Public vs. Private O ers in the Market for Lemons (PRELIMINARY AND INCOMPLETE)

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1 Public vs. Pivate O es in the Maket fo Lemons (PRELIMINARY AND INCOMPLETE) Johannes Höne and Nicolas Vieille Febuay 2006 Abstact We analyze a vesion of Akelof s maket fo lemons in which a sequence of buyes make o es to a long-lived selle endowed with a single unit fo sale. We conside both the case in which pevious o es ae obsevable and the case in which they ae not. When o es ae obsevable, tade may only occu in the st peiod, so that the esulting ine ciency may be wose than in the static model. In the unobsevable case, tade occus with pobability one eventually. Intoduction In this pape, we eamine the elationship between the outcome of tade with asymmetic infomation and the obsevability of past o es. Seach models typically assume that successive potential buyes neve lean anything ove time about the o eing of the selle, so that the distibution of o es faced by the selle is stationay. On the othe hand, bagaining models usually assume that potential buyes obseve the entie public histoy, including past o es that wee ejected. This a ects the buyes beliefs about the quality of the good that is being put on sale, and theefoe the o es that ae submitted. While most makets chaacteized by advese selection fall between those two etemes, they widely vay in this espect. In the housing maket, potential buyes typically know the time on maket, as well as the list pice that is quoted by the selle. Buyes of second-hand cas do not usually have any eliable infomation egading what o es and how many o es have been tuned down by the selle. Employes may obtain vei able infomation about the duation of unemployment of potential employees, but much less evidence egading o es they may have ejected in the meantime. In yet othe makets, it is up to the selle to decide e ante whethe his decisions will be public o pivate. We conside two vaiants of Akelof s maket fo lemons. In both vaiants, a su ciently patient single selle with pivate infomation faces a sequence of (shot-un) potential buyes who

2 submit o es. Buyes know how long the item has been put on sale. In the st vaiant, buyes also know the o es that wee ejected, while in the second those o es emain unobsevable. All that is known by a potential buye is that all the pevious o es must have been tuned down. Results contast shaply. In the case of public o es, only the st potential buye submits a seious o e. That is, only the st o e is accepted with positive pobability. In case this o e is tuned down by the selle, as does occu if his valuation is high enough, all late o es ae losing o es: fom that point on, the selle ejects all equilibium o es. Theefoe, meely allowing fo tade ove time does not solve the advese selection poblem. In fact, it may wosen it. Not so, howeve, in the case of pivate o es. In evey equilibium, tade occus with pobability one eventually, and thee always eist equilibia in which tade occus in nite time. This stiking contast can be undestood as follows. Conside the case of public o es. Suppose that an out-of-equilibium high o e is submitted. Because tuning it down is intepeted as vey favoable news by the net buye, whose o e following such a deviation will be seious, the selle accepts this high o e only if his valuation is especially low. That is, the pespective of selling at a highe pice tomoow eacebates the advese selection poblem today, making the high o e unattactive to the selle. While this intuition helps eplain why it is equilibium behavio fo all potential buyes but the st one to submit losing o es, it is woth pointing out that we show that this is the unique equilibium. Conside net the case of pivate o es. Fo the sake of contadiction, suppose that fom some point on, all equilibium o es ae losing. Because o es ae unobsevable, the behavio of futue biddes is not a ected by a deviation by the cuent bidde. Theefoe, a potential buye would stictly pefe an o e slightly below his wost-case valuation fo the good to his equilibium o e, as the fome always implies a stictly positive po t conditional on tade, and it is necessaily accepted by the selle with some positive pobability. In tems of actual payo s, compaisons ae less clea-cut. Fo instance, the dynamic game with public o es may actually be moe o less e cient than the static game, depending on the paametes. As we ague, the low pobability of tade in the dynamic model is diven by competition among potential buyes. Somewhat paadoically, if thee is a unique, long-lived and equally patient potential buye, the good is taded with pobability one in the unique equilibium outcome. To shed moe light on the elationship between the static game and the in nite-hoizon game, we povide a detailed analysis of the game with nite, but abitay, hoizon. While it is possible to eplicitly solve fo equilibium stategies in the case of pivate o es, the case of public o es is moe comple, and we povide only a patial chaacteization of the equilibium stategies. We show by means of speci c eample that equilibium multiplicity can occu, and we pove that, quite geneally, all potential buyes but the st and the last ones must use mied stategies. Eplicit solutions ae available in some special cases: (i) the case of two peiods, (ii) the case of two values, (iii) some paticula anges of paametes. 2

