Strategic Information Transmission under Reputation Concerns

Transcription

1 Strategic Information Transmission under Reputation Concerns Emilia Oljemark October 7, 03 Abstract This paper analyzes strategic information transmission in a repeated model of communication. The sender learns about the possible bias of the receiver by observing his chosen actions. Decisions dier in their importance for the two players across periods. I analyze and compare two regimes, one in which receiver's stakes are his private information, and an other in which the stakes are commonly known. Surprisingly, in a wide range of environments, disclosing receiver's stakes reduces all players' ex ante welfare. Because the stakes determine in equilibrium the behavior of the bad type, knowledge of them makes his action ex ante observable to the sender. If receiver is known to misbehave, sender engages in screening only if her own stakes are low, which undermines communication. Remaining ignorant about the stakes encourages sender to transmit informative reports in the hope of beneting from the bad type's reputation concerns. The ex ante gains from this are enough to compensate for the ex ante losses in case the biased type deviates immediately. JEL Classication: C73, D7, D8, D83 Keywords: Cheap talk, Two-sided incomplete information, Transparency, Reputation Department of Economics, Aalto University School of Business, P.O. Box 40, FI AALTO, Finland; and HECER (Helsinki Center of Economic Research). emilia.oljemark@aalto..

2 Introduction If a lobbyist, who transmits information to a politician, wants to fully benet from her private information and inuence decision-making, she needs to know the politician's objectives. How aligned the objectives of a politician are with those of the lobbyist determines how eciently information is shared; with a like-minded politician a lobbyist is safe in revealing her information perfectly. How, then, is a lobbyist to design her communication strategy when the objectives of a politician are unknown, and the future value of the partnership depends on today's play? Models of strategic communication in economic theory have been mostly applied to settings in which communication partners either know each other's preferences, or a sender has an informational advantage both regarding the underlying state of the world and his own payo-type. This paper analyzes formally the implications to communication of a setting in which a sender is no longer the more informed player. Building on the seminal cheap talk model of Crawford & Sobel (98), I characterize equilibria when a single sender transmits private information to a single receiver whose preferences are his private information. To the best of my knowledge, the current informational setup has not been formally analyzed in the cheap talk literature. However, the abundance of real life settings makes it a relevant scenario to look at. It seems natural to assume that senders, often labeled as advisers or experts, do not always know how their recipient will use the information conveyed. Such settings are prevalent for instance in politics where lobbyists interact with politicians or government ocials whose positions on various, often complex, political issues are not always known to the lobbyist. In a similar vein, nancial advisers must give stock recommendations to investors whose risk preferences may not be commonly known. The same applies also to numerous principal-agent relationships within rms in which an employee submits (verbal) reports to a manager who acts on the information transmitted. Moreover, these relationships are often longterm in nature, allowing both players' strategies to depend on the realized history of play. Specically, decision makers could prot from the uncertainty of the sender by disguising themselves as good types so as to benet in later periods where the precision of communication is decreasing in sender's perception about the actual conict of interest. To allow for learning, the communication relationship is a repeated one. This is also something that is widely observed. Decision makers tend to listen to the opinions

3 of the advisors who they have consulted earlier, provided that their recommendations have been correct before. This long-term structure of advising allows for learning to take place, both ways; sender learns about receiver's type but at the same time sender must take care of her own reputation as being a reliable source of information. With multiple decisions comes the fact that dierent decisions bear dierent weights, both to the decision maker and the advisor. It seems generally to be the case that all players possess some ranking for a given set of decisions, and these rankings may result either from purely subjective evaluations, or they may be based on some objective measure, for instance the monetary value of a given project. Ranking of decisions brings imperfect monitoring into the picture. Namely, a biased decision maker would nd it optimal to mimic an unbiased type if the rst period is not important to him. Pretending to be someone who implements advisor's recommendations as they come allows a biased type to cash in on this reputation at a later date when the decision is of importance to him. Investing in reputation may be worthwhile if it provides the biased type with more accurate information in the future when the sender is more condent that she is facing a like-minded receiver. By and large, under the presence of reputation concerns, a sender is better o by postponing the revelation of the bad type as long as possible and by reaping the immediate benets from his reputation building. The aim of this paper is twofold: rstly, to characterize equilibria of two alternative cheap talk models; one in which receiver's relative decision weight is his private information, and an other in which this information is publicly known. Secondly, I will compare the two models in terms of the ex ante welfare they yield to each player. The results suggest that, under the presence of imperfect information about the state of the world and about receiver's payo-type, disclosing the relative decision weight is ex ante welfare reducing for a wide range of environments. Ignorance is a bliss in the sense that by not knowing the precise motivations of the bad type, sender takes into account the positive probability that a bad type invests in reputation, and therefore, to prot from this, transmits more information than if it was known how the bad type will actually behave. Related literature The seminal paper on cheap talk by Crawford and Sobel (98) has been followed by numerous applications and variations. Most notably for the current paper, the full information assumption in Crawford and Sobel (98) concerning the preferences of the players has been subject to discussion in several studies. Sobel (985), Bénabou & Laroque (99), Morris (00), Morgan & Stocken (003) and Li & 3

