Behavioral Aspects of Arbitrageurs in Timing Games of Bubbles and Crashes

Transcription

1 CIRJE-F-606 Behavoral Aspects of Arbtrageurs n Tmng Games of Bubbles and Crashes Htosh Matsushma Unversty of Tokyo January 2009 CIRJE Dscusson Papers can be downloaded wthout charge from: Dscusson Papers are a seres of manuscrpts n ther draft form. They are not ntended for crculaton or dstrbuton except as ndcated by the author. For that reason Dscusson Papers may not be reproduced or dstrbuted wthout the wrtten consent of the author.

2 Behavoral Aspects of Arbtrageurs n Tmng Games of Bubbles and Crashes Htosh Matsushma 2 Faculty of Economcs, Unversty of Tokyo January 5, 2009 Frst Verson: March 25, 2008 Ths s a revsed verson of the manuscrpt enttled Effects of Reputaton n Bubbles and Crashes (Dscusson Paper CIRJE-F-560, Unversty of Tokyo, Department of Economcs, 2008). I would lke to thank Professor Marcus Brunnermeer for hs helpful comments. Ths research was supported by a Grant-n-Ad for Scentfc Research (Kakenh ) from the Japan Socety for the Promoton of Scence (JSPS) and the Mnstry of Educaton, Culture, Sports, Scence and Technology (MEXT). 2 Department of Economcs, Unversty of Tokyo, 7-3- Hongo, Bunkyo-ku, Tokyo , Japan. Fax: , E-mal: htosh@e.u-tokyo.ac.jp

3 2 Abstract We model a tmng game of bubbles and crashes a la Abreu and Brunnermeer (2003), n whch arbtrageurs compete wth each other to beat the gun n a stock market. However, unlke Abreu and Brunnermeer, nstead of assumng sequental awareness, the present paper assumes that wth a small probablty, each arbtrageur s behavoral and commtted to rde the bubble at all tmes. We show that wth ncomplete nformaton, even ratonal arbtrageurs are wllng to rde the bubble. In partcular, the bubble can persst for a long perod as the unque Nash equlbrum outcome. Keywords: Bubbles and Crashes, Tmng Games, Behavoral Arbtrageurs, Postve Feedback Traders, Reputaton, Characterzaton, Unqueness JEL Classfcaton Numbers: C720, C730, D820, G40

4 3. Introducton Ths paper demonstrates the theoretcal foundaton underlyng the wllngness of ratonal arbtrageurs to rde the bubble n a stock market. We model the stock market as a tmng game wth ncomplete nformaton among arbtrageurs; the model s nspred by that of Abreu and Brunnermeer (2003). 3 However, our model does not requre the assumpton of sequental awareness that has played a central role n the above paper. Instead, we assume that each arbtrageur s not necessarly ratonal,.e., he s behavoral wth a small probablty, n that he s commtted to rde the bubble at all tmes. Wth the assumpton of ncomplete nformaton, the present paper shows that even ratonal arbtrageurs are wllng to rde the bubble for a long perod. The effcent market hypothess n modern fnancal theory asserts that by reflectng all relevant nformaton, the stock prce s always adjusted to the fundamental value. However, there are consderable evdences that contradct ths hypothess: the stock prce sometmes ncreases beyond the fundamental value and contnues to ncrease untl t goes nto a free fall. Advocates of behavoral fnance, such as Shlefer (2000) and Shller (2000), argued that the bubble s drven by postve feedback traders, who ncorrectly beleve that the stock prce wll grow n perpetuty. The effcent market hypothess, on the other hand, clams that ratonal arbtrageurs quckly undo ths msprcng. The arbtrageurs sellng pressures then dampen the enthusasm of the postve feedback traders, mmedately burstng the bubble. In contrast to ths deal of ratonal arbtrageurs, actual professonal arbtrageurs who are mostly consdered to be ratonal generally do not thnk that the best strategy s 3 See also Abreu and Brunnermeer (2002) and Brunnermeer and Morgan (2006).

5 4 to undo msprcng n ths manner; nstead, they would lke to rde the bubble. On the bass of hstorcal experences, several authors such as Kndleberger (978) and Soros (994) even emphaszed a self-feedng aspect of bubbles and crashes: speculatve prce movements nvolve multple professonal arbtrageurs who contnuously drve the stock prce up and then sell out at the top to the postve feedback traders. However, we may dsagree wth ths vew, because the arbtrageurs may compete wth each other n order to ext the market by sellng out,.e., beat the gun or tme the market at the earlest. Ths phenomenon along wth the backward nducton method prevents the persstence of a bubble. Hence, n order for ths vew to be convncng, we need to demonstrate a further theoretcal foundaton, based on whch each ratonal arbtrageur s wllng to termnate ths chan reacton of competton and develop a reputaton among them to rde the bubble. On the bass of these arguments, the present paper models the stock market as a tmng game, where the stock market operates durng the tme nterval [0,], and each arbtrageur selects a tme to ext the market by sellng out hs share. As long as no arbtrageur has sold out, the bubble contnues to be drven by the postve feedback traders. Once any arbtrageur sells out, the other arbtrageurs ve wth each other to follow n the footsteps of ths arbtrageur. As soon as ther sellng pressures exceed a crtcal amount of shares, the postve feedback traders fal to support the stock prce, and then the bubble crashes. The key assumpton of ths paper s that every arbtrageur s not necessarly ratonal. Wth a small probablty, he s behavoral and commtted to rde the bubble at all tmes. We also assume ncomplete nformaton, n that whether each arbtrageur s behavoral or ratonal s unknown to the other arbtrageurs. On wtnessng the