3 . Related Liteatue Ou contibution is elated to thee stands of liteatue. Fist, seveal authos have aleady consideed dynamic vesions of Akelof s model. Second, ou model shaes many featues with the bagaining liteatue. Finally, a pai of papes have investigated the di eence between public and pivate o es in the famewok of Spence s educational signaling model. Janssen and Roy (2002) conside a dynamic, competitive duable good setting, with a ed set of selles. They pove that all tade fo all qualities of the good occus in nite time. While thee ae seveal inessential di eences between thei model and ous, the citical di eence lies in the maket mechanism. In thei model, the pice in evey peiod must clea the maket. That is, by de nition, the maket pice must be at least as lage as the good s epected value to the buye conditional on tade, with equality if tade occus with positive pobability (this is condition (ii) of thei equilibium de nition). This epected value is deived fom the equilibium stategies when such tade occus with positive pobability, and is assumed to be at least as lage as the lowest unsold value even when no tade occus in a given peiod (this is condition (iv) of thei de nition). This immediately implies that the pice eceeds the valuation to the lowest quality selle, so that tade must occu eventually, if not in a given peiod. Also elated ae Taylo (999), Hendel and Lizzei (999), and Hendel, Lizzei and Siniscalchi (2005). In the bagaining liteatue, the closest pape is Evans (989), which shows that, with coelated values, the bagaining may esult in an impasse when the buye is too impatient elative to the selle. Ou assumption of shot-un buyes is less geneal, since it implies that the buye is actually myopic. 2 Howeve, Evans consides the case of binay values. Moeove, thee is no gain fom tade in case of a low value. In ou set-up, Poposition holds instead quite geneally, and tade is always stictly e cient. In his appendi, Vincent (989) povides anothe eample of equilibium in which bagaining beaks down. As in Evans, thee ae only two possible values in his object. While thee ae gains of tade fo both values in his case, it is not known whethe his eample admits othe equilibia, potentially ehausting all gains of tade eventually. Othe elated contibutions include Camton (984), Gul and Sonnenschein (988) and Vincent (990). Othe elated contibutions include Camton (984), Gul and Sonnenschein (988) and Vincent (990). Nöldeke and van Damme (990) and Swinkels (999) develop an analogous distinction in Spence s signalling model. Both conside a discete-time vesion of the model, in which education is acquied continuously and a sequence of shot-un ms submit o es that the woke can eithe accept o eject. Nöldeke and van Damme consides the case of public o es, while Swinkels focuses mainly on the case of pivate o es. Nöldeke and van Damme shows that thee is a unique equilibium outcome that satis es the neve a weak best esponse equiement, and that Moe pecisely, equality obtains wheneve thee is a positive measue of goods qualities taded, since thee is a continuum of selles in thei model. 2 Thee is no di culty in genealizing Poposition to the case of an impatient, but not myopic buye, but we feel that thee is no much gain fom such geneality. 3

4 the equilibium outcome conveges to the Riley outcome as the time inteval between consecutive peiods shinks. With pivate o es, Swinkels poves that the sequential equilibium outcome is unique, and shows that, in contast to the public case, it involves pooling in the limit. While the set-up is athe di eent, the logic diving these esults is simila to ous, at least with public o es. Indeed, in both papes, when o es ae obsevable, ms (buyes) ae deteed fom submitting mutually bene cial o es because ejecting such an o e sends a stong signal to futue ms (buyes) that is so attactive that only vey low types would pefe to accept the o e immediately. 2 The model We conside a dynamic game between a single selle, with one unit fo sale, and a countable in nity of potential buyes, o buyes fo shot. Time is discete, and indeed by n = ; : : : ;. At each time n, one buye makes an o e fo the unit. Each buye makes an o e only at one time, and we efe to buye n as the buye who makes an o e in peiod n, povided the selle has accepted no pevious o e. Afte obseving the o e, the buye eithe accepts o tuns down the o e. If the o e is accepted, the game ends. If the o e is tuned down, a peiod elapses and it is the net buye s tun to submit an o e. The (esevation) value of the unit is selle s pivate infomation. The esevation value to the selle is c (), whee the andom vaiable is detemined by natue and unifomly distibuted ove the inteval [; ] ; 2 [0; ). We intepet as an inde, such as the quality of the good. The valuation of the unit to buyes is common to all of them, and is denoted v (). Buyes do not obseve the ealization of, but its distibution is common knowledge. We assume that c is stictly inceasing, positive and twice di eentiable, with bounded deivatives. We assume that v is positive, stictly inceasing and di eentiable, with bounded deivative. We set M c 0 = sup jc 0 j, M c 00 = sup jc 00 j, M v 0 = sup jv 0 j, M = ma(m c 0; M c 00; M v 0), and m = inf jv 0 j. Obseve that the assumption that is unifomly distibuted is with little loss of geneality, since few estictions ae imposed on the functions v and c. 3 We assume that gains fom tade ae always positive with := inf fv () c ()g > 0. In eamples and etensions, we shall often estict attention to the case in which v () = and c () =, with > 0, i.e. the esevation value to the selle is a faction 2 (0; ) of the valuation to the buyes. The selle is impatient, with discount facto <. We ae paticulaly inteested in the case in which is su ciently lage. To be speci c, we set = m=3m, and will always assume >. In each peiod in which the selle owns the unit, he deives a pe-peiod goss suplus of ( ) c (). Theefoe, the selle can always guaantee a goss suplus of c () by neve selling the unit. zeo. 3 In paticula, ou esults ae still valid if the distibution of has a bounded density, bounded away fom 4

5 Buye n submits an o e p n that can take any eal value. An outcome of the game is a tiple (; n; p n ), with the intepetation that the ealized inde is, and that the selle accepts buye n s o e of p n (implying that he ejected all pevious o es). The case n = coesponds to the outcome in which the selle ejects all o es (set p equal to zeo). The selle s von Neumann-Mogensten utility function ove outcomes is his net suplus: Xn ( ) i c () + n p n c () = n (p n c ()) ; i= when n <, and zeo othewise. An altenative fomulation that is equivalent to the one above is that the selle deives no pe-peiod goss suplus fom owning the unit, but incus a poduction cost of c () at the time he accepts the buye s o e. It is immediate that this intepetation yields the same utility function. Buye n s utility is v () p n in the outcome (; n; p n ), and zeo othewise. We de ne the playes epected utility ove lotteies of outcomes, o payo fo shot, in the standad fashion. We allow fo mied stategies on the pat of all playes. We conside both the case in which o es ae public, o obsevable, and the case in which pevious o es ae pivate, o not obsevable. It is woth pointing out that the esults fo the case in which o es ae public would also hold fo any infomation stuctue (about pevious o es) in which each buye n > obseves the o e made by buye n. A histoy h n 2 H n in case no ageement has been eached at time n is a sequence (p ; : : : ; p n ) of o es that wee submitted by the buyes and ejected by the selle (we set H 0 equal to f?g). A behavio stategy fo the selle is a sequence ( n S ), whee n S is a pobability tansition fom [; ] H n R in to faccept; Declineg mapping the ealized valuation v, the histoy h n, and buye n s o e p n into a pobability of acceptance. In the public case, a stategy fo buye n is a pobability tansition n B fom Hn to R. 4 In the pivate case, a stategy fo buye n is a pobability distibution n B ove R. Obseve that, whethe o es ae public o pivate, the selle s optimal stategy must be of the cut-o type. That is, if n S (; hn ; p n ) assigns positive pobability to Accept fo some v, then n S (0 ; h n ; p n ) assigns pobability one to Accept, fo all 0 >. The poof of this skimming popety can be found in Fudenbeg and Tiole (Chapte 0, Lemma 0.), fo instance. The in mum of such valuations is called the maginal valuation (at histoy (h n ; p n ) given the stategy po le). Since the speci cation of the action of the selle with maginal valuation does not a ect payo s, we also identify equilibia which only di e in this egad. Fo de niteness, in all fomal statements, we shall follow the convention that the selle with maginal valuation ejects the o e. Fo conciseness, we shall omit to specify that some statements only hold with pobability one. Fo instance, we shall say that the selle accepts the o e when he does so with 4 That is, fo each h n 2 H n, n B (hn ) is a pobability distibution ove R, and the pobability n B ()[A] assigned to any Boel set A R is a measuable function of h n 5

