lemma (Equlbrum bds. In equlbrum, a buyer wh value v who s one of n buyers on he marke bds b = b (v,n,where b ( v,n := v δw (n. ( Proof. Noe ha, snce ( Pr b < { max b j} ( = Pr b > { max b j}, we may rewre V (v,n as { ( max Pr b b > { max b j} [ { } E v δw (n max b j b > { max b j} ] } + δw (n. As he ralng δw (n n he above expresson s merely an addve consan, he maxmzaon problem above corresponds o ha of a second-prce aucon wh n bdders n whch each bdder s valuaon s gven by v δw (n. The sandard domnance argumen for prvae value second-prce aucons hen mples ha b (v,n = v δw (n. Gven hs bddng sraegy and he symmery of connuaon payoffs across buyers n a symmerc equlbrum, he probably ha wns he perod- aucon s smply Pr(v > max {v j }, and her surplus n hs case becomes v max {v }. Therefore, we may rewre V as j ( ( V v,n = Pr v > { max v j} [ E v { } max v j v > { max v j} ] + δw (n = ( Pr (n [ < v E v (n (n < v] + δw (n = G (n ( ( v v E[ (n (n = v] + δw (n, (3 where he fnal equaly follows from well-known properes of order sascs see, for example, Davd and Nagaraja (003, Chaper 3. Noe ha, ex ane, any one of he n buyers presen on he marke n any perod s equally lkely o have he hghes value amongs her compeors (and hence wn he objec. We may use hs fac, along wh he resul above, o show ha he ex ane expeced uly of a buyer when here s an objec avalable for sale s smply he sum of her probably of wnnng he objec mulpled by her expeced payoff (condonal on wnnng and he dscouned opon value of losng he objec and remanng on he marke n he nex perod. Formally, we are able o prove he followng resul. Lemma (Expeced aucon payoffs. The expeced payoff o a bdder from an aucon wh n parcpans s E [ V ( v,n] = Ŷ (n + δw (n. (4 Proof. Recall ha Eq. (3 provdes an expresson for V (v,n. Takng he expecaon of hs expresson wh respec o v yelds E [ V ( v,n] = ( = n (( [ (n x E (n ] = v (n G (x + δw (n f (x dx Noce, however, ha ( xg (n n (x dx xg (n (x dx E [ (n (n = v E [ (n (n = v ] (n g (x dx = n ] (n g (x dx + δw (n. ( E [ (n ] E [ (n Ths s exacly he quany prevously defned as Ŷ (n, mplyng (as desred ha ] = E [ (n ] E [ (n ]. E [ V ( v,n] = Ŷ (n + δw (n.

5 40 M. Sad / Games and Economc Behavor 73 ( Wh hs resul n hand, we may rewre Eq. ( for n nermsofw and Ŷ alone: δpq δ( p( q p( q W (n + = W (n δ( pq ( pq W (n p p( q Ŷ (n + Ŷ (n. δ( p δ( pq Recall, however, ha a sngle buyer remanng from perod wll always be joned by anoher buyer a me +. Thus, when n =, we have W ( = ( p Ŷ ( + δw ( pŷ ( + δ( pw ( + ( pδw ( =. δp Thus, a buyer s expeced payoff s gven by a soluon o he second-order nonhomogeneous lnear dfference equaon and boundary condon n he wo equaons above. Whle one could solve hs sysem drecly, he connuous-me lm s sgnfcanly more racable. Recallng ha δ = e r, p = λ, and q = ρ and akng he lm as he perod lengh goes o zero yelds W (n + = r + λ + ρ W (n λ (Ŷ (n + W (n for all n, (5 ρ ρ and W ( = W (. (6 We may hen rewre hs second-order dfference equaon as a frs-order sysem of dfference equaons. In parcular, we have, for all k > 0, ( [ r+λ+ρ W (k + ρ λ ]( ( ρ W (k + λ = + ρ Ŷ (k +. (7 W (k + 0 W (k 0 Noe ha, n general, here are an nfne number of soluons o hs dfference equaon, each of whch s paramerzed by he values of W ( and W (; mposng he boundary condon n Eq. (6 reduces hs o a connuum of possble soluons. However, we are able o rule ou soluons n whch expeced uly dverges o nfny as he number of buyers on he marke grows here exss a unque bounded (and hence sensble soluon o he dfference equaon. To characerze hs soluon, we defne ζ := r + λ + ρ (r + λ + ρ 4λρ ρ and ζ := r + λ + ρ + (r + λ + ρ 4λρ. ρ These wo consans are he egenvalues