6 5 persstence of a bubble, each ratonal arbtrageur s ncreasngly convnced that the other arbtrageurs are behavoral, whch ncentvzes hm to further postpone tmng the market. On the bass of ths reasonng, each arbtrageur s convnced n the early stage of the tmng game that the other arbtrageurs are subject to rde the bubble for a long perod, even f they are ratonal. In ths respect, we should menton the relevance to the reputaton theory n fntely repeated games wth ncomplete nformaton explored by Kreps, Mlgrom, Roberts, and Wlson (982). Wth the assumpton of ncomplete nformaton on whether players are ratonal or crazy enough to have blnd fath n mplct colluson, any ratonal player s wllng to mmc the crazy players collusve behavor. The present paper wll apply the basc concept of ths theory to the tmng games of bubbles and crashes. Wth restrctons that the growth rate of a bubble and the probablty of each arbtrageur s beng behavoral are not very small, t s shown that there exsts the unque Nash equlbrum. Ths equlbrum s symmetrc and s named the bubble-crash equlbrum, where a partcular pont of crtcal tme (0,) exsts, such that () the bubble never crashes before the crtcal tme, whereas () after the crtcal tme, the tme at whch the bubble crashes s randomly determned accordng to a constant hazard rate. Ths hazard rate s very hgh when the rato between the total amount of shares that the arbtrageurs possess and the crtcal amount of share s close to unty; n ths case, the bubble can persst for a long perod, even f all arbtrageurs are almost certan from the begnnng that the bubble wll crash just around a partcular fxed tme. Wthout these restrctons, another symmetrc Nash equlbrum may exst, whch s named the quck-crash equlbrum. In ths equlbrum, any ratonal arbtrageur certanly sells out at the ntal tme, and therefore, the bubble never perssts, except for

7 6 the case n whch all arbtrageurs are behavoral. The present paper provdes a general characterzaton of all symmetrc Nash equlbra. Ths characterzaton mples that whenever the bubble-crash and quck-crash equlbra coexst, there also exsts another symmetrc Nash equlbrum named the hybrd equlbrum. In t, each ratonal arbtrageur randomzes between the quck-crash equlbrum strategy and a modfed verson of the bubble-crash equlbrum strategy. In general, there exsts no other symmetrc Nash equlbrum apart from the bubble-crash, quck-crash, and hybrd equlbra. There exst prevous works that dentfed condtons under whch the bubble exsts (see the survey by Brunnermeer (2008)). Among these works, the paper by Abreu and Brunnermeer (2003) s partcularly relevant to the present paper. Ths paper modeled the stock market as a tmng game smlar to ours, and was the frst to present a theoretcal background that explaned that the reslence of the bubble stems from the nablty of arbtrageurs to coordnate ther sellng strateges. Abreu and Brunnermeer assumed that the arbtrageurs become sequentally aware that the bubble has developed. Instead of ths sequental awareness, the present paper assumes that each arbtrageur s not necessarly ratonal and provdes an alternatve explanaton wth regard to the nablty of arbtrageurs to coordnate ther sellng strateges, manly because they use mxed strateges. The rest of the paper s organzed as follows. Secton 2 defnes tmng games of bubbles and crashes, and Secton 3 nvestgates the quck-crash equlbrum. In Secton 4, we nvestgate the bubble-crash equlbrum. Secton 5 nvestgates the hybrd equlbra, and Secton 6 characterzes all symmetrc Nash equlbra. In Secton 7, we show a suffcent condton under whch the bubble-crash equlbrum s the unque

8 7 Nash equlbrum, where we take nto account all asymmetrc Nash equlbra. Secton 8 concludes the paper. 2. The Model Ths paper consders the trade n a company s shares durng the tme nterval [0,]. The fundamental value of ths company s consdered to be y 0. We assume that the market nterest rate s set equal to zero and no dvdends are pad. There exst n 2 arbtrageurs, each of whom decdes the tme to sell out hs stockholdng that s normalzed to a sngle share. Let N = {,..., n} denote the set of arbtrageurs. Fgure llustrates a typcal pattern of bubbles and crashes. At the ntal tme 0, the arbtrageurs recognze that the bubble has occurred, where the stock prce s set equal to + y. The bubble perssts as long as at most arbtrageurs have sold out ther shares, where s a fxed postve nteger and < n. The dfference between the stock prce and the fundamental value grows exponentally accordng to a constant rate ρ > 0 ; the stock prce per share s consdered to be e y t. 4 ρ t + at any tme [0,] [Fgure ] Once any arbtrageur sells out hs share, ths sellng pressure trggers all the other arbtrageurs to sell out mmedately, whch bursts the bubble because ther collectve 4 Unlke Abreu and Brunnermeer, not the absolute value of the stock prce but the dstance from the fundamental value grows exponentally.

