Information Acquisition and Transparency in Global Games


 Sharyl Walters
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1 Informaton Acquston and Transparency n Global Games Mchal Szkup y and Isabel Trevno New York Unversty Abstract We ntroduce costly nformaton acquston nto the standard global games model. In our setup agents have the opportunty to mprove the precson of ther prvate sgnal, at a cost, before playng the coordnaton game. We show that only symmetrc equlbra exst and we provde su cent condtons for unqueness. Under these condtons, we nd that the unque symmetrc equlbrum of the game s ne cent. Fnally, we study the e ects of ncreased transparency on nformaton acquston and coordnaton. We nd that, dependng on the cost of nvestment and on pror belefs, an ncrease n the precson of publc nformaton can have a bene cal or detrmental e ect on welfare. Key words: global games, nformaton acquston, coordnaton, transparency. Introducton Global games have been extensvely appled to model economc phenomena featurng coordnaton problems, such as currency crss (Morrs and Shn, 998), bank runs (Goldsten and Pauzner, 005), FDI decsons (Dasgupta, 007) or poltcal revolts (Edmond, 007). In a global game, the payo s of agents depend on the state of the economy and on the actons of others. However, agents only observe a nosy prvate sgnal about ths state and, n order to choose an optmal acton, they have to make nferences about ts true value and about the belefs that other agents hold. Ths perturbaton of the nformaton structure of the game gves rse to a very rch sequence of hgher order belefs, whch leads agents to coordnate on a unque equlbrum. Ths predcton contrasts the complete nformaton model, whch features multple equlbra. The nformaton structure s thus at the heart of the predctons of global games. Whle the orgnal models have been extended along many drectons, the precson of prvate sgnals We would lke to thank Tomasz Sadzk and Enno Stacchett for ther constant support and encouragement throughout ths project. We also thank Davd Pearce, Thomas Sargent, Laura Veldkamp, and partcpants n the Mcro Student Lunch Semnar at NYU for ther comments. y Correspondng author: m.szkupnyu.edu
2 has always been exogenously gven and typcally set to be dentcal across agents. It remans an open queston whether or not these two assumptons are wthout loss of generalty. In ths paper we shed lght on these ssues by studyng a twostage global game where each agent has the possblty to costly mprove the precson of hs prvate nformaton about the state of the economy, and, gven ths precson choce, he receves a sgnal about the state that wll determne hs acton n the coordnaton game. In addton to characterzng equlbra, we study the crtcal role that nformaton plays n the coordnaton game and carefully analyze the strategc motves n nformaton acquston. Our ndngs ndcate that under mld condtons there exsts a unque equlbrum n symmetrc strateges. Ths ndng gves support, n the context of global games, to the commonly made assumpton of exante dentcal agents. We de ne the value of nformaton n our setup and analyze how t s a ected by pror belefs, the behavor of other players, and the cost of nvestment. We characterze the crtcal role that nformaton plays n the coordnaton game by studyng the e ect that precson choces have on the ncdence of two types of mstakes n the coordnaton game (nvestng when nvestment s not pro table, or not nvestng when nvestment s pro table). Interestngly, we nd condtons for strategc complementartes n nformaton acquston to arse and we provde an example where these condtons are volated, so that the optmal precson choce of an agent s a nonmonotonc functon of the precson choces of the others. In addton, we contrbute to the growng lterature that focuses on the mportance of government transparency by studyng the welfare e ects of an ncrease n the precson of publc nformaton. We provde a novel perspectve on the ssue by analyzng the tradeo between prvate and publc nformaton, snce n our model publc nformaton a ects outcomes not only through agents actons n the coordnaton stage, but also by changng ther ncentves to acqure prvate nformaton n the rst stage. We thus characterze the emergence of ncreasng and decreasng d erences n the precson of publc and prvate sgnals. Our analyss also hghlghts the d erences between global games and the closely related famly of beauty contest models (n the sprt of Morrs and Shn, 00). Frst, we nd that whether an mprovement n publc nformaton s welfare enhancng or not depends crucally on the ex ante belefs about the state relatve to the cost of nvestment, whle n beauty contest models t depends on the relatve nformatveness of prvate and publc nformaton (Morrs and Shn, 00). Secondly, n beauty contest models wth endogenous prvate nformaton acquston an ncrease n the precson of publc nformaton always leads to a decrease n the precson of prvate nformaton (Tong, 007), whereas n our context prvate and publc nformaton can be complements. Fnally, we provde condtons under whch strategc complementartes n actons translate nto strategc complementartes n nformaton acquston. We provde an example where these condtons are volated and Beauty contest models are also coordnaton games of ncomplete nformaton, but d er from standard global games n many respects. In partcular, n beauty contest models choce sets are contnuous and agents have a quadratc utlty functon that depends on the average acton of the other players as well as on the underlyng state of the economy.
