Dynamic Information Revelation in Cheap Talk

1 Introduction

This paper focuses on the problem of strategic communication between a privately informed expert (the sender) and an uninformed decision maker (the receiver) à la Crawford and Sobel (1982), hereafter CS. In general, communication between the involved parties is characterized by two features. The first one is a conflict of interest. Different objectives create an incentive for the sender to misrepresent information in her favor, which results in the lower quality of disclosed information. The second feature is the imperfect primary information of the sender, since even the most knowledgeable experts may have imperfect information. However, while the difference in players’ preferences is usually exogenous, the quality of the sender’s information can sometimes be endogenized by the receiver. That is, though the sender’s information is still private, the precision of this information is determined by the receiver. ^[1] Moreover, if players interact in multiple periods, the receiver may affect the precision of the sender’s information and request a report from her in each period. In other words, the receiver selects a dynamic informational control policy – a sequence of tests – and makes a final decision, whereas the sender conducts each test and reports its result to the receiver.

Our major contribution is that we show how the receiver can use these instruments – controlling the quality of the sender’s private information (hereafter, informational control) and dynamic interaction – to extract the sender’s private information about an unknown state of nature. In particular, the paper offers a sequence of tests that allows the receiver to learn information with any precision over an interval of states which converges to the state space as the preference bias tends to zero. An implication of this finding is the Pareto improvement over single-stage informational control and, hence, communication with the perfectly informed sender. This result becomes even stronger given that single-stage informational control provides a higher ex-ante payoff to the receiver than other incentive tools such as delegation or mediation with the perfectly informed sender (Ivanov 2010). ^[2]

The sender’s learning process and players’ interaction are organized as follows. The sequence of tests consists of binary tests such that each test progressively reveals whether the state is above a certain cutoff. The cutoffs are monotonic across periods and independent of the results of previous tests. The sender privately observes the result of each test and sends a report to the receiver. Testing and communication continue until either the sender reports that the state is above the current cutoff or the limit on the number of tests is achieved. Then the receiver makes a decision on the basis of reported information.

There is the similarity between the receiver’s process of learning the sender’s information in our setup and a sequential sampling plan performed by a single decision maker (see Feldman and Fox 1991). This plan tests a hypothesis on the basis of a sample of a variable size. In particular, if the result of the test based on the current sample is negative, then the hypothesis is rejected and the testing procedure stops. Otherwise, if the test result is positive, then the sample size increases by one and an additional test is conducted (until the limit on the sample size is achieved). The key difference between the receiver’s learning in our setup and the sequential sampling plan is the conflicting preferences of players in our setting. Consider, for example, a positively biased sender whose optimal decision exceeds the optimal decision of the receiver given any information about the state. Because the sender tends to exaggerate her information in an attempt to induce higher decisions, her strategic motives can potentially harm communication and, hence, the efficiency of decision making.

We show that the constructed sequence of tests preserves truthtelling, which stems from the sender’s trade-off between the set of feasible actions in each period and future informational benefits. In particular, as the sender accumulates more precise information over time, the set of feasible decisions shrinks. In order to illustrate this trade-off in the case of a positively biased sender, consider a sequence of tests with decreasing cutoffs. In this case, the sender observes either a very precise signal about the state (when it is above the cutoff) which will not be updated in future periods or a very imprecise signal (when it is below the cutoff) which will be updated in future periods. Second, the receiver’s best response to the sender’s truthtelling has the “trigger” character. In particular, because the sender’s signal “the state is above the current cutoff” will not be updated in future periods, her report about such a signal induces the receiver’s decision (the current trigger decision), unconditionally on future reports. In contrast, reporting that the state is below the cutoff allows the sender to induce trigger decisions, which are strictly below the current trigger decision, in future periods. If the positively biased sender learns that the state is above the cutoff, her best feasible decision is the highest one, i.e. the current trigger decision. The sender induces it by reporting her information truthfully. Otherwise, if the sender learns that the state is below the cutoff, her current information is sufficiently imprecise. In this case, she prefers to learn more information and induce one of the feasible decisions in future periods by reporting her information truthfully. ^[3]

