Good signals gone bad: dynamic signalling with switching efforts
GGood signals gone bad: dynamic signalling withswitching efforts
Sander Heinsalu Research School of Economics, Australian National University, HW Arndt Building 25aKingsley St, Acton, ACT 2601, Australia.Email: [email protected], website: http: // sanderheinsalu. com/
Abstract
This paper examines signalling when the sender exerts effort and receives ben-efits over time. Receivers only observe a noisy public signal about the effort,which has no intrinsic value.The modelling of signalling in a dynamic context gives rise to novel equilib-rium outcomes. In some equilibria, a sender with a higher cost of effort exertsstrictly more effort than his low-cost counterpart. The low-cost type can com-pensate later for initial low effort, but this is not worthwhile for a high-cost type.The interpretation of a given signal switches endogenously over time, dependingon which type the receivers expect to send it.JEL classification: D82, D83, C73.
Keywords:
Dynamic games, signalling, incomplete information
1. Introduction
In many signalling situations, the sender exerts effort over time, and theobservation of that effort is noisy. For example, a politician may be a (relatively)honest or a corrupt type, and can signal honesty by following the law to theletter (paying taxes in full, refraining from speeding and bribe-taking). Whereasthe politician incurs the cost of obeying the law at all points of time, voters learnof low effort only after the realisation of a random event, such as a scandal. The The author is greatly indebted to Johannes H¨orner for many enlightening discussionsabout this research. Numerous conversations with Larry Samuelson have helped improve thepaper. The author is grateful to Eduardo Faingold, Dirk Bergemann, Juuso V¨alim¨aki, PhilippStrack, Willemien Kets, Tadashi Sekiguchi, Flavio Toxvaerd, Fran¸coise Forges, Jack Stecher,Daniel Barron, Benjamin Golub, Florian Ederer, Vijay Krishna, Sambuddha Ghosh, SergiuHart, Idione Meneghel, J¨orgen Weibull, Alessandro Bonatti, Zvika Neeman, Scott Komin-ers, Andrzej Skrzypacz, George Mailath, Joel Sobel, anonymous referees and participants ofmany conferences and seminars for comments and suggestions. Any remaining errors are theauthor’s. Financial support from Yale University and the Cowles Foundation is gratefullyacknowledged.
Preprint submitted to Elsevier June 29, 2018 a r X i v : . [ q -f i n . E C ] J u l onest type has a lower cost of obeying the law. The honest type acts in thepublic interest in important matters, but the corrupt type does not. The voterscare about decisions in important matters, but not directly about whether thepolitician obeys the law in everyday life.As a consequence of the multiple opportunities of exerting effort, novel dy-namics of behaviour arise. For example, there exist equilibria in which thehigh-cost type chooses a strictly higher effort level initially than the low-costtype.In the model, the players are a sender and a competitive market of receivers.The sender is either a high-cost or a low-cost type. The type is private informa-tion. Receivers share a common prior belief about the type. The sender contin-uously chooses his effort level. Receivers observe noisy public signals about theeffort, rather than the effort itself. The signal process is Poisson with intensitydecreasing in effort. The types only differ in their flow cost of effort. The senderderives a flow benefit directly from the posterior belief of the receivers.Attention is restricted to equilibria that are Markovian (the belief of thereceivers is the state variable) and stationary, which means that behaviour doesnot depend on calendar time. Such equilibria exist, because minimal effort byall types in all circumstances is a Markovian and stationary equilibrium andalways exists.In some parameter regions, there exist equilibria in which first one type ex-erts higher effort and then the other. These are called switched effort equilibria ,because initially the ordering of the efforts of the types is the opposite to thatfound in the previous literature on signalling. The concept of a switched effortequilibrium can be illustrated by the example of the politician given above. Inthis example, the politician can be honest or corrupt and can exert effort to obeythe law. Lawful behaviour decreases the frequency of scandals. A switched effortequilibrium can be described in terms of four regimes, which are referred to as early career , insider , scrutiny and tainted . Play starts in the early career, dur-ing which the corrupt type exerts positive effort and the honest type no effort.If no scandal occurs by a given time, then the politician becomes an insider,which means that the voters ignore scandals and the politician no longer exertseffort. If, instead, a scandal occurs in the early career, then scrutiny results.Under scrutiny, the honest type exerts maximum effort and the corrupt typenone. Under scrutiny, a scandal leads to a tainted reputation, which means thatvoters believe the politician to be corrupt with higher probability than in theother regimes, and the politician exerts no effort. The path of play in switchedeffort equilibria is depicted in Fig. 1 below.In the early career regime of switched effort equilibria, a signal has theopposite meaning to that in the other regimes. If the low-cost type is more One example of a scandal increasing popularity is Bill Clinton, whose approval rating rosefrom 63% on 15 Dec 1998 to 73% on 20 Dec. The House of Representatives passed two articlesof impeachment in that period. Toronto mayor Rob Ford’s approval rating rose from 39% on28 Oct 2013 to 44% on 1 Nov. Between these dates, police chief Bill Blair confirmed a videoof Ford smoking crack cocaine. no scandal
Scrutiny s c a nd a l no scandal Tainted scandal
Figure 1: A switched effort equilibrium. likely to generate a particular signal, then this signal is evidence of low cost,whereas if the high-cost type becomes more likely to send it, then the signalsuggests high cost to the receivers.The voters do not care directly about the part of the politician’s behaviourthat leads to scandals (sex life, drug use, tax evasion), but care about thepolitician’s unobservable decisions on issues important to the country. Thevoters assume that a politician’s tendency for scandalous behaviour in mattersthat have no direct impact on the public is positively correlated with a tendencyof corrupt decisions in areas where these cause real damage. These combinedtendencies are called a corrupt type. The voters cannot observe the decisionsthat affect them, so they use scandals to update their belief about the likelihoodof honesty in such decisions.The politician derives a flow benefit from the voters’ belief, which can be jus-tified by a randomly arriving election. This arrival is exogenous to the politicianif he is not the head of government and cannot call a snap election. Because ofthe chance of an election, belief at each future instant has a positive probabilityof mattering for the politician’s career. Alternatively, the politician may deriveego rents from being popular with the public.