3 nformaton choces are not strategc complements, whch s n contrast to the results of Hellwg and Veldkamp (009) for a beauty contest game. Endogenzng nformaton n a global games framework s not only an mportant endeavor from a theoretcal pont of vew but t s also economcally relevant. Global games have been appled extensvely to analyze d erent types of nancal decsons. It s a well known fact that partcpants n nancal markets have the possblty to mprove the nformaton they possess, at a cost, and they take advantage of ths possblty. Investment groups and ndvduals pay experts to extract more accurate nformaton about the state of the economy n order to maxmze pro ts, creatng a market for nformaton expertse and nancal advsng. Followng Dasgupta (007), one can thnk of an emergng economy that starts a lberalzaton program to attract Foregn Drect Investment. Potental nvestors have to make a dscrete decson to nvest or not nvest (e.g. buldng a plant to produce goods for the local market). For the pro ts to be postve there has to be enough nvestment so that the lberalzaton program takes o due to ncreasng returns to aggregate nvestment. Ths means that a potental nternatonal nvestor wll be more lkely to undertake nvestment n ths emergng economy f he beleves that other nvestors wll do so as well. Moreover, the returns of the project depend also on the state of the emergng economy n the form of aggregate demand, nfrastructure, and poltcal envronment  all of whch can be uncertan at the tme of the nvestment decson. In ths context, potental nvestors can buy reports from nternatonal agences that wll assess the pro tablty of ths nvestment decson,.e. they acqure more precse nformaton about the state of the emergng economy. Alternatvely, we can nterpret the cost of nformaton as an opportunty cost assocated wth collectng and analyzng data about the state of the economy. Ths example llustrates why ntroducng costly nformaton acquston s a relevant extenson to global games also from an appled perspectve. The paper s structured as follows. In secton we set up the model wth costly nformaton acquston and explan the assumptons we make to solve the model. In secton 3 we solve the model and carefully analyze the role that nformaton plays n our model. We also present three key results: nonexstence of asymmetrc equlbra, exstence of symmetrc equlbra, and condtons ensurng unqueness of the symmetrc equlbrum. In secton 4 we nvestgate notons of e cency of the unque equlbrum. In secton 5 we nvestgate f strategc complementartes n the coordnaton game translate nto strategc complementartes n nformaton choces. In secton 6 we ask whether an ncrease n the precson of publc nformaton s welfare enhancng or not. Secton 7 summarzes related lterature and secton 8 concludes. All the proofs (wth few exceptons) are relegated to the appendx. The Model We start wth a bref descrpton of the benchmark global games model (a model wthout nformaton acquston). We then ntroduce nformaton acquston nto the model and carefully explan the tmng, underlyng assumptons, and we hghlght the d erences between See Farmer and Guo (994), Hall at al (986), Hall (987), and Caballero and Lyons (99) for evdence of ncreasng returns to scale n nvestment. 3
4 the models wth and wthout nformaton acquston. There s a contnuum of nvestors n the economy ndexed by, where [0; ], who have to decde whether to nvest n a rsky project or not. The economy s characterzed by the strength of ts economc fundamentals ; whch a ects the return of the nvestment. Each nvestor faces a bnary decson problem: he has to decde whether to nvest (I) or not nvest (NI). If an nvestor nvests he wll ncur a cost T (0; ). The bene t to nvestng s uncertan and depends on the state and on p, the proporton of nvestors who choose to nvest. Investment s successful f p >,.e. f the proporton of nvestors who nvest s hgh enough. The return of a successful nvestment for each nvestor who nvests s, n whch case he wll get payo T. Otherwse, f nvestment s unsuccessful, hs payo wll be T. The payo to not nvestng s certan and normalzed to 0. The payo s are summarzed below: u (I; p; ) = T f p T f p < u (N; p; ) = 0 (a) (b) Whether ndvdual nvestment s successful or not depends on the state of the economy and on the number of ndvdual nvestments. One can nterpret ths need for enough aggregate nvestment as resultng from ncreasng returns to scale n nvestment. 3 If nvestors know the state of the economy,, then, dependng on the value of the game can have a unque or multple symmetrc purestrategy Nash equlbra:. If everyone nvests and nvestment s successful;. If < 0 no one nvests; 3. If [0; ) there are multple equlbra characterzed by selfful llng belefs. In contrast, when nvestors have ncomplete nformaton about the state of the economy, t s possble to select a unque equlbrum. The lterature on global games started by Carlsson and van Damme (993) and Morrs and Shn (998) assumes that nvestors do not observe and beleve that N ;. In ths case, each nvestor prvately observes x, a nosy sgnal of : x = = " where " N (0; ) s ::d: across nvestors and ndependent of the realzaton of, and s a postve constant. It follows that x j N ; 3 The payo s are chosen to make the game analytcally tractable. All the qualtatve results would stll hold f we allowed the bene t from nvestng to be an explct functon of both the state and aggregate nvestment. However, n that case we would need to resort to numercal solutons due to the extra complexty. 4
5 so that s the precson level of the prvate sgnals. Note that whle nvestors receve sgnals of d erent values, they share the same precson. In ths setup, Morrs and Shn (998) show that the equlbrum s characterzed by two objects, (x ; ), where x s the threshold value for sgnals such that f an nvestor observes a sgnal x x he wll nvest, and not nvest f x < x. Lkewse, corresponds to the crtcal level of fundamentals such that f then nvestment wll be successful, and f < nvestment wll fal. It has been shown that both x and depend crucally on the precson of the nformaton that nvestors receve ( and ). We present the followng result, due to Hellwg (00) and Morrs and Shn (004), characterzng the unque equlbrum of the game: Theorem If > p then there s a unque equlbrum of the above game. Ths equlbrum s n monotone strateges and s characterzed by a par of thresholds (x ; ). The above result assumes that all nvestors have access to nformaton wth the same precson ( s the same for all nvestors). In what follows we focus on relaxng ths assumpton by allowng nvestors to purchase the precson of ther prvate sgnal. Ths ntroduces strategc motves n nformaton acquston and generalzes the model to a case where the precson of sgnals d ers across nvestors.. Informaton Acquston We consder now a twoperod model where nvestors rst smultaneously and prvately choose the precson of ther prvate sgnal, and then move on to play the coordnaton game, as descrbed above. Investors are ntally endowed wth a sgnal wth precson 0 and they have the opportunty to mprove the precson of ther sgnal by choosng [ 0 ; ). Ths mprovement of precson s costly, whch results n a tradeo between nformatveness and cost of sgnals. Once nvestors have chosen, we move on to perod. At the begnnng of perod all nvestors receve prvate sgnals: x = = ", 8 [0; ] where s the precson level chosen by nvestor n the rst stage. As before, " N (0; ) s ::d: across nvestors and ndependent of the realzaton of. Note that n contrast to the model wthout nformaton acquston, we now allow nvestors to observe sgnals wth d erent precsons. After observng ther sgnals nvestors smultaneously decde whether to nvest or not. The payo s n the coordnaton game are the same as n the benchmark model, gven by equatons (a) and (b). Once all nvestors make ther nvestment decsons payo s are realzed.. Assumptons The rst set of assumptons consders the underlyng parameters of the game. 5
