Noisy signalling over time

Transcription

1 Noisy signalling ove time Sande Heinsalu Januay 8, 2014 Abstact This pape examines signalling when the sende exets effot and eceives benefit ove time. Receives only obseve a noisy public signal about effot, which has no intinsic value. Time intoduces novel featues to signalling. In some equilibia, a sende with a highe cost of effot exets stictly moe effot than his low-cost countepat. Noise leads to obust pedictions: pooling on no effot is always an equilibium, while pooling on positive effot cannot occu. Wheneve pooling is not the unique equilibium, infomative equilibia with a simple stuctue ae shown to exist. Keywods: Dynamic games, signalling, incomplete infomation, infomation economics. JEL classification: D82, D83, C73. sande.heinsalu@yale.edu, website: Depatment of Economics, Yale Univesity. 28 Hillhouse Ave., New Haven, CT 06511, USA. The autho is geatly indebted to Johannes Höne fo many enlightening discussions about this eseach. Numeous convesations with Lay Samuelson have helped impove the pape. The autho is gateful to Eduado Faingold, Dik Begemann, Juuso Välimäki, Philipp Stack, Willemien Kets, Tadashi Sekiguchi, Flavio Toxvaed, Fançoise Foges, Jack Steche, Daniel Baon, Benjamin Golub, Floian Edee, Vijay Kishna, Sambuddha Ghosh, Segiu Hat, Idione Meneghel, Jögen Weibull, Alessando Bonatti, Zvika Neeman, Scott Komines, Andzej Skzypacz and paticipants of the 24th Intenational Confeence on Game Theoy at Stony Book Univesity, EconCon 2013, the Yale mico theoy beakfast and lunch, the Yale Gaduate Summe Wokshop and the NYU Mico Student Lunch fo comments and suggestions. Any emaining eos ae the autho s. 1

2 1 Intoduction Most signalling situations involve noisy obsevation of the sende s effot. In many, effot is exeted ove time. Fo example, a politician may be a (elatively) honest o a coupt type, and can signal honesty by following the law to the lette (paying taxes in full, efaining fom speeding and bibetaking). The cost of abiding by the law is incued at all times. Votes lean of low effot only when some andom events, such as scandals occu. Novel dynamics appea. Fo example, thee exist equilibia in which the high-cost type exets stictly moe effot initially than the low-cost type. Intuition fom the pevious liteatue may thus be misleading. Despite the ichness, some esults ae obust acoss noise stuctues. Pooling on zeo effot is always an equilibium, while pooling on positive effot cannot occu. Thee ae always pio beliefs at which pooling is the unique equilibium. When pooling is not unique, simple infomative equilibia exist. The ich dynamics ae due to the multiple oppotunities fo exeting effot, while the obust featues ae diven by the noise. The envionment is the natual adaptation of Spence (1973). The playes ae a sende and a competitive maket of eceives. The sende is eithe a high-cost o a low-cost type. Type is pivate infomation. Receives shae a common pio belief about the type. The sende continuously chooses his effot level. Receives obseve a noisy public signal about the effot, athe than the effot itself. The signal is modelled eithe as a Poisson o a Bownian pocess. Effot eithe inceases the intensity of the Poisson pocess (this is called the good news case), o deceases it (bad news). The types only diffe in thei flow cost of effot. The sende deives a flow benefit diectly fom the posteio belief of the eceives. Attention is esticted to Makov stationay equilibia. Some featues ae common to the Poisson and Bownian models. Pooling on no effot is an equilibium fo geneal noise stuctues. A sufficient condition is that wheneve the eceives expect the sende s effot to be typeindependent, beliefs stay constant egadless of signals. If beliefs do not espond to signals, then effot povides no benefit to the sende. This makes both types switch to zeo effot to minimize cost. The peceding easoning can also be used to ule out pooling on positive effot. Pooling is the unique equilibium when the benefit of signalling is too small to incentivize the low-cost type to signal. The cost of signalling is the same fo all pio beliefs, while the benefit depends on the diffeence 2

3 between the posteio beliefs afte diffeent signals. This diffeence is smalle fo exteme beliefs, fo a given impefectly evealing signal stuctue. Thee ae always pio beliefs high o low enough to make the benefit smalle than the cost. Wheneve pooling is not the unique equilibium, infomative equilibia with a simple stuctue exist. The high-cost sende neve exets effot. The low-cost sende initially exets maximum effot, switching to zeo effot when the belief becomes high o low enough. When the sende s benefit is concave in the eceives belief, the high-cost sende stictly pefes pooling to any infomative equilibium. This is because in an infomative equilibium, the posteio belief has positive vaiance, and the high-cost type expects this belief to become less favouable on aveage ove time. Each of the thee models has unique featues. In the bad news model, the high-cost sende exets moe effot than the low-cost in some equilibia. To descibe such equilibia, ecall the example of a politician who can exet effot to obey the law. Lawful behaviou deceases the fequency of scandals. The equilibia with highe effot by the coupt type display fou egimes, efeed to as ealy caee, inside, scutiny and tainted. Play stats in the ealy caee, duing which the coupt type exets positive effot and the honest type no effot. If no scandal occus by a given time, then the politician becomes an inside, which means that the votes ignoe scandals and the politician no longe exets effot. If instead a scandal occus in the ealy caee, then scutiny esults. Unde scutiny, the honest type exets maximum effot and the coupt type none. Unde scutiny, a scandal leads to a tainted eputation: votes ae cetain that the politician is coupt and the politician exets no effot. In the good news and Bownian models, the high-cost sende exets no moe effot (at any belief) than the low-cost in all equilibia. The good news model most closely esembles Spence (1973). Fo example, the high-cost sende pefes pooling to all othe equilibia. This is tue even when the benefit is convex in the eceives belief (so that leaning has positive value). The eason is that when the equilibium effot of the high-cost type is less than the maximum, then by the lineaity of the cost and signal stuctue, the high-cost type is indiffeent between equilibium effot and no effot. No effot means no signals. By indiffeence, the equilibium payoff of the highcost type must be the same as in the absence of signals. In the absence of signals, belief difts down unde good news when the high-cost sende 3

4 is expected to exet less effot than the low-cost. Flow benefit inceases in belief, so a downwad dift in belief lowes the payoff to the sende. In the Bownian model, belief is bounded away fom cetainty in all equilibia, unlike in the Poisson models. Thee need not exist a most infomative equilibium fo a given pio. Moe pecisely, fo a given equilibium, define the signalling egion as the set of beliefs at which the low-cost type of the sende exets maximal effot and the high-cost type minimal and outside which neithe type exets effot. Two signalling egions coesponding to equilibia at the same paametes might intesect without thei union (o any set of beliefs containing the union) being an equilibium signalling egion. The two signalling egions cannot be anked by infomativeness. Many authos have mentioned the elevance of time (Weiss, 1983; Admati and Pey, 1987) and noise (Matthews and Miman, 1983) in signalling contexts. Continuous time signalling with Bownian noise is consideed in Daley and Geen (2012b), Gyglewicz (2009) and Dilme (2012). In all thee, the benefit of signalling is eceived in a lump sum when the sende decides to stop the game. In Daley and Geen (2012b), the signal pocess is exogenous. In Gyglewicz (2009), one type of the sende is a commitment type. In the pesent pape, the benefit is a flow, both types ae stategic and effot contols the signal pocess. The esults fo epeated noiseless signalling games of Kaya (2009) and Roddie (2012) esemble those of Spence (1973) and diffe fom the cuent pape in that pooling on positive effot is always an equilibium and infomative equilibia always exist. An oveview of Kaya (2009) is given in the online appendix. Moe distantly elated epeated noiseless models ae Nöldeke and van Damme (1990) and Swinkels (1999), in which the sende eceives the benefit upon stopping signalling. One-shot noisy signalling in a limit picing context is studied in Matthews and Miman (1983). Calsson and Dasgupta (1997) select equilibia in noiseless one-shot games that ae the limits of equilibia in noisy games as noise vanishes. They assume the eceive only has two actions. Daley and Geen (2012a) look at pefectly obsevable effot togethe with a noisy exogenous signal about the type in a one-shot model. A moe detailed discussion of the liteatue is defeed to Section