Signalling is used to explain phenomena as diverse as education (Spence,1973), conspicuous consumption (Veblen, 1899) and issuing equity (Leland andPyle, 1977). Many authors have mentioned the relevance of time (Weiss, 1983;Admati and Perry, 1987) and noise (Matthews and Mirman, 1983) in signallingcontexts. The conclusion of the previous literature is that in all equilibria, thehigh-cost type exerts weakly less effort than the low-cost. This result fails tohold only in models that depart from pure Spence signalling, e.g. by addingexogenous information revelation and more than two types as in Feltovich et al.(2002). Introducing noise (Matthews and Mirman, 1983) or repetition (N¨oldekeand van Damme, 1990; Swinkels, 1999) has not been shown to switch the effortsof the types in pure signalling. Similarly, such switching of efforts has notbeen found to arise in models that incorporate both dynamics and imperfectmonitoring (Daley and Green, 2012; Gryglewicz, 2009; Dilme, 2014).In Dilme (2014), signalling in continuous time is obscured by Brownian noise.The sender (an entrepreneur) decides how much costly effort to exert over time,as well as when to stop the game (sell the firm) and receive a final benefit. This3ontrasts with the present paper, in which benefit accumulates continuouslyand the sender cannot stop the receivers from learning. Dilme (2014) studiesthe efficiency of equilibrium effort provision and finds that effort is too low andstops too early. The present paper examines the endogenous switching of theefforts of the types, which is not alluded to in the previous literature, includingDilme (2014).In Daley and Green (2012), the uninformed traders receive information (ob-servations of a diffusion process) exogenously over time, and the informed traderdecides when to stop the game (execute the trade) and receive a final payoff.Gryglewicz (2009) examines limit pricing over time. The low-cost incumbent isa commitment type and the high-cost incumbent decides when to stop imitatingthe low-cost type. Unlike Daley and Green (2012), the present paper models anendogenous signal. Moreover, the present paper differs from Gryglewicz (2009)in that both types are strategic. The situation described is one in which payoffsaccrue continuously or a lump-sum arrives at an exogenous exponentially dis-tributed time. In contrast, the other papers model a stopping decision by thesender that triggers a payoff. Neither Daley and Green (2012) nor Gryglewicz(2009) refers to switched efforts or interpretations of a signal.The present paper is also related—albeit less closely—to the literature onrepeated noiseless signalling (N¨oldeke and van Damme, 1990; Swinkels, 1999).In these models, time is discrete, there is no noise, and the sender pays the sig-nalling cost first and receives the benefit only upon deciding to stop signallingforever. For example, the completion of a traditional education can be modelledin this way—the salary is received only after graduating. In the current paper,the benefit is received concurrently with the payment of the cost, as when aworker takes continuing education courses while being employed, or a firm ad-vertises while selling its product. N¨oldeke and van Damme (1990) find a uniqueinformative equilibrium. Using different informational assumptions, Swinkels(1999) finds a unique pooling equilibrium. The models in the current paperhave many informative equilibria and one pooling equilibrium. In N¨oldeke andvan Damme (1990) and Swinkels (1999), the low-cost type always exerts weaklymore effort than the high-cost. While this feature of their models is in line withprevious signalling research, it stands in contrast with the current work.Discrete-time repeated signalling is also studied in Kaya (2009) and Roddie(2012). In their models, there is no noise in the observation of the sender’saction, whereas in the present paper this observation is noisy. Kaya (2009)focuses on least-cost separating equilibria. Roddie (2012) provides general con-ditions for reputation effects to arise. Neither paper mentions the possibility ofswitched efforts, which are the focus of the current paper.It is, in fact, easier to construct switched effort equilibria without noise, onaccount of the availability of belief threats. The author is not aware, however,of any papers in which this has been done. Using belief threats or ad hocrefinements invites the suspicion that switched efforts are driven by unrealisticbeliefs. This concern is addressed in the current work by using noise, so thatBayes’ rule applies after all histories. Similarly, non-Markov strategies can beused to create strange behaviour, but this paper stacks the deck against unusual4esults by focussing on Markov stationary equilibria. This serves to strengthenthe result that effort switching is possible.Switched effort equilibria could not be studied with existing noisy signallingmodels, which use either discrete time or Brownian noise. Discrete time withnoise makes equilibrium calculation intractable. Brownian noise is incompatiblewith switched efforts and is less tractable than Poisson for the models in thispaper.Switched efforts are reminiscent of the countersignalling of Feltovich et al.(2002), but the mechanism driving them is quite different, as is the model.The switched effort equilibria of this paper use dynamics, while Feltovich et al.(2002) present a one shot model. In the current paper, there is no exogenousinformation revelation and there are only two types, in contrast to Feltovichet al. (2002). In countersignalling, the lowest-cost type can rely on the exogenoussignal to partly distinguish him from the highest-cost. Moreover, he may exertless effort than the medium type in order to differentiate himself from that type.The effort of the lowest-cost type cannot be less than that of the highest-costtype. In the present paper, it is the threat of future information revelation thatincentivises the high-cost type to signal. This threat is not as severe for thelow-cost type, leading to strictly less effort.Cripps et al. (2004) show that, in a wide class of repeated games, if a reputa-tion for behaviour is not an equilibrium of the complete information stage game,then that reputation is temporary, and the type must eventually be learned. Insome switched effort equilibria of the present paper, both types face a positiveprobability of acquiring a ‘wrong’ permanent reputation in the following sense:when signalling and belief updating stop, belief about the good type may belower than the prior and belief about the bad type higher. In expectation,beliefs move in the direction of the sender’s type, but mistakes have positiveprobability.
2. Setup
The players are a strategic sender and a competitive market. The senderhas a type θ ∈ { G, B } , with G interpreted as the good (low-cost) type and B asbad (high-cost). The sender knows his type, the market does not. The initiallog likelihood ratio l ∈ R of the types is common knowledge. Throughoutthis paper, log likelihood ratio l is used instead of belief Pr( G ) = exp( l )1+exp( l ) , asthis simplifies the formulas in the dynamic models to follow. All results canbe restated in terms of beliefs. A generic log likelihood ratio l is an elementof R = R ∪ {∞ , −∞} . The log likelihood ratio corresponding to Pr( G ) = 1 is l = ∞ and corresponding to Pr( G ) = 0 is l = −∞ .Time is continuous and the horizon is infinite. The sender chooses signallingeffort e t ∈ [0 ,
1] at each instant of time t . To avoid the technical difficulties The proof is available upon request. It uses a coupling argument on the continuous beliefpaths.
5f defining behavioural strategies in continuous time, mixing is not allowed.Effort e costs type θ sender c θ ( e ), with c θ continuously differentiable, strictlyincreasing, convex and c θ (0) = 0. Strong single crossing c (cid:48) G (1) < c (cid:48) B (0) isassumed, which implies the single crossing in type and cost that is standard inthe signalling literature. Strong single crossing means that the marginal cost ofany effort level for the good type is lower than the marginal cost of any effortfor the bad type.Effort benefits the sender via its effect on the signal process, which drivesthe market’s log likelihood ratio process ( l t ). The sender is assumed to deriveflow benefit β ( l ) directly from the market’s log likelihood ratio l . Unless notedotherwise, the function β is assumed to be strictly increasing, bounded andcontinuously differentiable. Denote the flow benefit from l = ∞ (correspondingto Pr( G ) = 1) by β max and from l = −∞ by β min .In order to provide microfoundations that explain why the market rewardsthe sender’s type rather than the effort it expects from the sender, supposethere are two kinds of effort: effort which is noisily observable and that which isunobservable. The senders with a low cost of partially observable effort preferto exert the unobserved effort, but the high-cost senders prefer not to exertit. Only the completely unobservable kind of effort matters to the market. Thetype is the level of this hidden effort. The partially visible effort is the signallingactivity, which the market does not care about directly.A pure Markov stationary strategy is a measurable function ( e B , e G ) : R → [0 , that maps the log likelihood ratio to the efforts of the types. The statevariable l is the log likelihood ratio of the market. Formally, the value of thestate variable at time t is the left limit l t − of the log likelihood ratio process( l t ), similarly to Yushkevich (1988). The log likelihood ratio l t is a function oftime. The left limit means that time approaches t from below. This assumptionensures that the signals are not anticipated by the log likelihood ratio of themarket. Intuitively, the market does not condition on the signal realisationin the ‘next instant’. The convention l − = l is used for the initial valueof the state variable. Henceforth, only pure Markov stationary strategies areconsidered and the ‘pure Markov stationary’ phrase is omitted.The signal process is Poisson with rate (1 − e t ) λ + d at time t . The parameter λ ∈ (0 , ∞ ) is interpreted as the informativeness of effort, while d ≥ A Poisson rate linear in effort is with some loss of generality, but less than at first seems.A change of variables ˆ e = f − ( e ) transforms cost to c θ ( f (ˆ e )) and the signal rate to (1 − f (ˆ e t )) λ + d , which may be nonlinear. A strictly increasing convex f preserves the convexityand the strong single crossing of cost. e ∗ = ( e ∗ G , e ∗ B ). This notation is also used for equilibrium strategies. If asignal occurs at log likelihood ratio l ∈ R , then the log likelihood ratio jumps to j ( l ) = l + ln (cid:18) λ (1 − e ∗ G ( l )) + dλ (1 − e ∗ B ( l )) + d (cid:19) . (1)Call | j ( l ) − l | the jump length . If d = 0, then the fraction in (1) may become for some strategy e ∗ that the market expects. In that case, updating uses thelimit of (1) as d (cid:38)
0. This implies that = 1, and if l = ±∞ , then l staysconstant regardless of signals. Note that updating is defined for any e ∗ , notjust the equilibrium e ∗ introduced below. The set of strategies available to thesender is independent of d . The limit d (cid:38) e ∗ separately todefine the overall updating rule that maps e ∗ and the signal history to l . Theuse of the d (cid:38) Lemma 1.