6 Assumpton (Concavty) We assume the followng: [ ; ]; < < < [ ; ]; < < < T [T ; T ]; 0 < T < T < 0 > maxf; g Under these assumptons we can show that the exante utlty functon s concave n the ndvdual precson choce. The detals of determnng can be found n clam A:3 n the appendx. 4 We also assume that 0 > to ensure unqueness of equlbrum n the coordnaton game at t =. The next assumpton pertans to the cost functon, C ( ): Assumpton (Cost functon) We assume that the cost functon C () s: strctly ncreasng n, C 0 () > 0 strctly convex n, C 00 () > 0 C 0 ( 0 ) = 0 lm C 0 ( ) = These assumptons mply that the cost functon s convex, a common assumpton n the lterature on nformaton acquston. We further assume that the n ntesmal mprovement of precson s costless, to ensure that the problem s nontrval and that nvestors always acqure nformaton. The last assumpton ensures that nvestors wll never choose to acqure perfect nformaton. 3 Solvng the Model We now solve the model wth costly nformaton acquston usng backward nducton. We start n perod, takng as gven the precson choces made by the nvestors n perod. Once we characterze the equlbrum outcome at t = we move to the rst stage to determne optmal nformaton choces. 4 Numercal smulatons suggest that to ensure concavty of the utlty functon t s su cent to have 0 >. 6
7 3. Solvng the Model: t = Let be a dstrbuton of precson choces n the economy, that s () s the proporton of nvestors who chose precson n the rst stage. To make hs decson, nvestor has to take nto account the dstrbuton of s n the economy ( ), hs own precson level ( ), hs sgnal (x ), and hs pror belef about. We rst show that for any dstrbuton of precsons,, there exsts a unque equlbrum n monotone strateges and we then establsh that ths s the only equlbrum n the second stage of the game. Let a (x ; ; ) be nvestor s strategy. Then a () s monotone f there exsts x ( ; ) such that Invest f x x ( ; ) a (x ; ; ) = Not Invest f x < x ( ; ) Assume that all nvestors follow monotone strateges. Note that the thresholds can d er across nvestors wth d erent precson levels and that they also depend on. We assume that all nvestors wth the same precson level,, follow the same monotone strategy (.e. they have the same threshold x ( ; )). As n the standard global games, the equlbrum n monotone strateges s characterzed by a Payo Ind erence and a Crtcal Mass condton. The d erence wth respect to the standard setup s that n our model each type has a d erent PI condton. Consder rst the Crtcal Mass (CM) condton, whch requres that at state the mass of nvestors that nvest s equal to the mass of nvestors needed for nvestment to succeed. Gven nvestors monotone strateges, the mass of nvestors that nvest at s gven by Z Pr (x x ( ; ) j) d ( ) At the same tme, n order for nvestment to be successful the proporton of nvestors that choose to nvest, p, has to be larger than. At the crtcal state, the mass of nvestors, p, s just equal to : Z Pr (x x ( ; ) j ) d ( ) = Ths condton can be rewrtten as Z x ( ; ) = d ( ) = From the above equaton we see that the crtcal state wll depend on the dstrbuton, henceforth we wll denote t by ( ). Consder now nvestor whose precson level s. The Payo Ind erence (PI) condton states that at the sgnal x the nvestor s nd erent between nvestng and not nvestng: Pr ( > ( ) jx ( ; )) T = 0 7
8 The sgnal that solves the above equaton s the optmal threshold x ( ; N x ; ( ) = we get ). Snce jx x ( ; ) = ( ) ( ) = (T ) () The above expresson s ntutve and llustrates how x depends on only through ( ). Recall that nvestment s successful f and only f ( ) and hence an ncrease n ( ) decreases the probablty of nvestment condtonal on any sgnal x. Ths n turn makes the nvestor less wllng to nvest and results n a hgher threshold x. Smlarly, a hgher means that hgher values of are more lkely to occur and hence t ncreases the probablty that nvestment wll be successful. Therefore, a hgher decreases x. Fnally, a hgher T ncreases the cost of nvestment, makng t less attractve, ncreasng x. Hence, an equlbrum n monotone strateges s characterzed by the PI and CM equatons: x ( ; ) ( ) = T; 8 [0; ] (3) ( ) = Z x ( ; ) ( ) = d ( ) = ( ) (4) Ths system of equatons can be smpl ed to one equaton n one unknown, ( ): Z ( ( ) = ) ( ) = (T ) d ( = ) = ( ) (5) The proposton below spec es the condtons under whch the above equaton has a unque soluton, mplyng that the coordnaton game has a unque equlbrum n monotone strateges. Furthermore, under the same condtons, the unque monotone equlbrum s the only equlbrum n the second stage of the game. Proposton For any, suppose that nf (supp( )) >. Then the coordnaton game has a unque equlbrum n whch all nvestors use threshold strateges and nvestment s successful f and only f ( ). Note that the above proposton s a generalzaton of the result stated n theorem. If s a degenerate dstrbuton wth a unque mass pont,.e.: f ( ) = 0 f < then our equlbrum condton reduces to the one n the benchmark model and the above condton on precsons reduces to the condton stated n theorem. Armed wth ths result we move on to the rst perod to analyze nvestors optmal choces of precson. 8
9 3. Solvng the Model: t = We now consder the rst stage of the game n whch nvestors choose the precson of the sgnal they wll observe at the begnnng of the second stage. We assume that each nvestor wll act optmally n the second perod and that he beleves that all other nvestors wll act optmally as well. 3.. Exante Utlty Denote by G () the pror belef of nvestors regardng and by F (xj) the condtonal dstrbuton of x gven and gven that the sgnal x has precson. Recall that all nvestors are exante dentcal,.e. they have the same exante utlty. The exante utlty of nvestor who chooses precson and faces a dstrbuton of precson choces, for any ( ; ; T ) s U ( ; ; ; ; T ) = Z = Z x x ( ; ) f ( )g T df (xj) dg () C ( ) (6) Whle the above expresson for exante utlty arses naturally from the problem, t can be rewrtten n a way that sheds more lght on the forces of the model: 5 U ( ; ) = Z Z x T df (xj) dg () Z Z Z x ( T ) df (xj) dg () ( T ) dg () C ( ) (7) The above expresson has an ntutve nterpretaton. The last term s smply the cost assocated wth the precson choce. Recall that nvestment s successful f and only f ( ) and f an nvestor nvests n ths case hs payo s T. Hence, the thrd term of the above expresson s the expected payo at tme t = for an nvestor who can perfectly observe n the second perod. However, for any < an nvestor s nformaton at t = s nosy. Ths means that the nvestor wll sometmes make mstakes, ether nvest when nvestment s unsuccessful (Type I mstake) or not nvest when nvestment s successful (Type II mstake). The rst two terms capture the expected cost of these two mstakes, respectvely. We denote ths cost for an nvestor wth precson who faces a dstrbuton of precson choces by M ( ; ). M ( ; ) = Z Z x T df (xj) dg () Z Z x ( T ) F (xj) dg () Rewrtng the utlty functon as n equaton (7) allows us to better understand how a hgher precson s bene cal to nvestors. Therefore, we abstract from the cost of precson 5 See secton A: of the appendx for dervatons. 9
