Information Acquisition, Noise Trading, and Speculation in Double Auction Markets*

Transcription

1 Ths Draft: October 2008 Informaton Acquston, Nose Tradng, and Speculaton n Double Aucton Markets* Tr V Dang Unversty of Mannhem Yale Unversty Abstract Ths paper analyzes nformaton acquston n double aucton markets and shows that for any fnte nformaton cost, f the number of traders and the unts a trader s allowed to trade are suffcently large, then an effcent equlbrum allocaton fals to exst. For a large set of parameter values any equlbrum wth postve volume of trade has the followng propertes. Ex ante dentcally nformed, ratonal traders evolve endogenously to nose traders, speculators, and defensve traders. ecause of defensve tradng the allocaton s neffcent,.e. not all gans from trade are realzed. ecause of endogenous nose tradng the prce s not fully revealng. Journal of Economc terature Classfcaton Numbers: G14, D82, D83 Key words: double aucton, endogenous lemons problem, nformaton acquston, nose tradng, speculaton * I thank Martn ellwg and Ernst udwg von Thadden for dscussons, and Markus runnermeer, Edde Dekel, Danel Krähmer, Albert Kyle, and enny Moldovanu for comments. Fnancal support from the Deutsche Forschungsgemenschaft (DFG) s gratefully acknowledged. Emal: trv.dang@yale.edu.

2 1. Introducton A prototype of a centralzed market s a large double aucton n whch traders submt lmt orders to buy and sell some unts of an asset. The buy and sell orders are ranked accordng to the bd and ask prces, respectvely, whch generates an aggregate demand and supply schedule. The market prce s set to equalze demand and supply. Two central questons arse. Does such a tradng mechansm lead to an equlbrum allocaton that s effcent,.e. do the traders wth the hghest valuatons of the asset obtan the asset? Does such a tradng mechansm lead to an equlbrum prce that s nformatonally effcent,.e. does the prce fully reveal the aggregate nformaton of all traders? For the case where the traders have exogenous prvate nformaton about the (common) value of the asset as well as prvate nformaton about ther own valuaton of the asset, Reny and Perry (2006) provde a postve answer to both questons. They show that a large double aucton market s allocatve and nformatonally effcent. The present paper does not assume exogenous prvate nformaton, but assumes that ex ante all traders have dentcal nformaton about the uncertan value of the asset and analyzes the mplcatons of endogenous nformaton acquston for allocatve and nformatonal effcency n small and large double aucton markets. The demand for fnancal analysts coverage, ratng servces, loomberg s and Reuters fnancal servces suggest that nformaton acquston s a prevalent actvty n fnancal markets. In secondary markets a seller does not necessarly posses better nformaton than a potental buyer but the traders can acqure nformaton about the rsky cash flow stream of the asset before they trade. Therefore, the analyss of double aucton markets wth endogenous nformaton can provde nsghts nto the functonng of real fnancal markets snce a double aucton mmcs the workng of a call market such as the overnght market on the New York Stock Exchange. 1 Ths paper shows that for any fnte nformaton cost, f the number of traders and the unts a trader s allowed to trade are suffcently large then an effcent equlbrum allocaton n double aucton markets fals to exst. For a large set of parameter values any equlbrum wth postve volume of trade has the followng propertes. Ex ante dentcally nformed, ratonal traders evolve endogenously to nformed speculators, unnformed defensve traders, and nose traders. ecause of defensve tradng the allocaton s neffcent,.e. not all gans from trade are realzed. ecause of endogenous nose tradng the prce s not fully revealng 1 Also, the openng prce and allocaton of the electronc tradng system, Xetra and floor tradng on Frankfurt Stock Exchange are determned by a double aucton type mechansm. 1

3 of the traders aggregate nformaton. Ths paper provdes a strategc foundaton for the Grossman and Stgltz (1980) mpossblty result of nformatonally effcent large (double aucton) markets as well as shows that equlbrum allocatons n such a market are not effcent f nformaton s endogenous. In the present model there are hgh and low valuaton traders of a rsky asset. 2 It s common knowledge that the asset s worth v+ to a hgh valuaton trader and v to a low valuaton trader, where s a constant, and v s a random varable and ether v or v. Each trader maxmzes hs expected payoff. To llustrate the strategc consequences of nformaton acquston, the paper frst analyzes a double aucton wth two traders. 3 The paper shows that f the nformaton cost s low, a trader s concerned about an endogenous lemons problem. For example, gven that a trader submts a prce around the expected value E[v] of the asset, then the best response of the other trader s to acqure nformaton and speculate. Trade only occurs wth probablty one f both traders are nformed. The motve of nformaton acquston s drven by the desre to trade wthout beng exploted. If the nformaton cost s larger than the total tradng surplus but smaller than the potental speculatve proft, then no pure strategy equlbrum wth effcent trade exsts although the traders mantan symmetrc nformaton n equlbrum and the gans from trade are common knowledge. Ths paper shows that an endogenous lemons problem,.e. the concern of sufferng a potental speculatve loss due to the mere possblty of nformaton acquston by the other trader, can already render effcent trade unattractve. 4 In any mxed strategy equlbrum n whch trade occurs wth postve probablty a trader randomzes between beng nformed or not. An unnformed trader s sometmes a defensve trader,.e. a hgh (low) valuaton trader only wants to buy (sell) at a prce around v (v ). An unnformed trader s sometmes a nose-type trader,.e. he submts an order at a prce around E[v], so that he may suffer a lemons problem when tradng wth an nformed 2 A low valuaton trader s a trader who has low lqudty (that s a need for cash) or hedgng reasons to sell. Also, portfolo rebalance needs, tax-nduced trades, or dvdend-captured trades gve rse to mutually benefcal transactons. See the dscusson n secton 6. If ratonal agents have the same prvate or margnal valuaton of the asset, then the No-trade Theorem apples. See Mlgrom and Stokey (1982). 3 A small double aucton or smultaneous offer barganng can be nterpreted as a model of over-the-countertradng. arganng s a standard feature n many decentralzed markets, such as those for mortgage-backed securtes, collateral debt oblgatons, dervatves, corporate and muncpal bonds. See Duffe et al. (2005). 4 Ths no trade result s dfferent from Myerson and Satterthwate (1983) because the gans from trade are common knowledge n the present model. Ths result s also dfferent from Akerlof (1970) and Gresk (1991) snce the traders possess symmetrc nformaton about the common valuaton n the no trade equlbrum. 2