9 8 sellng pressure exceeds the crtcal amount of shares. We assume that even f no arbtrageur sells out at or before the termnal tme, the bubble crashes just after the termnal tme for exogenous reasons. 5 Aganst the abovementoned background, t s mplct to assume the presence of many postve feedback traders who have psychologcal bases that lead them to engage n momentum tradng. They ncorrectly beleve that the stock prce wll grow n perpetuty and attempt to support the hgh stock prce. The moment or more arbtrageurs sell out ther shares, the postve feedback traders fal to support the stock prce, whch causes t to declne to the fundamental value y. In ths respect, lke Abreu and Brunnermeer (2002, 2003) and Brunnermeer and Morgan (2006), 6 our dynamc model shares aspects of coordnated attacks wth the statc models of currency attacks n nternatonal fnance, such as Obstfeld (996) and Morrs and Shn (998). These models assumed the necessty of speculators coordnaton to break a currency peg. 7 5 Ths paper assumes that short sellng s prohbted. Wthout ths assumpton, any sngle arbtrageur can burst the bubble alone; ths case s essentally the same as the case of n =, where the crtcal amount of shares equals one. Even n ths case, we could show that the bubble may survve; however, ths possblty s sgnfcantly lmted. 6 Abreu and Brunnermeer (2002, 2003) assumed that whether or not each arbtrageur has sold out s unobservable to the other arbtrageurs, whereas Brunnermeer and Morgan (2006) assumed that t s observable and all arbtrageurs rush to sell out once any arbtrageur tmes the market. The present paper follows Brunnermeer and Morgan. However, by addng rrelevant complextes, we can apply the basc concept of ths paper to the case n whch arbtrageurs transactons are observable, and endogenze the assumpton of ther rushng n the same manner as Brunnermeer and Morgan. 7 It mght be helpful to thnk about the game of muscal chars as a metaphor for our model: there exst n players and chars that are arranged n a crcle, where < n,.e., the number of chars s less than the number of players. Each player has a whstle n hs mouth. Musc starts at the ntal tme 0. Whle the musc s playng, the players walk n unson around

10 9 On the bass of the above arguments, we defne a strategy for each arbtrageur N as a cumulatve dstrbuton q :[0,] [0,] that s nondecreasng, rght contnuous, and satsfes q ( 0) =. Followng any strategy q, arbtrageur plans to sell out at or before any tme a [0,] wth the probablty of q( a) [0,]. Let Q denote the set of strateges for arbtrageur.we wll consder q = a to be a pure strategy f q ( ) = 0 for all [0, a ), and q ( ) = for all [ a, 0]. Sgnfcantly, ths paper assumes that any arbtrageur N s not necessarly ratonal, and therefore, does not necessarly follow any strategy n the set Q. Let us fx any arbtrary real number ε [0,]. Wth regard to the probablty of ε, arbtrageur s behavoral, n that he s commtted to rde the bubble,.e., sell out just after any other arbtrageur frst sells out. Hence, any such behavoral arbtrageur does not trgger the burst of the bubble of hs own accord. Wth regard to the remanng probablty of ε, arbtrageur s ratonal, and follows any strategy n Q. Whether each arbtrageur s behavoral or ratonal s ndependently determned and s not common knowledge among the arbtrageurs. In other words, each arbtrageur does not know whether the other arbtrageurs are ratonal or behavoral. Suppose that each arbtrageur N plans to sell out at tme a [0,]. Let us the chars. When a player blows the whstle, the musc s mmedately shut off, and then every player must race to st down n one of the chars. The players who could st down are t regarded as the wnners; each wnner obtans e ρ + y dollars, whle any loser obtans y dollars, provded some player blew the whstle and the musc was shut off at tme t [0,]. Even f no player blows the whstle, the musc s automatcally shut off at the termnal tme. By assumng that any player who blows the whstle has the advantage of rushng for a seat, we can regard ths game as beng the same as our model.

11 0 arbtrarly set any nonempty subset of arbtrageurs H N, and let us suppose that any arbtrageur H s ratonal, whle any arbtrageur N \ H s behavoral. Note that any behavoral arbtrageur N \ H never sells out at hs planned tme a ; he s nstead commtted to wat for any other arbtrageur to tme the market. Let us denote by [0,] the tme at whch any ratonal arbtrageur sells out frst, whch s defned as the earlest tme at whch the ratonal arbtrageurs plan to sell out,.e., = mn{ a H}. Let l {,..., H} denote the number of ratonal arbtrageurs who plans to sell out at ths earlest tme,.e., l = { H a = }. If l >, then, wth the probablty of n, any ratonal arbtrageur H who plans to l sell out at tme sells out before the crash of the bubble, and earns e ρ + y. Wth regard to the remanng probablty of, he fals to sell out before the crash and l earns only the fundamental value y. If l, then he certanly sells out before the crash. In ths case, l further arbtrageurs can sell out before the crash. Hence, even any arbtrageur who ether s behavoral or plans to sell out after tme has the opportunty to sell out before the crash wth regard to the postve probablty of n l n l On the bass of these observatons, we defne the expected earnng of any ratonal. 8 8 It s mplct to assume that the behavoral arbtrageurs have the followng advantage over the postve feedback traders: the behavoral arbtrageurs can sell out mmedately after some ratonal arbtrageur tmes the market, whle the postve feedback traders cannot sell out untl the stock prce declnes to the fundamental value.