10 and focus on the bene t n terms of expected utlty captured by the rst three terms of equaton (7). We de ne ths bene t as B ( ; ): B ( ; ) Z Z x T df (xj) dg () Z Z Z x ( T ) F (xj) dg () ( T ) dg () (8) An mprovement n precson makes t possble for nvestors to avod costly mstakes wthout alterng the overall coordnaton outcome. Ths s formalzed n the followng lemma. Lemma The bene t n terms of expected utlty of an ncrease n precson s equal to the reducton of the expected cost of mstakes due to ths ncrease. B ( ; ) = M ( ; ) (9) Fgure below llustrates how an ncrease n precson a ects the expected cost of mstakes. Each panel n gure shows the contour lnes of the exante jont dstrbuton of (; x ) for d erent precson levels. The shaded areas correspond to pars of states and sgnals, (; x ), for whch nvestors wll make ether Type I or Type II mstakes. The left panel depcts the stuaton when s low and the panel on the rght depcts the case when has ncreased. In partcular, an ncrease n decreases the expected cost of mstakes va two channels: () t changes the exante jont dstrbuton of (; x ) by shrnkng the contours towards the 45degree lne (gvng the mpresson of a clockwse rotaton together wth a stretch along the 45degree lne), and () by changng the threshold x. Fgure : E ect of an ncrese n An ncrease n algns the nvestor s sgnal x better wth the state through the rst channel, allowng hm to predct more precsely the behavor of the other nvestors. The 0
11 nvestor takes advantage of ths ncrease n the nformatveness of hs sgnal by adjustng hs threshold x n a way that wll allow hm to coordnate on the optmal acton more often. As a result, an ncrease n precson leads to a decrease n the the total expected cost of mstakes. We now establsh some propertes of the exante bene t functon that wll prove to be useful for further analyss. Lemma. B ( ; ) s strctly ncreasng n ;. lm B = 0; 3. For > ; B < 0. The rst part of the lemma states that a hgher precson s always bene cal for the nvestor, whch follows from the analyss above. The second part of the lemma shows that ths bene t goes to zero n the lmt,.e. the bene t of ncreasng precson s zero for an nvestor that holds almost perfect nformaton. The thrd part states that for su cently hgh precson levels the exante bene t s concave,.e. as an nvestor becomes better nformed, further ncreases n precson wll have a smaller bene t. Ths s due to the fact that for a well nformed nvestor addtonal nformaton s of lttle value, compared to an nvestor that has very mprecse nformaton. Havng characterzed the exante expected utlty and the channels through whch changes n precson a ect nvestor s behavor, we move on to solve for equlbrum of the full game. 3.3 Equlbrum at t = In ths subsecton we rst state the problem faced by the nvestor at the begnnng of the game and we provde a de nton of equlbrum for the twostage game. We show that no asymmetrc equlbrum can potentally exst, and gven ths result, we prove exstence of symmetrc equlbra and provde su cent condtons under whch the symmetrc equlbrum s unque. Consder any player and assume that he faces a dstrbuton of precson choces and beleves that all nvestors behave optmally at t =. Hs expected payo of choosng s U ( ; ) = B ( ; ) C ( ) Z where ( ) solves Pr (x x ( ; ) j ( )) d = ( ) and x ( ; ) = ( ) ( ) = (T ) We now state the de nton of a pure strategy Perfect Bayesan Nash Equlbrum of the twostage game:
12 De nton A pure strategy Perfect Bayesan Nash Equlbrum s a set of precson choces f ; [0; ]g, a decson rule for the second perod a (x ; ; ), and a dstrbuton functon such that:. Each nvestor s choce of precson s optmal gven : B ( ; ) C( ) B (b ; ) C(b ) 8b [; ). The dstrbuton mpled by the nvestors choces s almost surely equal to the dstrbuton 3. All nvestors behave optmally n the second stage: where a (x ; ; ) = ( f x x ( ; ) 0 f x < x ( ; ) x ( ; ) = ( ) ( = ) (T ) and ( ) solves: Z = ( ( ) ) ( ) = = (T ) d ( ) = ( ) We now show that there cannot exst an equlbrum such that s nondegenerate, that s, we show that t has to be the case that n any equlbrum of the above game all nvestors choose the same precson. 6 Theorem Suppose that assumptons (A) and (A) hold. Then there are no asymmetrc equlbra n whch nvestors choose d erent precson levels n the rst stage. Proof. Suppose that s nondegenerate. By proposton we know that for any there exsts a unque equlbrum n monotone strateges n the second stage of the game. Snce all nvestors are n ntesmally small, t follows that no nvestor can n uence the outcome of the second stage and hence all nvestors take the equlbrum outcome as gven. Moreover, under assumptons (A) and (A) each nvestor s problem at t = has a unque soluton. Snce all nvestors are exante dentcal, ths mples that they face the same decson problem 6 For ths result to be true we need quasconcavty of the exante utlty functon net of the precson cost and a unque equlbrum n the second stage. The assumptons made n secton. ensure these condtons are met. =
13 and that the optmal soluton s the same for all nvestors. It follows that the dstrbuton of nvestors precson choces s degenerate. We now turn our attenton to symmetrc equlbra. A symmetrc equlbrum s an equlbrum such that 8 [0; ] =, where [ 0 ; ). In what follows, we wll slghtly abuse notaton and wrte as a functon of rather than as a functon of the dstrbuton mpled by the symmetrc choce of precson levels. Hence, when we wrte ( ) what we really mean s ( ) where () = 0 for all < and () = for all. Smlarly, from now on, we wll wrte an nvestor s threshold x as a functon of. Theorem 3 Suppose that assumptons (A) and (A) hold. Then there exsts a symmetrc equlbrum of the nformaton game where. All nvestors choose the same precson n the rst perod;. The outcome of the game n the second perod s characterzed by thresholds ( ( ); x ( )). Proof. We showed above that regardless of the dstrbuton of precson choces there exsts a unque equlbrum n the second stage. Denote by the precson choce of all other nvestors and let () be the optmal precson choce of nvestor gven that all other nvestors choose precson. By the