4 trader. An nformed trader s always a speculator and only trades n hs preferred state,.e. a hgh (low) valuaton trader only buys (sells) at a prce around E[v] f the true state s v (v ). Consequently, trade only occurs under two crcumstances: () the two traders are nose traders or () one s a nose trader and the other one s nformed and the state of nature s the preferred one of the nformed trader. Therefore, the prce s always around E[v] and not unnformatve. The second part of the paper analyzes a double aucton wth many hgh and low valuaton traders and shows that for any fnte nformaton cost, f the number of traders and the unts a trader s allowed to trade are suffcently large, then an effcent equlbrum allocaton fals to exst. In decentralzed tradng, f the nformaton cost s large, the traders face no potental lemons problem and there exst equlbra where trade occurs wth probablty one. ut as the number of traders ncreases, the potental speculatve proft of an nformed trader ncreases because there are potentally more unnformed traders to explot. Therefore, the speculatve threat an unnformed trader faces exsts not only for low but also for large nformaton costs. In such a case only tradng equlbra n mxed strateges exst where some unnformed traders behave lke nose traders, some unnformed traders are defensve traders, and some traders become nformed speculators. ecause of endogenous nose tradng the prce s not fully revealng and because of defensve tradng the allocaton s neffcent. The remander of the paper s organzed as follows. The next secton relates ths paper to the lterature. Secton 3 ntroduces the model. Secton 4 analyzes nformaton acquston n a small double aucton. Secton 5 analyses nformaton acquston n large double auctons. Secton 6 dscusses the assumptons, and secton 7 concludes wth a dscusson of some market mcrostructure mplcatons. Omtted proofs n the text are gven n Appendx. 2. Relaton to the terature Ths paper s most closely related to Reny and Perry (2006) who analyze a large lmt double aucton market where the traders have exogenous prvate nformaton and show that such a market s allocatve and nformatonally effcent. They provde a strategc foundaton for a ratonal expectatons equlbrum (REE) outcome. Ths paper analyzes nformaton acquston n double aucton markets and shows that f the number of traders and the unts a trader s allowed to trade are suffcently large, then a double aucton market s nether allocatve nor nformatonally effcent. Reny and Perry (2006) have a much more general nformaton and valuaton structures. Yet the key economc reason why ther result n the double aucton stage does not 3

5 apply to ths settng s the followng. In the present model prvate nformaton s endogenous and an nformed trader has to cover hs nformaton cost. If the prce was fully revealng, then some traders would have proftable devatons: () An nformed trader chooses not to acqure costly nformaton snce there s no speculatve proft to make. () Snce there s no lemons problem, no unnformed trader submts defensve offers. Consequently, some of these traders devate to nose traders. On the other hand, f there are too many nose traders and very few nformed, then an nformed trader can move prces and make speculatve profts. Therefore, n an equlbrum wth postve volume of trade the prce s not fully revealng. ecause of the potental lemons problem some traders behave defensvely and the allocaton s not effcent. Ths paper provdes a strategc foundaton for a nosy REE outcome as well as the behavor assumptons n the nosy REE framework whch consttutes a workhorse n fnancal economcs. Market mcrostructure models typcally assume that there are three exogenous types of agents: () nformed traders (speculators), () unnformed traders wthout real tradng motves (such as market makers), and () unnformed traders wth real tradng motves or dfferent prvate valuatons of the asset (lqudty traders). An assumpton n REE models wth exogenous nose s that the tradng behavor of lqudty traders s nelastc. These agents just want to trade some exogenous unts of the asset rrespectve of prces. Grossman and Stgltz (1980), ellwg (1980), Kyle (1985, 1989), and Glosten and Mlgrom (1985) are nfluental papers that are based on the REE framework wth exogenous nose. Glosten (1989), Admat and Plederer (1988), Chowdhry and Kanda (1991), Spegel and Subrahmanyam (1992) replace the exogenous nose assumpton by the assumpton that the lqudty traders account for the exstence of speculators and adjust ther tradng behavor to the expected nformaton asymmetry by choosng dfferent amounts or dfferent markets to trade. Yet n these so-called endogenous nose tradng models, the lqudty traders do not consder or are not allowed to acqure nformaton. The key dfference between the present model and most papers on nformaton acquston n fnancal markets, such as Verreccha (1982), Jackson (1991), arlevy and Verones (2000), Mendelson and Tunca (2004), and Veldkamp (2006) s that these papers assume that a subset of traders (lqudty traders) s ether not maxmzng or not allowed to acqure nformaton. A standard feature of REE models wth both exogenous and endogenous nose s that the equlbrum prce s typcally not nformatonally effcent. Gven the nose assumpton, there s no meanngful dscusson of allocatve effcency n nosy REE models. The present model assumes that all traders can acqure nformaton. In some sense ths paper endogenzes both the number k of nformed speculators and the number n of 4