5 2 Poisson signalling This section tuns to the main model whee effot changes the intensity of a Poisson signal pocess. Both the good news and the bad news cases ae consideed, but fist the setup of the model is descibed. Time is continuous and the hoizon is infinite. Thee is a stategic sende and a competitive maket of eceives. The sende has two types, H and L, with initial log likelihood atio 1 l 0 R that is common knowledge. The sende knows his type, the eceives do not. A geneic log likelihood atio l is an element of R = R {, }. The log likelihood atio coesponding to P(H) = 1 is l = and coesponding to P(H) = 0 is l =. The sende has action set [0, 1] (endowed with the natual Boel σ- algeba) at each instant of time. The action 0 is intepeted as no effot of signalling and the action 1 as maximal effot. Effot e costs type θ sende c θ e, with c L > c H > 0. Effot benefits the sende via its effect on the signal pocess, which dives the maket s log likelihood atio l. This in tun detemines the flow payoff. Befoe descibing this benefit, the signal pocess, stategies and maket expectations must be defined. The signal is binay, with values in {0, 1}. The signal 1 occus at a Poisson ate, and in its absence the signal is 0. In the good news (beakthough) case, the ate of signal 1 is e t at time t. The paamete (0, ) is intepeted as the infomativeness of effot and e t denotes the effot at t. The intensity inceases in the sende s effot, so the occuence of the signal is good news about the sende. In the bad news (beakdown) case, the ate of 1 is (1 e t ), which deceases in effot. Note that zeo effot in the good news case o maximal effot e = 1 in the bad news case ensues no signals occu. Given that the values of the signal ae fixed, a ealization of the signal pocess is descibed by the times at which 1 occus. The eceives obseve the public signals, but not the sende s effot. Since the signal is public, the l it leads to is common to all eceives. A signal sequence is a sequence (τ k ) k=1 of signal times satisfying 0 = τ 0 < τ 1 < τ 2 < and having no accumulation points. The set of signal sequences H is endowed with the σ-algeba geneated by cylindes. An n-signal public histoy is a finite sequence (τ 1,..., τ n, t) satisfying τ 1 < < τ n < t, with 1 Thoughout this pape, log likelihood atio l is used instead of belief P(H) = exp(l) 1+exp(l), as fomulas simplify significantly in the dynamic models to follow. Thee is a one-to-one map fom log likelihood atio to belief, so all esults can be stated in tems of beliefs. 5

6 t (0, ). The set of n-signal histoies is H n. It inheits the σ-algeba fom H. The set of nonteminal public histoies is H = n N {0} H n. H inheits a σ-algeba fom {H n } and is a Boel space (Yushkevich, 1980). The teminal public histoies ae the signal sequences. The tuncation of a histoy h = (τ 1,..., τ n, t) to time s t is h s = (τ 1,..., τ m, s), with τ m < s. The notation τ k h means that unde histoy h, a signal occus at time τ k. A pue public stategy is a pai of measuable maps e = (e H, e L ) fom H into the action set [0, 1]. Thoughout the pape, only public stategies ae consideed. In this section, the focus is on pue stategies. Denote the stategy the maket expects by e = (e H, e L ). This notation is also used fo equilibium stategies. Some sets of histoies that ae used in defining the updating ule fo the log likelihood atio ae defined next. These consist of histoies in which a signal occus at a time at which the stategy expected fom the sende is such that the signal is pefectly infomative about the type. In the good news case, H g max(e ) = {h H : τ k h, e H(h τk ) > 0, e L(h τk ) = 0}, H g min (e ) = {h H : τ k h, e H(h τk ) = 0, e L(h τk ) > 0}, and in the bad news case, H b max(e ) = {h H : τ k h, e H(h τk ) < 1, e L(h τk ) = 1}, H b min(e ) = {h H : τ k h, e H(h τk ) = 1, e L(h τk ) < 1}. The updating of the maket s log likelihood atio is descibed next, stating with the esponse to signal 1. If the stategy the maket expects is e, signal 1 occus at time t, and at t the log likelihood atio is l t, then in the good news case the log likelihood atio jumps to j g (l t ) = { lt + ln ( ) e H (h t) e L (ht) (with 0 0 = 1) if h t / H g max(e ) H g min (e ), l t if h t H g max(e ) H g min (e ). In the bad news case the log likelihood atio jumps to j b (l t ) = { lt + ln ( ) 1 e H (h t) 1 e L (ht) (with 0 0 = 1) if h t / H b max(e ) H b min(e ), l t if h t H b max(e ) H b min(e ). 6

7 Two efinements ae built into the j(l) fomulas. Fist, if the effots expected fom the types at the time a signal occus ae such that the signal ate is zeo, then by the 0 = 1 assumption, the log likelihood atio does not espond 0 to the signal. Second, signals ae ignoed when thee is cetainty about the type (a pefectly infomative signal occued in the past). Definition 1. Unde good news, the log likelihood atio at t is l t = l 0 and unde bad news, it is l t = l 0 + t 0 t 0 e H(h s ) e L(h s )ds + e H(h s ) e L(h s )ds + n j g (l τk ), (1) k=1 n j b (l τk ), (2) whee (e H, e L ) is the stategy the maket expects and h = (τ 1,..., τ n, t) is the histoy up to t. Given a signal sequence, the solution to (1) is the log likelihood atio pocess (l t ) t 0 unde good news, and the solution to (2) is the pocess unde bad news. The integals in (1) and (2) ae uniquely defined, because e L, e H ae bounded and measuable in the σ-algeba of histoies, which contains singletons. Fixing a signal sequence, a histoy is detemined by its length t, so e L, e H ae measuable functions fom time to actions. A Makov stationay stategy is a public stategy measuable w..t. the log likelihood atio pocess. A Makov stationay stategy can be witten as a pai of functions (e L, e H ) : R [0, 1] 2. Subsequently only pue Makov stationay stategies ae consideed. To simplify the statements to follow, attention is esticted to e L, e H piecewise continuous 2 and at evey discontinuity, continuous fom the left o the ight. The state vaiable l is the left limit l t of the log likelihood atio (with the convention l 0 = l 0 ), so jumps ae not anticipated by a stategy. In this pape, l 0 is teated as a paamete, not a vaiable, so the stategies ae a function of l 0 in addition to l. This is to avoid discussion of log likelihood 2 A piecewise continuous stategy is undestood to have at most finitely many discontinuities on R. k=1 7