6 pobability one. Standad aguments also establish that buyes neve submit any o e that is stictly lage than c () = c, the highest possible esevation value to the selle. We use the pefect bayesian equilibium concept. In the public case, we will compute the belief of buye n afte a (possibly out-of-equilibium) histoy h n unde the assumption that the selle ejected pevious o es out of ational puposes. 5 Thus, the belief of buye n ove is the unifom distibution ove some inteval [ n ; ]. In the pivate case, the only infomation sets which ae eached with pobability zeo ae associated with stages fo which the pobability is one that the selle will have accepted some ealie o e. In such a stage, we assume that buye n s belief assigns pobability to =. Given some (pefect bayesian) equilibium, a buye s o e is seious if it is accepted by the selle with positive pobability. An o e is losing if it is not seious. Clealy, the speci cation of losing o es in a equilibium is to a lage etent abitay. Theefoe, statements about uniqueness ae undestood to be made up to the speci cation of the losing o es. Finally, an o e is a winning o e if it is accepted with pobability one. We bie y sketch hee the static vesion of the dynamic game descibed above: thee is only one potential buye, who submits a take-it-o-leave-it o e. The game then ends whethe the o e is accepted o ejected, with payo s speci ed as befoe (with n = ). The model consideed by Akelof (970) is not quite the static vesion of this game, as the maket mechanism adopted thee is Walasian. Much close is the second vaiant analyzed by Wilson (980), although he consides a continuum of buyes. Clealy, the selle accepts any o e p povided p > c (). Theefoe, the buye o es c ( ), whee maimizes (v (t) c ()) dt; ove 2 [; ]. Moe geneally, given t 2 [; ), let (t) denote the maginal valuation given the optimal o e when the distibution is unifom ove [t; ]. Obseve that (t) > t fo all t 2 [; ). 3 Obsevable o es 3. Main esult In this section, we maintain the assumption that o es ae public. Obseve that the equation Z (v (t) c) dt = 0 5 This is well-de ned as long as a selle with = would nd it optimal to eject all o es along h n. If this is not the case, the belief of buye n is assumed to assign pobability to =. 6

7 admits eithe no o eactly one solution in [; ) since its integand is stictly positive (negative) above (below) the unique oot of v (t) = c. Obviously, this solution eists if and only if: Z (v (t) c) dt < 0; that is, if it is unpo table fo the st buye to submit an o e that is accepted with pobability one by the selle. Moe geneally, given =: 0 > > > k, de ne k+ as the unique solution in [; k ), if any, of the equation k (v (t) c ( k )) dt = 0: Clealy, k k+ is bounded away fom zeo, hence this pocess must eventually stop. The esulting nite sequence f k g K k=0, k [; ], all k, is easy to compute fo special functions v and c. Fo instance, if c () = v () =, with > =2, one has k = (2 ) k. The sequence f k g plays an impotant ole in Poposition 3.. Poposition 3. Assume that K >, and >. Thee is a (essentially) unique equilibium, which is independent of. On the equilibium path, the st buye submits the o e c ( K ), which the selle accepts if and only if < K. If the o e is ejected, all buyes n > submit a losing o e. Fo each n >, buye n s stategy o es c( k ), afte any histoy h n with maginal type 2 ( k+ ; k ]. Poof: The poof is by induction ove K. The poof fo K = 0 is in most espects identical to the poof of the induction step, and we theefoe povide only the latte. We let an equilibium be given and assume that, fo evey n and afte any histoy h n on the equilibium path such that n =, buye n o es c( l ) wheneve 2 ( l+ ; l ] fo some l < k. We now pove that the same conclusion holds fo l = k. The poof is boken into the following fou steps: wheneve n = 2 ( k+ ; k ), no equilibium o e of buye n is accepted by some type s > k ; wheneve n = 2 ( k+ ; k ], if an equilibium o e of buye n is accepted by s = k, then all subsequent o es ae losing ones; besides, if n = k, the unique equilibium o e of buye n is c( k ); wheneve n = 2 ( k+ ; k ] is close enough to k, the unique equilibium o e of buye n is c( k ), which the selle accepts if and only his type is at most k ; wheneve n = 2 ( k+ ; k ], the unique equilibium o e of buye n is c( k ), which the selle accepts if and only his type is at most k. 7

