Information Acquisition in Auctions: Sealed Bids vs. Open Bids

Transcription

1 Informaton Acquston n Auctons: Seale Bs vs. Open Bs Ángel Hernano-Vecana Unversa e Alcante Uner Revson, Do Not Crculate Aprl 1, 2005 Abstract Ths paper stues the ncentves of a ber to acqure nformaton n an aucton when her nformaton acquston ecson s observe by the other bers before they b. Our results show that the ncentves are stronger n a seale b (secon prce) aucton than n an open (Englsh) aucton when the nformaton acqure refers to a common component of the value. However, the rankng s the opposte when the nformaton acqure refers to a prvate component of the value, at least for a large number of bers. Our results seem to be ue to fferences on the strength of the wnner s curse, although a more careful analyss show that the key force s rather the loser s curse. JEL Classfcaton Numbers: D41, D44, D82. Keywors: auctons, open nformaton acquston, wnner s curse, loser s curse. Aress: Departamento e Funamentos el Análss Económco, Campus e Sant Vcent el Raspeg, Unversa e Alcante, Apt. 99 E Alcante, Span. Emal: angel@merln.fae.ua.es. Web-page: Phone: ext. 3232, fax:

2 1 Introucton Most aucton theory moels assume that bers have some prvate nformaton. However, very lttle s known about the orgn of ths prvate nformaton an n partcular, the ncentves of bers to acqure t. Ths s not only an mportant theoretcal queston but also of practcal concern n aucton esgn. The reason s that more or less nformaton acquston affects to both the effcency of the allocaton mplemente n the aucton an the revenue generate. In ths paper, we shall stuy the ber s ncentves to acqure nformaton about the value that she may get from the object for sale. In partcular, we shall focus on the case n whch a ber s nformaton acquston ecson s observe by the other bers, what we call open nformaton acquston. Ths moel allows a rcher theoretcal problem than the moel n whch the ber s nformaton acquston ecson s not observe by the other bers. The reason s that n the latter moel nformaton acquston s not a strategc varable as t s n our case. Moreover, there are relevant real-lfe auctons n whch the open nformaton acquston moel s the most approprate. For nstance, bers that want to acqure nformaton n ol tract auctons use exploratory rlls that are easly vsble. There are also other cases n whch the auctoneer can control whether the bers nformaton acquston ecson s observable. Conser agan the example of ol tract auctons. It s qute common that bers that want to run exploratory rlls have to communcate t to the auctoneer who can ece whether to reveal t. In our paper, we shall compare the ncentves to acqure nformaton n two stanar aucton formats: a (secon prce) seale b aucton, an an open (Englsh) aucton. Both formats are smlar n the sense that, n both cases, the wnner pays the hghest losng b. However, they ffer n one mportant aspect. The seale b aucton s a statc game n whch the only nformaton reveale occurs when the aucton s over, whereas the open aucton s a ynamc game n whch there s a lot of nformaton revelaton along the game, partcularly all the losng bs. Thus, the comparson between these two aucton formats s nterestng not only by tself, but also because t solates the effect of the nformaton reveale n a ynamc aucton on the ncentves to acqure nformaton. We shall show that ths fference may have mportant effects on the ncentves to acqure nformaton. Our results show that n a moel n whch the ber s uncertanty about her value has prvate components an components common to all bers, the ncentves to acqure more nformaton about the common components n the seale b aucton are hgher than n the open aucton. However, the rankng s the opposte, at least f there are suffcently many bers, when the nformaton acquston s about the prvate component of the value. A reaer famlar wth aucton theory may conjecture an ntutve explanaton to the frst effect base on the wnner s curse. If one ber acqures prvate nformaton about a common component of the value, her b becomes more nformatve of the common value an thus, we may expect that t ncreases the wnner s curse on the other bers. Ths suggests that there s a strategc effect that makes the rvals bs lower an hence ncreases the ber s ncentves to acqure nformaton. 2

3 We may expect the former effect to be larger n the seale b aucton than n the open aucton. The reason s that there s less uncertanty about the value, an thus a greater wnner s curse, n the open aucton than n the seale b aucton because of the nformaton reveale along the game. We shall show, however, that we sa before s not completely correct an that the ntuton s slghtly more subtle. The key effect s a symmetrc force to the wnner s curse that has been calle the loser s curse. 1 Agan, a reaer famla wth aucton lterature can also conjecture an explanaton to the secon result base on the wnner s curse. Informaton acquston about the prvate value prouces a (mean preservng) sprea n the ber s contonal expecte prvate value, whch nuces a smlar sprea n the ber s bs. The fact that our ber hgh bs become hgher affect the hgh bs of the other bers. The reason s that, as Bulow, Huang, an Klemperer (1999) have argue, an ncrease n the bs of a ber ncreases the wnner s curse on the others, an hence, nuce them to b lower. 2 The opposte happens to low bs of the other bers because of symmetrc reasons, although these bs are less relevant when there are suffcently many bers snce the prce s more often fxe by hgh bs. Ths explans that the above strategc effect ncreases the ncentves to acqure nformaton about the prvate value, at least when there are suffcently many bers. The above effect shoul be greater n the open aucton than n the seale b aucton. The reason s that n the seale b aucton, the other bers have uncertanty of whether the ber that has acqure nformaton s bng low or hgh, whereas ths uncertanty oes not exst n the open aucton. We shall see that all these effects can be accurately explane usng the loser s curse. The ssue of bers open nformaton acquston n auctons have receve lttle attenton n spte of ts mportance, both practcal an theoretcal. There are only some some partal results as a by se prouct, for nstance, n the work of Engelbrecht- Wggans, Mlgrom, an Weber (1983), Hernano-Vecana (2004). Hernano-Vecana (2005) an Larson (2004) prove a careful analyss of the value of nformaton n auctons that has mplcatons on open nformaton acquston. The man two fferences wth our work s that they o not conser the seale b aucton 3 an that they only conser the value of nformaton about the common value. One possble explanaton for ths lack of attenton are the techncal ffcultes nherent to the analyss of what has been calle asymmetrc auctons. These are auctons n whch bers ffer from an ex ante pont of vew, or n other wors, where the entty of bers matters. Most of the work n aucton theory requres the stuy of sophstcate mathematcal moels that can only be solve explctly appealng to the anonymty assumpton. In fact, t s complex to prove contons that assure exstence of the equlbrum once asymmetres between bers are allowe, see for nstance Athey (2000). 1 The loser s curse was orgnally ntrouce n the aucton lterature by Pesenorfer an Swnkels (1997). 2 Bulow, Huang, an Klemperer (1999) also explan that there s a feeback loop because lower bs of the rval bers reuces the wnner s curse on our ber an hence t nuces even hgher bs. 3 At least when t s not strategcally equvalent to the open aucton, ths s when there are three or more bers. 3