In comparing the equilibrium in our setup to the CS equilibria for the leading uniform-quadratic specification (with the uniform distribution of states and quadratic payoff functions of players), it is worth noting that the sender reveals less information about high states in the CS equilibria. This is reflected in the fact that the lengths of subintervals of states which induce the same decision in any CS equilibrium are increasing in the state. Intuitively, because the sender has both the incentive and the possibility of exaggerating her information, the receiver’s credibility to the sender’s messages about high states is low. In contrast, the sender reveals more information about high states in our setup. As described above, the sender eventually learns precise information about the state only if it is high enough. At that moment, however, the sender lacks the opportunity to exaggerate her information since her best feasible decision is the receiver’s best response to the sender’s current information.

As an application of our results, consider the communication problem in defense procurement. While the military is an expert in evaluating characteristics of weapons, the budget for them is determined by the Congress (the receiver). Moreover, the parties’ interests are not aligned – it is argued that the military tends to be biased toward weapons with excessive costs (Rogerson 1990). Also, the Department of Defense has received many accusations of manipulating test results to yield the most favorable interpretation (U.S. General Accounting Office 1988, 1992). In this light, suppose that the issue of communication is the efficiency of a weapon, which monotonically depends on the unknown maximum operating temperature (the state), and the decision of the Congress is the budget spent on the weapon production. Assume that the optimal budget of the Congress monotonically depends on the state and is below the optimal budget desired by the military. Thus, the military is interested in exaggerating the actual state. In order to mitigate the resulting communication problem, our paper suggests that the receiver must determine a sequence of field tests of the weapon such that each test is performed by the military at a specific weather temperature. This specific temperature plays the role of the cutoff, since testing the weapon at this temperature allows the military to see whether the maximum operating temperature exceeds it or not. Also, the cutoffs must be decreasing over time, and the military must report about the result of each test.

Our work is related to the literature that investigates the role of the sender’s imperfect information and multi-stage strategic communication on the efficiency of decision making. First, Green and Stokey (2007) show that the receiver’s ex-ante payoff is not always monotone in the quality of the sender’s information. Austen-Smith (1994) and Fischer and Stocken (2001) demonstrate this result in the CS setup. Ivanov (2010) extends this result by showing that communication with an imperfectly informed sender can be more efficient to the receiver than delegating authority to the perfectly informed sender. This paper extends the works by Fischer and Stocken (2001) and Ivanov (2010) by allowing the sender to learn information over time. The work closest to this paper is Ivanov (2013), which shows that by determining the precision of the sender’s information and communicating in multiple rounds, the receiver can elicit perfect information from the sender. However, there are two crucial differences between the two papers. First, the procedure of acquiring information in Ivanov (2013) is predicated upon stronger assumptions. In particular, the precision of the sender’s future information is highly sensitive to her previous report(s). In other words, the type of the test which can provide additional information to the sender depends on her previous reports. In contrast, the sequence of tests in this paper is fixed from the beginning. Thus, though the receiver can stop the sequence at any moment or, equivalently, ignore the sender’s future messages, he cannot modify the initial sequence of tests. Because of this, the information acquisition procedure in our paper is simpler and more applicable. Second, our procedure is more robust to the sender’s prior information. In particular, if the sender is privately informed about whether the state is above or below some level, this information does not affect her incentives to communicate in our model. In contrast, this information can collapse informative communication in the setup of Ivanov (2013).