theorem of the Maxmum t follows that () s a contnuous functon of. Snce C 0 ( 0 ) = 0 we know that ( 0 ) > 0. At the same tme, by assumpton (A) we know that nvestors wll never nd t optmal to choose precson level > (see appendx, clam A:4). Therefore, we conclude that () s a contnuous functon mappng [ 0 ; ] nto tself. By Brouwer s Fxed Pont theorem we know that () has a xed pont, call t. Ths xed pont of () s a symmetrc equlbrum snce f an nvestor beleves that all other nvestors choose, hs best response s to choose hmself. The above theorem and proposton establsh exstence of a symmetrc equlbrum n whch all nvestors choose the same precson n the rst stage,, they follow a symmetrc monotone strategy wth threshold x ( ), and where nvestment s successful f and only f ( ). The followng theorem establshes su cent condtons for the exstence of a unque equlbrum n symmetrc strateges. Theorem 4 There exsts < such that f >, then there s a unque equlbrum n the nformaton acquston game, that s there s a unque [; ) that solves = ( ). Proof (Sketch). Denote by () the best response functon of nvestor. Ths best response functon s de ned mplctly by the rst order condton of nvestor s decson problem B ( () ; ) C ( ()) = 0 3
14 De ne the set E as the set of all equlbrum choces of : E = fj = ()g The full proof n the appendx shows that there exsts such that 0 > mples that for any E we have () j = < ; that s, the slope of the bestresponse functon at any symmetrc equlbrum s less than. Ths mmedately mples unqueness of a symmetrc equlbrum snce exstence of multple equlbra requres that there exsts at least one E such that () j =. Notce that the condton we mpose on 0,.e. that sgnals are precse enough, s n the same sprt as the standard condton to ensure unqueness of equlbrum n global games. 7 In what follows we assume that ths condton for unqueness s sats ed and we nvestgate the propertes of the unque equlbrum. 4 Spllover E ects and the Ine cency of Equlbrum The game above exhbts spllover e ects, n the sense that nvestors choces of precson a ect the decson of each ndvdual nvestor n a way that s not taken nto account by them. In partcular, an ncrease n the precson of all but one nvestor a ects ths nvestor s utlty through the mpact of these precson choces on. However, snce all nvestors take as gven they gnore ths e ect when choosng ther ndvdual level of precson. Ths, as we show below, leads to ne cency of the unque equlbrum. We de ne an e cent symmetrc precson choce as the one that maxmzes the exante expected utlty takng nto account these spllover e ects. De nton (E cency) Precson choce s e cent f max [;) B (; ) C() The above de nton says that the precson choce s e cent f t allows nvestors to acheve the hghest exante utlty when they coordnate ther actons n the rst stage. The d erence between the equlbrum precson and the e cent precson s that the rst one s chosen n a noncooperatve fashon, that s = ( ), whle the other one s chosen n a cooperatve fashon and hence s not necessarly a bestresponse to all other nvestors choosng. Indeed, we wll show that genercally, 6= ( ). Under assumpton (A) we show n clam A:6 n the appendx that the set of arguments that maxmzes the above expresson s nonempty and that <. The soluton to the above problem s ether a corner soluton, = 0, or t sats es the followng necessary rst order condton: B ( ; ) B ( ; ) C 0 ( ) = 0 Therefore, a necessary condton for the equlbrum precson choce to be e cent s that at, B ( ; ) = 0. The followng proposton establshes condtons under whch ths s ndeed the case. 7 See Theorem. 4
15 Proposton (Ine cency) Let s K( ( ); ; T ) ( ) (T ) p (T ) ( ) ( ). A necessary condton for the unque equlbrum to be e cent s that solves = K( ( ); ; T ).. For any par ( ; T ), such that T 6=, there exsts a unque that solves = K( ( ); ; T ) It follows that genercally the unque equlbrum s ne cent. 3. For T = the unque soluton to = K( ( ); ; T ) s =. The rst part of the above proposton provdes condtons for the followng to hold at the equlbrum precson choce : 0 = B ( ; ) C 0 ( ) = B ( ; ) B ( ; ) C 0 ( ) (0) Ths condton s necessary for e cency but not su cent for two reasons. Frst of all B (; ) C () may not be a concave functon of, hence the above condton s only necessary for to be the utlty maxmzng choce of precson. Secondly, the e cent precson level mght be 0. Ths can happen because a hgher precson level chosen by nvestors can lead to a narrowng of the regon where nvestment s successful (ncrease n ), therefore leadng to a decrease n exante utlty. In that case, snce we showed that the noncooperatve equlbrum precson s always greater than 0, t follows that the equlbrum s ne cent. The second part of the proposton states that when T 6= then for any combnaton of and T there s a unque for whch equaton (0) holds. Because the condton n the above proposton s very spec c, t follows that n general the unque equlbrum of the game s ne cent. An example where ths necessary condton s sats ed, for any, corresponds to the very partcular case where T = and =. Above we explaned why the game features spllover e ects. Formally, the spllover e ects are captured by the fact that the partal dervatve of the exante bene t functon wth respect to the precson choce of others s d erent from zero,.e. B ( () ; ) 6= 0. We now de ne our noton of global spllover e ects. De nton 3 Consder nvestor and let be the precson choce of all other nvestors.. If 8 > we have B. If 8 > we have B > 0 then we say that the game exhbts global postve spllovers < 0 then we say that the game exhbts global negatve spllovers 5
16 Global spllover e ects are of nterest because ther presence means that the equlbrum of the game s necessarly ne cent (see Cooper and John, 988). Moreover the presence of global negatve spllovers mples that nvestors overacqure nformaton whle the presence of global postve spllovers means that nvestors underacqure nformaton compared to the e cent nformaton choce. The result below provdes su cent condtons for the exstence of global spllover e ects. Lemma 3 Consder the nformaton acquston game.. If the cost of nvestment s hgh ( T > ) and nvestors are exante optmstc ( < ) then the game exhbts global postve spllovers. If the cost of nvestment s low ( T < ) and nvestors are exante pessmstc ( > ) then the game exhbts global negatve spllovers 3. If cost of nvestment s neutral (T = ) and nvestors are nether optmstc nor pessmstc ( = ) then the game exhbts no spllover e ects. The above lemma provdes condtons under whch nvestors wll overacqure nformaton (part ) or underacqure nformaton (part ). Under these condtons the unque equlbrum of the game s ne cent. On the other hand, part 3 of the above lemma, provdes an example where the unque equlbrum s e cent. Whle the above result may seem contradctory to the proposton above t s actually complementary. In partcular one can show that when the cost of nvestment s hgh ( T > ) then the unque that solves = K( ( ); ; T ) s larger than. Smlarly, when T < the unque that solves = K( ( ); ; T ) s smaller than. 5 Strategc Complementartes n Informaton Acquston We now nvestgate whether the strategc complementartes present n the coordnaton game translate nto strategc complementartes n nformaton acquston. 