4 But, when bers nformaton acquston ecsons are observe before the begnnng of the aucton game, one must necessarly conser asymmetrc auctons. Even f we stuy a symmetrc equlbrum n whch bers acqure the same level of nformaton n equlbrum, one must solve the aucton game for evatons of ths symmetrc equlbrum. In these evatons, a ber takes a fferent nformaton acquston ecson than the non-evatng bers. Snce the other bers observe the evatng ber choce before the aucton stage, we can no longer analyze the aucton as a symmetrc game. There exsts some other papers that have stue the problem of bers nformaton acquston when each ber s choce s unobserve by the other bers. For nstance, Matthews (1984) look to the queston of nformaton aggregaton when bers may choose how much nformaton to acqure; an Persco (2000) compare the bers ncentves to acqure nformaton n frst prce an secon prce auctons. A closely relate paper that les between the moels of open an hen nformaton acquston s that by Compte an Jehel (2002). They also compare the seale b aucton an the open aucton. However, ther paper ffers from ours n that they stuy a pure prvate value moel, n whch there s no ssue about the wnner s curse or loser s curse. The prvate value assumpton makes also rrelevant whether the nformaton acquston choces are observe or not as n ths case there are no strategc effects, at least for the aucton formats that Compte an Jehel (2002) stuy. The key n ther moel s that n the open aucton bers may acqure nformaton along the aucton an epenng on how the aucton evolves, an thus, the open aucton gves the opton of wat an see before takng the nformaton acquston ecson. In spte of the fferent assumptons, they obtan the same rankng as ours for nformaton acquston about the prvate value. The rest of the paper s organze as follows. Next secton proves the assumptons of the moel an the man results. Secton 3 scusses at a more or less ntutve level two ssues: how the nformaton acquston ecson of one ber may have strategc effects on the ncentves to change the b of the others, an how ths change n ncentves to b ffers across aucton formats an explans our rankngs. Secton 4 proves the equlbrum analyss that establshes our results an Secton 5 conclues. We also nclue an Appenx wth the most techncal proofs. 2 The Moel We stuy a moel n whch bers values have common an prvate components, an bers types are nformatve of both, the common an the prvate value. In our moel, we also allow the precson of the bers nformaton about the common an the prvate value component to vary. More precsely, we assume that there s a set 4 I = {1, 2,..., n} (n 3) of rsk neutral bers. A generc ber I puts a monetary value of T + j I Q j ( I) n the consumpton of the goo. Note that T (for taste) s a prvate value component 4 We assume that n 3 snce our open an seale b auctons are strategcally equvalent for n = 2. 4

5 as t only affects s preferences whereas Q (for qualty) s a common value component that affects all bers preferences. Note that we are assumng atve separablty of the utlty functon. Ths assumptons largely smplfes our problem as t allows us a straghtforwar applcaton of the technques evelope by Myerson (1981). Moreover, t smplfes enormously the comparson of the allocaton mplemente n each of the aucton games we stuy. We coul get smlar results assumng atve separablty only of the prvate an the common value. We also conjecture that a margnal verson of our results must also hol true when we relaxe ths assumpton. The reason s that any smooth functon can be approxmately lnearze locally. We assume that all the vector of ranom varables (T 1,..., T n, Q 1,...Q n ) are statstcally nepenent. From a techncal pont of vew, ths assumpton smplfes our problem snce we can apply Myerson s (1981) technques, but more mportantly, t allows us to abstract from the non-strategc ncentves to acqure nformaton. Uner ths assumpton, t may be shown that both the open aucton an the seale b aucton nuce the same ncentves to acqure nformaton covertly, whch s not the case when the bers prvate sgnals are correlate across bers. In fact, an aaptaton of the arguments gven by Persco (2000) shows that the seale b aucton gves larger ncentves to acqure nformaton than the open aucton n a partcular sense. We shall assume that Ber I observes a nosy sgnal nformatve of Q an T whch we assume to be an element of the famly {X η } η N. We assume that each of these sgnals s nformatve of both T an Q. We shall mpose some structure on ths famly of sgnals to ensure that the realzatons of the sgnals can be orere n the sense of hgher realzatons nuce hgher bng an that the famly of sgnals can be orere n the sense that a hgher η means a more nformatve sgnal. Uner our assumptons of atve separablty an nepenency of the bers types, t s easy to see that the utlty of a ber n any aucton game, an for some fxe strateges of the other bers, s lnear n the state, ths s n T an n Q. An applcaton of the results of Athey an Levn (2001)[Lemma 1] to such ecson problems shows that a suffcent conton to assure the exstence of an ncreasng best response s that E[V X η = x] s ncreasng n x, for V {T, Q }, an for any η. Note that n our partcular case, ths s not restrctve as we coul always reorer the sgnal s realzatons to obtan the requre monotoncty. To make our arguments clearer, we normalze the margnal strbuton of each of the sgnals {X η } η N to be unform on [0, 1]. To see why ths normalzaton s wthout loss of generalty, suppose that our orgnal sgnal X η ha a strbuton functon G that was not unform. We coul efne a new sgnal ˆXη G(X η ) that has unform strbuton functon. The new sgnal s equally nformatve of ether T or Q snce t s a monotone transformaton 5 of X η. Athey an Levn (2001) gves us a meanngful efnton of more nformatve sgnals for ecson problems that are lnear n the state. Uner the above normalzaton, ths efnton s as follows: 5 The argument we have gven only works for G contnuous an strctly ncreasng n the support. See Athey an Levn (2001) an Lehmann (1988) for the general case. 5