Let ( e ∗ G , e ∗ B ) be the strategy the market expects. Fix t ≥ . Supposesignals have occurred at τ , . . . , τ n , with ≤ τ ≤ . . . ≤ τ n ≤ t . Then the loglikelihood ratio at t is l t = l + λ (cid:90) t [ e ∗ G ( l s ) − e ∗ B ( l s )] ds + n (cid:88) k =1 [ j ( l τ k ) − l τ k ] . (2)Except for jumps, l evolves deterministically given the market expectations( e ∗ B , e ∗ G ). Given l at the time of a jump, the jump length is deterministic. Thelog likelihood process depends on the chosen effort only via the (random) timingof the signals τ , . . . , τ n . Lemma 1 applies even if multiple signals occur in thesame instant τ i , but this event has zero probability, because the signal processis Poisson.If for some ˆ l , the strategy features e ∗ G ( l ) > e ∗ B ( l ) for l ∈ (ˆ l − (cid:15), ˆ l ) and e ∗ G ( l ) < e ∗ B ( l ) for l ∈ (ˆ l, ˆ l + (cid:15) ), then ˆ l is a stasis point . A stasis point is describedin more detail in the appendix. It occurs at a point ˆ l that has the log likelihoodratio drift towards ˆ l from immediately above and below ˆ l . The log likelihoodratio does not drift away from a stasis point, but may jump away in eitherdirection.Given the strategy e ∗ = ( e ∗ B , e ∗ G ) that the market expects the sender tochoose, the payoff of type θ from actually choosing the effort function e θ ( · ) isthe expected discounted sum of flow payoffs J e θ l ( e ∗ ) = E e θ (cid:20)(cid:90) ∞ exp( − rt ) [ β ( l t ) − c θ e θ ( l t )] dt (cid:12)(cid:12)(cid:12)(cid:12) e ∗ , l t =0 = l (cid:21) , (3)where the expectation is over the stochastic process ( l t ) t ≥ , given e θ ( · ). Payoffsboth on and off the equilibrium path are given by (3), depending on whether ornot the e G , e B that maximise (3) for each type satisfy ( e G , e B ) = ( e ∗ G , e ∗ B ). Thediscount rate is r >
0. 7iven a strategy e ∗ that the market expects from the sender, the supremumof (3) over e θ is denoted V θ ( l ). If the market expects a Markov stationarystrategy, then every time a given l is reached, the continuation value V θ ( l ) oftype θ is well defined and independent of the path of ( l t ) that led to l . Thedependence of V θ ( l ) on e ∗ is suppressed in the notation. If e ∗ G ( l ) = e ∗ B ( l ), then( l t ) stays at l forever and V θ ( l ) = (cid:82) ∞ exp( − rt ) β ( l ) dt = β ( l ) r . Value V θ ( l ) isbounded above by β max r and below by β min r . Definition 1. A Markov stationary equilibrium consists of a strategy e ∗ =( e ∗ G , e ∗ B ) of the sender and a log likelihood ratio process ( l t ) t ≥ s.t.1. given ( l t ) t ≥ , e ∗ θ maximises (3) over e θ ,2. given e ∗ , ( l t ) t ≥ is derived from (2).Henceforth ‘equilibrium’ means a pure Markov stationary equilibrium.A pooling equilibrium is defined by e ∗ B ( l ) = e ∗ G ( l ) ∀ l . It is clearly Markovian(independent of past play conditional on l ) and stationary (independent of cal-endar time). A pooling equilibrium exists for all parameter values, because if e ∗ B ( l ) = e ∗ G ( l ), then the log likelihood ratio stays constant forever at l . If a loglikelihood ratio is unresponsive to effort, then there is no benefit to signalling.This implies that there is no incentive to exert effort at l . The unique bestresponse is e B ( l ) = e G ( l ) = 0. Existence is thus guaranteed for the equilibriumconcept in Def. 1.The focus in this paper is on equilibria in which e ∗ B ( l ) > e ∗ G ( l ) = 0 for anonempty open set of l . In the introduction, such equilibria were called switchedeffort equilibria . The introduction also described the path of play of such equi-libria, which is depicted in Fig. 1.A switched effort equilibrium cannot be unique, because the pooling equi-librium always exists. It will be shown below that if one switched equilibriumexists, then a continuum of such equilibria exist. Refinements cannot be usedto select an equilibrium, because Bayes’ rule applies everywhere, as explainedabove.
3. Switched effort equilibria
For some parameter values, there exist equilibria in which, for some loglikelihood ratios of the market, the B type exerts higher effort than G , despitethe uniformly higher marginal cost of effort. The result is reminiscent of thecountersignalling of Feltovich et al. (2002), but the mechanism is quite different.In this model, it is the threat of future information revelation that incentivises B to signal. This threat is not as severe for G .The switched effort pattern e ∗ B ( l ) > e ∗ G ( l ) = 0 is counterintuitive, becausethe flow benefit from a higher log likelihood ratio is the same for the types, but B has a higher marginal cost of signalling. The strong single crossing makes theresult more stark. The switched effort pattern requires V B to decrease at some l , because if e ∗ B ( l ) > e ∗ G ( l ), then j ( l ) > l and B is paying a cost to avoid jumps.8or this to be optimal, V B ( j ( l )) < V B ( l ) is necessary. The decrease of V B in l means that the discounted payoff of the B type decreases in the probabilitythat the receivers assign to him being G .A switched effort equilibrium is constructed as follows: first, assume that theappropriate strategy is employed; second, calculate the value functions; finally,check that the strategy is a best response at every l . A sketch of a switched effortequilibrium is in Fig. 2, in which the initial log likelihood ratio l is assumedto be in the interval ( l, l ). To maximise the duration of switched efforts, let l → l . Figure 2: Value functions and strategy in a switched effort equilibrium. Early career regime: e ∗ B > e ∗ G = 0, l ∈ ( l, l ], insider: e ∗ B = e ∗ G = 0, l = l , scrutiny: e ∗ G = 1 , e ∗ B = 0, l > l , tainted: e ∗ B = e ∗ G = 0, l < l . lV θ −∞ le ∗ B = e ∗ G = 0 e ∗ B > e ∗ G = 0 no signal l e ∗ G = 1 , e ∗ B = 0 no signalsignalsignal V B V B V G V G Prop. 2 below provides sufficient conditions for a switched equilibrium to ex-ist. These involve the value functions, which are endogenous, but all conditionscan be expressed using primitives, as shown subsequently in Prop. 3.For those values of l at which e ∗ B ( l ) = e ∗ G ( l ) (the insider and tainted statesin Fig. 1), incentives are trivial, because l does not respond to signals accordingto (1). The unique best response, therefore, is e B ( l ) = e G ( l ) = 0. The samereasoning was used above to show the existence of the pooling equilibrium.It remains to check the log likelihood ratio regions with e ∗ G ( l ) = 1, e ∗ B ( l ) = 0(scrutiny in Fig. 1) and e ∗ B ( l ) > e ∗ G ( l ) = 0 (early career in Fig. 1). There areincentive constraints at each l for both types, but this continuum can be reducedto just five inequalities, depicted in Fig. 3. There is one inequality for each typein the scrutiny region; one inequality for G in the early career; and a furthertwo inequalities for B in the early career.To check that the best responses are e G ( l ) = 1 , e B ( l ) = 0 when e ∗ G ( l ) = 1, e ∗ B ( l ) = 0, it is enough to verify two conditions: (i) G has an incentive toavoid jumps when the avoidance motive (value before minus after jump) isminimal; and (ii) B does not have this incentive when the avoidance motive ismaximal. The condition for G to exert maximal effort is Prop. 2 (a) below andthe condition for B to choose zero effort is Prop. 2 (b). These are also shownin Fig. 3. 9n the region where e ∗ B ( l ) > e ∗ G ( l ), it must be checked that G has no incentiveto avoid jumps and B has neither too much nor too little incentive, so that thebest response of B is interior. Define V >θ = sup { V θ ( l ) : e ∗ B ( l ) > e ∗ G ( l ) } ,V >θ = inf { V θ ( l ) : e ∗ B ( l ) > e ∗ G ( l ) } ,V θ = sup { V θ ( l ) : e ∗ G ( l ) = 1 , e ∗ B ( l ) = 0 } ,V θ = inf { V θ ( l ) : e ∗ G ( l ) = 1 , e ∗ B ( l ) = 0 } . The greatest temptation for G to avoid jumps in the early career occurs at themaximal value V >G when jumps go to just above l (see Fig. 3). If the differencebetween V >G and V G ( l ) = V G does not incentivise G to exert effort (condition(c) in Prop. 2), then neither do other V G ( l ) and V G ( j ( l )) in these regions.The equilibrium conditions that determine the effort of B in the early careerare λ [ V B ( l ) − V B ( j ( l ))] = c (cid:48) B ( e ∗ B ( l )) and j ( l ) = l + λ + dλ (1 − e ∗ B ( l ))+ d . The first isthe FOC for B , ensuring that B is optimising given the jump length. Thesecond condition determines the jump length given the equilibrium effort e ∗ B ( l ),using e ∗ G ( l ) = 0. A sufficient condition for B to choose the equilibrium level ofeffort when e ∗ B ( l ) > e ∗ G ( l ) = 0 is as follows: jumps at V >B going to V B do notincentivise B to exert maximal effort (Prop. 2 (d)) and jumps at V >B going to V B do motivate a positive level of effort (Prop. 2 (e)). Then by the Mean ValueTheorem, an interior effort level and jump length can be found such that thejump is derived from the effort using (1) and the effort is a best response to thejump. Figure 3: Incentives in a switched effort equilibrium with d = 0. Prop. 2 (a)–(e) are thevertical double-ended arrows. lV θ −∞ l l l ∞ V B V B V G V G (b)(a)(e) (c) (d) Proposition 2.