8 atios uneached in equilibium afte any deviation. Define the eachable set of log likelihood atios L(e ) = { l R : h H t R + s.t. l t = l }, whee l t is given in Def. 1. The eachable set depends on the stategy the maket expects and on l 0. Only behaviou on the eachable set is discussed subsequently. Since no deviation can take l outside the eachable set, behaviou thee can be abitay. Some l L(e ) can only be eached by deviating, not by following the equilibium stategy. Fo example, L cannot each l = in the good news case by following e L (l) = 0 l L(e ) and H cannot each l = in the bad news case by following e H (l) = 1 in the inteio of the eachable set. To specify behaviou in these cases, assume e L ( ) = 0 and e H ( ) = 0. Now that stategies and the log likelihood atio pocess have been descibed, the sende s payoff can be defined. The sende is assumed to deive flow benefit β(l) diectly fom the maket s log likelihood atio l. 3 The sende s flow utility fom effot e and the maket s log likelihood atio l is β(l) c θ e, whee β is assumed stictly inceasing, bounded and continuously diffeentiable. Denote the flow benefit fom l = (coesponding to P(H) = 1) by β max and fom l = by β min. Given the stategy e = (e L, e H ) the maket expects, the payoff of type θ fom the effot function e θ ( ) and the log likelihood atio pocess (l t ) t 0 is the expected discounted sum of flow payoffs J e θ l 0 (e ) = E e θ [ 0 exp( t)[β(l t ) c θ e θ (l t )]dt l t=0 = l 0 ], (3) whee the expectation is ove the stochastic pocess (l t ) t 0, given e θ. The discount ate is > 0. Except fo jumps, l evolves deteministically given the maket expectations (e L, e H ). The jumps occu at Poisson times. Given l at the time of a jump, the size of the jump is deteministic. The expectation in (3) is thus ove the jump times of the Poisson signal pocess induced by e θ ( ). 3 This can be micofounded by assuming that each eceive has a unique one-shot best esponse a (l) to each log likelihood atio l R. Since each eceive is infinitesimal, thei cuent action does not influence the futue, so in any equilibium each eceive must play the one-shot best esponse. The sende is then assumed to deive flow benefit β a (a (l)) fom the eceives action a (l). 8

9 Since l 0 is a paamete, the payoff stating at l 0 need not in geneal equal the continuation value fom l 0 on when stating at some ˆl 0 l 0. Howeve, if the stategy the maket expects is Makov stationay, then evey time l is eached, the continuation value of type θ fom l is well defined and is denoted V θ (l). 4 Lemma 1. V H (l) V L (l) l 0 R e l L(e ), with stict inequality if unde the optimal (e L, e H ) stating at l, thee is a positive pobability of eaching some ˆl β with e L (ˆl) > 0. min V θ (l) βmax, with stict inequalities if l R. All poofs omitted fom the text ae in Appendix A. Definition 2. A Makov stationay equilibium consists of a Makov stationay stategy e = (e H, e L ) of the sende and a log likelihood atio pocess (l t ) t 0 s.t. 1. given (l t ) t 0, e θ maximizes (3) ove e θ, 2. given e, (l t ) t 0 is deived fom (1) unde good news and (2) unde bad news. The definition implies that on the eachable set, behaviou is optimal fom any point on. Theefoe the equilibium concept could also be called Makov pefect. Hencefoth equilibium means a pue Makov stationay equilibium. Call an equilibium extemal when the equilibium effots only take values in {0, 1}. These ae analogous to pue equilibia, because the cost and the signal ate ae linea in effot. The extemal effots will be shown to imply that in an inteval of log likelihood atios, H exets maximal effot, and outside the inteval zeo effot, while L neve exets effot. Exeting effot initially and then potentially stopping can be intepeted as eputation building by H. A pooling equilibium is an equilibium in which e L (l) = e H (l) l L(e ). Lemma 2. Assume the updating ule satisfies l L(e ) t 0 s > 0 if e H (l) = e L (l) and l t = l, then l t+s = l t. Then (a) e θ (l) = 0 is the unique best esponse to e H (l) = e L (l) fo θ = H, L, (b) the pooling equilibium with e L (l 0) = e H (l 0) = 0 exists fo any l 0 R, 4 The dependence of V θ (l) on e and l 0 is suppessed in the notation. 9

10 (c) in any equilibium fo any l L(e ), e L (l) = e H (l) > 0 cannot occu. Poof. If e H (l) = e L (l) and l t = l imply l t+s = l t, then upon eaching any ˆl R with e H (ˆl) = e L (ˆl), l emains at ˆl foeve egadless of effot, thus thee is no benefit to signalling. Signalling is costly, so zeo effot is the unique best esponse fo both types. If both types ae expected to exet no effot at l 0, then zeo effot is the unique best esponse. This poves the existence of the pooling equilibium with e L (l 0) = e H (l 0) = 0. If fo some l L(e ), the eceives expect e L (l) = e H (l) > 0, then both types choose no effot at l. This ules out e L (l) = e H (l) > 0 in equilibium. Lemma 2 coves a vaiety of noise stuctues, including the Poisson (with the efinements descibed above) and the Bownian signal stuctues of this pape. Lemma 2 shows that an extemal equilibium always exists, because pooling on zeo effot is an extemal equilibium. Pooling is the unique equilibium if the benefit of signalling is low enough elative to the cost. It is poved below (Popositions 5 and 9) that if pooling is not the unique equilibium, then thee exists a continuum of nonpooling extemal equilibia. Poposition 3. Pooling is the unique equilibium l 0 R if βmax β min c H. The conditions fo the existence of an infomative equilibium have the same intuition in the Bownian and the one-shot cases as in the Poisson model: fo some initial log likelihood atio, the benefit to signalling when the eceives expect H to signal and L not to signal has to be high enough to incentivize H to signal. Then an l 0 can be found at which the benefit of signalling is low enough fo L not to imitate H, but high enough fo H to signal. Given e, the pooling egion is defined as P(e ) = {l L(e ) : e H(l) = e L(l) = 0}. Due to the assumptions that at l = and, the log likelihood atio does not espond to signals and e L ( ) = e H ( ) = 0, we have, P(e ) e. 10

11 It is intuitive that when the sende s benefit is concave in the eceives posteio belief, 5 then L always pefes the pooling equilibium to any othe equilibium. This is because fo L in an infomative equilibium the expected posteio is lowe than the pio, and with a concave benefit, the vaiance in the posteio is not beneficial. Lemma 16 in Appendix A states this fomally. The esult holds in all the signalling models discussed in this pape. 2.1 The bad news case In the bad news model, thee exist equilibia in which fo some beliefs of the eceives the L type exets highe effot than H, despite the unifomly highe maginal cost of effot. This distinguishes the bad news case fom the pevious liteatue on signalling. The esult is eminiscent of the countesignalling of Feltovich, Habaugh, and To (2002), but the mechanism is quite diffeent. In the bad news model, it is the theat of signalling being equied in the futue that incentivizes L to signal. This theat is not as sevee fo H whose signalling cost is lowe. Some effot pattens cannot occu in equilibium. Lemma 2 uled out e L (l) = e H (l) > 0. Lemma 4 shows that e L (l) > e H (l) cannot occu unde some conditions, e.g. when the jump j(l) lands in the pooling egion. Lemma 4. In equilibium, the following cannot occu l 0 R l L(e ): (a) e L (l) > e H (l) and pooling at j(l), (b) 0 < e L (l) < e H (l) = 1. The intuition of the poof of (a) is as follows. The value afte a jump is the same fo both types, so due to V H V L, the benefit of the jump is lage (o the loss is smalle) fo L than fo H, egadless of whethe V θ (j(l)) > V θ (l) o not. The cost of avoiding the jump is lage fo L, so it cannot be that L chooses a geate effot to avoid jumps than H. Fo (b), it is poved that V L is stictly inceasing in the egion whee 0 < e L (l) < e H (l) = 1. If L is taking inteio effot and H is taking maximal effot, then the jumps go to l =. Given that V L is stictly inceasing, L is indiffeent to the jumps 5 exp(l) β(l) is linea in the posteio belief if it has the fom k 1 1+exp(l) + k 2, with k 1 > 0, because the pobability coesponding to log likelihood atio l is. The function k 1 exp(l) 1+exp(l) + k 2 is convex fo l < 0 and concave fo l > 0. exp(l) 1+exp(l) 11