of he ranson marx n Eq. (7. I s sraghforward o show ha ζ ζ = λ ρ and 0 <ζ < <ζ for all r,λ,ρ > 0. Theorem (Unque symmerc Markov equlbrum. The unque symmerc Markov equlbrum of hs nfne-horzon sequenal aucon game s characerzed by he ex ane expeced payoff funcon gven by W ( = ζ ζ ζ and, for all n, k= Ŷ (k + (8 W (n = ζ n (ζ W ( + ζ n ζ n Ŷ (k + + ζ ζ n ζ ζ ζ ζ ζ ζ Proof. Noce ha we may rewre Eq. (7 as where w n+ = Aw n + y n, k= k=n Ŷ (k +. (9 w m := ( W (m +, W (m and y m := ( ζ ζ Ŷ (m +, 0 for all m N, and A s he ranson marx from Eq. (7. Elayed (005, Theorem 3.7 yelds he general soluon n w n = A n w + A n k y k. k=

6 M. Sad / Games and Economc Behavor 73 ( Snce ζ and ζ are he egenvalues of he marx A, s sraghforward o show ha A k = [ ζ k+ ζ k+ ζ k+ ζ ζ ζ k+ ] ζ ζ ζ k ζ k ζ k ζ. ζ ζ k Combnng hs wh he mposon of he boundary condon of Eq. (6 allows us o (afer some rearrangemen and smplfcaon wre he soluon o hs second-order sysem as W (n = ζ ( n n ( ζ W ( ζ ζ Ŷ (k + ζ ζ k= + ζ ( n n (ζ W ( + ζ ζ Ŷ (k +. (0 ζ ζ k= ζ Consder he second erm n he above expresson. Snce 0 <ζ < n, ζ ζ (ζ W ( approaches 0 as n ncreases. Moreover, ζ n ζ n Ŷ (k + = ζ n ζ k Ŷ (n k + ζ ζ ζ ζ k= k= ζ n ζ k ζ ζ σ < ζ ζ ζ k= ζ kσ < k= ζ ζ σ (ζ ζ ( ζ, where σ s he (assumed fne varance of he dsrbuon F. Ths follows from Arnold and Groeneveld (979, who show ha E [ (m ] [ E (m] mσ for all m >. m Recallng he defnon of Ŷ,wehenhave Ŷ (m = ( [ (m] [ (m] σ E E < σ m m for all m >, mplyng ha he second erm n Eq. (0 s bounded. The frs erm n Eq. (0, however, s mulpled by posve powers of ζ >, mplyng ha an approprae choce of W ( s crucal for ensurng he boundedness of our soluon. One such choce s o le W ( = w := ζ ζ ζ k= Ŷ (k +. Noe ha w s well-defned, as ζ > and Ŷ (m <σ for all m >. Moreover, snce ζ <, w > 0. The frs erm n Eq. (0 hen becomes ζ ζ n ζ ζ k=n Ŷ (k + = ζ ζ ζ k=n ζ n k Ŷ (k + = ζ ζ ζ k=0 Ŷ (n + k + <, where we agan rely on he fac ha Ŷ (m<σ for all m > when F has fne varance. Thus, seng W ( = w and mposng he boundary condon W ( = W ( leads o a bounded soluon (whch we wll denoe by W (n of he dfference equaon defned n Eq. (7. To show ha hs s he unque bounded soluon, fx any arbrary α R, and consder he soluon where W ( = W ( = αw. The soluon n Eq. (0 becomes W (n = ζ ( n n ( ζ αw ζ ζ ζ ζ k= = ζ ( n n ( ζ w ζ ζ ζ ζ + ζ n ζ ζ k= ( n (ζ w + ζ ζ Ŷ (k + + ζ ( n ζ ζ Ŷ (k + + ( ζ (α w k= = W (n + ζ n ( ζ + ζ n (ζ ζ ζ (α w. Ŷ (k + + (ζ (α w n (ζ αw + ζ ζ k= Ŷ (k +

7 4 M. Sad / Games and Economc Behavor 73 ( Snce ζ > >ζ > 0 and w > 0, he above expresson remans bounded f, and only f, α =. Thus, he unque bounded soluon o he sysem of dfference equaons n Eq. (7 sasfyng he boundary condon n Eq. (6 s pnned down by he condon W ( = w. The expresson n Eq. (9 follows mmedaely. Thus, buyers expeced payoffs n he unque symmerc Markov equlbrum of hs dynamc aucon marke are a dscouned sum of expeced payoffs from parcpang n each elemen of he nfne sequence of aucons wh dfferng numbers of bdders. The wegh on each aucon s a combnaon of pure me dscounng and he lkelhood of marke dynamcs leadng o he correspondng sae. Noe ha hs soluon s easly generalzed o radng nsuons oher han he sequenal second-prce aucon. By revenue equvalence, Ŷ (n s he ex ane expeced payoff of a buyer n any effcen one-sho aucon mechansm wh n buyers. Therefore, Eq. (9 connues o hold, as s, for markes n whch objecs are sold va, for nsance, sequenal frs-prce aucons whle buyers bds wll dffer, her expeced payoffs wll be as n our heorem above. On he oher hand, f rade occurs va a dfferen nsuon, hen replacng Ŷ (n by he approprae ex ane expeced payoff of a buyer n ha mechansm would yeld he