4 Madarász (008) among others introduce incomplete information about sender's type. Sobel (985) analyzes, in a discrete setting, a dynamic communication game where the receiver is uncertain about the sender's preferences. Extending the static cheap talk game over several periods brings about reputation concerns as the sender may wish to keep up a reputation as a trustworthy source of information. The model assumes that senders can be of two types, good and bad, where good senders are nonstrategic and always tell the truth while bad senders choose between truth-telling (investing in reputation) or lying (exploiting reputation). Moreover, bad senders' preferences directly conict with those of the receiver. Bénabou & Laroque (99) analyze a version of Sobel's model in which sender's private information is noisy. This hinders receivers' ability to learn whether sender's reports are honest which gives some senders an incentive to misreport repeatedly, without ever being fully found out. Morris (00) analyzes a model close to Bénabou & Laroque (99), the main dierence being in the bias distribution of the sender: sender has either a zero bias or a positive bias. Further, by letting both types behave strategically, the model endogeneizes the behavior of the good sender in Bénabou & Laroque (99). Morgan & Stocken (003) is a static analog of Morris (00) with the dierence that Morgan and Stocken assume a continuous state space and allow the sender to be perfectly informed of the true state. Moreover, whereas in the previous models the bias distribution of the sender is skewed in one direction (there are good advisers who are unbiased and bad advisers who have a non-zero bias in one direction), Li & Madarász (008), which builds on the previous work by Li (004), introduce uncertainty about the direction of the sender's bias. By allowing sender to have either a positive or a negative bias, they focus on studying whether the mandatory disclosure of sender's bias is welfare improving or not as compared to voluntary disclosure via cheap talk. They conclude that it never benets the receiver nor the sender to have mandatory disclosure of the bias of the sender. Cheap talk models with a possibly naive receiver come quite close to my model. Kartik, Ottaviani and Squintani (007) study a model in which receivers may be credulous and blindly believe the sender's message. By further imposing an unbounded state space and a cost for the sender from misreporting they show that a fully separating equilibrium exists in the game. Their results rely on the assumption that a strategic receiver can perfectly decode a message and that a fully naive receiver takes the action recommended without any reservation. In my model, in contrast, all receiver types 4

5 behave strategically. Other examples of models with naive receivers are Ottaviani & Squintani (006) and Chen (0). Two-sided asymmetric information in cheap talk games has been studied by Chen (009). In her model, however, receiver's private information concerns a noisy observation of the state variable, and diers in this sense from mine. Watson (996) deals with a model with two-sided asymmetric information in which a sender is confused about his information; he does not fully understand it because he lacks a decoding device which the receiver, in turn, possesses. Sender's confusion in Watson's model has the same implications as in my model: sender is uncertain about the reaction of the receiver to his information. However, in the model of Watson, sender's uncertainty is fundamentally about the state of the world while in my model, sender's uncertainty is about receiver's preferred way to use the information. Watson shows that when sender is suciently confused a fully revealing communication equilibrium exists. He further notes that the receiver may exploit sender's confusion by delaying decisions or clustering them in time. This might be relevant also in the present context. As to the issue of transparency, Prat (005), Matozzi & Merlo (007), Levy (007), and Bar-Isaac (0) are but a few examples about transparency in principal-agent relationships. However, transparency in their papers has been coined as the ability of the principal to either observe how the agent behaves and/or what the consequences of such behavior are. For example, principal may or may not observe agent's choice of eort. The transparency in the current paper deals with implicitly observing the action of the agent ex ante, before it is actually chosen. The agent's (receiver's) type is partly characterized by a random variable which in equilibrium will determine the action of the bad agent. As a result, if principal (sender) observes the realization of this variable, she knows the agent's action ex ante. The rest of the paper is structured as follows. Section formalizes the model where uncertainty concerns receiver's payo-type. Section 3 solves for the informative equilibria of the game. Section 4 covers a variant of the model in which uncertainty concerns both receiver's payo-type and his stakes. Section 5 compares the two models in terms ex ante welfare. Discussion and limitations of the model are covered in section 6, and section 7 concludes. 5

6 The Model There are two players, a sender (S) and a receiver (R) who interact twice, in time periods t =,. Before playing begins, Nature moves and draws three parameters once and for all. Firstly, Nature draws R's payo-type which is privately observed by R. His payotype is denoted by variable τ which may take one of two values, U or B, where U stands for unbiased and B for biased. The bias type determines R's preferences vis-à-vis the sender: unbiased R has preferences aligned with those of S, whereas a biased R has preferences in pure conict with S. The bias type is drawn from the following commonly known distribution: U wp. π τ =. B wp. π From here on, I take up the short-hand notation R U and R B for unbiased and biased receiver, respectively. Secondly, all players publicly observe the realizations of s and ω which are drawn from distributions H and G, respectively. Both distributions have a continuous support, S R +, and Ω R +. These parameters measure players' stakes in the rst period, that is, the relative importance of the rst period as compared to the second period. Specically, if stakes are high, the decision to be made in the rst period is more important than that in the second period. Conversely, low stakes mean that the rst period is not as important as the second period. The stake may also be used to capture discounting, but its role is mainly to measure any time-independent factors that aect the importance of a particular decision. There are many reasons why one decision may be more important than another, depending on the application. For example, politicians put more weight on decisions that have regional impacts on their home district, and a project manager cares more about the project which is more likely to have a larger impact on his career. Moreover, it is assumed that the distribution H is independent from the state of the world, and G is independent of τ. As a result, R's type is a twodimensional vector, (τ, ω ) {U, B} Ω, and there is asymmetric information about the rst dimension. In the repeated game, S hopes to learn τ in order to be able to use rst-best communication strategy throughout the rest of the game. With R U, this would be perfect communication, with R B this would be babbling. 6