6 unnformed lqudty traders n nosy REE models such as Spegel and Subrahmanyam (1992). In the present paper there are N hgh valuaton traders and N low valuaton traders. Out of the 2N ex ante unnformed traders, n equlbrum k traders become nformed speculators, and the remanng 2N k traders stay unformed and are comparable to the n lqudty traders n Spegel and Subrahmanyam (1992). In contrast to ther symmetrc lnear equlbrum n whch the n lqudty traders all behave dentcally, some of the unnformed traders n ths model become nose traders whle others become defensve traders and do not trade. The behavor of the traders and the expected number of the three types of traders are endogenous and depends on the nformaton cost, the severty of the lemons problem, and the gans from trade. 5 A second mportant dfference between ths paper and many market mcrostructure models concerns the tradng envronment. In ths model there s no market maker who observes the order flow and determnes the prce. The neffcency results of the present paper gves rse to the followng queston. Can the exogenous presence of a fourth type of traders, such as desgnated market makers who are forbdden to speculate, facltate allocatve and nformatonal effcency n double aucton markets? Secton 7 dscusses ths queston and further market mcrostructure mplcatons. Ths paper s also related to the aucton lterature. Mlgrom (1981), Matthews (1984), ausch and (1993), Persco (2000), and ergemann and Pesendorfer (2007) analyze nformaton acquston n auctons where only the buyers sde consders nformaton acquston. The seller s typcally non-strategc or nosy and just wants to sell the asset. In contrast, ths model assumes that all traders behave strategcally and can acqure nformaton. ecause of the endogenous lemons problem a strategc buyer (seller) may not want to buy (sell) and forgoes the tradng gan. The two-sded strategc behavor (even wth exogenous prvate nformaton) gves rse to some techncal dffcultes snce the random varables or socalled order statstcs are not afflated. Reny and Perry (2006) provde a new technque to solve for an equlbrum n the large lmt market. For the exstence of a mxed strategy equlbrum wth postve volume of trade n double auctons wth exogenous nformaton see Jackson and Swnkels (2005). They establsh the proof by ntroducng a nose trader and show that as the probablty for the exstence of the nose trader vanshes, there s stll some trade. The present paper assumes that nformaton s endogenous and shows that n a tradng equlbrum some strategc traders may endogenously 5 Trueman (1988) and Dow and Gorton (1997) provde a theory of nose tradng based on agency consderatons n a delegated portfolo management settng. In the present model, all agents trade on ther own behalf. 5

7 behave lke nose traders. Therefore, ths paper suggests that an equlbrum wth postve volume of trade also always exsts n double auctons wth endogenous nformaton. 3. The Model There are 2N rsk neutral traders n a market for a sngle asset. The frst N traders are hgh valuaton traders and the traders N+1 to 2N are low valuaton traders. It s common knowledge that the asset s worth v+ to a hgh valuaton trader and v to a low valuaton trader, where s the prvate value component and a constant, and v s the uncertan common value component of the asset. v s ether v or v wth equal probablty and 0< < 81 (v v ). 6 Each trader has a tradng need of one unt. If each of the N pars of hgh and low valuaton traders have traded one unt wth each other, then all traders have the same (margnal) valuaton of v for the next unts bought or sold. The total tradng gan s therefore 2N. Secton 6 argues that n an economy wth one rskless asset and one rsky asset, the specfc valuaton of v+ (v ) can be nterpreted as a shortcut for the margnal valuaton of a rsk averse trader wth low (hgh) endowment of the rsky asset. avng hedged ther postons and realzed the gans from trade, the traders have equalzed ther margnal valuatons and are only wllng to pay at most or accept at least v for further unts bought or sold. An alternatve story s dvdend and tax-motvated trade. A trader facng a low (hgh) dvdend tax rate has a hgh (low) valuaton of the asset. 2 represents the tax gans. A hgh valuaton trader s wllng to pay v+, whle a low valuaton trader s wllng to accept v. avng exhausted the tax gans, the traders have a valuaton of v for the next unts. The sequence of moves s as follows. In the frst stage, a trader can obtan a perfect sgnal about the true value of the asset by ncurrng the cost c>0. Informaton acquston s not observable by the other traders. In the second stage the traders play a double aucton,.e. they submt lmt orders to buy and sell up to m unts of the asset. Short sellng s allowed. The exact tradng, allocaton and prcng rule s specfed n the subsequent sectons. 6 Ths paper assumes that the prvate valuatons are common knowledge. Otherwse, one also has to deal wth strategc rent extracton dscussed n Chatterjee and Samuelson (1983) and the well-known Myerson and Satterthwate (1983) problem. Ths further uncertanty may cause addtonal allocatve neffcences. The focus here s on common value uncertanty whch mght be more mportant than prvate value uncertanty n fnancal transactons. The assumpton that < 1 8 (v v ) makes the problem nterestng. If the tradng gan s large, the potental lemons problem may have no adverse allocatve consequences whch wll become clear n the analyss. 6

8 b s A pure strategy of trader s denoted wth t = ( n, ( b,u ),( s,u )) where n {0,1} denotes nformaton acquston, b (s ) s the bd (ask) prce, and u ( u ) the sze of the buy b s (sell) order. If trader chooses n =0 and s unnformed, then b (n ) R + and b u (n ) {0,..,m}. For n =1, then b (n )=(b,b ) and u (n )=( u, u b b b ) are vectors wth two components, where b,b R + and u b, u b {0,..,m} and denote the bd prce and the sze of the buy order of trader when the true value of the asset s v and v. Analogously for s (n ) and u s (n ). A mxed strategy s a probablty dstrbuton over pure strateges and denoted wth. The followng examples llustrate ths notaton. 7 () n =0, b =v, u =1, s = u s =0 s a pure strategy where trader does not acqure nformaton and submts a bd prce of v to buy one unt and no sell order. () n =1, b =(v,e[v]), u =(1,m), s = u s =(0,0) s a pure strategy where trader acqures nformaton, submts a bd prce of v to buy one unt f v=v and a bd prce of E[v] to buy m unts f v=v, and no sell order. () A mxed strategy s e.g. a randomzaton that puts probablty 0.4 on the pure strategy (), probablty 0.6 on the pure strategy (), and zero probablty on any other pure strategy. Snce nformaton acquston s not observable, the soluton concept s ayesan Nash equlbrum (NE). A NE n pure strateges n ths game s a profle EU ( t * b b * 2N { t } = 1, such that * *,t ) EU ( t,t ), for all potental pure strateges t of trader where =1,..,2N. A NE n mxed strateges s a profle * 2N { } = 1 of probablty dstrbutons over pure strateges, * * * such that EU (, ) EU (, ), for all potental probablty dstrbutons of trader where =1,..,2N. Equlbrum always refers to a NE. Ths paper dscusses two types of effcences: allocatve effcency and nformatonal effcency. () Allocatve effcency has two notons n ths model. An allocaton s effcent f all low and hgh valuaton traders trade wth each other and the total tradng gan, 2N s realzed. The (overall) outcome s socally effcent f the tradng gan 2N s realzed and no rsk neutral agent acqures (socally useless) nformaton. The focus of the paper s on the frst 7 It s assumed that each trader has a tradng need of one unt. owever, n a large market f a trader becomes nformed, he may want to speculate and try to explot unnformed traders by tradng a lot of unts. Two allow for ths possblty, a trader can trade up to m unts n ths model. One result of ths paper shows that once a trader (or ntal hedger) s nformed he speculates rather than hedgng hs poston. 7