12 Ealy caee no signal Inside signal no signal Scutiny signal Tainted Figue 1: The fou egimes in an equilibium in which e L > e H. at most at a single point, so cannot be taking inteio effot ove a ange of l. Lemma 4 does not ule out e L > e H occuing in equilibium, as shown in Example 1. Both j(l) > l and j(l) < l ae possible, because if e L (l) > e H (l), then j(l) > l. The effot patten e L > e H is counteintuitive, because the benefit fom a highe log likelihood atio is the same fo the types, but L has a highe maginal cost of signalling. Afte the example, the set of extemal equilibia is chaacteized. Thee exist equilibia in which e H (l) (0, 1) and e L (l) = 0 fo some eachable l. This is shown in Lemma 17 in Appendix A. These equilibia can be found numeically, but cannot be solved fo in closed fom. Example 1. Take c H = 0.1, c L = 1.14, = 1, = 2, β(l) = exp(l), l 1+exp(l) 0 = 2 ɛ 2, ɛ (0, 1). Then Poposition 18 in Appendix A povides sufficient conditions fo the existence of an equilibium in which e L (0, 1) and e H = 0 ove the inteval (2 ɛ, l 0 ]. In the inteval [2, ), the effots ae e L = 0 and e H = 1. Elsewhee, the effots ae zeo. The play in this equilibium is illustated in in Figue 1. The initial egime is ealy caee, which in the absence of a signal eventually tansitions to inside, but afte a signal tansitions to scutiny. Unde scutiny, anothe signal takes the play to the tainted egime. The incentive fo L to exet highe effot in the ealy caee egime is ceated by the payoff diffeence between the inside and scutiny egimes (the inside egime has a highe payoff). The payoffs to both types fom the inside egime ae equal, but the payoff fom scutiny is lowe fo L. The diffeence between inside and scutiny is lage fo L, so L can be incentivized to positive effot while H takes zeo effot. The incentive fo initial signalling is povided by the theat of futue expectation of signalling. In the log likelihood atio space, the fou egimes ae depicted in Figue 2. 12

13 no signal e L = e H = 0 e L > e H signal signal no signal e H = 1, e L = 0 l Figue 2: The log likelihood atios in an equilibium in which e L > e H Extemal equilibia In the class of extemal equilibia, it is w.l.o.g. to conside only the ones in which e L 0 and e H (l) = 1 if l [l 0, l) fo some l R, with e H (l) = 0 othewise. The inteval [l 0, l) is called the signalling egion. The eason why extemal equilibia must take an inteval fom is as follows. Pooling on positive effot (e H (l) = e L (l) = 1) cannot occu. Maximal effot by L cannot occu, because with e L (l) = 1 > e H (l), the jumps go to l =, which is absobing and gives the maximal payoff. This makes L deviate to zeo effot. The log likelihood atio cannot dift acoss a egion on which e L = e H = 0 and if e H (l) = 1, e L (l) = 0, then j(l) =, so thee ae no jumps into anothe egion whee e L = 0 and e H = 1. The inteval [l 0, l) is assumed open at l fo thee easons. At l =, beliefs do not espond to signals, so both types will choose e = 0. Singleton intevals ae uled out by the assumption that e L, e H ae piecewise continuous and continuous fom the left o the ight at jumps. Nonsingleton closed intevals of finite length can be eplaced with intevals open on the ight without changing the esults. Next, the value functions of the types in extemal equilibia ae calculated. Afte that, bounds on the set of signalling egions ae povided, the existence of nonpooling extemal equilibia is poved if the condition in Poposition 3 fails, and finally compaative statics ae epoted. At l, the value functions of both types ae V θ (l) = β(l). The bounday l can be infinite. In [l 0, l), the value functions ae solved fo using Hamilton- Jacobi-Bellman (HJB) equations and a veification theoem (Theoem 4.6 in Pesman, Sonin, Medova-Dempste, and Dempste (1990) as modified fo the discounted case in Yushkevich (1988)) is used to check that the solutions 13

14 coincide with the value functions. The HJB equation of type θ is { [ ] } V θ (l) = β(l) + V θ(l) βmin + max (1 e) V θ (l) c θ e. e Afte eaching l, incentives ae tivial. In the signalling egion, type θ chooses e θ = 1 if [ β min V θ (l)] c θ 0. Reaanging this, one obtains the incentive constaints (ICs) c H + β min V H (l), c L + β min V L (l), (4) which must hold fo evey l in the signalling egion. These estict the set of possible signalling egions and must be checked afte solving fo the candidate value functions. To solve fo the candidate value functions, substitute the equilibium stategies e H = 1 and e L = 0 into the HJB equations of H and L. The HJB equations become the odinay diffeential equations (ODEs) V H (l) = β(l) c H + V H (l) and V L(l) = β(l) + V L (l) + β min V L (l). In the absence of a signal, the log likelihood atio ises continuously to l. Assume l is finite (the case l = is discussed afte solving the l < case). Then value matching gives lim l l V θ (l) = V θ (l) = β(l), which povides the bounday condition fo the ODEs. The solutions of the ODEs ae ( V H (l) = exp l l ) β(l) l + l ( V L (l) = exp ( + ) l l ) β(l) + l l [ β(z) + β min ] exp β(z) c H exp ( ( + ) z l ( z l ) dz, ) dz. These ae continuously diffeentiable on (l 0, l), with a ight deivative at l 0 and a left deivative at l, so by the veification theoem in Yushkevich (1988), they coincide with the candidate value functions. The ICs must be checked to confim that the candidate value functions ae indeed the value functions. If l =, then both types get benefit β(l) foeve. In addition, L has a flow ate of jumps to β min and H pays a flow cost c H foeve. The candidate (5) 14