8 Step : If buye n submits an o e p(s) with maginal type s 2 ( l+ ; l ] fo some l < k, the following o e is c( l ) by the induction hypothesis. Hence, p(s) must solve so that buye n s payo is p(s) c(s) = (c( l ) c(s)); Z s (v(t) c( l ) ( )c(s)) dt: As a function of s, the integal is twice di eentiable ove the inteval ( l+ ; l ], with st and second deivatives given by and v(s) c( l ) ( )c(s) ( )c 0 (s)(s ) v 0 (s) 2( )c 000 (s)(s ): Since (2M c 0 + M c 00)( ) < m, buye n s payo is stictly conve ove ( l+ ; l ]. Since buye n s payo is negative fo s = l, the claim follows. Step 2: We ague by contadiction. We thus assume that, fo some n and h n with n = 2 ( k+ ; k ), an equilibium o e p n by buye n with maginal type k is eventually followed, with positive pobability, by a seious o e. This implies p n > c( k ). Let p be the supemum of all such o es, whee the supemum is taken ove all, n and h n, and conside a buye and a histoy still denoted n and h n such that p n > ( )c( k ) + p. If instead, buye n deviates and submits a seious o e p(s) with maginal type s < k, then p(s) does not eceed p(s) ( )c(s) + p ( )c( k ) + p: By choosing s close enough to k, buye n s payo, fv(t) p(s)g dt, is thus highe than Z k the equilibium payo, fv(t) p n g dt a contadiction. We tun to the second assetion, and let n and h n 2 H n be given, with n = k. Since < K, thee must eist, along h n, a buye who submitted a seious o e with maginal type k. As we just poved, any such o e is necessaily followed by losing o es. In paticula, buye n s equilibium o e is c( k ). Step 3: Let n and h n 2 H n be given, with n = < k. Conside a potential o e p(s), with maginal type s. Obviously, Z p(s) c(s). Obseve also that p( k ) = c( k ) by Step 2. s Z s Hence, buye n s payo, fv(t) p(s)g dt, is at most fv(t) c(s)g dt, with 8 Z s

9 equality if s = k. The latte integal, as a function of s, is di eentiable, with deivative v(s) c(s) c 0 (s)(s ), which is positive as soon as s < m. Thus, fo close enough M c 0 Z s to k, the uppe bound, fv(t) c(s)g dt, is inceasing ove [; k ]. Hence, fo such, buye n s equilibium o e is c( k ). Step 4: Again, we ague by contadiction. We assume that, fo some n and h n with n > k+, buye n assigns positive pobability to seious o es with maginal type below k. Among all such n and h n, let ~ 2 ( k+ ; k ) be the supemum of n. Conside now any n and h n with = < ~. By de nition of ~, any o e p(s) with maginal type s > ~ is followed by an o e c( k ) fom the net buye, so that p(s) must satisfy and buye n s payo wites p(s) c(s) = (c( k ) c(s)); Z s fv(t) c( k ) ( )c(s)g dt: As in Step, the integal is stictly conve in s. Theefoe, the maginal type of any equilibium o e is eithe equal to k, o lies in the inteval [; ~]. In the fome case, buye n s o e is c( k ), and his payo is positive since ~ > k+. In the latte case, buye n s payo is at most (~ ) (v(~) c()), which is abitaily close to zeo, povided is close enough to ~. As a consequence, fo < ~ close to ~, the unique equilibium o e of buye n is c( k ), with maginal type k. This contadicts the de nition of ~. Fo completeness, let us bie y comment on the knife-edge case in which = K. Then as long as the maginal valuation is, any andomization ove the o es fc ( K ) ; c ( K )g is optimal, the payo of eithe o e being zeo. Because = K, equilibium consideations do not uniquely pin down the mitue, as is done in the poof above fo the case < K in which the maginal valuation is k, k K, afte an equilibium o e that is seious. Indeed, the only eason why the equilibium (as opposed to the equilibium outcome) fo the case < K is not unique is that nothing pins down the behavio when the maginal valuation is k, k K, following an out-of-equilibium o e. Beyond this indeteminacy, the case = K is identical to the case < K ; in paticula, along the equilibium path, the selle will eject all o es povided t K. The compaison to the static case is immediate: if is su ciently close to K, then the pobability of ageement is abitaily small, and the outcome is moe ine cient than in the static case. On the othe hand, if is su ciently small elative to K, then the pobability of ageement is lage than in the static case, as it must be that () < K, since the payo fom o eing c ( K ) can be chosen abitaily close to zeo. 9

10 3.2 Patient Single buye Poposition assumes that each buye makes only one o e. Howeve, its poof goes though with a single, long-lived buye povided the buye s discount facto is small enough, ing the selle s discount facto. On the othe hand, the esult is no longe valid if the long-lived buye and selle shae the same discount facto <. In that case, we know fom Vincent (989) that thee eists a (geneically) unique Pefect Bayesian Equilibium, and that, in this equilibium, bagaining ends afte a nite sequence of seious o es, one of which is accepted. Futhemoe, Vincent ehibits an eample with binay values in which delay does not vanish as the time inteval between successive o es tends to zeo. 6 Fo sake of illustation, we descibe hee the equilibium in the case in which c () = v () =. De ne the sequences: 0 = 0, n+ = + 2 n n ; z 0 =, z n = ny k= k k : We show in appendi that thee eists a unique equilibium of the game with a long-lived buye with discount facto. With pobability one, ageement is eached in nite time. If the maginal valuation is s 2 [z n+ ; z n ), the buye o es which the selle accepts if and only if: p = ( ) ( ) 2 n n s + n ; The epected payo of the buye is then: ( ) ( ) 2 t < 2 n n n n s: s 2 + n s 2 n : 2 We solve hee fo the case in which F is the unifom distibution. All poofs ae gatheed in Appendi. In fact, the maimal numbe of o es in equilibium, N, conveges as!, so that, as the time inteval between successive o es tends to zeo, ageement is immediate, contasting with the binay eample studied by Vincent (989). [To see this, obseve that, fo all n, the value of n tends to a well-de ned limit stictly lage than one, and theefoe z n+ z n tends to a stictly positive limit; given, it then follows that, fo lage enough, the duation of bagaining is independent of.] 6 In fact, Vincent (989) poves this esult moe geneally fo the case in which the buye is at least as patient as the selle. 0