7 When Nature has moved, we can take a look at the stage game. At the start of each period, S observes privately the realization of a state of the world, θ t Θ t {0, }, with Pr(θ t = 0) = Pr(θ t = ) = for t =,. The binary state of the world could concern, for example, whether or not a certain project should be carried out, whether a tax rate should be set high or low, whether or not to build a new nuclear power plant, or hire a job applicant, or whether the economy is forecast to grow or decline. After observing the state, S reports it to the receiver, by sending a message m t M t = {0, }. After observing m t, R chooses an action, a t, from A t = {0, } so as to maximize his expected payos. All messages and actions are publicly observed. Because I assume that (i) receiver's action is payo relevant to both players, (ii) sender's message is unveriable, and (iii) there is an expected conict of interests as long as π > 0, the information transmission that results is modeled as a cheap talk game between the players. At the end of each period, each player privately observes his or her payos. This implies that R is able to deduce the realization of θ t and hence determine whether S reported the state truthfully or not. This is crucial for the analysis of the game.. Payos The stage game payos for S are given by a function u S : Θ A R, and for the receiver by u R : Θ A {U, B} R. In addition, both players weight the rst period with s and ω, respectively. The stage-game payos of S and R U are maximized when a t = θ t. The payos of R B, in turn, are maximized when a t = θ t. That is, R B 's preferences are in pure conict with those of S. The stage-game payo vector (u S, u U, u B ) as a function of the state and the action taken is characterized by the following matrix. a = 0 a = θ = 0 0, 0, 0,, θ =,, 0, 0, 0 Since players' payos are symmetric in the state of the world, the realization of θ t is not of importance per se. Instead, what players care about is whether or not S's report was truthful, and whether or not R followed the report. Therefore, when analyzing the game, one can abstract away from the realization of θ t and the explicit choice of 7

8 message, m t. Finally, both players maximize the sum of expected stage-game payos which are given as U S (θ, θ, a, a : ; s) = su S (θ, a ) + u S (θ, a ) U R τ (θ, θ, a, a : ; ω) = ωu τ (θ, a ) + u τ (θ, a ), τ {U, B}.. Histories and strategies The history of play in period is the empty set h { }. The set of period public histories is given by H = (Θ M A), with a typical element h. There are 3 possible histories at the beginning of the second period. From S's point of view, histories can be divided into two categories, good or bad, with the good histories given by the set HS+ = {h a = m }, and the bad histories by the set HS = {h a m }. On the other hand, from R's point of view, good histories are only those where S has reported the state correctly. Therefore, denote the set of good histories from R's point of view by HR+ = {h m = θ }, and the set of bad histories by HR = {h m θ }. As a result, for the analysis of the game, what matters are the commonly perceived good or bad histories. Therefore, denote the commonly perceived set of good histories by H+ HS+ H R+ = {(0, 0, 0), (,, )}. All the remaining histories of play are categorized as bad, since along such a history either the receiver or the sender has misbehaved. A behavior strategy for S in period t =, is a function σ t : H t Θ S [0, ], where σ t (h, θ, s t ) gives the probability that she reports the state truthfully given history h t, the current state of the world, θ t, and her stake s t. By denition of s, the stakes at t = are equal to. The probability that S misreports the state, or lies, is given by the complementary probability σ t (h t, θ t, s t ). A behavior strategy for R of type (τ, ω t ) is a function µ t : H t M Ω {U, B} [0, ], where µ t (h t, m, ω t, τ) gives the probability that R follows S's message by taking action a t = m t given his type, message received, and the history of play. The probability that R deviates from S's message, by not following it, is then given by µ t (h t, m, ω, τ). Again, by denition of ω, the stakes at t = are equal to. 8

9 .3 Equilibrium concept The equilibrium concept applied is Perfect Bayesian equilibrium (PBE). This requires sequential rationality, that is, for any date t and any history h t the strategies being played from h t onwards constitute a Bayesian equilibrium of the continuation game. Formally, given the distribution for posterior beliefs, π t, and history h t, let V S ((σ, µ) h t, θ, s t, ω t ) be the expected continuation payo of sender of type θ under strategy prole (σ, µ) conditional on reaching history h t. Similarly, let V τ ((σ, µ) h t, ω t, s t, m) be the expected continuation payo of a receiver of type (τ, ω) under strategy prole (σ, µ) conditional on reaching h t. With the continuation payos dened, a PBE of the two-period game consists of a strategy prole (σ, µ U, µ B ) and posterior beliefs π t (h t ), f (θ m) such that i) for the sender, for each state of the world θ, alternative strategy σ, and for any history h t, V ( S (σ, µ U, µ B ) h t, θ, s t, ω t, π t (h t ) ) ) V ((σ S, µ U, µ B ) h t, θ, s t, ω t, π t (h t ). ii) for a receiver of type (τ, ω t ), for each message m, alternative strategy µ, and for any history h t, V ( τ (µ τ, σ) h t, m, ω t, s t, π t (h t ) ) ) V ((µ τ τ, σ) h t, m, ω t, s t, π t (h t ). iii) beliefs about the state of the world held by R, and beliefs about R's type 9