Fix l, l ∈ R with l < l and fix l ∈ ( l , ∞ ] . If(a) inf (cid:110) V G ( l ) − β ( j ( l )) r : l ≤ l < l (cid:111) ≥ c (cid:48) G (1) λ ,(b) sup (cid:110) V B ( l ) − β ( j ( l )) r : l ≤ l < l (cid:111) ≤ c (cid:48) B (0) λ ,(c) V >G − V G ≤ c (cid:48) G (0) λ , d) V >B − V B < c (cid:48) B (1) λ ,(e) V >B − V B ≥ c (cid:48) B (1 − d/λ − ( λ + d ) exp( l − l ) /λ ) λ ,(f ) lim l → l j ( l ) ≤ l ,then there exists an equilibrium in which e ∗ B ( l ) > e ∗ G ( l ) = 0 if l ∈ ( l, l ] , e ∗ B ( l ) = 0 , e ∗ G ( l ) = 1 if l ∈ ( l , l ) , e ∗ B ( l ) = e ∗ G ( l ) = 0 if l / ∈ ( l, l ) . The proof is in the appendix. The sufficient conditions in Prop. 2 hold evenwith the strong single crossing of cost (the marginal cost of B is uniformly higherthan that of G ), which makes the result more surprising.When d = 0 in Prop. 2, the upper bound l of the e ∗ G ( l ) = 1, e ∗ B ( l ) = 0 regionmust be ∞ . Otherwise V B ( l ) − β min r is large enough to induce B to exert effortat l . If, in the scrutiny region, effort is required forever, then the value of type B is lowered, which helps to restore incentives.Conditions (a)–(e) in Prop. 2 have a bound on the marginal benefit of avoid-ing a jump on the LHS and a bound on the marginal cost per unit of jumpfrequency on the RHS. The marginal benefit is the value difference between thelog likelihood ratios before and after a jump. The marginal cost c (cid:48) θ is evaluatedat a bound on the effort. The rate of jumps absent effort is λ , which is alsothe reduction in jump rate per unit of effort. The inequality and the bound ontype θ ’s effort in each condition ensure the required effort level. For example,in (a), the lower bound on the marginal benefit of avoiding a jump is largerthan the marginal cost to G at the maximal effort 1 in the scrutiny region. In(e), the lower bound on e ∗ B is not 0, because together with e ∗ G = 0, this wouldmake jump length zero. Instead, the lower bound 1 − dλ − λ + dλ exp( l − l ) on e ∗ B ensures that the jumps from the early career region land in the scrutiny region.The maximal jump length necessary for this is l − l . Condition (f) in Prop. 2ensures that jumps end in a region where the value function can be calculatedin closed form, which is technically convenient.Prop. 2 is useful for checking incentives after numerically calculating thecandidate value functions. In the region of l where e ∗ G ( l ) < e ∗ B ( l ), the valuefunctions cannot in general be found in closed form. Numerical simulations mustbe used. Fig. 4 displays a numerical example of a switched effort equilibrium.Belief µ = exp( l )1+exp( l ) is used on the x -axis instead of l , in order to display V θ ( l )at all l . The y -axis has log scale. The prior probability of G is 0 . l = ln(1 . µ hasfallen to 0 .
4, i.e. l = ln (cid:0) (cid:1) .Relaxing the assumptions on β to bounded and weakly increasing, for some β the value functions can be found in closed form. Such β can be chosen to11 igure 4: A switched effort equilibrium with β ( l ) = exp( l )exp( l )+exp(4 . , c G ( e ) = e , c B ( e ) = e , d = 0, r = , λ = 2. Blue dashed line: V G , purple line: V B , black line: e ∗ B . make the switched effort region unboundedly large (in belief space, from (cid:15) to1 − (cid:15) for any (cid:15) ∈ (0 , )). The next proposition provides sufficient conditions for a switched effort equi-librium in terms of primitives. The value functions are replaced with their closedforms when e ∗ B , e ∗ G ∈ { , } , but bounded by parameters otherwise. The bound-ing makes the results less tight than in Prop. 2. Parts (a)–(f) in Prop. 3 andProp. 2 correspond, in the sense that each inequality in Prop. 3 is a bound onthe respective inequality in Prop. 2. The intuition for Prop. 3 (and Prop. 6 tofollow) is thus the same as for Prop. 2, discussed above and illustrated in Fig. 3. Proposition 3.
Fix l, l ∈ R with l < l . If(a) (cid:82) ∞ l β ( z ) − c G (1) λ exp (cid:0) − r z − lλ (cid:1) dz − β min r ≥ c (cid:48) G (1) λ ,(b) β max r + λ + λβ min r ( r + λ ) − β min r ≤ c (cid:48) B (0) λ ,(c) β ( l ) r − (cid:82) ∞ l β ( z ) − c G (1) λ exp (cid:0) − r z − lλ (cid:1) dz < c (cid:48) G (0) λ ,(d) β ( l ) r − (cid:104) β max r + λ + λβ min r ( r + λ ) (cid:105) < c (cid:48) B (1) λ ,(e) β ( l ) r − (cid:82) ∞ l (cid:104) β ( z ) λ + β min r (cid:105) exp (cid:0) − ( r + λ ) z − lλ (cid:1) dz > c (cid:48) B (1 − exp( l − l )) λ ,(f ) d = 0 ,then there exists l ∈ ( l, l ) and an equilibrium in which e ∗ B ( l ) > e ∗ G ( l ) = 0 if l ∈ ( l, l ] , Example available upon request. ∗ B ( l ) = 0 , e ∗ G ( l ) = 1 if l ∈ [ l , ∞ ) , e ∗ B ( l ) = e ∗ G ( l ) = 0 if l / ∈ ( l, l ] ∪ [ l , ∞ ) . The proof is in the appendix.Prop. 3 requires d = 0, so there exists an effort level making the signal ratezero. Prop. 6 in the appendix covers the case d >
0, so jump length is uniformlybounded. The expressions become more complicated, but the idea is the sameas in Prop. 3.Whenever one switched effort equilibrium exists, there is a continuum ofthem: if the endpoints of the switched effort region are shifted slightly and if e ∗ B ( l ) is adjusted to ensure that (A.2) in the appendix holds, then the result isagain an equilibrium. More formally, the sufficient conditions in Prop. 3 andProp. 6 hold in a nonempty open set of parameters.If d >
0, then the log likelihood ratio never reaches ±∞ , so type is neverlearned with certainty. There is a positive probability that when learning ends,either the B type has reached l > l or G has reached l < l , so a possibilityexists of a ‘wrong’ permanent reputation. Equilibria in which B exerts minimal effort and G maximal in some inter-val, outside which both types exert minimal effort are called extremal effortequilibria . Extremal efforts occur in the scrutiny region of switched effort equi-libria. Sufficient conditions for the existence of extremal effort equilibria areparts (a),(b) of Prop. 2 or of the subsequent Propositions. Value functions aregiven in closed form in the appendix in (A.3).If d = 0, then the scrutiny region (where e ∗ B ( l ) = 0, e ∗ G ( l ) = 1) may extendfrom some l ∈ R to ∞ , as seen in Prop. 3 (a),(b). The larger the log likeli-hood ratio becomes, the greater the incentive to exert effort. This differs fromthe literature on signalling with Brownian noise, where efforts are highest forintermediate beliefs and go to zero as type becomes certain. The reason is thatbelief responds less to the Brownian signal when less uncertainty remains aboutthe type. In the current paper, the effort incentive is determined by the jumplength. Jumps go from l to −∞ when d = 0, e ∗ B ( l ) = 0 and e ∗ G ( l ) = 1.If d >
0, then the scrutiny region is bounded above and below, as shown inLemma 4, which is proved in the appendix. Efforts are greatest at intermediatebeliefs and go to zero as the market becomes certain of the type.