12 arbtrageur H by and K ρ v( H, a) = mn[, ] e + y f a l ( v Ha K l, ) max[,0] n l e ρ = + y f a =, >. Let Q = Q Qn, and let q= ( q,..., qn ) Q denote a strategy profle. The payoff functon u : Q R for each arbtrageur N s defned as follows. For every q Q, let us specfy u ( q ) as beng equal to the expected value of v ( H, a ) n terms of ( ah, ),.e., n H ε H ε. H N: N u ( q) E[ v ( H, a) ( ) q] A strategy profle q Q s sad to be a Nash equlbrum f u( q) u( q, q ) for all N and all q Q. A strategy profle q Q s sad to be symmetrc f q = q for all N. Let us ntroduce several notatons as follows. For each strategy profle q Q, let us denote the probablty that the bubble has crashed at or before any tme t [0,] by Dtq (; ) [ ε + ( ε){ q ()}] t. N n Note that D(; q) ε < whenever ε > 0, whch mples that the bubble does not necessarly crash durng the tme nterval of [0,], because t s wth the postve n probablty of ε > 0 that all arbtrageurs are behavoral. Let us defne the hazard rate at whch the bubble crashes at any tme t [0,] by Dtq (; ) θ (; tq) t. Dtq ( ; ) Moreover, for each arbtrageur N and each strategy profle q Q, let us denote the probablty that the bubble has crashed at or before any tme t, provded arbtrageur

13 2 never bursts the bubble of hs own accord, by D(; t q ) = [ ε + ( ε){ q ()}] t. j j N\{ } 3. Quck-Crash Equlbrum We denote by * q (0,...,0) the symmetrc strategy profle, named the quck-crash strategy profle, accordng to whch any ratonal arbtrageur plans to sell out at the ntal tme 0. Hence, accordng to * q, the bubble quckly crashes at the ntal tme 0. Fgure 2 llustrates * Dtq (; ), whch s kept equal to a constant value of n ε. [Fgure 2] Proposton : The quck-crash strategy profle * q s a Nash equlbrum f and only f () H N H, H φ ( n )! H H!( n H )! H + n H H n H ( ε) ε {mn[, ] max[0, ]} n ε ( e ρ ). Proof: For every N, * ( n )! H n H n u ( q ) = ( ε) ε mn[, ] + ε + y, H!( n H )! H + H N H, H φ and for every a (0,], ( n )! H (, * H n H ) ( ε) ε = max[0, ] H N H!( n H )! n H H, H φ u a q

14 3 n ρa + ε e + y. Hence, for every a (0,], u q * * ( ) u( a, q ) ( n )! H = H!( n H )! H + n H H N H, H φ n a + ε ( e ρ ). H n H ( ε) ε {mn[, ] max[0, } From e ρ e ρa for all a (0,], and from the above observatons, t follows that the nequalty of () s necessary and suffcent for * q to be a Nash equlbrum. Q.E.D. Note that * q s a Nash equlbrum f ε = 0,.e., t s certan that all arbtrageurs are ratonal. In ths case, the left-hand sde of () equals zero, whle ts rght-hand sde equals mn[, ] 0 n >, whch automatcally mples the nequalty of (). The assumpton of ε = 0 also mples the unqueness of the Nash equlbrum; any ratonal arbtrageur dslkes losng the opportunty to become the sngle wnner of the tmng game. Ths presses hm to hasten the tme to beat the gun slghtly earler than the others. Ths aspect of tal-chasng competton elmnates all equlbra other than * q. Ths logc, however, cannot be appled to the case of ε > 0. Even f any arbtrageur plans to sell out at the termnal tme, he stll has the opportunty of becomng the n sngle wnner wth regard to the postve probablty of ε > 0 ; the assumpton of ε > 0 may apply the brakes to ther tal-chasng competton. The rest of ths paper wll focus on the case of ε > 0.

15 4 4. Bubble-Crash Equlbrum Ths secton specfes a symmetrc strategy profle q = ( q ) Q, named the bubble-crash strategy profle, accordng to whch the bubble perssts for a long perod. Let us suppose that the growth rate of a bubble ρ > 0 and the probablty of each arbtrageur beng behavoral ε (0,) satsfes n (2) ε n e ρ,.e., ( n )lnε < 0, ρ whch mples that ρ and ε are not very small. Let us defne a partcular tme [0,) by ( n )lnε (3) +, ρ where the nequalty of (2) guarantees [0,). We specfy the bubble-crash strategy profle q as follows: for every N, and q ( a ) = 0 for all a [0, ), N (4) ρ( a ) ε exp[ ] q ( a) = n for all a [,]. ε Note from (3) that q ( ) = 0, whch mples that q s contnuous. Accordng to q, the bubble never crashes before the specfed tme. The followng theorem states that q s a Nash equlbrum. Theorem 2: Wth the nequalty of (2), the bubble-crash strategy profle q s a Nash equlbrum, and n n (5) u ( q) ε = e ρ + y.