10 held by S are updated according to Bayes' rule whenever possible. That is, π (h ) = Pr {τ = U a (m ) = a} = π µ (m, ω, s, U) π µ (m, ω, s, U) + ( π )µ (m, ω, s, B) Pr(θ t = k m t = k, h t ) = Pr(θ t=k)σ t (h t,k,s t) k Θ Pr(θt=k )σ t (h t,k,s t)..4 A special case without reputation To illustrate how sender's incentives to screen the receiver in the rst period of the game aect the informativeness of communication, consider the following example where ω =. That is, receiver puts equal weight on the decisions in periods and. Sender's stakes are given by some s 0. The game always holds a babbling equilibrium. I focus on characterizing all informative equilibria of the game. Denition. An equilibrium is said to be informative if, after receiving a message m, receiver's belief Pr(θ = k m = k) >, for k {0, }. As a result, in any informative equilibrium, receiver's expected payos E θ u(a(m), θ) are higher than in a non-informative, babbling, equilibrium. A feature that arises from the repeated structure of the game with publicly observed stage-game outcomes is that R U can exert pressure on sender by punishing her if she fails to report θ truthfully. This strategy can be implemented as follows: At t =, R U follows sender's report. At t =, after knowing the outcome of the rst period, if sender was honest R U follows her second report. But if S misreported the rst state, she is punished forever with a babbling equilibrium which yields her her max-min payos. Eectively, this translates to the communication relationship being cut out and sender losing her inuencing channel. Sender, on the other hand, follows a grim-trigger strategy that tells to babble whenever anyone, including herself, has deviated earlier in the game. 0

11 .4. Strategies in period Working backwards, the second and last period of the game is played like a static game. R B will play his myopic best response to S's message, and therefore deviates from any truthful message. As to R U, he conditions his play on sender's action in the rst period: if she has been truthful, R U follows the second report, but if she has lied earlier, R U plays according to the babbling equilibrium. The interesting case is the one in which the history of play is good. Denote in this case the prior in the second period by π +. Given the strategy of both receiver types, if S reports the second state truthfully, she obtains an expected payo of ( π +). On the other hand, if S deviates and instead lies about the state, keeping receivers' strategies unchanged, she obtains an expected payo of π +, due to R U following the misleading message. Truthful reporting in the second stage is optimal whenever π +. If π + =, sender is indierent between reporting the second state truthfully or lying about it. If π+ <, sender cares more about maximizing her payos against the biased receiver. However, since the preferences of sender and R B are in pure conict, the equilibrium can only consist of babbling..4. Strategies in period Turning to the rst period of the game, for a given prior π, if R B is to invest in reputation, it must maximize his expected payos. A direct application of the payos tells that mimicking R U 's action R B gets, and by separating from R U, R B would get an expected payo of ω +, where is his expected payo in a babbling equilibrium. Hence, R B would follow the rst message only if ω, that is, only if the decision in the second period is at least twice as important as the decision in the rst period. When both periods are equally important, the rst decision should not be wasted for reputation building, and R B deviates immediately. As a result, in equilibrium, π + =. Given the strategies of both types of R, for any realization of θ, if sender reports the rst state truthfully she obtains an expected payo of ( π ) ( s + ). By deviating and misreporting θ, S would obtain an expected payo of π s. As a result, truthful reporting is optimal in the rst period only when sender's stakes are low enough, as captured in Lemma below. Lemma. When ω >, all informative equilibria of the game are separating. Sender's

12 strategy at t = is characterized by the following cuto strategy. if s ŝ σ (θ) =, where sup(s) if π ŝ = π. ( π ) Lemma establishes that for priors below one half, S engages in screening only if her stakes are low enough. The cuto ŝ is increasing in the prior belief of R being unbiased. In addition, there always exists an informative equilibrium, for all positive priors, provided that the stakes are low enough. Notice that informative equilibria exist for priors below all due to the grim-trigger strategy followed by R U. Specically, sender is willing to screen R also for lower priors in order to preserve the communication channel if R turns out to be unbiased. Without the threat of punishment by the unbiased type, S would not send informative messages, for any s > 0, when π <. Corollary establishes a benchmark result when both players' stakes are equal to and therefore do not play a role in the model. Corollary. If both ω and s are equal to, sender reports the rst state truthfully for all priors π and babbles. 5 Proof. For priors below, ŝ = 3 Public regime equilibria π when ( π ) π. 5 In this section, I analyze the model laid out in the preceding section. Since ω is assumed to publicly observed at the beginning of the game I shall hereafter refer to the model as Public regime. This is to contrast the model with an alternative informational structure, discussed and analyzed in section 4. The alternative model is labeled as Private regime because of the underlying assumption that ω is private knowledge of R. The distinctive feature between the regimes is that under the Public regime, R's equilibrium action is ex ante known, since the realization of ω and m t will determine

13 the behavior of R B. The behavior of R U is always known since he always follows S's recommendations when they are truthful. However, under both regimes, if S observes that her rst message is followed, she cannot distinguish which type has taken it. Hence, both regimes feature imperfect monitoring by S. The key question is, then, whether it is benecial or not for the players that S is able to tell, before choosing her message, what action the biased type is going to take. Throughout the analysis, I abstract away from the mechanism of making ω public knowledge. The paper merely analyzes the dierence in ex ante welfares under two different regimes, where the regime is exogenously imposed and pre-existing. A question of its own would consider a potential transition from the Private regime to the Public one; whether the receiver could credibly communicate all or part of his private information to the sender prior to receiving advice from her. In some settings, the disclosure can be credibly obtained via legislation. A case in point would be the disclosure of elected candidates' campaign nances which in many countries has been made compulsory by law. As such, information about campaign contributions is not always a perfect proxy for a decision maker's stakes in two given issues, but in many instances it reveals enough to at least know if the relative importance of the rst decision is below or above the critical threshold that determines the legislator's action, which is the essence of this model. Because the sender is able to tell at the beginning of the rst period whether R B is going to follow m or not, her updated belief about R's payo type at the start of the second period is π if ω π (h ) =. 0 or if ω > Knowing that sender's report m is truthful only if π, R B is able to follow with probability only when π. If ω but sender's prior at the start of the game is less than, the only way for R B to follow is to randomize between his pure actions at t = so as to induce sender's updated beliefs to rise above the threshold. That is, for priors below all informative equilibria are in mixed strategies such that the probability µ B (m) < with which R B follows m in case ω π π + ( π )µ B (m) =. is determined from 3