11 3.3 Finite Hoizon Poposition implies that all buyes but the st one submit losing o es. Yet when the game has nite hoizon, this conclusion is blatantly false when <. In paticula, if the selle ejects all pevious o es with positive pobability, the last buye must submit a seious o e. Indeed, his poblem educes then to the static case, fo some speci c. Whethe ageement is eached with pobability one befoe the last buye, o the last buye submits a seious bid, the qualitative conclusions of the nite hoizon game seem to cast some doubt on the petinence of Poposition. Theefoe, the analysis of the game of the nite-hoizon is not only an impotant etension that includes the static benchmak as a special case, but also a obustness test: as the length of the hoizon incease, do the equilibia of the game with nite-hoizon convege to the in nite-hoizon equilibium? Fo simplicity, we state ou esults hee only fo the case in which v () = and c () =, with > 0, only thei etension to the case of geneal functions is immediate. Poposition 2: The equilibium stategies of the nite hoizon game convege pointwise to the equilibium stategies in the in nite-hoizon game, as the length goes to in nity. In paticula, the Pefect Bayesian equilibium is essentially unique, and is in pue stategy. In that equilibium, the stategy of buye i is associated with thesholds 0 < s 0 i < s i < < s k i i =. The stategy i has the following fom: if v i s 0 i, buye i o es a pice b i v i, and attacts all types up to c i v i fo some c i and b i. Thus, i (v i ) = c i v i. if v i 2 (s k i ; s 0 i ), i (v i ) = s l k i fo some l k : buye i o es a pice which does not depend on the speci c value of v i in that inteval, and attacts all types up to s l k 4 Unobsevable o es In this section, we maintain the assumption that all o es ae unobsevable, o pivate. Fo the sake of completeness, let s st sketch a poof that an equilibium eists. If no othe buye eve submits an o e above c, it is suboptimal fo a playe to submit such an o e. Hence, fo the pupose of equilibium eistence, we can limit the set of mied (o behavio) stategies to the set M([0; c]) of pobability distibutions ove the inteval [0; c], endowed with the weak-? topology. The set of stategy po les is thus the countable poduct M([0; c]) N. It is compact and metic when endowed with the poduct topology. Since the andom outcome of buye n s choice is not known to the selle unless he has ejected the st n o es, the buye n s payo function is not the usual multi-linea etension of the payo induced by pue po les. We howeve follow the standad poof. Let any buye n be given, and denote (p; ) the maginal type fo the o e p, given a stategy po le. It is jointly continuous in p and. Hence, the set B n () M([0; c]) i.

12 of best eplies of buye n to the stategy po le, is conve-valued and uppe hemi-continuous in. Using a ed-point theoem, the eistence of a (Nash) equilibium follows. We let an equilibium be given. The main esult is the following. Poposition 4. Tade eventually occus, with pobability one. Poof: Given 2 [; ], let F n () denote the (unconditional) pobability that the selle is of type t and has ejected all o es submitted by buyes i = ; : : : ; n. Suppose fo the sake of contadiction that tade does not occu with pobability one eventually, i.e. lim n! F n () 6= 0 fo some <. In paticula, the pobability that the selle will accept buye n s o e, conditional on having ejected the pevious ones, conveges to zeo as n inceases, hence the successive buyes equilibium payo s also convege to zeo. Let F = lim n! F n. Choose such that F () > 0 and v(t) c() df (t) > 0: () 2 Note that F () F n() is the pobability that type will accept an o e fom some buye F n () beyond n (conditional on having ejected all pevious o es). Since F () > 0, this pobability conveges to zeo, and the o e p n (s) with maginal type s thus conveges to c(). As a esult, p n (s) c(s) + fo all n lage and, using (), buye n s equilibium payo is bounded away fom 2 zeo a contadiction. Thus, o es that ae accepted by selle s types abitaily close to one ae eventually submitted. Poposition 4. holds iespective of. We now assume, without futhe notice, that >. We pove below that type actually accepts an o e in nite time with pobability one. Thus, thee is a buye who eventually o es c, which the selle accepts, iespective of his type. We intoduce some notation. As befoe, given some equilibium, F n () denotes the (unconditional) pobability that the unit is of inde t and that the selle has ejected all o es submitted by buyes i = ; : : : ; n. Set n := inf f : F n () > 0g. Buye n s stategy is a pobability distibution ove o es in [c(); c]. We denote by P n its suppot 7 and T n the coesponding (closed) set of maginal types. That is, if buye n s stategy has nite suppot, 2 T n if and only if thee eists an o e p n submitted by buye n with positive pobability that the selle accepts if and only if his type is less than o equal to. Poposition 4.2 Thee eists n, with 2 T n. Let N 0 denote the st of these stages. Then T n f N0 ; g, fo all n N 0. Fo each n > N 0, buye n s equilibium payo is zeo. 7 That is, the smallest closed set with pobability one. 2

13 It is convenient to discuss hee the case whee >. Note that a buye who is called to make an o e, gets a positive payo when o eing c. Hence, it follows fom Poposition 4.2 that the unique equilibium outcome of the game is such that the st buye o es c, which the selle accepts. Fo such values of, the equilibium outcome is the same in both vesions of the game (but it di es fom that of the static case). Fom now on, we will assume. Buye N 0 s equilibium payo is then also zeo. Thus, fom stage N 0 on, all equilibium o es ae eithe winning o es, o losing o es. By Poposition 4., one of these buyes eventually submits the winning o e c. Howeve, equilibium behavio afte buye N 0 is essentially indeteminate. Indeed, as the net esult states, any equilibium can be modi ed into an equilibium whee tade occus in bounded time, o into an equilibium in which all buyes tade with positive pobability. Poposition 4.3 Fo evey equilibium, Thee is an equilibium ~, with ~ n B = n B fo all n < N 0, and n = fo some n N 0. Thee is an equilibium ~, with ~ n B = n B fo all n < N 0, and n = N0 fo all n N 0. Moeove, in nitely many such equilibia eist. All these equilibia ae payo -equivalent, fo the selle and the buyes alike. Denote N 0 the last buye n who submits a seious o e below c with positive pobability. Thus, all buyes n = N 0 + ; : : : ; N 0 submit only losing o es. In the st phase of the equilibium, all buyes tade with positive pobability. Poposition 4.4 No buye n N 0 uses a pue stategy, ecept possibly buye. All buyes n N 0 submit a seious o e with positive pobability. Additional esults on the natue of equilibium stategies will be added late. In the public case, competition between buyes pevent buyes n 2 fom getting positive payo s. Howeve, the st buye buye gets a positive payo (independent of ). We do not know whethe a simila zeo-payo esult holds in the pivate o es vesion. In any case, all equilibium po ts ae tiny and vanish apidly, as the selle becomes in nitely patient. Poposition 4.5 The epected payo to buye n is at most 2 m 2 n+( ) 2 v n+ c n+ 2 : Thus, all equilibium po ts ae bounded by 2m ( ) 2 sup jv cj. Due to the tem ( ) 2, the discounted sum of all buyes po ts conveges to zeo, as the selle becomes moe patient. In addition, we pove that buyes with positive payo ae inceasingly ae, as the selle becomes moe patient. 3