8 In the context of beauty contest games, Hellwg and Veldkamp (009) have shown that n ths s ndeed the case. It turns out that n our model ths s not always the case. We wll provde condtons under whch strategc complementartes n nformaton acquston arse and we provde an example of a stuaton where they do not. There s a set of pars of underlyng parameters of the game ( ; T ) for whch strategc complementartes n actons ndeed translate nto strategc complementartes n nformaton acquston. For other pars of ( ; T ) we mght have regons where nvestor s ncentves to acqure a hgher level of precson decrease n the precson choce of hs opponents, resultng n a nonmonotone bestresponse functon. Recall from secton (3::) that the bene t of ncreasng the precson of prvate sgnals s equal to the reducton n the expected cost of mstakes. From equaton (9), t follows that 8 As shown by Vves (005), global games are games of strategc complementartes. 6
17 the game has strategc complementartes n nformaton choces f M( ;) < 0. One can show that: M ( ; ) = M (x ) (x ) () where M s a negatve constant, x s nvestor s threshold and x s the common threshold of all other nvestors. Equaton () mples that whenever (x ) and (x ) are of the same sgn precson choces are strategc complements. In general, nothng guarantees that (x ) and (x ) are both postve or negatve. However, when nvestors are exante optmstc (hgh mean of the pror) and the nvestment cost s low (small T ), or when nvestors are exante pessmstc and the nvestment cost s hgh, we have strategc complementartes n nformaton choces. Proposton 3 Suppose that one of the followng holds:. The cost of nvestment s hgh (T > ) and the nvestors are exante pessmstc ( < ). The cost of nvestment s low (T < ) and the nvestors are exante optmstc ( > ) Then there are strategc complementartes n the nformaton acquston game. 3. There exst pars of (T; ) such that there s lack of strategc complementartes. The fact that strategc complementartes do not arse for all combnatons of parameters s due to the fact that when makng hs choce of precson, an nvestor cares about the reducton n the expected cost of mstakes that hs precson choce wll lead to. For some parameters, the rate at whch the ncrease n precson decreases the total expected cost of mstakes s a decreasng functon of the other nvestors precson choces. Snce nformaton s costly, an ncrease n the precson of all other nvestors wll lead to a lower precson choce by nvestor, n whch case, there are no strategc complementartes n precson choces. In partcular, n the Appendx we show that we can nd pars of (T; ) for whch equaton () s postve, mplyng that the rate at whch the expected cost of mstakes decreases due to an ncrease n the nvestor s own precson s decreasng n the precson choce of others. Ths mples that there exst regons for whch a hgher precson acqured by other nvestors decreases the ncentves to acqure more precse nformaton by a sngle nvestor, makng precson choces strategc substtutes over these regons. Fnally, we provde a numercal example that supports ths ntuton. We set = 0, T = 0:, = 3 and we consder precson choces n the nterval [3; 0]. Fgure depcts the best response functon obtaned from smulatons performed wth these parameters. We see that the best response functon s nonmonotonc, con rmng the lack of global strategc complementartes. 7
18 Fgure : BestResponse Functon 6 Transparency and Welfare In recent years, the e ect of transparency on welfare has attracted a lot of attenton (see Morrs and Shn, 00, and the lterature that followed ther paper). Ths motvates us to study notons of transparency and welfare n the context of our model. Gong back to our example n the ntroducton, consder a government that tres to encourage foregn drect nvestment. We nterpret pror belefs as publc nformaton released by a governmental agency and an ncrease n government transparency as an ncrease n the precson of the pror,. We nvestgate the mpact that an ncrease n transparency has on the ncentves to acqure prvate nformaton, on coordnaton n the nvestment game, and on exante socal welfare. We rst focus on the nteracton between publc and prvate nformaton and provde condtons under whch they are substtutes. We also provde ntuton va an example of a stuaton where publc and prvate nformaton are complements. We then turn our attenton to the e ect that an ncrease n transparency has on coordnaton among nvestors. Fnally, we nvestgate welfare mplcatons of a change n the nformatveness of publc nformaton. Changes n the precson of publc nformaton determne the outcome of the game through a drect and an ndrect e ect. Frst, holdng ndvdual precson choces constant, t a ects the probablty of a successful nvestment n the second stage through ts e ect on (drect e ect). Second, t a ects the optmal choce of precson of prvate sgnals n the rst stage, whch wll have a further e ect on (ndrect e ect). 6. Tradeo between Publc and Prvate Informaton Publc nformaton s d erent from prvate nformaton. More precse prvate sgnals ncrease the amount of nformaton that an nvestor has n hand n the second stage of the game, allowng hm to make a better decson and avod costly mstakes (see secton 3::). On the other hand, more precse publc sgnals not only ncrease the amount of nformaton avalable to all nvestors when makng ther nvestment decson ( nformaton e ect ) but also change the probablty of successful coordnaton on nvestment through ts mpact on ( coordnaton e ect ). Whle the rst e ect always leads to substtuton between prvate and publc nformaton, ths addtonal coordnaton e ect can make publc and prvate 8
19 τ * (τ) τ * (τ) nformaton complmentary. Proposton 4 Suppose that one of the followng holds:. The cost of nvestment s hgh (T > ) and nvestors are exante very pessmstc ( < p ) or,. The cost of nvestment s low (T < ) and nvestors are exante very optmstc ( > p ) Then prvate and publc nformaton are substtutes. The ntuton for the above result s smple. Fx for all nvestors at some arbtrary level. At the extreme values of and T stated n the above proposton, nvestors correctly assgn a small probablty to the realzatons of for whch they mscoordnate. Because of ths, an ncrease n the precson of publc nformaton has a small mpact on, leadng to a small coordnaton e ect. On the other hand, because nvestors are coordnatng heavly on one of the two actons (nvestng when s hgh and T s low, or not nvestng when s low and T s hgh), further ncreases n the precson of any type of nformaton are of lttle value, compared to cases wth less extreme values of and T. Therefore, any ncrease n the precson of publc nformaton has a strong negatve e ect on the choce of precson of prvate nformaton, leadng to a strong nformaton e ect. Ths mples strategc substtutablty between prvate and publc nformaton. Interestngly, t s also possble for prvate and publc nformaton to be complements. Ths arses when the coordnaton e ect domnates the nformaton e ect. In partcular, ths can happen when both and T are ether hgh or low. 7 Best Response Functons µ=; T=0.3 6 Best Response Functons µ=0; T= τ 45 τ θ = τ θ =.5 τ θ = τ 45 τ θ = τ θ =.5 τ θ =.5 Fgure 3: E ect of a change n on the bestresponse functon Fgure 3 summarzes ths dscusson by showng on the left panel a stuaton of strategc substtutabltes, and on the rght panel one of strategc complementartes. In both panels 9