Our work also complements the literature on communication through multiple periods with a perfectly informed sender(s). Aumann and Hart (2003) consider two-person games with discrete types and two-sided asymmetric information. In their setup, one side is better informed than the other, and the players can communicate without time constraints. They demonstrate that the set of equilibrium outcomes in the dynamic setup can be significantly expanded compared to the static one. In the uniform-quadratic CS setup, Krishna and Morgan (2004) investigate multi-stage communication such that the sender and the receiver communicate simultaneously in every period. Golosov et al. (2013) consider a dynamic game consisting of repeating the CS model multiple times (i.e. in each period the sender sends a message, and the receiver takes an action). These papers demonstrate that two factors – multiple rounds of cheap talk conversation and active participation of the receiver in the communication process – can improve information transmission. ^[4] However, these papers consider the case of the perfectly informed sender and thus utilize multi-stage interaction for communication only. Such an interaction, however, is not effective enough for achieving the first-best outcome of the receiver. In contrast, this paper shows that using the dynamic setup for both information updating and communication allows the receiver to achieve the first-best outcome with any precision in a subinterval of states. ^[5]

The paper proceeds as follows. Section 2 presents the model. The analysis and motivating examples are provided in Section 3. Section 4 compares the receiver’s ex-ante payoffs in static and dynamic models of informational control. Section 5 concludes the paper.

2 The model

We consider a model of multi-stage communication in which two players, the sender and the receiver, interact during T+1<∞ periods. Players communicate about stateθ which is constant through the game and distributed on Θ=[0,1] according to a distribution function Fθ with a positive and bounded density fθ. The sender has access to imperfect information about θ, and the receiver makes a decision (or an action) a∈R that affects the payoffs of both players. The receiver’s and sender’s payoff functions are

respectively, where the bias parameter b>0 reflects the divergence in the players’ interests.

3 Dynamic informational control

Fischer and Stocken (2001) show that the receiver can increase his ex-ante payoff by restricting the quality of the sender’s information in the CS setup, in particular, by partitioning Θ into intervals Θkk=1N=θk,θk+1k=1N, such that the sender observes Θk∋θ. Intuitively, the preferences of the less informed sender are more closely aligned with those of the receiver. ^[9] As a result, the less informed sender communicates truthfully, which increases the receiver’s ex-ante payoff. In this section, we investigate the benefits of multi-stage updating of the sender’s information. Before focusing on the main setup, we consider a modified model in which the receiver commits to a communication schedule.

3.2 Truthtelling equilibria

Our main focus is truthtelling equilibria in which the sender truthfully reports her information in each period, i.e. mtht=st,∀ht,t=1,...,T. A communication schedule Itht0t=1T is incentive-compatible if there is a truthtelling equilibrium with such a schedule.

First, consider a game with single-stage communication, that is T=1. Define a function

Δwz,x=ωzx−ω0z,0≤z≤x≤1,

where ωθ_θˉ=Eθ|θ∈θ_,θˉ is a posterior mean of θ conditional on θ∈θ_,θˉ. (For θ_=θˉ, we put ωθ_θˉ=θ_.) Since fθ is bounded, Δwz,x is continuous in z,x. Note that ωqs is the receiver’s decision if the sender truthfully reports that θ∈q,s. Thus, Δwz,x determines the distance between posterior means obtained by splitting an interval 0,x into two subintervals, 0,z and z,x. Equivalently, it is the distance between the receiver’s decisions in the truthtelling equilibrium of the single-stage communication game with the prior distribution Fθ|θ∈0,x and the sender’s information structure given by the partition 0,z,z,x.

Suppose that b satisfies

[1]Δw1,1=1−Eθ>2b.

Intuitively, condition (1) guarantees that there is informative communication in the single-stage game. Consider a (degenerate) information structure that discloses only whether θ is 1 or not, and suppose that the receiver treats the sender’s messages as truthful. In this case, the sender has a choice between decisions ω11=1 and ω01=Eθ. Condition (1) states that the sender prefers to induce ω01 by truthful reporting upon learning that θ∈[0,1), since this decision is closer to the sender’s optimal interim decision ω01+b. Also, the sender prefers to induce ω11 after learning θ=1. By the continuity of Δz,1 in z, there is an informative equilibrium with the partitional information structure 0,z,z,1, where z is close to 1.