15 value functions ae V H (l) = V L (l) = l l β(z) c H [ β(z) + β min ( exp z l ) dz, (6) ( ( + ) z l ) dz. ] exp Fo evey ɛ > 0 thee exists ˆl R s.t. β max β(ˆl) < ɛ, which implies βmax c H V H (ˆl) < ɛ β(ˆl) and + β min V + (+) L(ˆl) < ɛ. Bounds on the set of signalling egions [l 0, l) fo which the incentive constaints (4) ae satisfied ae povided next. The maximal uppe bounday max l that an equilibium signalling egion can have must satisfy the incentive constaint lim l max l V L (l) β min + c L. The limit equals β(l) fo l finite and equals βmax + β min fo l =. The benefit to signalling is avoiding the + (+) bad signal, so the lage the diffeence between β min and V L (l), the geate the incentive of L to imitate H. Since V L is inceasing, the L type incentive constaint detemines the log likelihood atio above which signalling cannot continue. The minimal l 0 at which signalling can stat is detemined by V H, which depends on l. Denote the minimal lowe bounday given l by l 0 (l). This is finite, because is in the pooling egion by Def. 1. The incentive constaint lim l l0 (l) V H(l) β min + c H must hold with equality at l 0 (l). Since V H is stictly inceasing, if the IC fo H holds at l, then it holds at all ˆl > l. Thee exists an infomative extemal equilibium if the condition in Poposition 3 fo pooling to be the unique equilibium fails. Poposition 5. Suppose βmax β min > c H. Then l 0 R ɛ > 0 s.t. thee exists a extemal equilibium with signalling egion [l 0, l 0 + ɛ). The intuition fo Poposition 5 is that if the jumps fom to stictly incentivize H to signal, then thee is an l 0 lage enough s.t. jumps fom l 0 to do so as well. In that case thee exists a extemal equilibium with signalling egion [l 0, l 0 + ɛ). If thee exists one extemal equilibium with a nonempty signalling egion [l 0, l), then thee is a continuum of such equilibia, each coesponding to a paticula l (l 0, l]. Compaative statics ae the final item discussed in this section. Welfae is defined as W (l) = exp(l) V 1+exp(l) H(l)+ 1 V 1+exp(l) L(l), because the eceives fom a 15

16 competitive maket, so thei payoff is zeo in any equilibium. Based on (5), V L (l) and V H (l) ae infinitely diffeentiable in l,,, c H, c L fo any l L(e ), so deivatives can be used fo compaative statics. Changing l changes e and theefoe L(e ). In that case, the compaison is between payoffs at an l that is in the eachable set both befoe and afte changing l. Poposition 6. If l R, then (a) dv L(l) dl (b) dv H(l) dl > 0 iff β (l) β(l) + β min > 0, > 0 iff β (l) c H > 0, (c) dw (l) dl > 0 iff exp(l) [ β (l) c H ] + β (l) β(l) + β min > 0. Poof. The poof is by taking the appopiate deivatives. The condition fo dv L(l) dl > 0 holds when β(l) = ( 1+exp(l)) n, n N, n 2, l < ln(n 1). The effects of aising l on V L ae a highe payoff upon eaching l (the β (l) tem), but a lowe chance of eaching it (the β(l) tem) and a highe chance of jumping to l =. If β (l) β(l) + β min > 0 as l l 0, then an infomative equilibium yields a highe V L (l 0 ) than pooling. If β (l) β(l) + β min 0 fo all l > l 0, then pooling maximizes the payoff of L. This is the case if β(l) = exp(l) 1+exp(l) linea in the belief). (the benefit fom the eceives belief is If the condition fo dv H(l) > 0 holds as l l dl 0, then an infomative equilibium gives H a highe payoff than pooling. If the condition fails at all l > l 0, then pooling maximizes the payoff of H. The intepetation of the condition is that inceasing l inceases the payoff upon eaching l at a ate β (l) and inceases the time duing which the signalling cost is paid. Whethe V H inceases o deceases in l depends on which effect dominates. The effect of inceasing l on welfae is a combination of the effects on H and L, with a weight exp(l) on the payoff of H and a weight 1 on L. 2.2 The good news case The esults about the good news model ae pesented next. The L type always pefes the pooling equilibium unde good news, even when the flow benefit fom the eceives log likelihood atio is convex. This is eminiscent 16

17 of L pefeing pooling to any sepaating equilibium in noiseless models, but diffes fom the othe noisy models consideed in this pape. Some peliminay obsevations about the equilibium effots ae collected in the following lemma. Since the lemma ules out e L > e H, the log likelihood atio can only jump up: j(l) l in the good news case. Lemma 7. l 0 R l L(e ), the equilibium effots satisfy e L (l) = e H (l) = 0 o e L (l) < e H (l). Moeove, if e L (l) = 0 < e H (l), then e H (l) = 1. Some equilibia with e L (l), e H (l) (0, 1) at some l L(e ) can also be uled out, as shown in Lemma 19 in Appendix A. Resticting attention to extemal equilibia, 6 Lemma 7 implies that in equilibium the only possible effot combinations ae e H = e L = 0 and e H = 1, e L = 0. An extemal equilibium must theefoe be an inteval of log likelihood atios (l, l 0 ] on which e H = 1, e L = 0 and outside which e H = e L = 0. 7 The lowe bounday of the inteval can be infinite. The uppe bounday is finite, as shown in Lemma 8. Lemma 8. l 0 R ˆl R s.t. in any equilibium l L(e ) [ˆl, ], e H (l) = e L (l) = 0. As in the bad news case, the value functions ae calculated fist. Then bounds on the set of signalling egions ae discussed, followed by the existence of infomative extemal equilibia. Compaative statics ae deived at the end of the section. Outside (l, l 0 ], the value functions of both types ae V θ (l) = β(l). In (l, l 0], the value functions ae solved fo using the HJB equation and a veification theoem. The HJB equation fo type θ is { V θ (l) = β(l) V θ(l) + max e e [ βmax ] } V θ (l) c θ. 6 Focussing on extemal equilibia is a estiction. Thee exist equilibia whee e L (0, 1) and e H = 1, as shown in Lemma 20 in Appendix A. 7 It is w.l.o.g. to conside only one inteval, because l cannot dift acoss a egion whee e H = e L = 0 and if e H = 1, e L = 0, then jumps ae to l =. The inteval is open at l, because e H, e L ae piecewise continuous and continuous fom the left o the ight at jumps. This ules out singleton intevals. The inteval [, l 0 ] cannot be an equilibium, because beliefs do not espond to signals at l =, so both types will choose zeo effot thee. 17

18 In the pooling egion, incentives ae tivial. At evey l in the signalling egion, the incentive constaints [ ] [ ] βmax βmax V H (l) c H 0, V L (l) c L 0 must be satisfied in ode fo H to choose e H (l) = 1 and L to choose e L (l) = 0. These incentive constaints estict the set of possible signalling egions and must be checked afte solving fo the candidate value functions. The constaints have a simple intepetation: the maginal benefit of an incease in effot is the inceased pobability of jumping to l = and getting β max foeve instead of the cuent value V θ (l). The pobability inceases with effot at ate. The maginal cost of effot is c θ. If maginal cost minus maginal benefit is positive, type θ chooses e = 1, othewise e = 0. To solve fo the candidate value functions, substitute the equilibium stategies e H = 1 and e L = 0 into the HJB equations of H and L. The HJB equations become the ODEs V H (l) + ( + )V H(l) = β(l) + βmax c H and V L (l) + V L(l) = β(l). In the absence of a signal, the log likelihood atio falls continuously to l. Fo l >, the value matching condition lim l l V θ (l) = V θ (l) = β(l) holds, because close to l, eaching it is likely and a jump to anothe value unlikely. The limit gives the bounday condition V θ (l) = β(l) fo the ODEs. The case whee l = is discussed afte solving the l > case. The solutions of the ODEs ae ( V H (l) = exp ( + ) l l ) β(l) l [ β(z) ch + + β ] max exp l ( V L (l) = exp l l ) β(l) l + l ( ( + ) l z β(z) ( exp l z ) dz, (7) ) dz. The solutions to the HJB equations of the types ae continuously diffeentiable on (l, l 0 ), with a ight deivative at l and a left deivative at l 0, so by the veification theoem in Yushkevich (1988), the solutions ae the candidate value functions. If the ICs ae satisfied in (l, l 0 ], then V H, V L ae the value functions. Next, the signalling egions with l = ae discussed. Type L has no chance of a jump and gets the discounted flow benefit foeve, so L s 18