14 To see that this is the equilibrium condition, consider the case that it holds as a strict inequality, and π >. In that case, R B with ω could increase µ B (m) by a small amount and still satisfy the condition for truthful communication at t =, given by the weak inequality π. To give enough incentives for R B to randomize at t =, sender must randomize between her pure actions at t =. This means that she would reward R from following with probability less than. Let r P ub < be the probability with which she reports θ truthfully if the history is good, and let the complementary probability r P ub give the probability with which she babbles about θ. The subscript P ub refers to the Public regime and it is used to dierentiate r from that under the Private regime, to be derived in the next section. In equilibrium, r P ub must solve the indierence condition that R B faces in order to randomize between following or not, given by 0 + r P ub + ( r P ub) = ω +. Sender's incentive compatibility constraint determines a lower bound for the existence of a mixed-strategy equilibrium. For all priors below that, R B has no way to protably follow in the rst period. Therefore, the rst period equilibrium will be separating, and sender reports θ truthfully only if her stakes are low enough for her to engage in screening. Lemma characterizes the informative equilibrium of the game depending on the level of sender's initial prior. Lemma. The following proles of strategies constitute a PBE of the game under Public regime. Case. If π, Alternatively, the equilibrium could be constructed such that S lies with probability ( r). This would not change the expected payos for any player since r would in this specication adjust and equal + ω, which is the same probability of telling the truth than under the current specication once we insert r = ω into r + ( r). 4

15 σ (θ) = s 0 µ B (m) = if ω 0 if h = h σ (θ, h + ) = µ B (m, 0 if h = h h + ) =, Case. If π [ 4, ), µ U (m) = µ U (m, if h = h h + ) =. if ω σ (θ) = if ω > and s ŝ P ub r + ( r) if ω and h = h + σ (θ, h ) = if ω >, s ŝ P ub, and h = h + π µ U (m) = µ B (m) = if ω π 0 µ U (m, if h = h h + ) = µ B (m, 0 if h = h h + ) = where ŝ P ub = Case 3. If π < 4, π, and r = ω ( π ) 5

16 µ U (m) = if s ŝ P ub µ U (m, if h = h h + ) = where ŝ P ub = Proof: In the appendix. if s ŝ σ P ub (θ) = if h = h σ (θ, h + ) = π. ( π ) µ B (m) = 0 if s ŝ P ub µ B (m, h ) = for all h H The equilibrium of the game is visualized by the graphs in gure below [Take out the middle graph, has no value added]. It assumes that S = [0, ] where the upper bound of the interval is chosen merely for exposition purposes. As the rst graph shows, ŝ P ub reaches the upper bound at π = 4 9. The third graph displays R B's strategy in the rst period in the nontrivial case when ω <. For intermediate priors [, ), R 4 B randomizes his action in the rst period by following with probability π [, ). π 3 6

17 Figure. Public regime equilibrium.5 Sender s cutoff strategy, t= ŝ.5 σ (θ) = σ (θ) = if ω σ (θ) = 0.5 σ (θ) = if ω > π.5 Strategy of R B when ω µ B (m) π 4 Private regime and its equilibria Consider now the case where ω is the receiver's private information that he learns at the beginning of the game together with his payo-type. The public history of play is now given by just H = (Θ M A). Moreover, Bayesian updating becomes slightly more complex now that S only knows the equilibrium cuto ˆω but not the realization of ω itself. Sender's Bayesian updating of R's type is now formulated as π (h ) = Pr {τ = U a (m ) = a} = π Ω a (m,ω,u)=adg(ω) π Ω a (m,ω,u)=adg(ω) + ( π ) Ω a (m,ω,b)=adg(ω). In other respects, the model is as described in section 3. We look for cuto strategies for both S and R B with the following structure: sender 7

18 reports the rst state truthfully if s ŝ and misreports it, and R B follows the rst message if ω ˆω and deviates from it. When S's stakes are publicly known, and s > ŝ, the only equilibrium will consist of babbling because S cannot be able to lie protably in any equilibrium. Therefore, the interesting case is when s ŝ where we must check that S has the incentive to report the rst state truthfully given R B 's strategy. Analyzing the game goes along the same lines as in the Public regime game, but now that ω is observed by receiver only, the probability that ω is less or more than becomes relevant. First of all, working backwards, we know that S reports the second state truthfully if and only if π (h +). Using Bayes' rule, this gives us a condition that S's initial prior must satisfy for there to exist a sequentially rational equilibrium in which S reports the second state truthfully after a good history of play. That is, π must satisfy π π + ( π )G( ) π G( ) + G( ) π SR, where the subscript SR refers to sequential rationality. Suppose that π π SR, and consider S's incentives to report the rst state truthfully. Sender's cuto strategy requires that a sender of type ŝ P r is indierent between reporting the rst state truthfully and lying about it, where the subscript P r refers to Private regime, to distinguish the cuto from that of the Public regime. That is, given receiver's strategy, for the type ŝ P r and for any realization of θ, expected payos from truthful reporting, π 0 + ( π ) { G( ) + ( G( )) ( ŝ P r )} must be higher than expected payos from lying, ( π + ( π )G( )) ŝ P r. If s was private knowledge of S, lying could be part of an equilibrium. See section 6 for a short mention on such equilibria. 8