Consider θc∈0,1 defined as

[2]θc=maxx|Δwx,x=2b=maxx|x−Eθ|θ≤x=2b,

which exists because function Δwx,x=x−Eθ|θ≤x is continuous in x and Δw0,0=0<2b<1−Eθ=Δw1,1. ^[11] The interpretation of θc is as follows. (For simplicity, let θc be a unique solution to Δwx,x=2b.) Suppose that both players believe that θ∈0,x, where 0<x<1. Then, θc is the smallest interval 0,x such that there exists informative single-stage communication with the partitional information structure [0,x),x. In other words, θc is a measure of the lowest sender’s uncertainty about θ relative to her bias, such that the sender is willing to disclose whether θ=x or θ<x.

3.2.1 Approximately full information revelation in θc,1

We restrict the analysis to a particular class of communication schedules such that the sender’s information structure in each period t=1,...,T is the two-interval partition Θ0t,Θ1t=0,θt,[θt,1] for any history ht0. That is, the sender’s signal st∈Θ0t,Θ1t is such that θ∈st,,t=1,...,T. Equivalently, a communication schedule in this class can be determined by the sequence of cutoffs θtt=0T+1, where we put θ0=1 and θT+1=0.

Consider a decreasing communication schedule, i.e. a schedule with a strictly decreasing θtt=0T+1. Such a communication schedule is characterized by two important properties which can be clearly seen if T is large and cutoffs θt are distributed uniformly over Θ, i.e. the distance between any adjacent cutoffs θt−1−θt>0 is small. The first property reflects the quality of the sender’s current information – in each period her information is either very precise or very imprecise. As long as the sender observes the lower interval 0,θt, the quality of her information is low because shrinking the set of possible states from Θ0t−1=0,θt−1 to Θ0t−1∩Θ0t=0,θt updates the sender’s information insignificantly. However, if the sender observes the higher interval θt,1, she infers that θ is in the set Θ0t−1∩Θ1t=θt,θt−1. This substantially updates her previous information. The second property determines the sender’s future informational benefits. If the sender’s current information is precise, it will not be updated in the future. In contrast, if her information is still vague, then it will surely be improved in the future periods. An example below illustrates how a combination of these factors allows the receiver to extract all available information from the risk-averse sender.

Example 1

Suppose that θ is distributed uniformly on 0,1 and let b=3/14. Consider the two-period communication schedule {Θ01,Θ11}= [0,6/7],[6/7,1]},Θ02,Θ12=0,3/7,3/7,1 and the decision rule am0,m0=a00=3/14,am0,m1=a01=9/14,am1,.=a1=13/14 depicted in Figure 1.

Figure 1

Two-stage communication

Suppose that the sender observes the higher subinterval Θ11 in the first period. Hence, her information is not updated in the second period. Then, the truthful message m1 induces the decision a1 unconditionally on the sender’s message in the second round. In contrast, reporting m0 allows the sender to induce two decisions, a00 and a01. Because these decisions are strictly inferior to a1 given Θ11, the sender reveals her information truthfully. ^[12] If the sender observes Θ01, her information will be updated in the second period. Then, distorting information by sending message m1 induces the decision a1 unconditionally on the future messages. This deprives the sender of all future informational benefits because new information does not affect the receiver’s decision. As a result, the sender’s interim payoff Va1|Θ01=−1/7 is lower than that of −3/49 in the case of truthful reporting in both periods. ^[13]

Now, suppose that the sender observes the subinterval Θ12 in the second period after observing Θ01 in the first period. She infers that θ∈Θ01∩Θ12=3/7,6/7. In this period, however, only decisions a01 and a00 are feasible, where a01 is strictly preferable to a00. ^[14] That is, the sender induces a01 by truthful communication. If the sender observes Θ02 in the second round, she deduces that θ∈0,3/7. Given this information, a00 and a01 results in the same interim payoff to the sender, so that the sender (weakly) prefers to communicate truthfully. Also, the induced decisions are the receiver’s best response to the sender’s truthtelling strategy. Finally, note that the receiver’s ex-ante payoff is approximately −1/75, which exceeds that of −1/48 in the most informative equilibrium in the single-stage informational control. ^[15]

The following theorem generalizes the above example and is the main result of the paper.