9 noton of allocatve effcency. 8 () The prce s nformatonally effcent or fully revealng f t reflects the jont nformaton of the traders. Remark 1 There always exst no trade equlbra. A set of no-trade equlbra s gven by the followng strateges: No trader acqures nformaton and all hgh valuaton traders (potental buyers) only choose to buy at very low bd prces (e.g. b v ), whle all low valuaton traders (potental sellers) only choose to sell at very hgh ask prces (e.g. s v + ). 4. Informaton Acquston n a Small Double Aucton Ths secton analyzes the two trader (N=1) case. In order to fnd a tradng equlbrum, one can focus on strateges where the low valuaton traders (potental seller) only submts an ask prce to sell one unt and the hgh valuaton trader (potental buyer) only submts a bd prce to buy one unt. To save on notaton, n ths secton a pure strategy of the buyer and seller s just denoted wth t =(n,b) and t S =(n S,s), respectvely. avng made ther nformaton acquston decson, n {0,1}, the buyer submts a bd prce b and the seller submts an ask prce s smultaneously (for tradng one unt). If b s, then the asset s traded at the prce p= 21 (b+s), the surplus 2 s realzed, U =v+ p and U S =p (v ). Otherwse no trade occurs and the payoffs are normalzed to zero. If nformaton s acqured, the nformaton cost c s subtracted from the payoff. The effcent outcome s trade wth probablty one and wthout costly nformaton acquston. If both traders are unnformed, then the set of mutually acceptable prces s p [E[v], E[v]+ ]. Therefore, a set of potentally best responses wthout nformaton acquston s (0,b) and (0,s) wth b=s=e[v]+k and k [, ]. In such a (k-sharng) outcome the buyer gets EU = k, and the seller gets EU S = +k. When do these strateges consttute best responses? Gven the above strategy (0,s) of the seller, suppose the buyer acqures nformaton and speculates. In state v he chooses a bd prce b <s and no trade occurs. In state v he chooses b =s and makes some speculatve profts snce he pays less than the true value of the asset. Ths response yelds EU = 1 2 [(v + ) (E[v]+k)] c= 41 (v v )+ 21 ( k) c. So f 41 (v v )+ 21 ( k) c> k, speculaton s the best response, and the seller suffers an endogenous lemons problem snce EU S = 21 ( +k) 8 If nformaton s bought from an nformaton provder, then the payment of nformaton cost s only a transfer. 8

10 41 (v v )<0. (It s assumed that < 81 (v v ).) Analogously, f 41 (v v )+ 21 ( +k) c> +k, the seller s best response to (0,b) wth b=e[v]+k s to choose (1,s,s ) wth s =b and s >b. Consequently, a k-sharng outcome wthout nformaton acquston can only be establshed as a NE n pure strateges, f c>max{π 21 ( k), π 21 ( +k)}=π 21 ( k ) where π 41 (v v ). 9 owever, f c<π 21 ( k ), then no effcent k-sharng NE n pure strateges exsts. Ths condton has a smple economc nterpretaton. π s the expected speculatve proft an nformed trader makes and 21 ( k ) s the expected opportunty cost of speculaton. If the buyer acqures nformaton and speculates, he does not trade n state v and he forgoes the surplus ( k) wth probablty 0.5. If the seller speculates, hs opportunty cost of speculaton s 21 ( +k). So f the nformaton cost s smaller than the speculatve proft net the opportunty cost of speculaton, then trade at a prce p=e[v]+k wth k [, ] s not an equlbrum outcome snce one trader has an ncentve to speculate. The next proposton characterzes for the full range of nformaton costs, when a pure strategy NE wth trade exsts. 10 Proposton 1 (a) If c 41 (v v ) 21, there exst a set of pure strategy NE where trade occurs wth probablty 1 and where no nformaton s acqured. (b) If <c< 41 (v v ) 21, there exsts no pure strategy NE wth postve probablty of trade. (c) If c, there exst pure strategy NE where trade occurs wth probablty 1. In any such NE both traders acqure nformaton and the prce fully reveals the traders nformaton (to a thrd party). Proof (a) See the analyss above. (b) It remans to show that there s also no pure strategy NE wth one-sded or two-sded nformaton acquston. It s easy to see that f c>, then no pure strategy equlbrum exsts 9 If c 1 4 (v v ), any k-sharng outcome s attanable as a NE. As n a standard double aucton, a contnuum of tradng equlbra exsts. The set of effcent equlbra shrnks as the nformaton cost decreases. If c= 1 4 (v v ) 1 2, only the equal-splt (k=0) outcome s attanable as an effcent equlbrum,.e. the effcent NE s unque. 10 Note, the assumpton < 1 8 (v v ) mples that < 1 4 (v v )

11 n whch both traders acqure nformaton. Suppose only the seller acqures nformaton. The assumpton < 81 (v v ) mples that v >E[v]± >v +. A standard lemons argument shows that gven the seller s nformed, the best response of an unnformed buyer s to offer at most v +. Trade only occurs n state v, and the seller s payoff s at most EU S = c<0. In such a case, no trader acqures too expensve and non-explotable nformaton, but because of the endogenous lemons problem the buyer proposes b v + and the seller proposes s v. So no pure strategy NE wth trade exsts. (c) See Appendx. Proposton 1(c) shows that f the nformaton cost s low, the traders face an nformaton acquston dlemma. Snce the nformaton cost can be covered by the tradng gans, the desre of the agents to trade wthout beng exploted nduces both traders to acqure nformaton. The prce fully reveals the two traders nformaton (to a thrd party). 11 Remark 2 If c, there also exst pure strategy equlbra n whch trade occurs wth probablty 0.5. In any such NE one trader acqures nformaton and the unnformed trader accounts for the lemons problem. When trade occurs the prce s fully revealng. Proposton 1 (b) shows that f the nformaton cost s n an ntermedate range, then no pure strategy NE wth trade exsts. Ths neffcency result s dfferent from Chatterjee and Samuelson (1983) and Myerson and Satterthwate (1983) snce the tradng gans are common knowledge. It s also dfferent from Akerlof (1970) and Gresk (1991) snce there s no asymmetrc nformaton about the common valuaton n equlbrum. The concern about a potental lemons problem due to the mere possblty of nformaton acquston by the other trader can cause no trade. Dang (2008) shows that ths result also holds n ultmatum and alternatng offer barganng. efore proceedng to the analyss of mxed strategy equlbra the followng terms are defned. Defnton () A trader plays a defensve strategy, f he chooses (0,b) wth b v + or (0,s) wth s v h. Such a trader s called a defensve trader. 11 Jackson (1991) shows that wth mperfect competton fully revealng prces exst despte costly nformaton. In hs model the seller s nosy,.e. hs behavor s nsenstve to prces. 10