19 candidate value is V L (l) = l β(z) ( exp l z ) dz. Since β is bounded, fo any ɛ > 0 thee exists ˆl R s.t. β(ˆl) β min < ɛ, which implies V L (ˆl) β min < ɛ. Theefoe lim l V L (l) = β min. The tem β(l) = β min does not appea in the V L (l) expession, because it takes an infinite time to each l = and, as can be seen fom (7), the discounting then makes the (finite) β(l) vanish. Type H has a constant ate of jumps to l = and pays a flow cost c H foeve, in addition to getting the flow benefit. The candidate value of H is l [ β(z) ch V H (l) = + β ] ( max exp ( + ) l z ) dz. The limiting value is lim l V H (l) = β min c H + than β min iff βmax β min c H. + βmax. The RHS is geate (+) The signalling egions (l, l 0 ] fo which the incentive constaints ae satisfied can now be chaacteized. Due to V L > 0, if V L(l) βmax c L holds at l, then it holds fo all ˆl > l. If L is deteed fom imitating H at l, then L is also deteed at all ˆl > l, because the benefit to imitation is the diffeence between the payoff of being believed to be the H type and the cuent value. The highe l, the highe the cuent value, so the lowe the incentive to exet effot. Since l difts down in the absence of a signal, the L type incentive constaint detemines the log likelihood atio below which signalling must stop. The minimal lowe bounday min l that an equilibium signalling egion can have must satisfy the incentive constaint lim l min l+ V L (l) βmax c L, whee lim l min l+ V L (l) = β(l) and β( ) = β min. If the expectation was fo signalling to continue at min l, then L would deviate to e = 1. The maximal initial log likelihood atio at which signalling can stat is detemined by V H, which depends on l. Due to V H > 0, if the incentive constaint V H (l) βmax c H holds at l, then fo all ˆl < l we have V H (ˆl) < β max c H. If H is incentivized to signal at l, then H is also incentivized at all ˆl < l, because the benefit to signalling is the diffeence between the payoff of being believed to be the H type and the cuent value. Since l difts down in the absence of a signal, the H type incentive constaint detemines the log likelihood atio l 0 (l) above which signalling cannot stat. If the expectation 19

20 was fo signalling to stat above l 0 (l), then H would deviate to e = 0. By Lemma 8, l 0 (l) < fo any l. The paamete values fo which thee exists an infomative extemal equilibium ae given in Poposition 9. Poposition 9. Suppose βmax β min > c H. Then l 0 R ɛ > 0 s.t. thee exists an extemal equilibium with signalling egion [l 0, l 0 + ɛ). If thee exists one extemal equilibium with a nonempty signalling egion (l, l 0 ], then thee is a continuum of such equilibia, each coesponding to a paticula l [l, l 0 ). Compaative statics of extemal equilibia ae pesented next. Based on (7), V H (l) and V L (l) ae infinitely diffeentiable in l,,, c H, c L fo all l L(e ), so deivatives can be used fo compaative statics. Changing l changes e and theefoe L(e ). In that case, the compaison is between payoffs at an l that is in the eachable set both befoe and afte changing l. Poposition 10. If l R, then (a) dv L(l) dl > 0, (b) dv H(l) dl > 0 iff β (l) β max + β(l) + c H > 0, (c) dw (l) dl > 0 iff exp(l) [ ] β (l) β max + β(l) + c H + β (l) > 0. Poof. The poof is by taking the elevant deivatives. Compaing extemal equilibia, dv L(l) > 0, so pooling always gives L the dl highest payoff. This holds even when the benefit fom the eceives log likelihood atio is abitaily convex. The eason is that e L = 0 in extemal equilibia, so L neve eceives good signals. In infomative equilibia, thee is a downwad dift in l, which lowes V L below pooling. The esult that pooling always gives L the highest payoff in the good news model holds not just fo extemal equilibia. By Lemma 7, a nonextemal equilibium must featue e L (0, 1) in the signalling egion, i.e. L is indiffeent to eceiving good signals and paying the signalling cost. In that case, e = 0 is still a best esponse fo L, so V L is unchanged if the chosen action of L is switched to 0, keeping expectations e L, e H equal to the equilibium stategies. In othe wods, the payoff of L in an infomative equilibium is the same as it would be without good signals and with zeo 20

21 signalling cost. Due to the downwad dift of l, this is lowe than the pooling payoff. If the condition fo dv H(l) > 0 holds at evey l < l dl 0, then pooling gives H the highest payoff. If the condition fails as l l 0, then an infomative equilibium gives H a highe payoff than pooling. The condition has a staightfowad intepetation. Inceasing l inceases the payoff upon eaching l at a ate β (l), lowes the chance of jumping to l = (the β max tem), inceases the chance of eaching l (the β(l) tem) and educes the time duing which the signalling cost is paid. The balance of these effects detemines whethe V H inceases o deceases in l. The condition fo welfae to incease in l holds when β(l) = exp(l), so 1+exp(l) pooling gives the highest welfae when the sende s benefit fom the eceives belief is linea. The condition fo welfae to incease in l is a combination of the effects on the payoffs of H and L. The initial exp(l) multiplie is a weight on the payoff of H. The weight inceases in the pobability that the sende s type is H. The tem in the squae backets is the effect of inceasing l on the payoff of H, which is discussed above. The final tem β (l) (with a weight of 1) is positive and descibes L s benefit fom an incease in l, namely that the payoff upon eaching l is lage. 2.3 Final emaks on the Poisson model Some of the estictive assumptions in the Poisson games consideed above ae elaxed in the online appendix. Including both good and bad news in the game, with good signals occuing at ate g e and bad signals at ate b (1 e), the set of extemal equilibia is simila to the good news case when g > b and simila to the bad news case when g < b. If g = b, then the log likelihood atio stays constant in the absence of signals and jumps when a signal occus. Adding a small positive ate of bad news to the good news model does not change the esult that L always pefes pooling. The set of extemal effot equilibia emains simila to the case with only good news. A small positive ate of good news in the bad news model does not significantly affect extemal equilibia. The assumption that zeo effot in the good news case and maximal effot in the bad news case ensue the absence of signals is athe stak. Howeve, as shown in the online appendix, a low Poisson intensity ɛ > 0 of signals even with zeo effot in the good news case o with maximal effot in the bad news case does not qualitatively change the set of extemal equilibia. 21