Lemma 4. If d > , then sup { l : e ∗ B ( l ) = 0 , e ∗ G ( l ) = 1 } < ∞ and inf { l : e ∗ B ( l ) = 0 , e ∗ G ( l ) = 1 } > −∞ . Equilibria in which e ∗ G ( l ) = e ∗ B ( l ) > l do not exist, because thelog likelihood ratio does not respond to signals when e ∗ G ( l ) = e ∗ B ( l ). Both typeswill then save cost by minimising effort. This completes the characterisationof equilibria in which e ∗ G ( l ) , e ∗ B ( l ) ∈ { , } . Such equilibria are either extremaleffort or pooling on 0. Equilibria with interior efforts e ∗ G ( l ) ∈ (0 ,
1) or e ∗ B ( l ) ∈ ,
1) are more difficult to characterise. The results about these (other thanswitched effort equilibria) are summarised next.There are no equilibria in which e ∗ G ( l ) < e ∗ B ( l ) in a positive-length interval[ l a , l b ], and e ∗ G ( l ) = e ∗ B ( l ) = 0 for l > l b . The endpoint l b must be finite, becausethe incentives require V B to decrease over the course of the jump. If jumps upcontinue indefinitely, then the flow benefit keeps increasing, so V B increases.The incentives require the value functions after a jump to be strictly ordered: V G ( j ( l )) > V B ( j ( l )). If the jump ends at j ( l ) with e ∗ G ( j ( l )) = e ∗ B ( j ( l )) = 0, asoccurs from close to the right endpoint l b of [ l a , l b ], then V G ( j ( l )) = V B ( j ( l )) = β ( l ) r , a contradiction. The scrutiny region of switched effort equilibria is thusindispensable.If the cost functions are linear, then there are no equilibria in which 1 >e ∗ G ( l ) > e ∗ B ( l ) ≥ l a , l b ] of positive length, with e ∗ G ( l ) = e ∗ B ( l ) = 0elsewhere. Linearity of cost requires indifference between efforts 0 , l for e ∗ θ ( l ) ∈ (0 , l is then the same as when the chosen effort e θ ( l ) iszero, but the receivers expect the equilibrium efforts 1 > e ∗ G ( l ) > e ∗ B ( l ) ≥
0. Thisholds at all l ∈ [ l a , l b ]. If the chosen effort is zero everywhere, then the payoffsto the types are identical everywhere. Therefore V G ( l ) = V B ( l ) at all l . Inparticular, V G ( j ( l )) − V G ( l ) = V B ( j ( l )) − V B ( l ). Then the strong single crossingof the cost functions precludes both types from being indifferent between efforts0 , d = 0 and the cost functions are linear, then there are no equilibria inwhich e ∗ B ( l ) ∈ (0 ,
1) and e ∗ G ( l ) = 1 for l in an interval ( l a , l b ) of positive length.The log likelihood ratio would jump to −∞ from ( l a , l b ), so if B is indifferentbetween efforts 0 , l c ∈ ( l a , l b ), then B strictly prefers 0 at any l < l c and 1at any l > l c . Switched effort equilibria also exist for some parameter values when thePoisson rate of jumps is λ (1 − e θ )+ d θ for type θ , where d G (cid:54) = d B and d G , d B > l still moves. Pooling equilibria may not exist, because l responding tosignals may motivate the sender to exert effort. The possibility of switchedefforts with exogenous information revelation is less surprising than in the puresignalling case, because the countersignalling result of Feltovich et al. (2002)uses exogenous info. The mechanism of countersignalling relies on there beingat least three types. This paper has two types, so the mechanism for switchedefforts is distinct from countersignalling.The value functions will be proved jointly continuous in d G and d B , so ifthe assumptions of Prop. 2 hold strictly, then there is a nonempty open set of d G , d B for which there exists a switched effort equilibrium. Proposition 5.
If Prop. 2 (a)–(f ) hold strictly, then there exists δ > s.t. forany < d G < d B with | d θ − d | < δ , there exists a switched effort equilibrium. d G < d B and e ∗ G ( l ) = e ∗ B ( l ), then l drifts upand jumps down.A similar result to Prop. 5 can be derived with d G > d B . This is a lessintuitive assumption, because the G type is disadvantaged in the signal, butadvantaged in the cost. If d G > d B and e ∗ G ( l ) = e ∗ B ( l ), then the drift of l goesdown and the jumps up. Switched effort equilibria are still continuous in | d θ − d | . A continuity argument analogous to Prop. 5 shows that switched effort equi-libria exist when the signal structure is a L´evy process in which the dominantcomponent is the Poisson process considered in the benchmark model in Sec-tion 2.Adding more types with a two-peaked distribution is expected to yield resultssimilar to the two-type case. Again, this follows from the continuity of the valuefunctions in the prior. The state variable (distribution over types) then becomesmultidimensional, which complicates the analysis.If the benefit depends to a small extent on the true type, the signal, the actualeffort or the effort the market expects from the sender, then the equilibrium isagain continuous in the extent of the influence of these factors.If the environment is modified so that the benefit depends partly on theeffort the market expects, then the model becomes somewhat similar to thecareer concerns situation considered in Holmstr¨om (1999). The remaining con-ceptual difference between the present paper and Holmstr¨om (1999) is that, inthe present paper, the sender knows his type. If the sender does not know histype, then all types choose the same effort, so switched efforts are impossible.A more radical departure from the current model is to make the benefitdepend only on the effort the market expects. Suppose the market expects astrategy consisting of switched efforts on ( l, l ], efforts e ∗ G ( l ) > e ∗ B ( l ) for ( l , l ),and pooling elsewhere. This strategy is similar to the one in Prop. 5. If themarket expects this strategy, then the benefit at any l ∈ ( l, l ) increases in d B − d G , and is zero when d B = d G . With a large enough d B − d G , it maybe possible to construct a switched effort equilibrium according to the patternused above, but its existence cannot be guaranteed based on the continuity of V θ alone. This existence question is left for future research.