12 () A trader plays a nose-type strategy, f he chooses (0,b) or (0,s) wth b,s [E[v],E[v]+ ]. Such a trader s called a nose trader. () A trader plays a speculatve strategy, f he chooses (1,b,b ) wth b v + and b [E[v],E[v]+ ] or (1,s,s ) wth s [E[v],E[v]+ ] and s v. Such a trader s called an nformed speculator. In other words, a trader s called a defensve trader f he s unnformed and hs offer accounts for the potental lemons problem. A trader s called a nose trader f he s unnformed and proposes a prce around the expected value of the asset. A trader s called an nformed speculator f he only buys (sells) at a prce around E[v] when the true state s v (v ). The next proposton shows that dependng on the outcome of the equlbrum randomzaton, a trader may become a nose trader, a defensve trader, or an nformed speculator. Proposton 2 Suppose <c< 41 (v v ) 21. (a) In any mxed strategy NE, n whch trade occurs wth postve probablty, the traders randomze over defensve strateges, nose-type strateges, and speculatve strateges. (b) The outcome n any such NE has the followng propertes. () Trade does not occur f both traders are nformed, or at least one trader s a defensve trader. () The prce s not fully revealng. () oth traders have zero expected payoffs. The followng example hghlghts the ntuton behnd Proposton 2. Suppose the traders are only allowed to choose three offer prces, namely b,s {v,e[v],v }. Appendx shows that under ths assumpton n the unque (non-degenerated) mxed strategy equlbrum the buyer randomzes over the strateges (0,v ), (0,E[v]) and (1,v,E[v]). The seller randomzes over the strateges (0,v ), (0,E[v]) and (1,E[v],v ). Trade only occurs n the followng crcumstances. () oth traders do not acqure nformaton and choose E[v]. () The seller s unnformed and chooses s=e[v] and the buyer s nformed and the true state s v. () The buyer s unnformed and chooses b=e[v] and the seller s nformed and the true state s v. The 16c² probablty of trade s (v v )² 4 ², and the prce s p=e[v] and not fully revealng. (a) Why s there no trade f both traders are nformed? 12 Equvalently, why does an nformed buyer never chooses b=v n state v,.e. why does he choose (1,v,v ) wth 12 Ths s n contrast to Proposton 1(c) where trade only occurs wth probablty 1, f both traders are nformed. 11

13 probablty zero? If he chooses b=v n state v then trade also occurs n the event where the seller s nformed snce an nformed seller choose s=v n state v. The problem s that f the buyer s ndfferent between the strateges (1,v,E[v]) and (1,v,v ), then both strateges wth nformaton acquston s strctly domnated by the strategy (0,v ). If the nformed buyer trades at the prce v n state v, then hs expected payoff net nformaton cost s negatve. So not acqurng nformaton would be a best response. In other words, f a trader acqures nformaton n a mxed strategy equlbrum, he speculates snce he expects to meet a nose trader wth postve probablty. (b) In the mxed strategy equlbrum an unnformed trader proposes the offer prce E[v] wth postve probablty. In other words, he proposes an offer whch s prone to speculaton and may suffer a speculatve loss. Although hs behavor exhbts nose tradng, hs equlbrum payoff s non-negatve snce he meets an unnformed trader wth postve probablty. In such a case he realzes the tradng gan wthout sufferng a speculatve loss. (c) Why s the equlbrum payoff non-postve? Put t dfferently, snce the mnmum prce the seller demands s s=e[v], why does the buyer choose (0,v ) wth postve probablty? In order to make the seller ndfferent between acqurng and not acqurng nformaton, an unnformed buyer chooses not to trade (.e. bds v ) frequently enough so as to dscourage too much nformaton acquston by the seller. Snce U (0,v )=0 and the buyer s ndfferent between ths and other strateges, hs expected payoff s zero. (d) In contrast to Proposton 1(c), why s the prce not fully revealng? There are three reasons. () There s no trade between two nformed traders. () There s no trade f one trader plays a defensve strategy. () Suppose the seller does not acqure nformaton and observes trade at p=e[v]. In ths case he does not know whether the buyer has chosen (0,E[v]) or (1,v,E[v]). Although hs posteror belef for v=v ncreases, t s strctly below one. Otherwse he would know for sure that he has made a bad deal and ths cannot be an equlbrum outcome. (e) The probablty that a trader becomes a defensve trader, a nose trader, and an nformed speculator s gven as follows: 4c 4c 16c² 1 v 2, v v + 2, (v v )² 4 ². that an unnformed trader chooses the offer prce E[v] decreases. 12 v (f) The dfference v v captures the rskness of the asset and the mportance of the endogenous lemons problem. As the asset becomes more uncertan, ths exerts two effects. () The nformaton cost range whch mples no pure strategy tradng equlbrum ncreases,.e. even for hgh nformaton costs there s an endogenous lemons problem and no trade may 16c² occur. () The equlbrum probablty of trade (v v )² 4 ² decreases because the probablty