22 The positive lowe bound on the signal ate in the good news model does ovetun the esult that L always pefes pooling. 3 Signalling with Bownian noise The Poisson signalling game coesponds to an envionment whee infomation is evealed by ae and significant events. Thee ae situations in which a gadual and continuous infomation evelation is moe ealistic. This section tuns to the gadual infomation evelation case and models the signal pocess as a Bownian motion. In this model, the union of equilibium signalling egions need not be an equilibium signalling egion. The union would mean the eceives expect too much signalling, which induces one of the types to deviate. An analogous esult holds in the one-shot noisy signalling model in the online appendix, whee fo some paamete values an equilibium with e L = 0, e H = 1 exists iff the initial log likelihood atio belongs to one of two disjoint finite intevals. The expectation of too much signalling between these intevals induces L to imitate. The setup is simila to the Poisson model. The sende s effot pocess (e t ) t 0 now contols the dift of a signal pocess (X t ) t 0 given by dx t = e t dt + σdb t, whee X 0 R is a given paamete, B t is standad Bownian motion and σ > 0. Denote the filtation geneated by (X t ) t 0 by (F t ) t 0. The eceives at time t obseve (X τ ) τ [0,t], but not the sende s type o pesent o past actions. Based on the signal, the eceives update thei log likelihood atio. The log likelihood atio pocess (l t ) t 0 is adapted to (F t ) t 0. The flow utility of a sende of type θ fom action e [0, 1] and the eceives log likelihood atio l is β(l) c θ e. Assume β is bounded, Lipschitz, stictly inceasing and twice continuously diffeentiable on R. A pue public stategy of the sende is a pai of andom pocesses (e L, e H ), each taking values in [0, 1] and adapted to (F t ) t 0. A Makov stationay stategy of the sende is a pue public stategy measuable w..t. the log likelihood atio pocess (l t ) t 0. It can be witten as a pai of measuable functions (e L, e H ) : R [0, 1] 2. The state vaiable is the eceives log likelihood atio l. Hencefoth, stategy is undestood as a Makov stationay stategy. 22

23 Given the initial log likelihood atio l 0 R and the stategy (e L, e H ) expected fom the sende, the eceives update the log likelihood atio 8 dl t = σ 2 (e H(l t ) e L(l t ))[dx t 1 2 e H(l t )dt 1 2 e L(l t )dt]. (8) The initial log likelihood atio l 0 is a paamete, so a stategy is a function of l 0 in addition to l. The eachable set of log likelihood atios is L(e ) = { l R : t R + a path of l t s.t. l t = l }, whee l t is defined in (8). Only behaviou on the eachable set is discussed subsequently. Behaviou outside L(e ) can be abitay, because no deviation can take l thee. Definition 3. A Makov stationay equilibium consists of a stategy (e H, e L ) and a log likelihood atio pocess (l t ) t 0 s.t. 1. given (l t ) t 0, e θ solves [ ] sup E e θ exp( t) [β(l t ) c θ e θ (l t )] dt l t=0 = l 0, e θ ( ) 0 whee the expectation is ove the pocess (l t ) t 0, 2. given (e H, e L ) and l 0, the pocess (l t ) t 0 is deived fom Bayes ule (8). Given e, the pooling egion is P(e ) = {l L(e ) : e H(l) = e L(l)}. The complement of the pooling egion in L(e ) is called the signalling egion. The pooling equilibium whee e H = e L 0 always exists by Lemma 2. A nonpooling equilibium exists if the sende is patient enough. The poof is postponed to Poposition 15. Since l 0 is a paamete, the payoff stating at l 0 need not in geneal equal the continuation value fom l 0 afte stating at ˆl l 0. Howeve, if the stategy the maket expects is Makov stationay, then evey time l is eached, the continuation value V θ (l) of type θ fom l on is well defined. Some obsevations about the value functions ae fomalized in the following lemma. 8 The updating ule fo the log likelihood atio is deived fom the continuous time Bayes ule fo pobability (Liptse and Shiyaev (1977) Theoem 9.1) using Itō s fomula. 23

24 Lemma 11. V H (l) V L (l) l 0 R e l L(e ), with stict inequality if unde the optimal (e L, e H ) stating at l, thee is a positive pobability of eaching some ˆl β with e L (ˆl) > 0. min V θ (l) βmax, with stict inequalities if l R. V θ is stictly inceasing. The esults in Lemma 11 hold moe geneally. The fact that V H (l) V L (l) is independent of the signal stuctue and the cost function it only needs that each effot level costs less fo H than fo L. The value functions ae bounded independently of the signals and costs. The stict monotonicity of V θ does not cay ove to the Poisson case, but it holds wheneve the paths of the l pocess ae continuous fo any stategy expected o chosen. Equilibium stategies ae patially chaacteized in the following lemma. If L is expected to take highe effot than H, then a highe signal lowes the log likelihood atio. With inceasing V θ this implies both types optimally choose 0. This esult holds wheneve the signal is such that the paths of the l pocess ae continuous. Lemma 12. Fix l 0 R. In any equilibium, l L(e ) satisfying e L (l) = e H (l) > 0 o e L (l) > e H (l). It is clea fom (8) that once the log likelihood atio pocess eaches the pooling egion, l stays constant foeve, so signalling must occu in an inteval of l containing l 0. Stating inside the inteval, if the l pocess eaches the bounday, then in the next instant it entes the pooling egion due to the apidly vaying Bownian motion diving l. Fo this eason, it is w.l.o.g. to conside only open signalling egions. In light of this and Lemma 12, it is w.l.o.g. to conside only equilibia in which outside an inteval of log likelihood atios (l, l), both types choose action 0 and inside that inteval, e L < e H. The subsequent focus of this section is on extemal equilibia. It can be shown that if effot only takes values zeo and one, then outside an inteval of log likelihood atios (l, l) l 0, both types choose action 0 and inside that inteval, e L = 0, e H = 1. In (l, l), the l pocess is a simple Bownian motion with dift eithe 1 o 1, depending on whethe the sende s chosen action 2 2 is e = 1 o 0. The value functions ae calculated next, using the HJB equations and a veification theoem. Payoff compaisons eadily appaent fom the value functions ae noted while deiving the value functions. Necessay conditions fo equilibium ae deived. Afte that, the existence of infomative extemal 24

25 equilibia is poved, followed by compaative statics, which ae mostly numeical. It is clea that fo ˆβ(µ) = β(ln µ ) concave in µ (the eceives belief), 1 µ L pefes pooling to any othe equilibium, because the expected posteio is lowe than the pio and the vaiance in the posteio does not incease the payoff. The concavity of ˆβ in µ implies the condition in Poposition 13 that is sufficient fo L to pefe pooling. If pooling gives the highest payoff to L, then in the class of extemal equilibia, L s payoff is highe in an equilibium with a smalle signalling egion. Poposition 13. If β (ln (z)) is concave in z, then V L is highe in pooling than in any extemal equilibium. In that case if V L2 is the value of L in an equilibium with signalling egion (l 2, l 2 ) and V L1 is the value of L in an equilibium with signalling egion (l 1, l 1 ) (l 2, l 2 ), then V L1 (l) V L2 (l) fo all l (l 1, l 1 ). If β (ln (z)) is convex in z, then V L is lowe in pooling than in any extemal equilibium and V L1 (l) V L2 (l) fo all l (l 1, l 1 ). The idea of the poof is to tansfom the l pocess into a zeo-dift pocess f(l) by an inceasing tansfomation f using Itō s lemma. The benefit function β is simultaneously tansfomed by the invese of f. Pooling gives L a highe payoff than an infomative extemal equilibium iff the tansfomed benefit function is concave in f(l). It tuns out f(l) is the likelihood atio exp(l). ( An example whee L pefes pooling has β(l) = n, 1+exp(l)) n 2 and l ln n 1. 2 Next, the HJB equations ae solved and a veification theoem is used to check that the solutions of the HJB equations coincide with the value functions. The HJB equation of type θ is V θ (l) = β(l) + max { ( c θ e + V θ(l)σ 2 e 1 )} V θ (l)σ 2. Given the signalling egion (l, l) the eceives expect, the optimal stategy of type θ is to choose { 1 { c θ + V θ e θ (l) = (l)σ 2 0} if l (l, l), 0 if l / (l, l). 25