19 After rearranging, ŝ P r can be solved from [ π ( G( )) ( π ) ] ŝ P r = [ π G( )( π ) ], where the term in the brackets on the LHS is positive if π G( ) ( G( )). In addition, the term in the brackets on the RHS is positive if π G( ) +G( ). This latter condition holds by assumption. Therefore, if π G( ) ( G( )) π IC, where IC is short-hand notation for incentive compatibility, there is no type ŝ P r who would be indierent. As a result, all types s 0 would report θ truthfully. Dene now π max { π SR, π IC } to determine which of the thresholds is more demanding. Sender reports the rst state truthfully for all s 0, and rewards following by reporting the second state truthfully if and only if π π, where π SR, if G( π = ) 3 π IC. On the other hand, if π [ π SR, π IC ), which may occur if G( ) <, S reports the rst 3 state truthfully and rewards following if and only if s ŝ P r [ π G( )( π ) ] ( G( )) ( π ) π. () This cuto is increasing in π, and increasing in G( ) when π, but decreasing in 4 G( ) if π < 4. Now, if G( ) 3, R B with ω is able to follow freely if π π SR. For lower initial priors, the informative equilibrium would need to be in mixed strategies where R B with ω randomizes between his pure actions at t =, and sender in turn randomizes between her pure actions at t = in the same manner as under the Public regime. However, since the usual mixed equilibrium would require that the probability with which S reports the second state truthfully depend on the realization of ω, which she does not know, the same outcome must be implemented dierently. Namely, from the sender's point of view, the probability µ B (m)g( ) is eectively the same as the probability G(ˆω), where ˆω <. Hence, equilibrium requires that only R B with ω ˆω 9

20 follow m. Moreover, π G(ˆω) = or ( ) π π ˆω = G. π The equilibrium probability r P r with which S reports θ truthfully is solved from the cuto type's indierence condition to follow or not at t =, and it results to r P r = ˆω. Given this, R B with ω ˆω would indeed follow, and those with ω > ˆω would deviate immediately. Again, sender's incentive compatibility condition determines a lower bound for the priors that support a mixed-strategy equilibrium as described. For priors lower than this, no type of R B is able to follow protably, and the sender reports θ truthfully and screen R only if her own stakes are low. On the other hand, if G( ) <, R 3 B with ω follows freely if π π IC. For initial priors less than this, R B is still able to follow freely since π is still above π SR, but sender starts to be more cautious. She only experiments with R is her stakes are lower than a cuto ŝ P r as given by (). When π < π SR <, there is no longer room 4 for a mixed-strategy equilibrium so that the only informative equilibrium for low priors is one where R B separates from R U immediately, and given this the sender screens R only if her stakes are suciently low. Lemma 3. If G( ), the following proles of strategies constitute a PBE of the 3 game under Private regime. Case. If π π SR = G( ) +G( ), σ (θ) = s S, if h = h σ (θ, h + ) = µ U (m) = µ B (m) = if ω 0 µ U (m, if h = h h + ) = µ B (m, 0 if h = h h + ) = 0

21 Case. If π [ 4, π SR), σ (θ) = s S r + σ (θ, h ) = ( r) if h = h + µ U (m) = µ B (m) = if ω ˆω 0 µ U (m, if h = h h + ) = µ B (m, 0 if h = h h + ) = where ˆω = G ( Case 3. If π < 4, ) π π, and r = ˆω. µ U (m) = if s ŝ P r µ U (m, if h = h h + ) = where ŝ P r = Proof: In the appendix. if s ŝ σ P r (θ) = if h = h σ (θ, h + ) = π. ( π ) µ B (m) = 0 if s ŝ P r µ B (m, h ) = for all h H Lemma 4. If G( ) <, the following proles of strategies constitute a PBE of the 3 game under Private regime. Case. If π π IC = G( ) ( G( )),

22 σ (θ) = s S, if h = h σ (θ, h + ) = µ U (m) = µ B (m) = if ω 0 µ U (m, if h = h h + ) = µ B (m, 0 if h = h h + ) = Case. If π [ π SR, π IC ), if s ŝ σ P r (θ) = if h = h σ (θ, h + ) = µ U (m) = if s ŝ P r µ U (m, if h = h h + ) = where ŝ P r = [π G( )( π )] ( G( ))( π ) π. Case 3. If π < π SR, µ B (m) = if ω and s ŝ P r 0 µ B (m, 0, if h = h h + ) =

23 µ U (m) = if s ŝ P r µ U (m, if h = h h + ) = where ŝ P r = Proof. In the appendix. if s ŝ σ P r (θ) = if h = h σ (θ, h + ) = π. ( π ) µ B (m) = 0 if s ŝ P r µ B (m, h ) = for all h H At this point, it is instructive to take a short look at comparative statics. Notice that π SR is increasing while π IC is decreasing in G( ). As a result, when G( ) 3, and π is given by π SR, an increase in G( ) results in an increase in π thereby reducing the set of priors for which there exists a rst-stage equilibrium in pure strategies while at the same time increasing the range of priors for which there exists a mixed-strategy equilibrium. The opposite happens if there is a decrease in G( ). At the extreme, if G( ) =, π SR =, and equilibrium is in mixed strategies for all π [, ) in both 4 regimes. When G( ) =, π 3 SR = π IC =, the equilibrium is at its most informative, with an 4 equilibrium in pure strategies for priors above, and informative communication for 4 low stakes for priors below that. Any further decreases in G( ) will result in ˆπ, now equal to π SR, decreasing, and π IC increasing, thereby creating a wedge between π SR and π IC. As stated in Lemma 3, for priors in the range, [ π SR, π IC ), the equilibrium is in pure strategies but S's cuto ŝ P r becomes eective. At the extreme, if G( ) = 0, and R B is sure to deviate from any message sent by S, π SR = 0, π IC =, and information is transmitted for all positive priors as long as s is low enough, with the cuto given by ŝ P r = ŝ P ub = π for all ( π ) π [0, ). Before turning to the welfare analysis, a short look at the dierence in S's cutos between the regimes is in order as it will in part determine the relative ranking of the regimes in terms of ex ante welfare. For priors π 4, ŝ P r > ŝ P ub, and the former 3