14 5. Informaton Acquston n arge Double Auctons Ths secton analyzes the 2N trader case where N>1. Each trader can submt one bd prce to buy up to m unts of the asset as well as one ask prce to sell up to m unts. The buy and sell orders of the traders are ranked accordng to the bd and ask prces, respectvely. Ths generates an aggregate demand and supply schedule. The market prce s set to equalze aggregate demand and supply. () If there are multple-market clearng prces, the mean prce of these prces s chosen. () If there s excess demand (supply) at the market clearng prce, the orders wth the hghest bd prces (lowest ask prce) are executed frst. The remanng unts are allocated wth equal probablty to the traders who propose the same offer prce. Ths type of tradng rules s adopted from Reny and Perry (2006, secton 4.1). A natural queston s whether enlargng the sze of the market can mtgate the neffcences n the small double aucton. Ths secton shows that a common market place does not resolve the neffcences. In contrast, the next emma shows that a large double aucton market performs worse n the followng sense. In the small (N=1) double aucton, there exsts a crtcal nformaton cost c crt = 41 (v v ) 21, such that f c c crt, then a tradng equlbrum wthout nformaton acquston exsts. As both the number N of traders and the unts m a trader s allowed to trade ncrease, c crt also ncreases. In other words, n a large market a tradng equlbrum wthout nformaton acquston fals to exst even though the nformaton cost s large. emma Suppose N>1. Defne q=mn[m,n] and c crt = 21 q(v v ) 21. If c<c crt, then there exsts no pure strategy NE wth trade and where no nformaton s acqured. Proof Case 1: q=1. The proof for q=1 follows almost drectly from Proposton 1. Suppose the -traders (-traders) only submt buy (sell) orders and the offer prce profles =(b 1,.,b N ) and S=(s N+1,.,s 2N ) yeld p=e[v]. Ths prce s less prone to speculaton because the opportunty cost of speculaton s hghest for both the - and -traders. Snce there are addtonal traders, an nformed trader can potentally make more proft n a large market although he can only buy or sell one unt. For example, an nformed -trader becomes a speculator n state v and buys one unt of the asset whch s not possble n a small (N=1) double aucton. In state v, as n a small double aucton, he sells one unt of the asset. Therefore, hs expected payoff wth nformaton acquston s 1 2 (v v )+ 21 c. If 13

15 c< 21 (v v ) 21, ths strategy domnates trade at p=e[v] wth expected payoff, and no pure strategy tradng NE wthout nformaton acquston exsts. Case 2: q>1. The proof s based on three arguments. (a) If a pure strategy tradng NE wthout nformaton acquston exsts, then N unts are traded,.e. all traders are satsfed. (b) If a pure strategy tradng NE wthout nformaton acquston exsts, then trade s executed at the prce p=e[v]. (c) No pure strategy tradng NE wthout nformaton acquston exsts where trade s executed at the prce p=e[v]. The followng arguments prove clam (a). Suppose all traders do not acqure nformaton, and each -trader submts an order to buy one unt and no sell orders, whle each -trader submts an order to sell one unt and no buy orders. Suppose the bd and ask profles =(b 1,.,b N ) and S=(s N+1,.,s 2N ) yeld a market clearng prce, p (E[v], E[v]+ ). Suppose b <p and -trader does not get the asset. Gven (,S), -trader can do better by choosng b p and gets one unt wth postve probablty and EU>0. Any -trader or -trader who does not get to buy or sell one unt of the asset at the resultng prce, has not played a best response. (If p=e[v], unsatsfed -trader wll devate. If p=e[v]+, unsatsfed - trader wll devate.) Ths reasonng mples that f a tradng equlbrum wthout nformaton acquston exsts, then all traders are satsfed,.e. N unts are traded. Therefore, a canddate offer profle (,S) for beng part of a pure strategy NE must have b p and s j p for =1,..,N and j=1+n,..,2n, where p s the resultng market prce gven (,S). The proof of clam (b) s as follows. Suppose each trader trades one unt and the bd ask profle (,S) gves rse to the prce p [E[v],E[v]+ ]. Case () Suppose p<e[v]. Consder -trader who chooses b and gets one unt of the asset. For example, a proftable devaton s to choose the same offer prce but submts an order of q unts. The expected payoff s EU=(E[v]+ p)+prob(trade)(q 1)(E[v] p)>. Therefore, f p<e[v], then a -trader who gets one unt has not played a best response. In some sense there s ncentve to buy more and overbd the other buyers. () If p>e[v], then a -trader who sells one unt has not played a best response. There s ncentve to sell more and underbd the other sellers. Consequently, only f p=e[v], then an unnformed trader who trades one unt has no proftable devaton. The proof of clam (c) s smlar to the proof of Proposton 1. Suppose that no trader acqures nformaton and the bd ask profle (,S) yelds the market prce p=e[v] and all traders trade one unt each. Then some traders have a proftable devaton. For example, - trader acqures nformaton. In state v, he chooses b =E[v]+γ b to buy q unts where γ b s chosen such that b s larger than the q-th hghest bd prces gven =(b 1,..,b N ). The (addtonal) demand of -trader may ncrease the market clearng prce from p=e[v] to at 14

16 most p=e[v]+γ b. Suppose p=e[v]+γ b. s payoff n ths state s (v + )+(q 1)v qp c = 21 q(v v 2γ b )+ c. 13 In state v, he forgoes the surplus and chooses b =E[v] γ s to sell (short) q unts where γ s s chosen such that s s smaller than the q-th lowest ask prces gven S=(s N+1,..,s 2N ). Suppose p=e[v] γ s. s payoff n ths state s 21 q(v v 2γ s ) c. Snce q N, the prce mpact s zero n the followng sense. For a gven bd ask profle (,S) and the market clearng prce p=e[v], there always exsts the same market clearng prce such that -trader gets to buy or sell q unts. 14 Therefore, the expected payoff of -trader wth nformaton acquston s EU = 21 q(v v )+ 21 c. Speculaton s the best response f 1 2 q(v v )+ 21 c>. Analogously for a -trader. Consequently, f c< 21 q(v v ) 21, there exsts no pure strategy tradng equlbrum wthout nformaton acquston. QED Proposton 3 For any fnte nformaton cost c, there exsts an nteger q*, such that f mn[m,n] q*, then () an effcent equlbrum allocaton fals to exst and () the prce s typcally (.e. for all c q ) not fully revealng. A noton of a compettve or close to compettve market s that there are many traders and a trader can trade as many unts of the asset as he lkes wthout havng much prce mpact. Proposton 3 establshes the man result of the paper and states that f nformaton s endogenous and costly, there exsts no effcent equlbrum allocaton n such large double aucton markets. More precsely, for any fnte nformaton cost, f the both the number of traders and the unts a trader s allowed to trade are suffcently large, then there s no equlbrum n a double aucton n whch the total tradng gans of 2N are realzed. In a large market there are addtonal ncentves effects that are not present n the small double aucton. In contrast, to Proposton 1(c) whch shows that f the nformaton cost s low, then there exsts a pure strategy NE n whch the tradng gan 2 s realzed. In a large market, even for low nformaton costs, there exsts no NE where the full tradng gans are realzed. The reason s free-rdng. In a blateral double aucton an unnformed trader must account for the lemons problem and ths reduces the probablty of trade. If he wants trade to 13 The -trader s valuaton for the frst unt bought s v+ and v for the other unts. 14 Consder the two extreme cases. () All -traders choose E[v] and all -traders choose E[v]+ and the prce s E[v]. If the true value s hgh, an nformed -trader chooses E[v]+ +ε and buys q unts. For q<n, there exsts a market clearng prce p=e[v]. () All b=s=e[v]. If a -trader chooses E[v]+ +ε, a market prce p=e[v] exsts. 15