26 The incentive constaints fo H to choose e H (l) = 1 and L to choose e L (l) = 0 in the signalling egion ae V H(l) c H σ 2, V L(l) c L σ 2. (9) Call these IC H and IC L. Afte finding the candidate equilibium, it must be veified that the ICs hold at evey point in the signalling egion. Set the chosen actions equal to the equilibium actions. The HJB equations become the pai of linea second-ode ODEs V H (l) = β(l) c H V H(l)σ V H(l)σ 2, V L (l) = β(l) 1 2 V L(l)σ V L (l)σ 2. This is whee using the log likelihood atio instead of the belief is helpful with belief, the ODEs do not have constant coefficients. Afte solving the ODEs fo V L, V H, the ICs as well as the smoothness conditions fo the veification theoem must be checked at evey point in the signalling egion. The solutions V θ to the ODEs ae the sum of the geneal solution C θ1 y θ1 + C θ2 y θ2 of the homogeneous equation and a paticula solution y θp of the inhomogeneous equation. The geneal solutions fo H and L espectively ae ( C H1 exp l 1 ) ( 1 + 8σ 2 + C H2 exp l 1 + ) 1 + 8σ 2, 2 2 C L1 exp ( l σ 2 2 ) + C L2 exp ( l σ 2 2 Using d Alembet s method, the paticula solutions ae y Hp = c H ( 2σ 2 + exp l 1 ) ( 1 + 8σ 2 β(l) exp l 1 + ) 1 + 8σ 2 dl 1 + 8σ ( 2σ 2 exp l 1 + ) ( 1 + 8σ 2 β(l) exp l 1 ) 1 + 8σ 2 dl, 1 + 8σ ). 26

27 y Lp = ( 2σ 2 exp l 1 ) ( 1 + 8σ 2 β(l) exp l 1 + ) 1 + 8σ 2 dl 1 + 8σ ( 2σ 2 exp l 1 + ) ( 1 + 8σ 2 β(l) exp l 1 ) 1 + 8σ 2 dl, 1 + 8σ whee the integals ae nonelementay even fo simple functional foms of β, e.g. fo β(l) = exp(l), which descibes linea benefit fom the eceives 1+exp(l) belief. Imposing the bounday conditions V θ (l) = β(l) and V θ (l) = β(l), the constants in the geneal solution fo H ae C H1 = y H2(l)[ β(l) y Hp (l)] y H2 (l)[ β(l) y Hp (l)], y H1 (l)y H2 (l) y H2 (l)y H1 (l) C H2 = y H1(l)[ β(l) y Hp (l)] + y H1 (l)[ β(l) y Hp (l)]. y H1 (l)y H2 (l) y H2 (l)y H1 (l) The constants fo L ae detemined by a simila expession, eplacing the H subscipts with L. Now that all components of the solutions of the HJB equations have been found, it can be veified that they coincide with the candidate value functions. The ICs emain to be checked. Lemma 14. The solutions V H, V L of the HJB equations equal the candidate value functions in the signalling egion. The Makov contols fo the HJB equations maximize the candidate value functions. Based on the candidate value function expessions, the signalling egion depends on and σ 2 only though thei poduct σ 2. Closed fom compaative statics esults ae not available fo paametes othe than c L due to the complexity of the V θ expessions. Numeical simulations will be used instead. As to c L, the LHS of IC L in (9) does not contain c L, so thee exists ĉ L s.t. fo c L < ĉ L, IC L fails and fo c L ĉ L, IC L holds. Befoe tuning to numeics, the conditions fo the existence of nontivial extemal equilibia ae povided. The intuition of the conditions is simila to the Poisson case an infomative equilibium exists iff the maximal benefit fom signalling is lage enough to incentivize H to signal. To see the similaity, conside σ 2 analogous to 1 in the Poisson models. 27

28 Poposition 15. If l R with β (l) > c H σ 2, then l 0 R contained in the signalling egion of an extemal equilibium. If l R with β (l) c H σ 2, then pooling is the unique equilibium. The idea of the poof is that the slopes of the value functions ae close to the slope of β fo small signalling intevals. If the slope of the benefit function is high enough at some point l to incentivize H to signal at l (povided the eceives expect e L (l) = 0, e H (l) = 1), then thee exists anothe point l 0 with β (l 0 ) low enough not to incentivize L to signal, but high enough to still incentivize H. An infomative extemal equilibium can then be constucted with a signalling inteval containing l 0. It is clea that fo any l R, σ 2 > 0, c L > c H > 0 and stictly inceasing β(l), thee exists (0, ) that makes the sufficient condition in Poposition 15 hold. One can always find a level of patience fo a nontivial extemal equilibium to exist. Numeical compaative statics on the set of signalling egions ae pesented next. As in the Poisson signalling game, fo some initial log likelihood atios thee is a continuum of infomative equilibia. Until the end of this exp(l) 1+exp(l) section, it is assumed that β(l) =, so the sende s benefit fom the eceives belief equals the belief. The figues to follow depict signalling intevals (l, l) as points on a plane, with the x-coodinate of the point equalling l and the y-coodinate equalling l. Fo c H = 0.1, c L = 0.24 and = σ 2 = 1, the egion whee the ICs hold is depicted in panel (c) of Figue 3 as the shaded aea. Panel (a) shows the aea whee IC H holds and panel (b) the aea whee IC L holds. The shaded aea on panel (c) is the intesection of panels (a) and (b). The disconnectedness of the set of equilibium signalling egions is in pat due to esticting attention to extemal equilibia. The effect of inceased patience o educed noise on the ICs is shown in Figue 4, whee c H = 0.1, c L = 0.24, = 1 and σ 2 = 0.5. Note the diffeent scale of the axes compaed to Figue 3. Since and σ 2 affect the ICs only though thei poduct, educing σ 2 by half has the same effect as educing by half. Thee need not exist a signalling egion containing all othes. Such a signalling egion is the point at the uppe left cone of the shaded aea of panel (c), i.e. a point that is simultaneously at maximal hoizontal and vetical distance fom the diagonal. Figue 5 shows that fo c H = 0.15, c L = 0.28 and = σ 2 = 1, a highe l pemits a highe l fo a signalling 28

29 (a) IC H holds (b) IC L holds (c) Both ICs hold Figue 3: Region whee ICs hold (shaded) fo c H = 0.1, c L = 0.24 and = σ 2 = 1. 29

30 (a) IC H holds (b) IC L holds (c) Both ICs hold Figue 4: Region whee ICs hold (shaded) fo c H = 0.1, c L = 0.24, = 1 and σ 2 =

31 egion. Theefoe the union of two equilibium signalling egions need not be an equilibium signalling egion. This distinguishes the game with Bownian noise fom the Poisson signalling game, the epeated noiseless game and the one-shot noisy and noiseless games. (a) IC H holds (b) IC L holds (c) Both ICs hold Figue 5: Region whee ICs hold (shaded) fo c H = 0.15, c L = 0.28, = 1 and σ 2 = 1. In a given infomative equilibium, the H type payoff can be highe o lowe than the pooling payoff fo diffeent log likelihood atios. exp(l) (1+exp(l)) exp(l) fo (1+exp(l)) This is illustated in Figue 6, whee V H is stictly highe than l ( 1.5, 0.2) and stictly lowe fo l ( 0.2, 3). In this equilibium, the compaison of infomative equilibium and pooling payoffs of H accods 31