correspondng soluon for ha nsuon. For example, suppose ha each seller employs some (fxed mul-laeral barganng game for allocang her objec. Leng Ỹ (n denoe he ex ane expeced payoff o each of n buyers from parcpang n he one-sho barganng game, equlbrum n he nduced dynamc marke s hen characerzed by he analogue o Eq. (9 where Ŷ s replaced by Ỹ. In addon, s possble o generalze he model o he case n whch marke dynamcs and he dsrbuon of buyers values evolve accordng o a (known Markov process. Ths corresponds o, for nsance, he case n whch here are buyer s and seller s markes wh relavely hgher arrval raes of sellers and buyers, respecvely. The analyss s analogous o he case we consder above, bu wh a coupled sysem of dfference equaons lnkng he expeced payoffs across he wo saes of he world. A smlar unqueness resul and characerzaon of expeced payoffs follows mmedaely. One addonal advanage of fndng he closed-form soluon descrbed n our heorem s ha enables clean comparave sacs. In parcular, we are able o analycally characerze he response of payoffs o changes n each of he man parameers. Corollary (Comparave sacs. For all n, W(n>W (n +, W r (n<0, W λ (n>0, andw ρ (n<0, wherew x ( denoes W ( x. Proof. Avalable on reques. As s o be expeced, buyers expeced payoffs are decreasng n he compeveness of he marke he arrval of an addonal buyer leads o a decrease n each buyer s surplus. Smlarly, expeced payoffs are decreasng n he arrval rae of new buyers (as hs ncreases ancpaed compeveness, bu are ncreasng n he dscoun rae (as buyers become more paen, he poenally nfne sream of aucons becomes more appealng and he arrval rae of sellers (as aucons become more frequen, compeon wll be hnned ou more quckly. I s easy o see ha he reverse s rue n each of hese cases when consderng sellers expeced revenue Concluson Ths paper characerzes he manner n whch curren marke condons, as well as ancpaed fuure condons, creae an endogenous opon value for bdders n a dynamc marke. Snce buyers mus rade off purchasng n he presen agans parcpang n he fuure, he value of hs fuure opon s crucal for curren-perod bddng; however, he value of he opon s self deermned by equlbrum bddng behavor. We prove ha here exss a unque symmerc Markov equlbrum, and show ha he endogenous opon value assocaed wh s, n fac, he expeced dscouned sum of he poenal payoffs from ndvdual ransacons n he nfne sequence of possble saes of he world, each dfferenaed by he poenal number of buyers presen on he marke a ha me. When he radng nsuon s an aucon mechansm, buyers are herefore wllng o bd her rue values less hs dscouned opon value of parcpang n he fuure sequence of aucons. There are several drecons for exendng our analyss. One possbly s droppng he assumpon of sochasc equvalence and endowng each buyer wh a fxed prvae value for obanng each objec. Zehammer (00 explores hs queson when he values may dffer across objecs. There are several echncal dffcules n conducng such an analyss n a model wh sealed-bd aucons when values are perfecly correlaed across objecs, however. These complcaons are dscussed n Sad (0. In parcular, he sequenal second-prce aucon s no effcen n he presence of buyer arrvals, so nsead we examne a model n whch objecs are sold usng he ascendng aucon forma. Anoher poenally neresng lne of research nvolves allowng for mulple smulaneous aucons, or allowng sellers o reman on he marke for several 4 Recall ha a buyer wh value v who s one of n buyerspresenwllbdb (v, n = v W (n. Ths mples ha a seller facng n buyers receves an expeced revenue of Π(n := E[ (n ] W(n. The saemen n he ex follows mmedaely from hs observaon.

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