6 Defnton: We say that sgnal X η s more nformatve of V, for V {T, Q }, than sgnal X η f an only f, E[V X η x] E[V X η x] for any x [0, 1]. We also say that X η s as nformatve of V, for V {T, Q }, as sgnal X η f an only f, X η s more nformatve of V than X η an vceversa. An applcaton of Theorem 1 of Athey an Levn (2001) mples that our efnton s such that a sgnal X η s more nformatve than another sgnal X η f an only f, any ecson maker wth a utlty functon lnear n the state can acheve hgher expecte utlty wth sgnal X η than wth sgnal X η for a gven pror. In ths sense, our efnton assures that the expecte utlty of a ber that acqures a more nformatve sgnal ncreases when the nformaton acquston ecson s not observe by the other bers. In ths sense, we entfy more a nformatve sgnal wth a more valuable sgnal. Note that the ntutve meanng of our efnton s that the more nformatve a sgnal s, the more pessmstc becomes the expecte value contonal on ba news n the sense of X η x. It may also be note that our efnton of more nformatve sgnal s equvalent to a secon orer stochastc sprea of the expecte value contonal on the sgnal. To have more clear-cut results, we shall ntrouce a refnement n our efnton of more nformatve sgnals. We shall ncate how our results woul change ha not we ntrouce ths refnement. Defnton: We say that sgnal X η s monotoncally more nformatve of V, for V {T, Q }, than sgnal X η f an only f, X η s more nformatve of V than X η an, 6 E[V X η = x] E[V X η x] > E[V X η = x] E[V X η x], for any x (0, 1]. The above conton s mple by our efnton of more nformatve sgnal only for the case x = 1. Ths refnement on the efnton s also less restrctve than the refnement that for x > x, E[V X η = x] E[V X η = x ] s ncreasng the more nformatve the sgnal s, a refnement that has been use, for nstance, by Hageorn (2004). Note that the nequalty that efnes monotoncally more nformatve sgnals s equvalent to: E[V X η x, X η x] E[V X η x] > E[V X η x, X η x] E[V X η x], whch may be nterprete that a more nformatve sgnal makes the contonal expecte value becomes more optmstc contonal on goo news n the sense of X η x even when we conton atonally on X η x. We also ntrouce some symmetry assumptons across bers. In partcular, we assume that for all η N the jont strbuton of the vector (T, Q, X η ) s the same 6 The strct nequalty s assume to avo some trval problems n Lemma 4. 6

7 across bers. Ths mples that the contonal expecte values E[T X η = x] an E[T X η = x] are the same across bers, for a any η N. We also ntrouce some regularty assumptons. We assume that E[T X η = x] an E[Q X η = x] have a contnuous ervatves an are strctly ncreasng n x for any η N. We also assume that the functon µ η (x) E[Q X η = x] E[Q X η x] s ncreasng n x, for any 7 η. Next example llustrates our assumptons: Example: Suppose that Q an T, I, follow an nepenent unform strbuton wth support [0, 1], an that X η = Q wth probablty η T (η), an X η = T wth probablty η Q (η), an wth probablty 1 η T (η) η Q (η) an nepenent ranom varable wth unform strbuton, where η T an η Q are ncreasng functons. To see that hgher values of η mean n ths example monotoncally more nformatve sgnals, note that E[T X η = x] = η T (η)x + (1 η T (η)) 1 2, E[T X η x] = η T (η) x 2 + (1 η T (η)) 1 2, E[Q X η = x] = η Q (η)x + (1 η Q (η)) 1 2, an E[Q X η x] = η Q (η) x 2 + (1 η Q(η)) 1 2. We are ntereste n stuyng the bers ncentves to acqure nformaton openly. To o so, we have to solve the bng game for fferent vectors of bers nformaton precsons, a problem that n general may be cumbersome. To make thngs smpler, we shall focus on stuatons n whch all the bers sgnals but one have the same nformaton precson. Note that these are the only cases that we nee to conser f we want to stuy symmetrc equlbra of games wth an nformaton acquston stage. To keep coherency wth the former story we shall call the ber wth fferent nformaton precson the evatng ber, an we shall enote ts nex by I. We shall refer to the other bers as non-evatng bers. We also use η an η to refer to the nformaton precson of the evatng ber an the non-evatng bers, respectvely. We shall compare the bers ncentves to acqure nformaton n two aucton formats, an open aucton 8 (O) an a seale b (secon prce) aucton 9 (S). Our equlbrum analyss wll be base on the (perfect) Bayesan equlbrum, an for the sake 7 A suffcent conton for ths regularty assumpton s that the cumulatve strbuton functon of the ranom varable ρ(x η ), for ρ(x) = E[Q X η = x], s log-concave, an assumpton satsfe by many strbuton functons, see Bagnol an Bergstrom (1989), for nstance any strbuton functon of the form F (q) = q r, r 1, an any truncate exponental, normal, logstc, extreme-value, ch-square, ch, an Laplace strbutons. Ths s also equvalent to the assumpton that φ s log-concave. 8 We assume that the aucton proceure s as follows, at every moment of tme there are two types of bers: actve bers an nactve bers. Bers are actve untl they manfest that they want to become nactve. Once a ber has ece to become nactve her ecson s rreversble. The entty of the actve bers s publcly observable along the aucton. The prce s also publcly observable an ncreases contnuously from zero. At each moment n tme bers can ece to become nactve. The prce stops ncreasng whenever there s no more than one actve ber. In ths case, the remanng actve ber gets one unt. If there s no actve ber when the prce stops, the goo s ranomly allocate (wth equal probablty) among the bers that qut at the last prce. The prce pa by all the wnners s the prce at whch the aucton stoppe. 9 In ths aucton set-up, all bers submt smultaneously one b each. The ber that have submtte the hghest b gets the goo at the prce of the secon hghest b. If two bers submt the hghest b, the prce equals ths b an the goo s allocate ranomly among all the bers that submtte the hghest b, whereby all such bers have the same probablty of beng selecte. 7