16 5 Proof: Snce the bubble never crashes before the tme, t follows from the contnuty of q that ρa u( a, q ) = e + y for all a [0, ). Snce q s contnuous, t follows u (, q ) = e ρ + y > u ( a, q ) for all a [0, ). For every a [,], ρa ρt u( a, q ) = e { D( a; q )} + e dd( t; q ) + y. n The specfcaton of q mples and D( t; q ) = 0 for all t [0, ), a t= (6) n ρ ( t ) n n n D(; t q ) = ε e for all t [,]. From (6), the followng frst-order condtons hold for all a [,] : (7) ρa ρt u( a, q ) = [ e { D( a; q )} + e dd( t; q )] a a n t= a ρa ( ; ) ρa D a q = ρe { D( a; q )} ( ) e n a ρa n ( ; ) ρa D a q = ρe { D( a; q )} e n a n ρ ( a ) ρa ( ; ) n n n n ρa D a q = ρe ε e e = 0, n a where, from (6), we have derved n D( ; ) ( a ) n a q n ρ n = ρε e, a n whch mples the last equalty of (7). Hence, we have proved that

17 6 u (, q ) = u ( a, q ) for all a [,]. From the above arguments, we have proved that q s a Nash equlbrum,.e., u( q ) u( q, q ) for all q Q. From (3), ρ n n ρ u(, q ) e y ε = + = e + y, whch along wth u ( q ) u (, q ) mples the equalty of (5). Q.E.D. From the specfcaton of q, the probablty Dtq ( ; ) that the bubble has crashed at or before any tme t [0,] s gven by and Dtq (; ) = 0 f 0 t <, n ρ ( t) n n n Dtq (; ) = ε e f < t. See Fgure 3, whch llustrates Dtq ( ; ). [Fgure 3] and From the specfcaton of q, the hazard rate θ ( tq ; ) at any tme t [0,] equals θ (; q ) = 0 f 0 t <, nρ n n. θ (; q ) = f < t After the crtcal tme, the tme at whch the bubble crashes s randomly determned accordng to a constant hazard rate nρ. By slghtly postponng the tme to beat the n ρa gun from tme a to a +, arbtrageur obtans the gan ρe { D( a; q )}

18 7 from the ncrease n stock prce, whereas he suffers the loss ρa D( ; ) a q ( ) e n a from the decrease n wnnng probablty. Snce any tme choce s the best response n [,], the above gan and loss must be balanced, whch mples the equaltes of (7),.e., the frst-order condton. Note From the specfcaton of q, t follows q () t lm t = 0 t q ( t) q () t ρ ρ( t) exp[ ] t = n n. q ( ) ( t) t ρ exp[ ] n, whch mples that around the crtcal tme, any ratonal arbtrageur almost certanly postpones tmng the market. Note also q () t t lm t q ( t ) =+, whch mples that as the termnal tme s drawng near, any ratonal arbtrageur s n a great hurry to tme the market. Hence, the reason why even ratonal arbtrageurs have ncentve to rde the bubble s as follows, whch bears an analogy to the reputaton theory n fntely repeated games: () As the termnal tme s drawng near, any arbtrageur s almost convnced that the other arbtrageurs are behavoral. () At any tme around the crtcal tme, any arbtrageur s convnced through hs ratonal reasonng that the other arbtrageurs are lkely to rde the bubble even f they are ratonal. Note that the hazard rate after the crtcal tme dverges to nfnty as the rato between the total amount of shares that the arbtrageurs possess and the crtcal amount of shares n n approaches to unty. Note also that the crtcal tme depends on, not

19 8 ths rato, but ( n )ln ε. Ths mples that the bubble can persst for a long perod even f all arbtrageurs are almost certan that the bubble wll crash just around a partcular fxed tme,.e.,. 5. Hybrd Equlbra Ths secton specfes another symmetrc strategy profle named the hybrd strategy profle, accordng to whch any ratonal arbtrageur sells out at the ntal tme 0 wth a probablty that s postve, but less than unty. We arbtrarly set a tme (0,), where we assume ( n )lnε (8) max[0, + ] < <. ρ Let us specfy k (0,) by (9) ρ( ) ε exp[ ] k n, ε where the nequalty of (8) guarantees k (0,). Assocated wth (0,), let us specfy the hybrd strategy profle = as follows: for every N, q ( q ) N Q ( q a ) = k for all a [0, ), and ρ( a ) ε exp[ ] q ( a) = q ( a) = n for all a [,], ε where specfcaton (9) of k mples k = q ( ),.e., q s contnuous. Accordng to q, any ratonal arbtrageur N plans to sell out at the ntal tme 0 wth the

20 9 probablty of k > 0. Wth regard to the remanng probablty of k > 0, the ratonal arbtrageur plans to rde the bubble up to the tme, and he later follows the same strategy as the bubble-crash strategy q. The followng proposton shows the necessary and suffcent condton under whch q s a Nash equlbrum. Proposton 3: The hybrd strategy profle (0) ε q ε ( ε ) k n ρ n { ( ) k} { e ( ) } H N: H, H φ s a Nash equlbrum f and only f ( n )! = [ {( ε) k} { ( ε) k} H!( n H )! H n H H {mn[, ] max[0, ]}]. H + n H Proof: See the Appendx. When the hybrd Nash equlbrum s played, the mert from rdng the bubble s severely lmted,.e., u q = u q < + y must hold, because the tme choce of 0 ( ) (0, ) s a best response. The nequalty of > mples that once the bubble takes off, t tends to grow further than the bubble nduced by the bubble-crash equlbrum (See Fgure 4, whch llustrates Dtq (; )). [Fgure 4] From the specfcaton of q, t follows that