14 Poposition 4.6 Let n < n 2 be two buyes with positive equilibium po t. One has n 2 n ln + m : 2M c 0 The e ciency of the tading mechanism is diectly elated to the delay in tade that is, to E [ ], whee is the st buye o eing c. As we now show, any equilibium must ehibit a non-tivial, but nite, delay. Poposition 4.7 Thee eist constants 0 < c < c 2 < and C > 0 such that, fo all and all equilibium: N 0 C=( ); c E [ ] c 2. c > 0 implies that the discounted delay emains nite. c 2 < implies that the discounted delay is non-tivial, even as the selle becomes moe patient. Non-tivial delay makes it di cult to compae equilibium payo s of the selle acoss vesions. Howeve that the po t of the selle is bounded away fom zeo in the pivate case. By contast, the selle s payo is zeo if = K+, and depends continuously on. Hence thee ae cases in which the selle s payo is unambiguously highe in the pivate vesion. Whethe o not this holds in geneal is an open question. 4. Conjectues and Numeical Evaluations We have been unable to eplicitly solve fo the equilibia of the game, ecept in special cases discussed below. As shown above, any equilibium can be patitioned into two phases. Povided that the st playe s payo fom submitting an o e accepted with pobability one is negative, thee must be a st buye (say, playe N > ) who must be pecisely indi eent between submitting a losing o e o submitting an o e accepted with pobability one. Fom this point on, all buyes andomize ove those kinds of o es, and as long as the latte o e is not submitted with pobability one by some buye, the conditional beliefs of the selle do not change any moe. The st phase (up to playe N ) is moe complicated. In paticula, all buyes but the st must use a mied stategy, with stictly positive pobability assigned to at least two o es. While it is possible to ule out many con guations with simple consideations, a wide ange of possibilities emain; in fact, simple eamples of multiple equilibia can be constucted. Thee ae many popeties of equilibia that one may conjectue. One is (i) to look fo an equilibium in which evey buye submits a losing o e with positive pobability; a second is (ii) to look fo an equilibium in which the epected o e is inceasing ove time; a thid is (iii) to look fo an equilibium in which evey buye andomizes ove at most two equilibium o es. 4

15 It is simple to show that it follows fom eithe (i) and (ii) o fom (i) and (iii) that the equilibium should be as follows: each buye andomizes ove two o es. The lowe one is a losing o e (fo all buyes but possibly fo the st buye) while the highe one is accepted by all types up to i, whee i is stictly inceasing in i fo i < N. (This andomization must be stict up to buye N ). Fom numeical simulations, it appeas that such equilibia eists fo all and, but they equie that be su ciently high. See Figue. [Hoizontal dotted lines ae suppot of distibutions, solid cuves ae payo s as the function of the maginal type accepting an o e; all Figues omit buyes beyond N 0 ] Figue ( not to scale) Fo lowe values of, this does not wok, because buye 2 stictly gains fom submitting an o e accepted with small but positive pobability. This poblem can be emedied by assuming instead that buye 2 s lowe o e is seious as well, so that only the low o e of buyes n 3 is a losing o e (in this evised conjectue, buye 2 s payo is still zeo). Such equilibia eist, and indeed, they can be constucted fo lowe values of than is consistent with the st conjectue. Howeve, it is again necessay that be su ciently high, fo othewise the same poblem aises with buye 4. 5

16 It seems theefoe natual to amend the conjectue futhe, by consideing the case in which, fo a subsequence of the buyes in the st phase, the low o e is seious, while it is losing fo the othes. (It is easy to see that no two consecutive buyes can belong to that subsequence). Unfotunately, the esulting systems of equations is quite untactable. In the special case in which ( ) > =2 > 2, we could constuct an equilibium as depicted in Figue 2, which is consistent with a moe geneal conjectue along the lines of Figue Figue 2 ( not to scale) 6

17 Figue 3 : Conjectue ( not to scale ) ( ) > 2 > ( 2 ) 5 Poofs 5. Poof of Poposition 4.2 The net lemma is epeatedly used in the sequel. Lemma 5. If n+2 > n+, buye n submits no o e in the inteval ( n ; n+ ). In paticula, n+ = n, and buye n s equilibium payo is zeo. Poof: //in pogess// Fo > n, let p n () denote the o e that buye n must submit fo type to be the maginal type. Fo < n+2, the hypothesis of the lemma implies that p n () = ( ) c () + C fo some constant C. The (unconditional) payo of buye n fom 7