8 of smplcty, we restrct attenton to symmetrc equlbrum n the sense that all the non-evatng bers use the same strategy. In most of the paper we shall focus on two cases that are specally clear-cut: Defnton: [Common value nformaton moel] Suppose that for any elements of N such that η > η, the sgnal X η s monotoncally more nformatve of Q than the sgnal X η. Suppose also that Xη s as nformatve of T as the sgnal X η. Defnton: [Prvate value nformaton moel] Suppose that for any elements of N such that η > η, the sgnal X η s monotoncally more nformatve of T than the sgnal X η. Suppose also that Xη s as nformatve of Q as the sgnal X η. 3 Strategc Effects Assocate to Informaton Acquston In ths secton, we analyze the effects that a ber s nformaton acquston ecson may have on the other bers ncentves to change ther b. To ths am, we start wth the stuy of these ncentves. In both aucton formats, the seale b aucton an the open aucton, a ber s b only etermnes whether the ber wns or losses the aucton, but not the prce that she pays when she wns. Ths s equal to the hghest b of the other bers. A margnal ncrease (or ecrease) n the b makes the ber pass from losng to wnnng (respectvely, from wnnng to losng) but only f she was tyng wth the hghest b of the other bers. Consequently, a ber s ncentves to change her b margnally wll epen on the expecte utlty of wnnng the aucton contonal on the event tyng wth the hghest b of the other bers. If ths expecte utlty s postve, the ber has ncentves to ncrease her b, an f t s negatve the ber has ncentves to ecrease her b. The event escrbe above s the ntersecton of two other events that we use below to unerstan a ber s ncentves to change her b. The frst one s that the hghest b of the other bers must be greater than the prce, say p. Note that t s goo news about the common value, an hence nuces greater ncentves to ncrease the b. Ths event has been calle the loser s curse as a ber who gnores ths event wll b too low an regret losng. The secon event s that the hghest b of the other bers must be less than p. Note that t s ba news about the common value, an hence nuces lower ncentves to ncrease the b. Ths event has been calle the wnner s curse as a ber who gnores ths event wll b too hgh an regret wnnng. The open aucton s a complcate ynamc game. However, the fnal prce an the wnner of the aucton, an hence the ncentves to change the b, only epen n general on what happens n nformaton sets wth two bers actve. Moreover, snce our nterest s on the expecte utlty of the evatng ber, we only nee to conser nformaton sets n whch the evatng ber s one of the two bers that reman bng. The nformaton that the non-evatng ber can learn from the loser s curse s that the b of the the evatng ber must be greater than p, whereas the 8

9 wnner s curse conveys that the b of the evatng ber must be less than p. To unerstan the latter note that, n equlbrum, the uncertanty of a remanng ber has been reuce to the type of the other remanng ber. Ths s because the other bers types can be nferre from the prces at whch they qut. We next formalze the ncentves of a non-evatng ber, say Ber, wth type x to change margnally the b aroun p to llustrate the loser s curse an the wnner s curse n the open aucton: E T + Q j Xη = x, b O (Xη }{{ ) p, b O } (Xη ) p, X p, (1) }{{} loser s curse wnner s curse where b O enotes the b functon use by the evatng ber n the above nformaton sets, an X the nformaton nferre along the equlbrum path. The nterpretaton of the events loser s curse an wnner s curse for a non-evatng ber s slghtly more complex n the seale b aucton snce ths ber has uncertanty about whether the hghest b of the other bers s the b of the evatng ber or the b of another non-evatng ber. In general, the non-evatng ber wll put a postve probablty on each of the two cases. Ths mples that the loser s curse means that wth some probablty the b of the evatng ber s greater than p an wth the complementary probablty the hghest b of the non-evatng bers s greater than p. The wnner s curse s more straghtforwar. It always means that all the other bers b less than p. Agan, to llustrate the loser s curse an the wnner s curse, we formalze the ncentves of a non-evatng ber wth type x to change her b margnally aroun a prce p: (1 ρ) E T + ρ E T + Q j X η Q j X η = x, b S n (Xη j }{{ ) p, b S } (Xη ) p, {bs n (Xη l ) p} l, + }{{} loser s curse wnner s curse = x, b S (Xη }{{ ) p, b S } (Xη ) p, {bs n (Xη l ) p} l, p, (2) }{{} loser s curse wnner s curse where b S enotes the b functon of the evatng ber, bs n the b functon use by all the other non-evatng bers, an 10 ρ the probablty that the b of the evatng ber s the hghest b of the other bers gven that the latter b s equal to p. Conser frst the common value nformaton moel. If the evatng ber acqures a more nformatve sgnal, t ncreases both the loser s an the wnner s curse of the non-evatng bers, an hence affects the non-evatng bers ncentves to change ther bs. But the ncrease of the wnner s curse affects both aucton formats 10 To make our arguments clearer we assume ρ to be constant, but as we shall see n the next secton, ρ wll epen n general on b S, b S n an p. 9

10 wth the same magntue, whereas the ncrease of the loser s curse s stronger n the open aucton than n the seale b aucton. The reason s that n the latter case t only affects wth probablty ρ. Consequently, we may expect that the strategc effect of nformaton acquston wll favor more nformaton acquston n the seale b aucton than n the open aucton. Conser next the prvate value nformaton moel. In ths case, there s no rect effect. However, a more nformatve sgnal nuces a more sprea (n the sense of secon orer stochastc omnance) of the evatng ber s contonal expecte prvate value an, hence, of her bs. We shall argue that ths effect has the opposte consequences on the non-evatng bers bs, ths s, t makes them less sprea. The fact that the evatng ber makes even hgher her hgh bs affects the non-evatng bers ncentves to submt hgh bs. The reason s that hgher bs by the evatng ber means that the ba news of the wnner s curse becomes worse an the goo news of the loser s curse not so goo. The combnaton of both effects shoul nuce the non-evatng bers to make ther hgh bs lower. 11 Symmetrc reasons explan that the fact that the evatng ber makes even lower her low bs nuces the non-evatng bers to maker ther low bs hgher. The consequences of these strategc effects of nformaton acquston on the evatng ber are unclear. Nevertheless, f the number of bers s suffcently large, we may expect ths strategc effect to favor the evatng ber, as she competes more often wth the types of the non-evatng bers that submt hgh bs. Agan the change n the wnner s curse affects equally to both aucton formats, but the change n the loser s curse s stronger n the open aucton than n the seale b aucton. These means that the postve strategc effect of nformaton acquston that appears when the number of bers s suffcently large shoul be stronger n the open aucton than n the seale b aucton. As a consequence, we may expect more nformaton acquston n the former aucton than n the latter. 4 Equlbrum Analyss In ths secton we prove the equlbrum analyss of our two aucton games. Ths analyss wll show that the effects stue n the former secton translate nto some equlbrum results that support the rankng of auctons wth respect to the ncentves to acqure nformaton that we suggeste. To prove our equlbrum, we start proposng an allocaton that satsfy some equlbrum restrctons, an later we use ths allocaton to construct the equlbrum strateges. In the case of symmetrc equlbrum (n whch the goo s always sol), an allocaton can be escrbe smply wth a functon, that we call allocaton functon, that maps types of the evatng ber nto types of the non-evatng bers. The 11 Ths change n the b of the non-evatng bers may nuce a smlar effect but of opposte recton n the evatng ber s bs. Moreover, ths new change of the evatng ber s bs shoul renforce the change n the non-evatng bs creatng a feeback loop. As we menton n the Introucton ths effect was alreay ponte out by Bulow, Huang, an Klemperer (1999) for the open aucton. 10