21 20 n ( ρ ) n n (; ) = ε e and Dtq θ (; tq ) = 0 f 0 < t <, and n ρ ( t) n n nρ Dtq (; ) = ε e and θ (; tq) = f < t. n n Snce the frst-order condton s the same between the hybrd equlbrum q and the bubble-crash equlbrum q at any tme t [,], t follows that the assocated hazard rate s the same between q and q,.e., θ nρ = θ = n (; tq) (; tq) for all t (,]. The followng proposton shows that f the quck-crash equlbrum q * and the bubble-crash equlbrum q coexst, there also exsts (,) such that the related hybrd strategy profle q s another Nash equlbrum. 9 Proposton 4: If the nequaltes of () and (2) hold wthout equalty, then there exsts (,) such that q s a Nash equlbrum. Proof: For every h [0,], let us defne Bh ε h e ε ( ε ) h n ρ n ( ) { ( ) } { ( ) } ( n )! [ {( ε) h} { ( ε) h} H!( n H )! H N: H, H φ H n H H {mn[, ] max[0, ]}]. H + n H 9 Proposton 4 does not mply the unqueness of (,); the hybrd strategy profle whch s related to ths proposton, s a Nash equlbrum. q,

22 2 Note that B( h ) s contnuous, B (0) > 0, and B () < 0, whch mply that there exsts h (0,) such that Bh ( ) = 0. Ths mples the equalty of (0). Q.E.D. 6. Characterzaton of Symmetrc Nash Equlbra The followng theorem characterzes all symmetrc Nash equlbra, whch states that there exsts no other symmetrc Nash equlbrum apart from the quck-crash equlbrum * q, bubble-crash equlbrum q, and hybrd equlbrum q. Theorem 5: If any strategy profle q Q s a symmetrc Nash equlbrum, then ether q =, or * = q, q q q= q, where satsfes (0). Proof: We set any symmetrc Nash equlbrum q Q arbtrarly, where we assume q * q. We show that ( ) q s contnuous n [0,]. Suppose that q ( ) s not contnuous n [0,],.e., there exsts > 0 such that lm q( ) < q( ). Snce l mn[, ] max[0, ] > 0 for all l {0,..., n }, t follows from the symmetry of l+ n l q that by selectng any tme that s slghtly earler than tme, any arbtrageur can drastcally ncrease the probablty of hs wnng the tmng game. Ths mples that no arbtrageur selects tme, whch s a contradcton. Let q q = max{ (0,]: ( ) = (0)}.

23 22 We show that q ( ) s ncreasng n [,]. Suppose that q ( ) s not ncreasng n [,]. From the contnuty of q and the defnton of, we can select, [,] such that <, q( ) = q( ), and the tme choce s a best response. Snce no arbtrageur selects any tme n (, ), t follows from the contnuty of q that by selectng tme nstead of, any arbtrageur can ncrease the wnner s gan from ρ e ρ + y to e + y wthout decreasng hs wnng probablty. Ths s a contradcton. Note that any tme choce [,] s a best response, because ( ) q s ncreasng n.e., [,]. Ths mples the followng frst-order condtons for all u (, q ) n (; ) e { D( ; q )} e dd q ρ ρ = ρ = 0, n d ρ λ D(; q ) = Ce, [,]: where C s a postve real number. Snce q s symmetrc and contnuous, t follows that () ρ λ n (; ) = = ε, D q Ce and (2) ρ λ D ( ; q ) = Ce = D (0; q ). From (), t follows that ρ n ε λ e C =, and therefore, (3) ρ ( ) n λ D (; q ) = ε e for all [,]. Suppose q (0) = 0. Then, the symmetry of q mples D(0; q ) = 0, whch along wth (2) and (3) mples ( n )lnε = +. ρ

24 23 Hence, t follows from (3) that =, and therefore, q= q. Suppose q (0) > 0. Let k (0) = q. From (2) and (3), ε ( n )ln( ) ( ε ) k = +, ρ whch along wth (9) mples =,.e., q= q. Snce q= q s a Nash equlbrum, t follows from Proposton 3 that = must satsfy (0). Q.E.D. The outlne of ths proof s as follows. Fx any symmetrc Nash equlbrum q Q arbtrarly (see Fgure 5, where t was supposed that Dtq ( ; ) s dscontnuous at tme t > 0 ). Note that any arbtrageur can drastcally ncrease hs wnng probablty by sellng out slghtly earler than tme t,.e., at tme t. Ths contradcts the Nash equlbrum property. Hence, Dtq ( ; ),.e., q must be contnuous. [Fgure 5] See Fgure 6, where t was supposed that Dtq ( ; ) s constant n the nterval [, ], and the tme choce s a best response. Snce no arbtrageur sells out n the ρ nterval (, ), any arbtrageur can ncrease the wnner s gan from e + y to ρ e + y wthout decreasng hs wnng probablty. Ths contradcts the Nash equlbrum property. Hence, t follows that there must exst a tme such that Dtq (; ) s constant n [0, ], whereas Dtq ( ; ) s ncreasng n [,]. [Fgure 6]