18 submitting an o e fo which the maginal type is is given by: (v (t) p n ()) df n (t) : Clealy, if v () = p () fo some > n, o eing p () esults in a stict loss, so in ode to nd the maimizes of this payo, we can estict attention to such that v () > p (). Since c 0 is bounded and v is stongly inceasing, this implies that we can estict attention to above some theshold, povided is su ciently close to one. Since F n is conve (given the ceam-skimming popety), it is piecewise continuously di eentiable on ( n ; min S n+ ), with deivative, on each subinteval, given by: df n () dt (v () p ()) ( ) c 0 () F n () ; which is stictly inceasing fo all su ciently lage, since v is stongly inceasing, while F n is conve. Theefoe, the payo is stictly conve on each of those subintevals, a esult that holds moe geneally ove (; min S n+ ) since at all points this inteval; D F n () < D + F n (). Theefoe, this payo is st stictly negative and then stictly conve in, so that it admits no maimum in this inteval. Poof of Poposition 4.2: Fom Poposition 4., we know that lim n F n () = 0. Fi some N such that F N () < "=c 0, whee c 0 > 0 is an uppe bound on the deivative of c ove [; ]. Let: ~V n () := fv (t) c ()g df n (t) : Obseve that ~ V n () is an uppe bound to the (unconditional) payo buye n obtains when submitting an o e fo which is the maginal type, with equality if and only if eithe it is a losing o e (F n () = 0), o a winning o e ( = ). We st pove that 2 T n, fo some n 2 N. Let s ague by contadiction. In paticula, the highest o e s N := ma n<n ma S n is lowe than, and the function F N () is di eentiable on the inteval [s N ; ], with deivative equal to one. Hence, on that inteval, ~ V N is di eentiable and its deivative equals: ~V 0 N () = v () c () F N () c 0 () ; which is stictly positive. Theefoe V ~ N is stictly inceasing on (s N ; ), so that buye n s payo cannot be maimized on this open inteval. Hence, any such buye can only submit eithe a winning o e which is uled out by assumption o o es fo which the maginal type is less than T N, implying that such all maginal types ae bounded away fom one, contadicting Poposition 3. Conside now the second statement. Fo the sake of contadiction, suppose that some buye n N 0 submits w.p.p. an o e in c( N0 ); c. Then, conditional on submitting a winning o e 8

19 c, the epected value of the unit is highe fo buye n + than fo buye n. Thus, buye n + s payo is positive. By Lemma 5., it cannot be the case that buye n + assigns pobability one to c. Hence, buye n + 2 s equilibium payo is also positive a contadiction to Lemma Poof o Poposition 4.3 Denote the (andom) st buye who o es c, and de ne n 2 N by n+ E[ ] < n. Conside the stategy po le ~ in which (i) ~ n B = n B fo all n < N 0, (ii) all buyes n = N 0 ; : : : ; n submit a losing o e, (iii) buye n o es a winning o e c with pobability E[ ] n+, and a losing o e othewise, (iv) all buyes n n n + o e c with pobability one. By constuction, fo all buyes n < N 0, and afte any histoy h n 2 H n, the o e p n (s) with maginal type s is the same unde the two po les and ~. Hence, fo any such n, n B is a best-eply to ~. It emains to check that no buye n = N 0 ; : : : ; n nds it po table to submit a seious o e in the open (c( N0 ); c). Fo any such n, the epected payo that type obtains fom ejecting the o e is (c c())e n j > n. This continuation payo is highe when E n j > n is computed using ~ then when using. Hence the maginal type fo an o e in (c( N ); c) is lowe unde ~ than it is unde. Such a deviation cannot be po table unde ~, fo it would also be po table unde. We tun to the poof of the second statement. We assume that the equilibium is such that T n = fg fo some n, fo othewise the conclusion holds tivially. As above, we adjust the pobability of submitting c, while keeping the optimal decision ule of the selle in stages n = ; : : : ; N 0 unchanged. Set N := inffn N 0 : T n = fgg, and denote by the pobability assigned to c by buye N. We st check that this de nition is meaningful when N = N 0, and conside buye N. By Lemma 5., T N0 f N0 ; g. By de nition of N 0, it must be that buye N 0 assigns pobability one to a losing o e. Hence, = 0 in that case. Conside the stategy po le ~ in which (i) ~ n B = n B fo all n < N, (ii) buye N assigns pobability ~ to c, and submits a losing o e othewise, (iii) all buyes n N submit c with pobability, and a losing o e othewise. The paametes ~ and ae chosen so that (i) the epected discounted time E [ ] is the same unde both po les and ~, and (ii) no buye n N would nd it po table to submit a seious o e in (c( N ); c). Condition (i) wites + ( ) = ~ + ( ~) ( ) : (2) Z s Condition (ii) holds as soon as the payo fv(t) N p n (s)g df N (t) is conve in s. A su cient 9

20 condition is m 2M c 0 > 0: ( ) Povided is close enough to one, the condition is satis ed and the solution ~ to (2) is in (0; ). 5.3 Poof of Poposition 4.4 We stat with the second assetion. Assume that, fo some n N 0, buye n submits a losing o e with pobability. Then, n < N 0 by the choice of N 0, and buye n s equilibium payo is zeo. Denote n > n the st buye afte n who submits a seious o e with positive pobability. Denote p n (s) = ma P n the highest o e of buye n, with maginal type s. Obseve that s <, hence p n (s) < c. As a consequence, the pice p n (s) that buye n needs to submit so that s is indi eent between accepting and declining is stictly smalle than p n (s). Obseve also that the epected value of the unit, conditional on an o e being accepted by all selle s types less than s is the same fo buye n and n. Theefoe, buye n gets a stictly positive payo fom submitting such an o e, a contadiction. We now deal with the second assetion. Assume that, fo some n N 0, buye n s stategy is pue. By the pevious paagaph, buye n s unique o e must be seious. Denote < the maginal type fo this o e. By Lemma 5., buye n cannot submit any o e with maginal type in ( n ; ). This implies that the epected value conditional on the maginal type being equal to is the same both fo buye n and buye n +. Since buye n s o e is seious, buye n must thus submit a losing o e with positive pobability, and theefoe have a zeo equilibium payo. Howeve, the o e he must submit such that the maginal type equals is stictly less than the unique o e submitted by buye n (because of discounting), a contadiction. 5.4 Poof of Poposition 4.5 Fo n 2 N, and > n, we denote q n (s) the epected value of the unit fo buye n, conditional on an o e with maginal type being accepted: q n () = F n () n v(t)df n (t): Plainly, n q n (). Besides, q n () q n+ (), with equality if and only if buye n submits no seious o e with maginal type in ( n ; ). Poof of Poposition 4.5: We let hee n be any buye with positive po t. Note that n N 0. Since buye n s payo is positive, n+ > n. conside buye n s lowest o e, p n ( n+ ), with maginal type n+. Fo notational concision, we abbeviate n+ to. The pobability that 20