11 goo goes to the hghest type of the non-evatng bers f ts mage s hgher than the evatng ber s type. Otherwse, the goo goes to the evatng ber. Clearly, for a gven equlbrum, the corresponng allocaton functon maps types of the evatng ber nto types of the non-evatng bers, f any, that submt the same b. Hence, we may compute the allocaton functon from equlbrum restrctons that relate these types. In partcular, our equlbrum conton wll look at the ber s ncentves to change her b margnally. These ncentves were escrbe n the former secton. We conclue that f the expecte utlty of wnnng contonal on tyng wth the hghest b of the other bers s postve, the ber has ncentves to ncrease her b, an f t s negatve the ber has ncentves to ecrease her b. As a consequence, n equlbrum the former contonal expecte utlty must be equal to zero. Ths s the equlbrum conton we shall use to euce our allocaton functons. 4.1 The Open Aucton Agan, to euce the allocaton functon mplemente n the open aucton t wll suffce to look at the nformaton sets n whch only the evatng ber an a non-evatng ber, say Ber, reman bng n the aucton. These nformaton sets mpose suffcent restrctons to euce the allocaton functon. Denote by x an x the types of the non-evatng ber an the evatng ber, respectvely, that b n a gven equlbrum the same prce p n the above nformaton sets. Denote also by X the nformaton that these two bers can nfer along the equlbrum path about the types of the other bers. For the reasons explane n the former secton, our equlbrum conton for the non-evatng ber s: E T + Q j = x, X η = x, X p = 0, (3) Xη an smlarly our equlbrum conton for the evatng ber s: E T + Q j Xη = x, X η = x, X p = 0. (4) If we subtract both equatons, an after some smplfcatons, we get the followng equaton that relates x an x : E[T X η = x ] E[T X η = x ] = 0. (5) Uner our assumptons, the left han se of Equaton 5 s strctly ncreasng n x an strctly ecreasng n x. Thus, there must exsts an nterval [a, b] [0, 1] an an mplct functon φ O that maps types of the evatng ber x [a, b] to types of the non-evatng ber x [0, 1] that solve the above equaton. However, t may be that [a, b] cannot cover [0, 1],.e that the above conton cannot be use to efne φ O n the oman [0, 1]. In such cases, we shall call the extenson of φ O n the oman 11

12 [0, 1] to a functon that we enote by φ O (x ), that equals φ O (a) (0, n ths case) f x [0, a), φ O (x ), f x [a, b], an φ O (b) (1, n ths case) f x (b, 1]. Smlarly, we enote by φ O the nverse of φ O an by φ the extenson of φ O to [0, 1]. Next proposton shows that there s an equlbrum that mplements the allocaton functon φ O. The reaer may fn n the proof of the next proposton the equlbrum strateges that support our allocaton functon. We o not nclue them n the man text snce they are a straghtforwar generalzaton of the equlbrum propose by Mlgrom an Weber (1982). Proposton 1. There exsts an equlbrum of the open aucton that mplements the allocaton functon φ O, an n whch the evatng ber wth type 0 gets expecte utlty: 12 φo (0) where. 0 ( E[T X η = 0] E[T X η = x] ) x n, Proof n the Appenx. Note that n the propose equlbrum, a evatng ber wth mnmum type may get strctly postve expecte utlty. Ths s what happens when a evatng ber wth ths type wns wth postve probablty,.e. when φ O (0) > The Seale B Aucton We now procee wth the analyss of the seale b aucton. Denote by x an x the types of the non-evatng ber an the evatng ber, respectvely, that b n a gven equlbrum the same prce p. Our equlbrum conton for the non-evatng ber s: (1 ρ) E T + Q j X η ρ E T + = X η j = x, X η x, {X η l x } l,,j + Q j X η = x, X η = x, {X η l x } l, p = 0, (6) where, ρ s the probablty that tes wth the evatng ber gven that ber tes wth the maxmum b of the other bers at prce p, ths s, ρ x n 2 b (x ) x n 2 b (x ) x b (x ) xn 3 + (n 2) 12 Note that the expresson that follows s never negatve, snce E[T X η = 0] E[T X η = x] for all x [0, φ O (0)] by efnton of φ O., 12

13 where b S n an bs enote the equlbrum b functons of the non-evatng ber an the evatng ber, respectvely. Smlarly, our equlbrum conton for the evatng ber wth type x s n ths case: E T + Q j = x, X η = x, {X η j x } j, p = 0. (7) Xη Suppose now that φ S (x) s the allocaton functon mplemente n the seale b aucton. Ths means that b (x) = b(φ S (x)), whch mples that φ S (x) = b (x) b (φ S (x)). If we use ths fact, an combne Equaton 6 an Equaton 7 elmnatng p, we get after some algebra the followng equaton: φ S(x) ( E[T X η = x] E[T X η = φ S (x)] + µ (x, η ) µ (φ S (x), η) ) = [ ] φs (x) (E[T X η = φ S (x)] E[T X η x(n 2) = x]), (8) where µ (x, η ) E[Q X η = x] E[Q X η x]. The rght han se of Equaton (8) correspons to the case n whch the hghest losng b s the evatng ber s, ths s when ρ = 1, as t happens n the nfference conton of the open aucton. In fact, ths rght han se equals to zero f φ S (x) = φ O (x) n the oman of φ O n spte of contonng on atonal nformaton n the open aucton. Ths s because of our assumpton of atve separablty an nepenency of the bers types. From the left han se of Equaton (8) we can get the followng equaton n x an x : E[T X η = x ] E[T X η = x ] + µ (x, η ) µ (x, η), (9) that can be use to efne mplctly a new functon φ. Note that ths s possble because uner our assumptons, the left han se of Equaton (9) s ncreasng n x an ecreasng n x. Agan, t may be the case that Equaton (9) oes not efne φ an ts nverse φ n the nterval [0, 1]. In ths case, we efne n the same way we wth φ O the extenson of φ an φ on [0, 1] that we enote by φ an φ respectvely. Clearly, at any pont x an ncreasng soluton to Equaton (8) must be between φ O (x) an φ (x). Next lemma shows that such a soluton exsts. Lemma 1. There exsts a strctly ncreasng soluton, φ S, to the fferental equaton n (8) wth oman an range two ntervals n [0, 1] an that splts [0, 1] 2 nto two sets. Moreover: 1. φ S (x) [mn{ φ (x), φ O (x)}, max{ φ (x), φ O (x)}]. 2. φ S (x) [mn{ φ (x), φ O (x)}, max{ φ (x), φ O (x)}]. 3. If φ S (1) exsts, then φ S (1) φ (1), an f φ (1) exsts, then φ(1) φ O (1). S S 13