25 24 Durng the nterval [,], the frst-order condton must hold,.e., the hazard rate θ (, tq) must equal n ρ n = mples q * = q ; and (see Fgure 7; t must be noted that < < mples q= q, where = mples q= q ; = ). [Fgure 7] 7. Unqueness In order for ether the quck-crash strategy profle q * or the hybrd strategy profle q to be a Nash equlbrum, the tme choce of 0 must be a best response. Ths along wth Theorem 5 mples that whenever the tme choce of 0 s a domnated strategy, the bubble-crash strategy profle q s the unque symmetrc Nash equlbrum. The followng theorem states that the unqueness holds even f we take all asymmetrc Nash equlbra nto account. Theorem 6: The bubble-crash strategy profle q s the unque Nash equlbrum f (4) n e ρ ε >. Proof: We wll show that q s the unque symmetrc Nash equlbrum. Note that u(0, q ) and > for all q Q. n u(, q ) ε e ρ Ths along wth the nequalty of (4) mples that the tme choce of 0 s domnated by the tme choce of,.e.,

26 25 u (, q ) > u (0, q ) for all q Q. Hence, any symmetrc Nash equlbrum q Q must satsfy q (0) = 0, whch along wth Theorem 5 mples q = q. Snce the nequalty of (4) mples the nequalty of (2), we have proved that q s the unque symmetrc Nash equlbrum. We wll show that q s the unque Nash equlbrum even f all asymmetrc Nash equlbra are taken nto account. We set any Nash equlbrum q Q arbtrarly. Frst, we show that q ( ) must be contnuous n [0,] for all N. Suppose that q ( ) s not contnuous n [0,]. Then, there exsts > 0 such that l lm q( ) < q( ) for some N. Snce mn[, ] max[0, ] > 0 for all l+ n l l {0,..., n }, t follows that any other arbtrageur can drastcally ncrease hs wnnng probablty by selectng any tme that s slghtly earler than tme. Hence, any other arbtrageur never selects any tme that s ether the same as, or slghtly later than, the tme. Ths mples that arbtrageur can ncrease the wnner s gan by postponng tmng the market further wthout decreasng hs wnnng probablty. Ths s a contradcton. Second, we show that D(; q) must be ncreasng n [,], where we denote = max{ (0,]: q ( ) = q (0) for all N}. Suppose that D(; q) s not ncreasng n [,]. Hence, from the contnuty of q, we can select, (,] such that <, D( ; q) = D( ; q), and the tme choce s a best response for some arbtrageur. Snce no arbtrageur selects any tme n (, ), t follows from the contnuty of q that by selectng tme nstead of, ρ any arbtrageur can ncrease the wnner s gan from e ρ + y to e + y wthout decreasng hs wnnng probablty. Ths s a contradcton.

27 26 Thrd, we show that q must be symmetrc. Suppose that q s asymmetrc. Snce the nequalty of (4) mples that the tme choce of 0 s a domnated strategy, t follows > 0, and (5) q ( ) = 0 for all N and all [0, ]. Snce q s contnuous and D(; q) s ncreasng n [,], from the supposton > 0 and the equalty of (5), t follows that there exst 0 such that >, >, and N q () t = q () t for all j N and all t [0, ], j (6) and D(; q) Dh(; q) t > mn t D( ; q) h D ( ; q) h for all t (, ), (7) D( ; q) Dh( ; q) t = mn t > 0, D( ; q) h D ( ; q) h where the last nequalty was derved from the ncreasng property of D( ; q) n [,]. Snce D(; q) s ncreasng n [,], any tme choce t n (, ) must be a best response for any arbtrageur j N who satsfes Dj(; t q) Dh (; t q) t = mn t. D ( t; q) h D ( t; q) j h qj () t Snce ths equalty mples > 0, t follows from the contnuty of q that the t followng frst-order condton holds for arbtrageur j ; for every t (, ),.e., uj(, q j) ρ ρ ddj( ; q j) = ρe { Dj( ; q j)} e = 0, n d

28 27 Hence, from (6), Dh (; t q) n ρ = mn t. h D ( t; q) D (; t q) n ρ < t, D( t; q) h whch mples that the frst-order condton does not hold for arbtrageur for every t (, ), where u(, q ) ρ ρ dd( ; q ) = ρe { D( ; q )} e < 0. n d Ths nequalty mples that arbtrageur prefers tme rather than any tme n (, + ε), and therefore, D (; q) = 0 for all (, + ε), t where ε was postve but close to zero. Ths s a contradcton, because the nequalty D ( ; q) of (7) mpled > 0. Hence, we have proved that any Nash equlbrum q t must be symmetrc. Q.E.D. The bref sketch of ths proof s as follows. We can prove the contnuty of q and the ncreasng property of q n a manner smlar to the proof of Theorem 5. In order to show that any Nash equlbrum q must be symmetrc, we have used the nequalty of (4) as follows. The frst-order condton mples (8) D (; t q) t n = ρ D( t; q) for all N and all t [,]. The nequalty of (4) mples (9) q = for all N, ( ) 0

29 28 whch along wth (8) mples that q = q for all N,.e., q s symmetrc. Wthout the nequalty of (4), however, we may not be able to show ths symmetry; we may not be able to exclude the possblty that any asymmetrc Nash equlbrum q exsts, such that q (0) > 0 and q(0) q (0) for some N and some j N \{}. j 8. Concluson Ths paper modeled the stock market as a tmng game wth ncomplete nformaton, where t was assumed that each arbtrageur s not necessarly ratonal, and s commtted to rde the bubble wth a small but postve probablty. We showed a suffcent condton under whch there exsts the unque Nash equlbrum, where ths equlbrum nduces the bubble to persst for a long perod even f all arbtrageurs are ratonal. We also characterzed all symmetrc Nash equlbra n general. It s mportant to generalze our model n several drectons. For nstance, the present paper assumed that each arbtrageur has a sngle share n common. If we drop ths assumpton and permt heterogenety among arbtrageurs n terms of ther shareholdngs, t mght be conjectured that larger shareholders are more lkely to nsst on rdng the bubble than smaller ones. Ths type of nterestng but careful analyss s, however, beyond the purpose of ths paper, and s regarded as a pendng problem for possble future researches.