4. Noiseless switched efforts
Equilibria in which B exerts more effort than G for some belief can beconstructed in noiseless discrete time signalling using belief threats. This is lesssurprising than the existence of a switched effort equilibrium with full-supportnoise, because the noise forces Bayes’ rule to apply everywhere. To the author’sknowledge, the possibility of switched efforts in pure signalling models (that haveno exogenous information revelation) has not been suggested in the literature.Such equilibria may exist in other models, but other authors have not mentionedit. 15n this model, time is discrete and the horizon infinite. The types are θ = G, B . The initial log likelihood ratio is l . Each period, the sender chooseseffort e ∈ R + , which the receivers observe and use to update their log likelihoodratio. The per-period payoff is β ( l ) − ˆ c θ · e for type θ , with β ( l ) = exp( l )1+exp( l ) (riskneutral sender) and ˆ c B > ˆ c G >
0. The discount factor is δ .In the equilibrium constructed, during the first period, B chooses e > G puts probability q G ∈ (0 ,
1) on e and 1 − q G on 0. After e is publiclyobserved in the first period, both types choose e = 0 forever. After 0 is seen inthe first period, B exerts zero effort forever, but G chooses e > e . After 0 in the firstperiod, as long as e has occurred every period, l = ∞ . A deviation from e after 0 in the first period is followed by l = −∞ and zero effort forever. Thisis the belief threat that sustains effort by G under scrutiny. Any actions notspecified above are punished with l = −∞ forever.After 0 is seen in the first period, the constraints for G to choose e and B to choose 0 are β max − ˆ c G e ≥ β min ≥ β max − ˆ c B e . The constraint for B toexert e in the first period is11 − δ q G exp( l )1 + q G exp( l ) − ˆ c B e ≥ β max + δ − δ β min . For G to be indifferent between 0 and e in the first period, it must be that11 − δ q G exp( l )1 + q G exp( l ) − ˆ c G e = β max + δ − δ ( β max − ˆ c G e ) . Take δ = , c G = 1, c B = 2, l = 0. The above constraints hold for e = , q G = and e = . This constitutes a switched effort equilibrium. Suchequilibria exist for a nonempty open set of parameter values.In this model, it is not possible that B exerts more effort than G withprobability one, because then high effort would reveal B , yielding the minimalflow payoff thereafter. Then B would deviate to the lowest effort level that themarket expects from G . Full separation cannot occur via switched efforts.Adding noise makes discrete time repeated signalling intractable, becausethe state variable (log likelihood ratio) then takes values in an infinite discreteset that is not a regular grid. Neither differential nor difference equations canbe used.
5. Conclusion
Repeating pure signalling permits equilibria in which the high-cost typeexerts more effort than the low-cost for some beliefs of the receivers. Theseequilibria are numerous and occur both with and without noise, fully revealingsignals or exogenous information revelation. To the author’s knowledge, theliterature has not alluded to the possibility of such equilibria in a pure signallingcontext. 16igher signalling effort by the high-cost types can be interpreted as a signof insecurity—they are trying to avoid future information revelation and effort.Intuitively, the weak act tough to deter attack, because they know they couldnot handle it. Similarly, the guilty avoid an investigation. The low-cost typesknow that they can compensate in the future for current low effort, should suchcompensation be necessary. They also know that future information revelationis likely to be good for them.Switched effort equilibria overturn one of the key intuitions from previoussignalling models. A good signal can go bad and then become good again,meaning that the same observation may raise belief at one point of the gameand lower it at another. Single crossing in type and cost does not carry overto single crossing in type and action. In fact, at some point in the game, effortlevels are ordered in the exact reverse order to that commonly found in theliterature. This result does not appear in countersignalling, where the hightypes pool with the low and separate from the medium types.Effort and cost are spread more equally across types in a switched effortequilibrium than with standard separation, because the bad type also exertspositive effort. If signalling has a positive externality (outside the scope of thecurrent model, e.g. education is good for civil society) and the social benefitfrom the externality outweighs its cost, then it may be encouraged as a matterof policy. Fairness considerations may then imply a preference for switchedefforts, e.g. to lead the low-ability workers to acquire at least a minimum ofeducation. Switched effort equilibria spread the cost more evenly than extremalefforts, but less evenly than pooling. However, the zero education in poolingmay be undesirable, leaving switched efforts as the best compromise.
Appendix A.
Proof of Prop. 2.
Condition (f) implies that if e ∗ G ( l ) = 1, e ∗ B ( l ) = 0, then e ∗ G ( j ( l )) = e ∗ B ( j ( l )) = 0. This implies V θ ( j ( l )) = β ( j ( l )) r .The Hamilton-Jacobi-Bellman (HJB) equation for type θ is rV θ ( l ) = β ( l ) + λ [ e ∗ G ( l ) − e ∗ B ( l )] V (cid:48) θ ( l ) (A.1)+ max e { [ λ (1 − e ) + d ] [ V θ ( j ( l )) − V θ ( l )] − c θ ( e ) } . The best response solves λ [ V θ ( l ) − V θ ( j ( l ))] = c (cid:48) θ ( e ) if interior. Corner solutionsfor the best response are given by e θ ( l ) = (cid:40) λ [ V θ ( l ) − V θ ( j ( l ))] ≤ c (cid:48) θ (0) , λ [ V θ ( l ) − V θ ( j ( l ))] ≥ c (cid:48) θ (1) . (A.2)A verification theorem (Theorem 4.6 in Presman et al. (1990) as modified forthe discounted case in Yushkevich (1988)) is used to check that the solutionsof (A.1) coincide with the value functions.In the e ∗ G ( l ) = 1, e ∗ B ( l ) = 0 region, condition (a) implies V G ( l ) − V G ( j ( l )) ≥ c (cid:48) G (1) λ for all l , because V G ( l ) ≥ V G and V G ( j ( l )) = β ( j ( l )) r . In turn, V G ( l ) − G ( j ( l )) ≥ c (cid:48) G (1) λ implies e G ( l ) = 1 based on (A.2). Therefore condition (a)suffices for e G ( l ) = 1 ∀ l in the region with e ∗ G ( l ) = 1, e ∗ B ( l ) = 0.Condition (b) implies V B ( l ) − V B ( j ( l )) ≤ c (cid:48) B (0) λ , because V B ( l ) ≤ V B and V B ( j ( l )) = β ( j ( l )) r . In turn, V B ( l ) − V B ( j ( l )) ≤ c (cid:48) B (0) λ implies e B ( l ) = 0 basedon (A.2). Therefore condition (b) suffices for e B ( l ) = 0 ∀ l in the region with e ∗ G ( l ) = 1, e ∗ B ( l ) = 0.If e ∗ B ( l ) > e ∗ G ( l ) = 0, then lim e ∗ B ( l ) → j ( l ) = l + lim e ∗ B ( l ) → ln d − e ∗ B ( l )+ d ≥ l by (1) and (f). In the equilibrium constructed, e ∗ B ( l ) is chosen in the e ∗ B ( l ) >e ∗ G ( l ) = 0 region to ensure the jumps end in the e ∗ G ( l ) = 1, e ∗ B ( l ) = 0 region(formally e ∗ G ( j ( l )) = 1 , e ∗ B ( j ( l )) = 0). Condition (c) implies V G ( l ) − V G ( j ( l )) ≤ c (cid:48) G (0) λ , which in turn implies e G ( l ) = 0 based on (A.2). Therefore condition (c)suffices for e G ( l ) = 0 ∀ l in the region with e ∗ B ( l ) > e ∗ G ( l ) = 0.To show that (d) and (e) imply the existence of an equilibrium level of e ∗ B ( l )at each l in the e ∗ B ( l ) > e ∗ G ( l ) = 0 region, first the candidate value functions inthe e ∗ G ( l ) = 1, e ∗ B ( l ) = 0 region are calculated. Substituting e ∗ B ( l ) = e B ( l ) = 0and e ∗ G ( l ) = e G ( l ) = 1 into each type’s HJB equation and solving the resultingordinary differential equations (ODEs) yields V G ( l ) = exp (cid:18) − ( r + d )( l − l ) λ (cid:19) V G ( l )+ (cid:90) ll (cid:20) β ( z ) − c G (1) λ + β ( j ( z )) drλ (cid:21) exp (cid:18) − ( r + d )( z − l ) λ (cid:19) dz, (A.3) V B ( l ) = exp (cid:18) − ( r + λ + d ) l − lλ (cid:19) V B ( l )+ (cid:90) ll (cid:20) β ( z ) λ + ( λ + d ) β ( j ( z )) rλ (cid:21) exp (cid:18) − ( r + λ + d ) z − lλ (cid:19) dz, where j ( z ) = z +ln (cid:16) dλ + d (cid:17) . If l is finite, then value matching gives lim l → l V θ ( l ) = V θ ( l ) = β ( l ) r , which provides the boundary condition for the ODEs in (A.3).By (A.3), V θ is strictly increasing in l , so V θ = V θ ( l ) and V θ = lim l → l V θ ( l ).The limit is relevant when l = ∞ , in which case lim l → l V θ ( l ) < β max r , as can beseen from (A.3).Given conditions (a)–(c) and (f), e ∗ B ( l ) in the region with e ∗ B ( l ) > e ∗ G ( l ) = 0is part of an equilibrium iff λ (cid:20) V θ ( l ) − V θ (cid:18) l + λ + dλ (1 − e ∗ B ( l )) + d (cid:19)(cid:21) = c (cid:48) θ ( e ∗ B ( l )) . (A.4)This condition merely says e ∗ B ( l ) is a best response to e ∗ G ( l ) = 0 and itself.Note that the left hand side (LHS) of (A.4) strictly decreases in e ∗ B ( l ) and theright hand side (RHS) increases. The LHS is continuous, because l is fixedand V θ (cid:16) l + λ + dλ (1 − e ∗ B ( l ))+ d (cid:17) is continuous by (A.3). The RHS is continuous byassumption. 18ondition (d) implies V B ( l ) − V B < c (cid:48) B (1) λ and (e) implies V B ( l ) − V B ≥ c (cid:48) B (ˆ e ) λ for ˆ e solving l + ln (cid:16) λ + dλ (1 − ˆ e )+ d (cid:17) = l . These bounds, the continuity of (A.4)and the Mean Value Theorem imply that there exists e ∗ B ( l ) ∈ [ˆ e,
1) s.t. (A.4)holds.