14 Proof n the Appenx. Note that as t happens wth φ O an φ, the functon φ S erve from the equlbrum contons an ts nverse φ S may not be efne n all the nterval [0, 1]. In ths case, we efne n the same way we wth φ O an φ the extenson of φ S an φ S on [0, 1] that we enote by φ S an φ S. The contons prove n the thr tem of Lemma 1 wll be use later n the proof of the next proposton to guaranty that we can exten the equlbrum b functons out of the omans of φ S an φ S, partcularly n the cases n whch ether φ S (1) < 1 an (1) < 1. φ S Proposton 2. There exsts an equlbrum of the seale b aucton that mplements the allocaton functon φ S (x) an n whch the evatng ber wth type 0 gets expecte utlty: φs (0) ( E[T X η = 0] E[T X η = x] ) x n, where. 0 Proof n the Appenx. 4.3 Equlbrum Incentves to Acqure Informaton We shall now conser a ber s ncentves to acqure nformaton. We ntrouce the followng efnton that captures the concept of ncentves: Defnton: [Incentves to acqure nformaton] We say that an aucton a gves hgher ncentves to acqure nformaton than another aucton a f an only f the followng two nequaltes hol: U a (η, η) U a (η, η) U a (η, η) U a (η, η), for η > η. U a (η, η) U a (η, η) U a (η, η) U a (η, η), for η < η. Where U a (η, η) s the evatng ber s expecte utlty n aucton a when she has nformaton precson η an all the other bers have nformaton precson η. The frst pont of the former efnton says that the evatng ber s expecte utlty ncreases more n aucton a than n aucton a when she evates acqurng more nformaton from the stuaton n whch all the bers have the same nformaton precson, ths s, the evatng ber have greater ncentves to evate upwars n aucton a than n aucton a. Smlarly, the secon pont says that the evatng ber s expecte utlty ecreases more n aucton a than n aucton a when she evates acqurng less nformaton from the stuaton n whch all the bers have the same nformaton precson, ths s, the evatng ber have less ncentves to evate ownwars n aucton a than n aucton a. 14

15 Inepenency of the bers types mply, see Myerson (1981), that the allocaton functons are suffcent to etermne the bers expecte utlty an hence, ther ncentves to acqure more or less nformaton. Moreover, the atve separablty of the bers utlty functon makes specally smple the expresson of the ber s expecte utlty. Next lemma shows these clams. Lemma 2. Suppose that there exsts an equlbrum for a gven aucton mechansm n whch allocaton functon φ s mplemente. Then, the ex-ante expecte utlty of the evatng ber n ths equlbrum of the aucton mechansm s equal to: 1 0 (1 x)φ(x) n E[T + Q X η = x] x x plus the expecte utlty that the evatng ber gets when she has type 0. Proof. Straghtforwar aaptaton of the arguments by Myerson (1981). From ths lemma, t s easy to see that when all bers have the same level of nformaton precson, a kn of revenue equvalence theorem hols n the sense that both the seale b an the open aucton gve bers the same expecte utlty. To see why note that n a symmetrc equlbrum, the allocaton functon s the entty n both aucton formats. One consequence s that to prove that an aucton have hgher ncentves, we only nee to show that the evatng ber s expecte utlty s hgher n the former aucton than n the latter when η > η an vceversa, f η < η. Moreover, Lemma 2 show that t s suffcent to look at the allocaton functons. We can now llustrate the results we antcpate for the common value nformaton moel wth an equlbrum analyss: Lemma 3. In the common value nformaton moel, we have that φ O (x) = x, an f η < η, φ S (x) x, an f η > η, φ S (x) φ O (x) = x, for any x [0, 1]. Proof. The efnton of φ O n Equaton (5) mples that φ O s nvarant to changes n η n the common value nformaton moel. Moreover, our symmetry assumpton mples that φ(x) = x for all x [0, 1]. We next conser the case η > η, the case η < η s symmetrc. In the common value nformaton moel an by efnton of monotoncally more nformatve µ(x, η ) > µ(x, η), whch means that, E[T X η = x] E[T X η = x] + µ (x, η ) µ (x, η) > 0. Snce E[T X η = x] + µ (x, η) s ncreasng n x, then φ (x) x to satsfy Equaton (9). The applcaton of the frst tem of Lemma 1 conclues the proof. Corollary 1. In the common value nformaton moel, the seale b aucton gves hgher ncentves to acqure nformaton than the open aucton. Ha we mae use only of more nformatve sgnals nstea of monotoncally more nformatve sgnals, we coul prove the above result, but only when the number of bers s suffcently large. Next, we look at the prvate value nformaton moel. 15