17 occur wth probablty one, he must become nformed f the other trader s nformed. Proposton 1(c) shows that the prce s fully revealng any such tradng equlbrum. ut n a large market where there s a sngle prce for all transactons and f that prce s fully revealng, there s free rdng,.e. some traders do not want to acqure costly nformaton. Proposton 1(a) shows that f c 41 (v v ) 21, then an effcent allocaton exsts. In a large market there are potentally more unnformed traders to explot. Therefore, even f the nformaton cost s large, some traders have an ncentve to speculate so that unnformed traders are concerned about the endogenous lemons problem. Ths mere concerns suffces to destroy the effcent equlbrum allocaton. For any fnte nformaton cost, f the number of traders submttng orders around the expected value of the asset s large, some traders have an ncentve to speculate. Free rdng and the ncreased ncentves to speculate (or the ncreased concern about the potental lemons problem) are renforcng reasons why for any fnte nformaton cost, there exsts no pure strategy equlbrum f the market s suffcently large. Suppose that some traders are nformed and the prce s fully revealng, then both an nformed trader and a defensve trader have proftable devatons. () An nformed trader chooses not to acqure costly nformaton snce there s no speculatve profts to make. () Snce there s no lemons problem, no unnformed trader submts defensve offers. Consequently, some of these traders devate to nose traders. On the other hand, f there are a lot of nose traders and very few nformed, then an nformed trader can move prces and make speculatve profts. Therefore, a fully revealng prce s typcally (.e. for all c q ) not consstent wth equlbrum behavor. Techncally speakng, any profle of pure strateges leads to a market clearng prce that s fully revealng. Proposton 3() s a strategc verson of Grossman and Stgltz (1980) mpossblty result of nformatonally effcent (double aucton) markets. In addton, snce the prce s not fully revealng, there s a potental lemons problem, some traders behave defensvely and the allocaton s neffcent. The next result restates the non-exstence of an effcent equlbrum n term of the rskness of the asset. Corollary For any gven set of parameter values (c, N, m, ), f the asset s suffcently rsky (.e. v v s large), then no equlbrum exsts n whch the allocaton s effcent. 16

18 6. Dscusson Rsk averson The followng arguments show that rsk neutralty s not a crucal assumpton for all qualtatve mplcatons of the paper. Suppose there s a rskless asset S, and a rsky asset R, and the agents have concave utlty functons u and endowments ω=(ω S,ω R ) of the assets. In general, f ω S >ω js and ω R <ω jr, then agent and j can realze gans from trade by reallocaton of rsks because agent has a hgher margnal valuaton of asset R than agent j. To smplfy the analyss ths paper assumes that agent and j have a valuaton of v+ and v for the frst unt of asset R traded. 15 The strategc ncentve of nformaton acquston under rsk neutralty also arses under rsk averson because a rsk averse agent also has to evaluate the gans from trade (or hedgng of rsks), the potental speculatve loss of beng unnformed, as well as the potental speculatve gans from beng nformed. In partcular, f the agents are rsk averse, the nformaton cost s low, and the sgnal perfectly reveals the true value of the asset, then s no trade at all because perfect nformaton prevents hedgng of rsks. So the Proposton 1(c) ceases to exsts. 16 Many heterogenous agents Suppose there are three types of traders wth valuatons u =v, u M =v, and u =v+. If agent and M trade wth each other, then the total tradng gan s and none of the qualtatve mplcatons change. For example, Proposton 1(b) would state that f 21 <c< 41 (v v ) 41, then no pure strategy NE wth trade between agent and M exsts. Also, n a large market any of these agents can become a speculator. Costless Informaton If c=0, then there exsts a NE wth the followng propertes. All traders acqure nformaton and trade one unt each. () If q=1, the equlbrum prce s p [v,v+ ]. () If q>1, then the equlbrum prce s p=v. 15 For example, the lqudty traders n Mendelson and Tunca (2004) have a smlar utlty functon. Duffee et al. (2005) assume that the traders have hgh or low valuatons of the form: v and v where >. 16 Smlarly, f the sgnal s nosy, ths only changes the expected speculatve proft and the crtcal value of the nformaton cost for the dfferent types of equlbra to arse, but not the qualtatve mplcatons. 17