Proof of Prop. 3. (a) V G = (cid:82) ∞ l β ( z ) − c G (1) λ exp (cid:0) − r z − lλ (cid:1) dz when l = ∞ basedon (A.3). When e ∗ G ( l ) = 1, e ∗ B ( l ) = 0 and d = 0, then j ( l ) = l + ln = −∞ by (1), so V θ ( j ( l )) = β min r . By (A.2), condition (a) suffices for e G ( l ) = 1 for any l s.t. e ∗ G ( l ) = 1, e ∗ B ( l ) = 0.(b) lim l →∞ V B ( l ) = β max r + λ + λβ min r ( r + λ ) when l = ∞ based on (A.3). By (A.2),condition (b) suffices for e B ( l ) = 0 for any l s.t. e ∗ G ( l ) = 1, e ∗ B ( l ) = 0, because j ( l ) = −∞ . Equilibrium behaviour in the e ∗ G ( l ) = 1, e ∗ B ( l ) = 0 region is thusguaranteed.(c)–(e). In the e ∗ B ( l ) > e ∗ G ( l ) = 0 region, in the absence of signals, l driftsdown by (2). At ˆ l ∈ ( l, l ], the probability of reaching l is exp (cid:16)(cid:82) ˆ ll λ (1 − e ∗ B ( z )) − λe ∗ B ( z ) dz (cid:17) for B , because at l , signals occur at rate λ (1 − e ∗ B ( l )) and l drifts λ [ e ∗ G ( l ) − e ∗ B ( l )]per unit of time. For G , the probability is exp (cid:16)(cid:82) ˆ ll λ − λe ∗ B ( z ) dz (cid:17) , smaller than for B because of lower effort of avoiding jumps. The payoff conditional on reaching l is β ( l ) r and conditional on not reaching, bounded by β min r and β max r . The discountrate is r and the time it takes to drift from l to l is (cid:12)(cid:12)(cid:12)(cid:82) ll dzλ [ e ∗ G ( z ) − e ∗ B ( z )] (cid:12)(cid:12)(cid:12) . Thereforefor any l ∈ ( l, l ],exp (cid:32)(cid:90) ll − dze ∗ B ( z ) (cid:33) exp (cid:32)(cid:90) ll − rdzλe ∗ B ( z ) (cid:33) β ( l ) r + (cid:34) − exp (cid:32)(cid:90) ll λdz − λe ∗ B ( z ) (cid:33)(cid:35) β max r ≥ V θ ( l ) ≥ (A.5)exp (cid:32)(cid:90) ll − dze ∗ B ( z ) (cid:33) exp (cid:32)(cid:90) ll − rdzλe ∗ B ( z ) (cid:33) β ( l ) r + (cid:34) − exp (cid:32)(cid:90) ll λdz − λe ∗ B ( z ) (cid:33)(cid:35) β min r . To construct the e ∗ B ( l ) > e ∗ G ( l ) = 0 part of the equilibrium, first conjecture e ∗ B ( l ) ≥ − exp ( l − l ) ∀ l ∈ ( l, l ) so that e ∗ G ( j ( l )) = 1, e ∗ B ( j ( l )) = 0, secondshow that V θ is continuous at l , third use (c) to ensure e ∗ G ( l ) = 0, and fourthuse (d),(e) to ensure e ∗ B ( l ) ∈ [1 − exp ( l − l ) , e ∗ B ( l ) ≥ (cid:15) > ∀ l ∈ ( l, l ], then by (A.5) and the Squeeze Theorem,lim l → l + V θ ( l ) = β ( l ) r . This and (c) imply the existence of l ∈ ( l, l ) s.t. for all l ∈ ( l, l ], V G ( l ) − V G ( j ( l )) ≤ c (cid:48) G (0) λ , which by (A.2) implies e G ( l ) = 0.Condition (d), lim l →∞ V B ( l ) = β max r + λ + λβ min r ( r + λ ) and lim l → l + V θ ( l ) = β ( l ) r implythe existence of l ∈ ( l, l ) s.t. for all l ∈ ( l, l ], V B ( l ) − V B ( j ( l )) < c (cid:48) B (1) λ , whichby (A.2) implies e B ( l ) < l ∈ ( l, l ) s.t. for all l ∈ ( l, l ], V B ( l ) − V B ( j ( l )) ≥ c (cid:48) B (1 − exp( l − l )) λ , which by (A.2) implies e B ( l ) ≥ − exp ( l − l ). 19et l := min { l , l , l } . As in the proof of Prop. 2, the bounds from con-ditions (d),(e) and the Mean Value Theorem imply that there exists e ∗ B ( l ) ∈ [1 − exp ( l − l ) ,
1) s.t. (A.4) holds.
Proposition 6.
Fix l, l , l ∈ R with l < l < l . If d > and(a) min l ∈ [ l ,l ] (cid:26) V G ( l ) − β ( l +ln ( dλ + d )) r (cid:27) ≥ c (cid:48) G (1) λ ,(b) max l ∈ [ l ,l ] (cid:26) V B ( l ) − β ( l +ln ( dλ + d )) r (cid:27) ≤ c (cid:48) B (0) λ ,(c) β ( l ) r − V G ( l ) < c (cid:48) G (0) λ ,(d) β ( l ) r − β ( l ) r < c (cid:48) B (1) λ ,(e) β ( l ) r − V B ( l ) > c (cid:48) B (1 − d/λ − ( λ + d ) exp( l − l ) /λ ) λ ,(f ) l + ln (cid:16) dλ + d (cid:17) < l ,where V θ is given in (A.3), then there exists l ∈ ( l, l ) and an equilibrium inwhich e ∗ B ( l ) > e ∗ G ( l ) = 0 if l ∈ ( l, l ] , e ∗ B ( l ) = 0 , e ∗ G ( l ) = 1 if l ∈ [ l , l ] , e ∗ B ( l ) = e ∗ G ( l ) = 0 if l / ∈ ( l, l ] ∪ [ l , l ] . The proof is omitted, because it is similar to that of Prop. 3: value func-tions or bounds on them are substituted into the conditions of Prop. 2 to checkincentives. Condition (d) in Prop. 6 always holds, but is added for better com-parability to the other propositions.
Proof of Prop. 5.