16 Lemma 4. In the prvate value nformaton moel, there exsts an ɛ > 0 such that f η > η, then φ O (x) > φ S (x) for all x ( φ O (1) ɛ, φ O (1)), an f η < η, then φ O (x) > φ S (x) for all x (1 ɛ, 1). Proof. We only stuy the case η > η, the other one s symmetrc. Snce E[T ] = x E[T X η x] + (1 x) E[T X η x], E[T X η x] an n partcular E[T X η = 1] ncreases wth η. Ths means that E[T X η = 1] E[T X η = 1] > 0 because of symmetry. As a consequence, φ O (1) < 1, an, E[T X η = φ O (1)] E[T X η = 1] + µ (φ O (1), η ) µ (1, η) < 0. Snce E[T X η = x] + µ(x, η) s ncreasng n x, then φ O (1)) < 1 to satsfy Equaton (9), whch mples that φ (φ O (1)) < φ O(φ O (1)). An applcaton of the frst tem of Lemma 1 conclues the proof by contnuty. (φ Corollary 2. In the prvate value nformaton moel, the open ascenng aucton gves hgher ncentves to acqure nformaton than the seale b aucton f n s large enough. Ths last result oes not make any use of our assumpton that sgnals are orere n the sense of monotoncally more nformatve sgnals, we only nee to assume that they are orere n the sense of more nformatve sgnals. 5 Conclusons In ths paper, we have stue the strategc effects assocate to open nformaton acquston. Ths strategc effects are orgnate because a ber s nformaton acquston ecson affects the other bers b behavor. In partcular, we have shown that these strategc effects are such that a ber has greater ncentves to acqure nformaton about the common value n a seale b aucton than n an open aucton. However, we have also shown that f the nformaton acquston s about the prvate value, the ncentves are hgher n an open aucton than n a seale b aucton, at least when the number of bers s suffcently large. Certanly, t s not ffcult to use our results to gve some rankngs of whch aucton nuces more nformaton acquston once we a a stage before the aucton n whch whch bers may acqure atonal nformaton at some cost. Clearly, the results woul be that the seale b aucton nuces more nformaton acquston f the atonal nformaton s about the common value, an the open aucton nuces more nformaton acquston f the atonal nformaton s about the prvate value. The latter result when the number of bers s suffcently large. The results suggeste n the paragraph are qute nterestng snce n the symmetrc equlbrum of an aucton, the more nforme about the prvate value the bers have the more effcent the allocaton s. We coul then conclue that the seale b aucton mplements a more effcent allocaton than the open ascenng aucton n the moel of the former paragraph. Smlarly, we also know that the more nforme the bers are the greater s the expecte revenue generate n the aucton, at least when the number of bers s 16

17 suffcently large. See for nstance Ganuza (2004), Ganuza an Penalva-Zuast (2004) an Hageorn (2004). Snce uner our assumptons our two aucton formats are revenue equvalent for a gven level of nformaton of the bers, we may conclue that epenng on whether the bers can acqure nformaton about the common value or the prvate value, the seale b aucton wll gve more or less expecte revenue than the open aucton. 17

18 APPENDIX Ths Appenx has four parts. In Appenx A we prove an equlbrum for the open ascenng aucton that proves Proposton 1. Appenx B shows that there exsts a soluton to the fferental equaton of Lemma 1. Appenx C proves an equlbrum for the seale b aucton that proves Proposton 2. Appenx A: Proof of Proposton 1 We start proposng some strateges. We procee sequentally, frst, nformaton sets n whch noboy has left the aucton yet: Devatng ber s b functon: b 0 (x) E T + Q j = x, Xη = φ { O (x), X η j = φ } O (x) Xη Non-evatng ber s b functon : b 0 } (x) E T + Q j = φ O (x), Xη = x, {X η j = x Xη In nformaton sets n whch k bers have left the aucton an where p l s the prce at whch the l-th ber n eclarng nactve has qut, an p l s her entty. Frst, when the non-evatng ber s not among the k bers that have left the aucton. Devatng ber s b functon: j, j,.. [ ] b k t {(p l, j l )} k l=1 E T + Q j {X = x, η j = φ } O (x) Xη j I\{j l } k l=1 [ ], {b l jl X η l {(p q, j q )} l q=1 } k = p l. l=1 Non-evatng bers b functon, : [ ] b k t {(p l, j l )} k l=1 E T + Q j = Xη φ O (x), { X η j = x } j I\{j l } k l=1 [ ], {b l jl X η j l {(p q, j q )} l q=1 } k = p l. l=1 18

19 An now, when the evatng ber s among the bers that has left the aucton, : b k [ ] t {(p l, j l )} k l=1 E T + Q j } {X η j = x j I\{j l } k l=1 [, {b l jl X η j l j l ] {(p q, j q )} l q=1 } k = p l, l=1 where recall that η l = η for any l. To prove that the above strateges form an equlbrum, note frst that the evatng ber s expecte value of the goo contonal on the vector of bers types (x 1, x 2,..., x n ) s equal to: E T + Q j Xη { = x, X η j = x j } j I\{} Moreover, f all the non-evatng bers follow the propose strategy, the prce that the evatng ber pays f she wns s equal to the b of the ber wth hghest type among the non-evatng bers, say Ber l, ths s: E T l + Q j Xη { } = φ O (x l), X η j = x j. j I\{}, After some algebra, we may show that the fference between these two values s equal to: E[T X η = x ] E[T l X η l = x l ] + E[Q X η = x ] E[Q X η. = φ O (x l)], (10) whch s non negatve f an only f x φ o (x l ), ths s φ(x ) x l. These are exactly the cases n whch the evatng ber wns f she follows the propose strategy, an thus, she cannot mprove by followng a fferent strategy. The proof that the nonevatng bers o not have ncentves to evate s smlar, an hence, we o not nclue t. Note next that the propose strateges mplement φ O. Frst, all non-evatng bers use the same strctly ncreasng b functon, an secon the evatng ber wns the aucton f an only f φ(x ) s greater than the maxmum type of the nonevatng bers. Fnally, straghtforwar computatons show that the evatng ber s expecte utlty wth type 0 s as state n the Proposton. Appenx B: Proof of Lemma 1 INCOMPLETE We may wrte the fferental equaton n (8) wth the help of the followng functon: Φ(x, φ) φ x(n 2) E[T X η = φ] E[T X η = x] E[T X η = x] E[T X η = φ] + µ (x, η ) µ (φ, η), (11) 19