24 is increasing in G( ). In contrast, for priors π <, ŝ 4 P r < ŝ P ub, and the former is decreasing in G( ). At the extreme, when G( ) = 0, the cutos are identical for all π. The reason why the cuto is higher under Private regime is that S accounts for the possibility that R B invests in reputation, which increases her incentives to report truthfully in the rst period. Figure below plots both cutos when G( ) = Both cutos are multiplied by for computational convenience, but the absolute value of them is not of importance here. Given the value of G( ), π IC = G( ) ( G( )) 0.4 but ŝ P r reaches the supremum of S (now at 4 ) already at around As for ŝ P ub, it reaches at.notice further 5 that for π < π SR 0.4, ŝ P r = ŝ P ub, which is not apparent from the gure. 4 Figure. Sender s cutoffs when G( ) = s.5 ŝ Private 0.5 ŝ Public π 3ŝ P r is only relevant for cases where G( ) < 3. 4 sup(s) = due to the use of a Beta(α, β) distribution for G. Beta distribution is supported on a unit interval. 4

25 5 Welfare comparison Having characterized the equilibria under the two regimes, the next natural question is which regime yields higher welfare. For example, let S and R be a lobbyist and an elected politician, and suppose that the campaign contributions, made by supporters of the elected candidate during the most recent parliamentary elections, are either public knowledge or only known by the politician. Transparency in any decision-making is usually coined as an invariably good goal, and governments have generally committed themselves to increasing the transparency of political decision-making in all of its tiers. In many countries, recently also in Finland, the generally held concern for policies being on sale has lead to the result that all elected candidates must report their campaign nances to a national authority who maintains a public record of them. Knowing who have sponsored a candidate's electoral campaign can serve as a proxy of the decision weights; a candidate who has received signicant donations from entrepreneurs in the north of the country is likely to care more about issues aecting the business environment in the north than in the south of the country. An other proxy for decision weights could be the parliamentary committees that a certain MP is a member of. The mere knowledge of the membership of a given committee is not necessarily information about the bias-type of the MP as committees may be very heterogeneous in composition. However, it gives an indication as to which decisions the MP is likely to prioritize over others. Or, in local government, the residential address of a given member is likely to tell which areas are closer to the heart of that member over other areas, without necessarily revealing anything about that member's bias-type - whether he or she is driven by environmental or business concerns, for instance. For the welfare analysis, I have computed ex ante expected payos for all players under both regimes, and compared them. Interestingly, results suggest that for a wide range of environments it does not strictly benet, and sometimes strictly hurts at least some players, if R's stakes are public information. In particular, there can be shown to exist a nontrivial subset of priors and stakes for which all players strictly prefer the Private regime. Comparison of the regimes is done separately for high priors (above ), low priors (below or π 4 SR), and intermediate priors (between and ). For high and low priors, 4 results are unambiguous; they hold for all distributional assumptions, and for all values of s and ω. For intermediate priors, results depend on distributional assumptions for both s and 5

26 ω. Therefore, to ease the exposition and to obtain analytical results, I look at players' ex ante payos at the time point where s is known but the realization of ω is not yet known. Treating s as a known parameter allows for a comparison of welfares in dierent environments, characterized by whether s falls below or above the cutos in the two regimes. There exist environments in which all players strictly prefer the Private regime over the Public one, but also those in which the reverse holds. The next subsections will provide results for high, low, and intermediate priors, respectively. Ex ante payos before the realization of s would be obtained by integrating over s. A few numerical examples where both s and ω follow a Beta(α, β) distribution with some, not necessarily the same, parameters α and β, are provided in the appendix. (Not provided now, though, and not sure if there is a need to make the appendix any longer than it already is). 5. High priors Proposition. When π, regimes are equivalent in strategies and expected pay- os. Proof. Straightforward once ex ante payos are known. The Private regime has a rst-period equilibrium in pure strategies for all π π, with π regardless of the shape of the distribution G. Under the Public regime, the rst-period equilibrium is in pure strategies for all π. Proposition establishes that when S is condent enough in facing an unbiased R, she will report the state truthfully regardless of the stakes and continue reporting truthfully as long as her messages are followed. Since the probability that R is biased is low, S does not bother paying attention to R B 's incentives. Because S ignores information about ω, it does not make any dierence whether it is publicly known or not. 5. Low priors Propositions and 3 dene a benchmark for priors less than 4. Proposition. If G( ), regimes are identical in strategies and payos for 3 all π <. Disclosing receiver's stakes does not help nor hurt any player. 4 6