20 of gure 3 we plot the e ect of an ncrease n the precson of publc nformaton on the best response functons (; ). The left panel of gure 3 shows that a hgher precson of the pror leads to a downward shft of the best response functon (substtutabltes). On the other hand, the rght panel depcts how a hgher shfts the best response functon upwards (complementartes). 6. E ects of Increasng Publc Informaton on Coordnaton In the prevous subsecton we analyzed the relatonshp between an ncrease n and nvestors behavor n the rst stage of the game,.e. the mpact of on. In ths subsecton we rst analyze the e ect of an ncrease n on, keepng xed (the drect e ect). Then we compute the total e ect of an ncrease n the precson of publc nformaton on the probablty of a successful nvestment (ths encompasses both the drect and ndrect e ects). The next result shows that, dependng on parameters of the model, an ncrease n has d erent e ects on. 9 Lemma 4 Assume that the condton for a unque equlbrum s sats ed. Then:. If the cost of nvestment s hgh (T ) and nvestors are exante pessmstc ( ), then an ncrease n the precson of the pror leads to a decrease n the probablty of a successful nvestment, holdng constant ( 0).. If the cost of nvestment s low (T ) and nvestors are exante optmstc ( ), then an ncrease n the precson of the pror leads to an ncrease n the probablty of a successful nvestment, holdng constant ( 0). These condtons hold wth strct nequaltes f ether or T are d erent from. Combnng lemma 3 wth proposton 4 we have the followng result on the probablty of a successful nvestment: Proposton 5 Suppose that ncreases.. If T < and > ncreases ( #).. If T > and < decreases ( ") then the exante probablty of a successful nvestment then the exante probablty of a successful nvestment Consder the rst case n whch T < and > p. We establshed that an ncrease n the precson of the publc sgnal for these values of parameters leads to a decrease n, holdng xed. Moreover, for ths set of parameters publc and prvate nformaton are substtutes, so an ncrease n decreases, whch further decreases. Hence the e ect on 9 Ths result s a drect corollary of a more general result establshed n lemma A:3 n the appendx. 0
21 s unambguously negatve. Snce the probablty of a successful nvestment s smply the probablty that, we conclude that n ths case a hgher precson of publc nformaton leads to an ncrease n. The opposte s true when T > and < p. In that case both the drect and ndrect e ects of an ncrease n lead to an ncrease and hence the exante probablty of a successful nvestment decreases. 6.3 Welfare Consequences of a Hgher Snce all nvestors are exante dentcal and play a symmetrc equlbrum, t s enough to analyze the exante utlty of a sngle nvestor to determne welfare consequences of an ncrease n the precson of. Gven that the nvestor plays a symmetrc equlbrum wth precson choce ; hs exante utlty s gven by U ( ; ) = Z Z x ( ) Z Z x ( ) T df (xj) dg () ( T ) df (xj) dg () Z ( T ) dg () C ( ) The total mpact of a change n the precson of publc nformaton s gven by: d d U ( ; ) = Z Z Z x T d d (f (x j) g ()) dxd ( T ) d d (g ()) d Z Z x d d g ( ) Pr (x x j ) ( T ) d (f (xj) g d ()) dxd C 0 () d () d Equaton () mples that the e ect of a change n on welfare depends on four factors: () the change n the exante expected cost of Type I mstake ( rst term), () the change n the exante expected cost of Type II mstake (second term), (3) the change n the probablty of a successful nvestment (thrd term), and (4) the change n the cost of the equlbrum precson choce (fourth term). Recall that n secton 6: when analyzng ncentves of nvestors to acqure prvate nformaton we ntroduced the drect and ndrect e ects of an ncrease n on the outcome of the game, and we further decomposed the ndrect e ect nto a coordnaton e ect and an nformaton e ect. We can see from equaton () that the same e ects play a role when analyzng the total e ect of a change n the precson of publc nformaton on the exante utlty functon. Because of the complexty of our model we are not able to determne analytcally the total welfare e ect of an ncrease n transparency Hence, n the next secton, we focus on numercal exercses Numercal Results In ths secton we evaluate numercally the e ect of an ncrease n on the exante utlty for d erent parameter values. For the purposes of ths exercse we assume that 0 = 3,
22 Ex Ante Net Utlty Cost of Mstake Ex Ante Net Utlty Cost of Mstake C ( ) = = ( 00 = 0 ) 0, and [; ]. 0 We nvestgate sx d erent cases. The rst case s ( ; T ) = ( 0:5; 0:6) and corresponds to the stuaton where an ncrease n leads to a reducton n the probablty of a successful nvestment. The second case s ( ; T ) = (:5; 0:4) where an ncrease n leads to an ncrease n the probablty of a successful nvestment. The other four cases consder stuatons where ether both and T are greater than or both are smaller than. Fgure 4 below depcts the rst two cases. In each panel we observe the exante net utlty as well as the expected cost of mstakes as changes. In both cases the cost of mstakes decreases wth the precson of the publc sgnal. 8 x Cost of Mstakes Cost of Mstakes Ex Ante Net Utlty Ex Ante Net Utlty τ θ τ θ Case Case Fgure 4 Consder the rst case. By proposton 4, for ths set of parameters prvate and publc nformaton are strategc substtutes, that s, an ncrease n wll lead to a lower equlbrum precson. Whle n general one cannot predct the e ect of these changes n precsons on the total expected cost of mstakes, we see n the left panel of gure 4 that for the parameters chosen the cost of mstakes decreases due to a strong nformaton e ect of an ncrease n the precson of publc nformaton. At the same tme, from proposton 5, we know that a hgher leads to a negatve coordnaton e ect,.e. as the publc sgnal becomes more precse, the probablty of successful nvestment decreases, va an ncrease n. Ths strong negatve coordnaton e ect outweghs the bene t of the decrease n the total expected cost of mstakes and of the reducton of the cost assocated to a lower equlbrum precson. Therefore, the total e ect of an ncrease n the precson of publc nformaton leads to a decrease n the exante utlty. In contrast, n the second case (rght panel of gure 4), we see that an ncrease n the precson of the publc sgnal leads to an ncrease n welfare even f, by proposton 4, we know that prvate and publc nformaton are strategc substtutes. Ths s due to the strong 0 The cost functon s chosen n such a way that t sats es assumpton (A) and ensures that agents wll choose to acqure a hgh precson of prvate sgnals.