21 this lowest o e is accepted is F n (). Conditional on it being accepted, buye n s suplus is q() p n (). Hence, buye n s equilibium payo is F n () (q n () p n ()). We st bound the conditional suplus. Since buye n s payo is positive, buye n + s payo must be zeo, hence p n+ (), fo othewise buye n + would obtain a positive payo when submitting a low, seious o e. Thus, This yields q n () p n+ () p n () ( )c(): q n () p n () ( ) (v () c ()) : (3) We net bound the pobability F n () that the lowest o e is accepted. We will ely on the inequality q n () v() + ( )c(): It is convenient to obseve that F n () = ejects the o es fom all buyes k < n. Intoduce the optimization poblem P: sup n f n (t)dt, whee f n (t) is the pobability that type t n f(t)dt; whee the supemum is taken ove all non-deceasing, [0; ]-valued functions f, such that n v(t)f(t)dt (v() + ( )c()) n f(t)dt: We will pove that the value v of P satis es v 2=m( )(v() c()). Togethe with (3), and since F n () v, this will conclude the poof. We analyze P by intoducing an auiliay poblem P! : sup R n R n v(t)f(t)dt ; f(t)dt whee the supemum is taken ove all non-deceasing, [0; ]-valued functions f that satisfy f(t)dt! (! n ). The poblem P! is some kind of dual to P. Eistence of an n optimal solution to P!, and continuity of the value function V (!) follow fom standad aguments. Plainly, one has v! as soon as V (!) v() + ( )c() indeed, any optimal solution to P! is then a feasible point fo P. Hence, v = supf! : V (!) v() + ( )c()g: (4) 2

22 Any solution f to P v must satisfy the feasibility constaint f(t)dt v with equality, fo n othewise V (! 0 ) would be equal to V (v ) fo! 0, close enough to v. Hence, f is also a solution to the poblem sup v(t)f(t)dt, subject to the constaint f(t)dt = v. Since v is stictly n n inceasing, f is a step function, with f (t) = 0 fo t <, and f (t) = fo t. The value of is detemined by the constaint n f(t)dt = v : v = V (v ) = v(t)dt:, and Note that v(t) v() ( t)m, hence V (v ) v() ( 2 m ) = v(). on the othe 2 hand, by (4), V (v ) = v() + ( )c(), hence as claimed. 5.5 Poof of Poposition 4.6 v 2 ( )(v() c()); m The poof of Poposition 4.6 elies on the following inequality. Lemma 5.2 Fo evey a < b in [; ], and evey non-deceasing function h : [; ]! [0; ], with h(t)dt > 0, one has h(b) R b a R b a h(t)dt Z b a v(t)h(t)dt + m 2 Z b a h(t)dt v(b)h(b): (5) Poof. The inequality is homogenous in h. We may thus assume that h(b) =. We pove the statement fo functions H that take nitely many values. The esult will then follow using a density agument. The poof goes by induction ove the numbe of values in the ange of H. To stat the induction, assume that h(t) = fo each t 2 [a; b]. Since v(t) v(b) + m(t b) fo each t 2 [a; b], one has Z b m(b a) 2 v(t)dt v(b)(b a) : 2 a m Hence, the left-hand side of (5) is at most v(b) (b a) + m (b a) = v(b). 2 2 Assume that (5) holds fo evey a < b, povided the ange of h contains at most n points. nx Let h = k [yk ;y k+ )(), whee 0 0 n = and a = y 0 < y < < y n = b. k=0 22

23 Fo given ; : : : ; n ; y 0 ; : : : ; y n, the left-hand side if (5) is a function of 0, which we denote ( 0 ). We pove below that is conve. This will imply that ( 0 ) ( 0 ) (0) + 0 ( ). Since, fo both = 0 and =, the ange of h contains at most n points, one has (0) v(b) and ( ) v(b), hence the esult follows. The second tem in ( 0 ) is linea, hence we only pove the conveity of Z P b n ( 0 ) := R b h(t)dt i=0 v(t)h(t)dt = R yi+ i y i v(t)dt P n a a i=0 i(y i+ y i ) : Obseve that 0 ( 0 ) = whee D( 0 ) = ( nx Z y Z yi+ D 2 i (y ) i+ y i ) v(t)dt (y y 0 ) v(t)dt ; ( 0 ) i=0 y 0 y i nx i (y i+ y i ). i=0 Since, v is inceasing, one has y y 0 Z y hence 0 ( 0 ) is inceasing in 0. y 0 v(t)dt < y i+ y i Z yi+ y i v(t)dt; fo i ; Lemma 5.3 Let n 2 N, and > n in T n be given. Then D + p() m 2 : Poof. Wite F n () = n obtains a payo equal to n f n (t)dt. If he submits an o e p n (s) with maginal type s, buye Z s n f n (t)(v(t) p n (s))dt: Since f n is ight-continuous, and p n is conve, the integal has a ight-deivative at, equal to f n ()(v() p()) D + p()) n f n (t)dt: Since 2 T n is a seious o e, the ight-hand side is non-positive and v() p n () q n (). Hence, R! v(t)f n(t)dt n f n () v () D + p() f n (t)dt 0: f(t)dt n R n The conclusion follows, using Lemma 5.2. Bibliogaphy 23

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