27 Proof. Straightforward from Lemmas and 4. Proposition 3. If G( ) < 3, and π [0, π SR ), regimes are identical in payos for all players. If π [ π SR, ), and 4 s ŝ P r, Sender strictly prefers the Public regime, R B strictly prefers the Private regime, and R U is indierent between the regimes; s (ŝ P r, ŝ P ub ], all players strictly prefer the Public regime; s > ŝ P ub, all players are indierent between the regimes. Proof: In the appendix. When sender's prior is low, below 4 or π SR depending on the value of G( ), R B is not able to invest in reputation and must deviate immediately from sender's reports. Given this, sender engages in experimentation only is her stakes are low, she nds it optimal to babble because that minimizes her loss in the relatively important rst period and this expected gain compensates for the higher loss in the second period due to babbling with certainty. If G( ) <, there is a range of priors, [ π 3 SR, ) for which Private regime still includes 4 an equilibrium that allows R B to follow in the rst period. This clearly benets R B who is able to maximize his payos by following when ω. For s ŝ P ub, sender prefers the Public regime because her stakes are so low that her expected payos in case R B follows, ( π ) are lower than what she would get under the Public regime where R B deviates immediately and S obtains ( π )(s + ) in expectation. From sender's point of view, it would be much more benecial to use the rst period for screening R and learn his type immediately in order to optimize communication in the second period where the decision to be made is an important one. 5.3 Intermediate priors As for priors in the range [, ), which regime yields higher expected welfare depends on 4 distributional assumptions imposed on G and H. This range of priors is also the most interesting one since it includes priors for which S is not condent enough in facing R U nor pessimistic enough to simply babble. Put dierently, this is also the range of priors 7

28 for which the it matters whether R B 's action is ex ante known or not. As a result, the regimes feature dierent equilibria, leading to a divergence in the expected payos they yield to players. Propositions 4., 4., and 4.3 summarize welfare results separately for S and R when G( ). Later on, Proposition 4.4 builds on these to draw results for total welfare. 3 Proposition 4.. (Sender) If G( ) 3, and π [ π SR, ), (i) (ii) If s ŝ P ub : Private regime yields strictly higher expected payos when s >. If s <, Public regime yields strictly higher expected payos; If s > ŝ P ub : Private regime yields strictly higher expected payos; π [ 4, π SR), (i) (ii) If s ŝ P ub : Public regime yields strictly higher expected payos; If s > ŝ P ub : Private regime yields strictly higher expected payos. Proof: In the appendix. Proposition 4.. is illustrated in gure 3. 8

29 4 Ranking of the regimes when G( ) 3 Sender Private.5 s.5 Indifference ŝ Public Indifference ŝ Public 0.5 Public π Considering rst the range [ π SR, ), where the Private regime consists of pure experimenting (termi?) by sender: she reports the truth in the rst period and lets R take the action whichever suits him best. If R B follows, S will learn his type only at t = but avoids any losses in the rst period. If R B deviates immediately, S learns his type right away which implies losses in the rst period but allows S to tailor her second report optimally. These scenarios must be compared to those that prevail under the Public regime: if ω, sender obtains mixed-strategy payos which may allow her to learn R's type immediately but also may allow her to avoid losses in the rst period if R B follows. On the other hand, if ω >, her payos are the same as under Private regime as long as s ŝ P ub. What matters then, when s ŝ P ub is how sender's expected payos compare in the case ω. If s <, the stakes are so low that it is best to use the rst period for screening R than experimenting with him, and Public regime is optimal. But if s, the rst period is important enough for S to be caring about its outcome, or rather, the second period is not all too important to be used for screening R. If s (ŝ P ub, ŝ P ub ], sender's stakes are high enough so that in case ω expected payos are maximized if R B is able to follow freely, rather than randomize between 9

30 following or not. On the other hand, the stakes are not too high so that in case ω > sender's expected payos are maximized under the Private regime where she obtains ( π )(s + ) as compared to babbling payos, s that she obtains under the Public regime. This wedge (ŝ P ub, ŝ P ub ] where Private regime is strictly optimal in both cases ω and ω > arises because under Public regime, when ω >, sender would like to lie about θ. Since lying cannot be part of an equilibrium in which s is publicly known, the only equilibrium must consists of babbling which, however, yields sender strictly less than if she was able to commit to truthful reporting at t =. There is a small set of stakes s such that ŝ P ub < s <. Although sender incurs expected losses under Private regime as compared to Public regime when ω, the ex ante losses remain smaller than the ex ante gains in case ω > where experimenting dominates babbling. Finally, if s > ŝ P ub, babbling would be optimal in case ω > and R B deviates immediately. However the ex ante gains of Public regime in this case always (when G( ) 3 ) fall short of the ex ante expected gains of Private regime in case ω when sender is able to avoid all losses in the important rst period due to R B following with certainty instead of randomizing. Proposition 4.. (Receiver) If G( ), and 3 π [ π SR, ), both receiver types obtain strictly higher expected payos under Private regime. Proof: Both receiver types strictly prefer the regime which minimizes the probability of babbling which always yields them strictly lower payos than any informative communication. Both receiver types strictly prefer the Private regime because sender reports the state truthfully for sure and rewards following fully. This allows both players to obtain their maximum payos: R U gets 0, and R B can either follow in the rst period and obtain, or deviate immediately and obtain ω +. When s ŝ P ub, regimes are equivalent in expected payos in case ω >, but Private regime yields strictly higher expected payos in case ω because Public regime's mixed strategy equilibrium involves a positive probability for babbling. When s > ŝ P ub, Private regime is strictly better both in case ω and in case ω > since in the latter, Public regime features only a babbling equilibrium. For ex ante expected welfare, refer to the appendix. Proposition 4.3. (Receiver) If G( ) 3, π [ 4, π SR), and 30