23 Ex Ante Net Utlty Cost of Mstake Ex Ante Net Utlty Cost of Mstake Ex Ante Net Utlty Cost of Mstake Ex Ante Net Utlty Cost of Mstake postve coordnaton e ect that arses from the ncrease n the probablty of successful nvestment, va a decrease n (see proposton 5). The above numercal examples consdered economes n whch nvestors are exante pessmstc (low ) and face a hgh cost of nvestment (hgh T ), or the opposte, exante optmstc (hgh ) facng a low cost of nvestment (low T ). The next four examples consder other nterestng cases. The rst two cases depct stuatons wth low but also a low cost of nvestment T ( = 0:4 and T f0:; 0:4g), whle the other two cases consder a hgh but also a hgh T ( = 0:8 and T f0:6; 0:8g) E x A nte Net Utlty 0.66 E x A nte Net Utlty Cost of Mstakes Cost of Mstakes τ θ τ θ 0.6 = 0:4 and T = 0: = 0:4 and T = 0: Cost of Mstakes E x A nte Net Utlty E x A nte Net Utlty Cost of Mstakes τ θ = 0:8 and T = 0:6 = 0:8 and T = 0:8 Fgure 5 Note that f the cost of nvestment s su cently hgh then more precse publc nformaton has a detrmental e ect on exante utlty. Ths s due to the fact that publc nformaton allows nvestors to coordnate better n ther actons  n the case of a hgh cost of nvestment nvestors coordnate on not nvestng. The opposte s true when the cost of nvestment s low wth respect to the mean of the pror. Fgure (5) depcts these four cases. From the above examples we conclude that publc nformaton s welfare mprovng f the cost of nvestment s su cently small compared to the expected value of fundamentals. 7 Related Lterature Our work s related to three strands of lterature: global games, nformaton acquston and transparency. Global games were ntroduced by Carlsson and van Damme (993) n ther 3 τ θ
24 semnal work as an equlbrum re nement and further extended by Frankel et al (003). Ths technque was rst appled by Morrs and Shn (998) to the context of currency crses and snce then t has been extensvely used to model economc phenomena featurng coordnaton problems (see Dasgupta, 007, Edmond, 007, Goldsten and Pauzner, 005, or Morrs and Shn, 004, among others). Whle the orgnal global games models were statc, that s, they featured only oneshot coordnaton games, several authors extended these models to multstage games (Angeletos et al, 007, and Dasgupta, 007, among others). We contrbute to ths lterature by consderng a model n whch nvestors have the choce to acqure more precse nformaton before playng the standard oneshot global game. Unlke these papers, n our model nvestors make choces n the rst perod that n uence the structure of the game they play n the second perod, whereas n the above papers nvestors play repeatedly a statc global game. In ths respect our work s most closely related to Angeletos and Wernng (005) and Chassang (008). However, none of these studes consders costly nformaton acquston and ts mpact on the coordnaton game. Fnally, Szkup and Trevno (0) consder a dscrete verson of our model and test ts predctons expermentally. Our model s also related to the lterature on the role of nformaton n beauty contest models. In ths type of games, nvestors payo s depend on how closely ther acton s to the average acton taken by others and to the unknown state. In the context of ncomplete nformaton games wth prvate and publc sgnals, these models were rst analyzed by Morrs and Shn (00), who study the tradeo s between publc and prvate nformaton. They show that publc nformaton may be detrmental to welfare f nvestors have access to very precse prvate nformaton. In contrast, n a global games setup, we nd that whether publc nformaton s welfare enhancng or reducng depends on the exante optmsm or pessmsm of nvestors and on the cost of nvestment (see secton 5:3). In a beauty contest game, Tong (007) analyzes the mpact of an ncrease n the dsclosure of publc nformaton on the ncentves to acqure prvate nformaton. He shows that an ncrease n the precson of publc nformaton always decreases nvestors ncentves to acqure prvate nformaton and leads to a lower precson of prvate nformaton n equlbrum. In contrast, we nd that n our model publc and prvate nformaton can be complements (see secton 5:). Ths follows from the fact that n the beauty contest models there s no clear dstncton between the nformaton and coordnaton e ects. More precsely, due to the contnuous nature of the beauty contest model, nvestors do not make mstakes (there s no wrong acton). Fnally, Hellwg and Veldkamp (009) show that n a beauty contest model complementartes n actons translate nto complementartes n nformaton acquston. Whle ths s true for a wde range of parameters n our model, we provde a numercal example n whch ths result does not hold (see secton 5). Hence, our work emphaszes mportant d erences between global games and beauty contest models. Fnally, the noton of transparency n the context of speculatve attack models has been addressed n the global games lterature by Henemann and Illng (00) and Banner and Henemann (005). There are several key d erences between these studes and our work. Frst of all, ther noton of transparency s very d erent from ours. They nterpret an 4
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