The equilibrium with jump rate λ (1 − e θ ) + d θ will be con-structed using the same l, l , l as in Prop. 6. If the Poisson rate is λ (1 − e θ ) + d θ for type θ , then (1) becomes j ( l ) = l + ln (cid:18) λ (1 − e ∗ G ( l )) + d G λ (1 − e ∗ B ( l )) + d B (cid:19) . (A.6)In the absence of jumps, the log likelihood ratio drift is λ [ e ∗ G ( l ) − e ∗ B ( l )] − d G + d B .Jumps reach l from l in the e ∗ B ( l ) > e ∗ G ( l ) region iff e ∗ B ( l ) ≥ − d B λ − λ + d G λ exp( l − l ), which is implied by Prop. 6 (f) holding strictly and by | d θ − d | < δ .If d G < d B , then l is a stasis point : to the left of l and at l , the drift of the loglikelihood ratio is positive and to the right of l , negative. After reaching l , the l process oscillates around it with infinite frequency and zero amplitude, spendingfraction w := d G − d B + λ lim l → l + e ∗ B ( l ) λ lim l → l + e ∗ B ( l ) of any time interval infinitesimally to the left20f l and 1 − w infinitesimally to the right. The drift at l is a mixture of drifts tothe left and to the right of it, with weight w on the left-side drift. A fraction w of the time, jumps to j − ( l ) := l + ln (cid:16) λ + d G λ + d B (cid:17) occur for type θ at rate λ + d θ . Afraction 1 − w of the time, jumps to j + ( l ) := l + ln (cid:16) λ + d G λ (1 − lim l → l + e ∗ B ( l ))+ d B (cid:17) > l occur for G at rate λ + d θ and for B at rate λ ∗ B := λ (1 − lim l → l + e B ( l )) + d B .The values of the types at l are V G ( l ) = β ( l ) r + λ + d G + λ + d G r + λ + d G [ wV G ( j − ( l )) + (1 − w ) V G ( j + ( l ))] ,V B ( l ) = β ( l )( λ + d B ) w + λ ∗ B (1 − w ) + ( λ + d B ) w ( λ + d B ) w + λ ∗ B (1 − w ) V B ( j − ( l )) (A.7)+ λ ∗ B (1 − w )( λ + d B ) w + λ ∗ B (1 − w ) V B ( j + ( l )) . Value is continuous at l by the same argument as in the proof of Prop. 3. Forany (cid:15) > δ > | d θ − d | < δ , then w > − (cid:15) and | l − j − ( l ) | < (cid:15) .Then by (A.7) and the continuity of V θ , β ( l ) r + λ + d G + λ + d G r + λ + d G (cid:15) β max r ≥ V G ( l ) − ( λ + d G )(1 − (cid:15) ) r + λ + d G [ V G ( j − ( l )) + (cid:15) ] ,V G ( l ) − ( λ + d G )(1 − (cid:15) ) r + λ + d G [ V G ( j − ( l )) − (cid:15) ] ≥ β ( l ) r + λ + d G + λ + d G r + λ + d G (cid:15) β min r , so by the Squeeze Theorem, lim d G ,d B → d V G ( l ) = β ( l ) r . A similar argument showslim d G ,d B → d V B ( l ) = β ( l ) r .At any l s.t. e ∗ B ( l ) = e ∗ G ( l ) = 0, for any T, (cid:15) > δ > | d θ − d | < δ , then with probability 1 − (cid:15) , the flow payoff remains within (cid:15) of β ( l ) for at least T units of time. This is because by (A.6), lim d θ → d j ( l ) = l , andwhen e ∗ B ( l ) = e ∗ G ( l ), the drift of the log likelihood ratio is − d G + d B . Thereforelim d G ,d B → d V θ ( l ) = β ( l ) r .Using (A.6) and the drift λ [ e ∗ G ( l ) − e ∗ B ( l )] − d G + d B in the HJB equation (A.1)and solving for V θ in the e ∗ B ( l ) = 0, e ∗ G ( l ) = 1 region, (A.3) becomes V G ( l ) = exp (cid:18) − ( r + d G )( l − l ) λ − d G + d B (cid:19) V G ( l )+ (cid:90) ll β ( z ) − c G (1) + d G V G ( j ( z )) λ − d G + d B exp (cid:18) − ( r + d G )( z − l ) λ − d G + d B (cid:19) dz, (A.8) V B ( l ) = exp (cid:18) − ( r + λ + d B )( l − l ) λ − d G + d B (cid:19) V B ( l )+ (cid:90) ll β ( z ) + ( d B + λ ) V B ( j ( z )) λ − d G + d B exp (cid:18) − ( r + λ + d B )( z − l ) λ − d G + d B (cid:19) dz. From e ∗ B ( l ) = e ∗ G ( l ) = 0, it follows that lim d G ,d B → d V θ ( l ) = β ( l ) r in (A.8). From e ∗ B ( j ( l )) = e ∗ G ( j ( l )) = 0, it follows that lim d G ,d B → d V θ ( j ( l )) = β ( j ( l )) r in (A.8).Based on these, the limit of (A.8) as d θ → d is (A.3).21ince Prop. 6 (a)–(f) hold strictly and V θ ( l ) is continuous in d G , d B for any l , there exist l ∈ ( l, l ) and δ > | d θ − d | < δ , then Prop. 2 (a)–(f) aresatisfied. This suffices for the existence of a switched effort equilibrium. Proof of Lemma 4. If { l : e ∗ B ( l ) = 0 , e ∗ G ( l ) = 1 } = ∅ , then the result is vacu-ously true. Suppose there exists l s.t. e ∗ B ( l ) = 0, e ∗ G ( l ) = 1. Define j n ( l ) := j ( j n − ( l )) and j ( l ) := j ( l ) = l + ln (cid:16) dλ + d (cid:17) . Clearly j ( l ) − l ∈ ( −∞ , n ∗ := λ ( β max − β min ) rc (cid:48) G (1) . By (A.2), e ∗ G ( l ) = 1 requires λ [ V θ ( l ) − V θ ( j ( l ))] ≥ c (cid:48) θ (1) . Since V θ ( l ) ∈ (cid:104) β min r , β max r (cid:105) , we havesup (cid:110) n ∈ N : ∃ l s.t. l, j ( l ) , . . . , j n ( l ) ∈ (cid:110) ˆ l : e ∗ B (ˆ l ) = 0 , e ∗ G (ˆ l ) = 1 (cid:111)(cid:111) ≤ n ∗ . (A.9)Pick l < l s.t. for any ˆ l ∈ ( l , l ), e ∗ B (ˆ l ) = 0, e ∗ G (ˆ l ) = 1. By (A.9), l − l ≤ n ∗ / ln (cid:16) dλ + d (cid:17) < ∞ . Raise l maximally, so for any η > e ∗ B ( l + η ) = e ∗ G ( l + η ) =0. In this case, by continuity of the V θ given in (A.3), lim l → l V θ ( l ) = β ( l ) r .By definition of n ∗ , if e ∗ B ( l ) = 0 and e ∗ G ( l ) = 1, then there exists ˆ n ∈ N ,ˆ n < n ∗ + 1, s.t. e ∗ B ( j ˆ n ( l )) = e ∗ G ( j ˆ n ( l )) = 0. Let n be the minimal such ˆ n and let l ∗ = l − ν for ν > β and β (cid:48) >
0, there exist
N, (cid:15) > | l | > N ,then β ( l ) − β ( j n ( l )) < r(cid:15) . Take (cid:15) = c (cid:48) G (1) λ . Then there exists N > l > N , then V θ ( l ∗ ) − V θ ( j n ( l ∗ )) < c (cid:48) G (1) λ . Since V θ is strictly increasingin the e ∗ B ( l ) = 0, e ∗ G ( l ) = 1 region, this implies the failure of (A.2) at all l = l ∗ , j ( l ∗ ) , . . . , j n − ( l ∗ ). Thus there is no incentive for G to exert maximaleffort. This shows l ≤ N for all intervals ( l , l ) on which switched efforts occur.To show l ≥ − N , note that V θ ( l ) < V θ ( l ) = β ( l ) r for any l ∈ ( l , l ). If V θ ( l ) − β min r < c (cid:48) G (1) λ , then (A.2) fails at any l ∈ ( l , l ). It was shown above that l − l ≤ n ∗ / ln (cid:16) dλ + d (cid:17) , so for G to have an incentive to exert maximal effort, itis necessary that l ≥ β − (cid:16) rc (cid:48) G (1) λ + β min (cid:17) − n ∗ / ln (cid:16) dλ + d (cid:17) . References
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