20 Thus, our fferental equaton can be wrtten as φ s(x) = Φ(x, φ s (x)). Defne also the set S as the set of ponts n (x, φ) [0, 1] 2 such that φ [mn{ φ (x), φ O (x)}, max{ φ (x), φ O (x)}]. Lemma 5. There exsts a unque soluton to the fferental equaton φ (x) = Φ(x, φ(x)) at passes by any pont (ˆx, ˆφ) n the nteror of S. Moreover, the soluton s strctly ncreasng an wth boune ervatve at (ˆx, ˆφ). Proof. Φ(x, φ) s contnuous n x an contnuously fferentable n φ at any pont n the nteror of S, thus t satsfes a Lpschtz conton n all ths set. Consequently, the frst part of the lemma s a rect applcaton of Congton an Levnson (1984)[Theorem 2.2, pag. 10]. To prove the secon part, note that at any pont n the nteror of S, ether φ O (x) < φ (x) or φ (x) < φ O (x). In the frst case, φ ( φ O (x), φ (x)) f (x, φ) n the nteror of S, whch mples that the numerator an the enomnator of the expresson that efnes Φ are strctly postve. Ths s because the numerator s equal to zero at φ = φ O (x) an t s strctly ncreasng n φ, moreover, the enomnator s equal to zero at φ = φ (x) an t s strctly ecreasng n φ. The proof for the secon case s smlar. Conser frst two cases: () φ (x) > φ O (x), an () φ (x) < φ O (x). In the frst case, we can contnue our local soluton to the left. Snce Φ(x, φ) goes to nfnty as φ goes to φ (x), an Φ(x, φ O (x)) = 0, the contnuaton soluton remans n the nteror of S untl ether x or φ reach zero, or the soluton converges to a pont n whch φ (x) = φ O (x). We can apply smlar arguments to () an contnue our soluton to the rght untl ether x or φ reach one, or the soluton converges to a pont n whch φ (x) = φ O (x). Fgure 1 llustrates our arguments for the case (). Conser now the case φ O (x) = φ (x). We call these values of x crossng ponts. Clearly, φ S (x) = φ O (x) = φ (x) s soluton to our fferental equaton f x s a crossng pont. If to the left of the crossng pont, t hols that φ O (x) > φ (x), we know from the arguments n the former paragraph that we can contnue to the rght any local soluton n the nteror of S an t wll converge to the crossng between φ O (x) an φ (x) as x tens to the crossng pont. Somethng smlar happens f to the rght of the crossng pont t hols that φ O (x) < φ (x). In ths case, we can contnue local solutons to the left. The two remanng cases, ether φ O (x) < φ (x) to the left of a crossng pont or φ O (x) > φ (x) to the rght of a crossng pont, are more complcate. We prove exstence for the latter case. The former case can be prove wth a smlar argument. Denote n what follows by x c the crossng pont wth the property that for x close x c an to the rght φ O (x) > φ (x). We construct frst a sequence of functons {φ τ } τ T, where T s a strctly ecreasng sequence that converges to x c. These functons are efne as follows: φ τ (x) = φ O (x) for x [x c, τ], an t equals to a soluton of the fferental equaton (8) wth ntal conton φ τ (τ) = φ O (τ) for x > τ. To see why ths soluton exsts, note that Φ(x, φ) s contnuous n x an contnuously fferentable n φ when φ = φ O (x) φ (x), thus t satsfes a Lpschtz conton an by Congton an Levnson (1984)[Theorem 2.2, pag. 10] the soluton exsts an t s unque. Moreover, snce Φ(x, φ O (x)) = 0 f φ O (x) φ (x) the slope of our soluton at the ntal conton 20

21 Fgure 1: Contnuaton of the soluton of φ (x) = Φ(x, φ(x)) n S. s zero, an hence when we move to the rght, the soluton enters n the nteror of S an by Lemma 5, t can be contnue. Once n the nteror of S, Lema 5 assures that the the solutons to our fferental are unque. Ths means that the functons φ τ an φ τ cannot cut n the nteror of S. We can thus euce that the sequence {ph τ (x)} T s ecreasng an hence the functon φ τ converges pontwse. We enote φ(x) ts pontwse lmt. We can also efne another fferental equaton: x 1 (φ) = Φ(x(φ),φ). Ths new fferental equaton has very smlar propertes to φ (x) = Φ(x, φ(x)) an n fact we can prove an equvalent verson of Lemma 5. We can also construct a sequence of functons {x ζ } ζ Z, where Z s a ecreasng functon that converges to φ O (x c ) = φ S (x c ). Each of the functons x ζ s efne as follows: x ζ (φ) = φ (x) for x [φ (x c ), ζ], an t equals to a soluton of the fferental equaton x 1 (φ) = Φ(x(φ),φ) wth ntal conton x ζ(φ) = φ (φ) for φ > ζ. The soluton exsts an t s unque by the same arguments use above for φ τ. Moreover, 1 snce = 0 f φ O (φ) φ (φ) the slope of our soluton at the ntal conton Φ(φ (φ),φ) s zero, an hence the soluton also moves towars the nteror of S an hence can be contnue. Once n the nteror of S by Lemma 5, x ζ must be strctly ncreasng. Snce x ζ s also strctly ncreasng for φ < τ, x ζ s nvertble. Denote by φ ζ the nverse of x ζ. Moreover, n the nteror of S, an for the same arguments that we use for φ τ, the functons {x ζ } Z cannot cut an hence, {x ζ (φ)} Z s a ecreasng sequence. Ths means that {φ ζ (x)} Z s an ncreasng sequence, an hence that t s convergent. We enote by φ the pontwse lmt of {φ ζ } Z. The functons φ an φ cannot across as otherwse, there shoul exsts functons 21

22 φ ζ (x) an φ ζ (x) that also cross, an ths s ncompatble wth the unqueness of the solutons state n Lemma 5. For the same reasons any soluton to our orgnal fferental equaton wth ntal conton (ˆx, ˆφ) such that ˆφ [φ(ˆx), φˆx] cannot cross nether φ nor φ. Any such soluton s then a soluton to our fferental equaton that remans n S untl x reaches x c. Appenx C: Proof of Proposton 2 We start proposng some bers strateges an we show that they form an equlbrum. In our constructon we use the functon φ S of Lemma 1 n the Appenx. Devatng ber s b functon: b S (x) E T + Q j Xη Non-evatng ber s b functon : = x, Xη = φ { S (x), X η j φ } S (x) j,. If 13 x > φ S (1), then: b S n (x) E T + Q j Xη If x [ φ S (0), φ S (1)], then: b S n (x) E T + Q j Xη = X η j = x, Xη 1, { X η j x } = x, X η { } = φ O (x), X η j x j,,j j,.. If x < φ S (0), then: b S n (x) = E T + Q j Xη } = 0, Xη = x, {X η j x We shall prove that the above strateges form an equlbrum showng that the bers wn when t s proftable for them to wn an losses otherwse. We start wth the evatng ber. Her expecte value of the goo contonal on her type x, an the maxmum of the other bers types, say Ber, x s equal to: E T + Q j Xη { = x, X η = x, X η j x } j, j, 13 Note that x > φ S (1) only f φs (1) < 1 an n that case φ S (1) = φ S (1). Somethng smlar happens below wth x < φ S(0)... 22