19 Proof: () Suppose k [, ]. It s easy to see that the followng strategy profle consttute a NE: All -traders choose n=1, b=(v +k, v +k), choose n=1, b= u b =(0,0), s=(v +k, v +k), b u =(1,1), s= u s =(0,0); and all -traders s u =(1,1). The prce s p=v+k and EU = k and EU S = +k. () For q>1, n the unque pure strategy tradng NE the -traders choose n=1, b=(v,v ), b u =(1,1), s= u s =(0,0); the -traders choose n=1, b= u b =(0,0), s=(v, v ), s u =(1,1). The outcome s p=v and EU =EU S =. Note f p v, then a trader may want to trade more unts. There s overbddng and underbddng whch can not be an equlbrum. 17 Observablty of nformaton acqustons The next proposton shows that f nformaton acqustons are observable pror to the tradng stage, then endogenous nformaton acquston has no adverse allocatve consequences. Proposton 4 Suppose N=1, nformaton acquston s observable, and k [, ]. Any effcent k-sharng outcome s attanable as a perfect NE rrespectve of nformaton cost. The ntuton s as follows. 18 Snce nformaton acquston s observable, a trader can also condton hs offer strategy on the fact whether the other party s better nformed or not. Suppose ex ante the traders agree to trade at p=e[v], but the buyer acqures nformaton. In the aucton stage the seller knows that the buyer s nformed. So the seller does not submt the prce s=e[v] anymore, but demands a hgh prce. Snce the buyer antcpates the lemons problem he hmself creates by acqurng nformaton, hs best response s not to acqure nformaton. No trader has an ncentve to acqure more nformaton than the counter party. owever, f nformaton acquston s not observable, the traders cannot target ther offer strateges approprately and are concerned about the endogenous lemons problem. Prvate nformaton acquston s unlkely to be publcly observable n a large market. 17 If there s asymmetrc demand and supply, then the agents on the short sde of the market capture the full surplus. Suppose there are more -traders than -traders, then the unque equlbrum prce s p=v+. 18 Formally, t s easy to show that for k [, ], r k, and z k, the strateges t =(0,b) wth b=e[v]+k f n S =0, and b=v +r f n S =1; and t S =(0,s) wth s=e[v]+k f n =0, and s=v +z f n =1 consttute a perfect NE wth EU = k and EU S = +k. Analogously, one can show that for N>1, a tradng equlbrum exsts n whch no trader acqures nformaton, and all traders chooses E[v] n the aucton stage f no trader has acqured nformaton. Otherwse the unnformed traders choose a defensve strategy. 18

20 7. Concluson Ths paper analyses nformaton acquston n double aucton markets populated wth hgh and low valuaton traders of an asset and shows that for any fnte nformaton cost, an effcent equlbrum allocaton fals to exst f the number of traders and the unts a trader s allowed to trade are suffcently large or the asset s suffcently rsky. There s a large set of parameter values where n any equlbrum wth postve volume of trade the traders play mxed strateges and ex ante dentcally nformed, ratonal traders evolve endogenously to nose traders, speculators, and defensve traders. ecause of defensve tradng the allocaton s neffcent,.e. not all gans from trade are realzed. ecause of endogenous nose tradng the prce s not fully revealng. Ths paper provdes a strategc foundaton for the Grossman and Stgltz (1980) mpossblty result of nformatonally effcent (double aucton) markets and derves some mplcatons. Mght the behavor one observes n large double aucton markets be understood as the realzaton of a mxed strategy equlbrum? Ths paper provdes a ratonale for the restrcton of short sellng n fnancal markets and hghlghts a potental beneft of over-thecounter markets. 19 In decentralzed tradng, f the nformaton cost s ether low or hgh, then there exst equlbra where trade occurs wth probablty one. These equlbra ceases to exst n a large market. owever, ths paper gnores the potental cost of fndng a trader wth the opposte tradng need. An mportant functon of a large market may be the bundlng of lqudty. In addton, ths paper analyzes a one-perod tradng game and hghlghts potental neffcences n a statc large double aucton market. A second mportant functon of a centralzed market s the transmsson of nformaton through prces and sequental tradng. Ths can mtgate the duplcaton of costly nformaton acqustons. 20 An nterestng extenson s to allow the traders to endogenously choose whether to trade n a centralzed or decentralzed market. From an mechansm desgn pont of vew, one can analyze an optmal mult-stage (drect) mechansm where a trader frst announces hs buy or sell preference, then whether he has acqured nformaton, and fnally hs nformaton about the asset value. Alternatvely, the 19 Common stocks are traded n centralzed markets wth hgh prce transparency and hgh daly tradng volume, whle many other fnancal nstruments are traded n decentralzed dealershp markets wth low transparency and relatvely low daly tradng volume. For example, the tradng of mortgage-back securtes, dervatves, structured debt products, credt default swaps, and corporate bonds n over-the-counter markets are sad to be opaque. 20 For example, f the traders are heterogeneous then traders facng large tradng gans or a contnuous nflow of tradng needs (such as the executon of ncomng orders of costumers) may be the ones who acqure nformaton and determne prces. The small traders wat, observe the prce and trade wthout costly nformaton acquston. 19

21 traders may wrte (complex) state contngent contracts. owever, as mentoned, the smple lnear utlty functon of the traders should be regarded as a shortcut for the margnal utlty of rsk adverse traders possessng dfferent endowments of the rsky asset. If a trader receves or has to make addtonal post transacton payments after the realzaton of the cash flow, then ths type of arrangements undermnes the dea of rsk sharng n fnancal markets. Market mcrostructure models typcally assume and real fnancal markets often have desgnated market makers. These agents are supposed to provde lqudty and explctly forbdden to speculate. Madhavan and Panchapagesan (2000) provde an emprcal analyss of the role of market specalsts for prce dscovery n the overnght market on the NYSE, and state that there s strong evdence that the NYSE's desgnated dealer (specalst) sets a more effcent prce than the prce that would preval n a pure call market usng only publc orders. (p.656) 21 It s nterestng to explore the mplcaton for ths model of the exogenous presence of a fourth category of players, desgnated market makers who attempt to earn profts from exclusvely facltatng the encounter of buyers and sellers, as well as to dentfy the set of condtons under whch such players would endogenously emerge n equlbrum. In other words, can market makers facltate allocatve and nformatonal effcency n double aucton markets wth endogenous nformaton? Further research may provde addtonal nsghts on the workng of dfferent tradng nsttutons as well as the competton between tradng platforms when all traders are strategc and can acqure nformaton before they trade. 21 Madhavan and Panchapagesan (2000) also analyse n ther nosy REE type settng wth exogenous nformaton a sngle-prce call aucton where a market maker sets the openng prce after observng the lmt order book and they state (p.657) The process by whch nvestors' latent demands are translated nto realzed prces and volumes s a hghly complex process that we are only now startng to understand. Our results add to a growng body of evdence that hghlghts the crucal roles of nformaton and market structure n determnng prce effcency, but there are stll many mportant questons to be answered before we fully understand the nner workngs of the black box of tradng mechansms. 20