30 29 References Abreu, D. and M. Brunnermeer (2002): Synchronzaton Rsk and Delayed Arbtrage, Journal of Fnancal Economcs 66, Abreu, D. and M. Brunnermeer (2003): Bubbles and Crashes, Econometrca 7, Brunnermeer, M. (2008): Bubbles, n The New Palgrave Dctonary of Economcs, 2 nd Ed. Durhauf, S. and L. Blume (eds.), London: Macmllan. Brunnermeer, M. and J. Morgan (2006): Clock Games: Theory and Experments, mmeo. Kndleberger, C. P. (978): Manas, Pancs and Crashes: A Hstory of Fnancal Crses, London: Macmllan. Kreps, D., P. Mlgrom, J. Roberts, and R. Wlson (982): Ratonal Cooperaton n the Fntely Repeated Prsoners Dlemma, Journal of Economc Theory 27, , Morrs, S. and H. Shn (998): Unque Equlbrum n a Model of Self-Fulfllng Currency Attacks, Amercan Economc Revew 88, Obstfeld, M. (996): Models of Currency Crses wth Self-Fulfllng Features, European Economc Revew 40, Shller, R. J. (2000): Irratonal Exuberance, New Jersey: Prnceton Unversty Press. Shlefer, A. (2000): Ineffcent Markets An Introducton to Behavoral Fnance, New York: Oxford Unversty Press. Soros, G. (994): The Alchemy of Fnance, New York: Lescher & Lescher.

31 30 Appendx: Proof of Proposton 3 From the specfcaton of q, t follows that u q equals (0, ) (A-) H N: H, H φ ( n )! ε ε H!( n H )! H + H n H {( ) k} { ( ) k} mn[, ] { ( ) k} n + ε + y, and for every a (0,], u a q equals (, ) (A-2) H N: H, H φ ( n )! H ε k ε k H!( n H )! n H H n H {( ) } { ( ) } max[0, ] n ρa + { ( ε ) k} e + y f a (0, ), and (A-3) H N: H, H φ ( n )! H ε k ε k H!( n H )! n H H n H {( ) } { ( ) } max[0, ] ρa k + e { D( a; q )} a ρt k + e dd(; t q ) + y n t= 0 f a [,]. Moreover, from the specfcaton of q, = ε + ε k for all t D(; t q ) { ( )( )} n [0, ), and (A-4) n ρ ( t ) n n n ε e D(; t q ) = for all t [,]. Snce q s contnuous, u q u a q = k e e ρ > for all a (0, ). (, ) (, n a ) { ( ) } ( ρ ε ) 0 From (A-) and (A-2), for every N and every a (0, ),

32 3 u q u a q (0, ) (, ) ( n )! H = ε k ε k H!( n H )! H + n H H N: H, H φ H n H {( ) } { ( ) } {mn[, ] max[0, ]} n a + { ( ε ) k} ( e ρ ) > 0. From the equaltes of (9) and (0), t follows that for every N, u q u q (0, ) (, ) ( n )! H = ε k ε k H!( n H )! H + n H H N: H, H φ H n H {( ) } { ( ) } {mn[, ] max[0, ]} n + { ( ε ) k} ( e ρ ) = 0, where the last equalty was derved from (9) and (0). From (A-3), the followng frst-order condton holds for every a [,] ; (A-5) a ρa ρt u( a, q ) = [ e { D( a; q )} + e dd( t; q )] t= 0 a a n = ρe D a q e ρa ( ; ) n n ρa D a q { ( ; )} n a n ρ ( t) ρa ( ; ) n n n ρa D a q = ρe ε e e = 0, n a where, from (A-4), we have derved D( a; q ) n = ρε a n n ρ ( t) n n whch mples the last equalty of (A-5). Hence, e, u q = u a q for all a [,], (, ) (, ) and therefore, we have proved that u q for all q Q and all N. ( ) u( q, q )

33 32 Fgure Stock Prce e ρ + y e ρ + y + y y : Fundamental Value 0 : Tme Crash Tme

34 33 Fgure 2 * Dtq (; ) n ε 0 Tme

35 34 Fgure 3 Dtq (; ) n ε n nρ( t) ε exp[ ] n 0 t Tme

36 35 Fgure 4 Dtq (; ) n ε n nρ( t) ε exp[ ] n n nρ( ) k ε exp[ ] n 0 t Tme

37 36 Fgure 5 Dtq (; ) n ε Wnnng Probablty Increases 0 t t Tme

38 37 Fgure 6 Dtq (; ) n ε Wnner s Gan Increases 0 t t Tme

39 38 Fgure 7 Dtq (; ) n ε n nρ( ) ε exp[ ] n 0 Tme Frst-Order Condton: nρ θ (; tq) = n n