Theorem 1

For a decreasing communication scheduleθtt=0T+1, such thatθT≥θc, there is a truthtelling equilibrium in the game withθtt=0T+1.

An implication of this result is that if there are no exogenous restrictions on the communication horizon T, the receiver can learn θ∈θc,1 with an arbitrary precision.

Corollary 1

For anyε>0, there isT<∞and a decreasingθtt=0T+1such that: θT=θc,maxt=1,...,Tθt−1−θt<ε, and there is a truthtelling equilibrium in the game withθtt=0T+1.

It is worthy to note that dynamic informational control demonstrates some similarity to the model of communication investigated by Krishna and Morgan (2004). In both cases, the main incentives for the risk-averse sender to provide more information stem from her uncertainty about the future receiver’s actions or the informational benefits, which are affected by the sender’s current message. However, the mechanisms of information extraction in the two models are different. In Krishna and Morgan’s model, the uncertainty stems from the random outcome of simultaneous communication in the first round which is generated by the jointly-controlled lottery. In particular, if the outcome of communication is “success”, the sender can update her information in the next period. Thus, even though the sender might not be allowed to update a report in the second round, the uncertainty about future interaction affects the sender’s incentives in the first period. In addition, in the case of “success”, the sender reveals more information in the second round for sufficiently high states. Together, these factors provide an overall improvement over CS communication. In the informational control case, uncertainty arises directly from updating the sender’s information over periods. The important feature of informational control is a possibility of generating a smaller number of sender’s types than that in the models with the perfectly informed sender, say, two versus a continuum. This substantially simplifies the problem of aligning the sender’s incentives with the receiver’s ones. As a result, although the number of sender’s types is smaller, each type fully reveals herself in each round.

3.2.2 Partial information extraction in 0,θc

Condition (2) states that if the sender learns that θ<θc at t=T, so she can induce two actions, ωT+1=Eθ|θ≤θc and ωT=Eθ|θ∈θc,θT−1, then she prefers to induce ωT+1 by truthfully reporting the lower interval. However, if θT−1 is close to θc and so is ωT, splitting the interval 0,θc into two subintervals, 0,z and z,θc, by specifying θT−1=θc and θT=z can violate the sender’s incentives to report truthfully about the lower interval. This is because the action ωc=Eθ|θ∈z,θc is feasible in the last communication period and is below θc. Hence, condition (2) seems to be too restrictive for informative communication about θ∈0,θc. Nevertheless, this logic is imperfect, and the receiver can extract partial information about θ even if it is below θc. In fact, if θ<z, then partitioning the interval 0,θc updates the sender’s information in the last communication period, which decreases her posterior valuation of θ from ωT+1 to ωz=Eθ|θ≤z<ωT+1. Hence, the sender’s optimal action decreases from ωT+1+b to ωz+b. This suppresses the sender’s incentive to overstate information. We use this observation in order to show the following result.

Lemma 2

For a decreasingθtt=0T+1such thatθT−1≥θcandΔwθT,θT−1≥2b, there is a truthtelling equilibrium in the game withθtt=1T+1.

According to Lemma 2, given a decreasing T−1 – period communication schedule θtt=0T such that θT−1≥θc and Δwz,θT−1≥2b for some z∈0,θT−1, the receiver can extract partial information in 0,θT−1 by specifying the decreasing T – period communication schedule θˆtt=0T+1 which replicates the original one in periods t=1,...,T−1 and is such that θˆT=z. Also, the condition ΔwθT,θT−1≥2b in the Lemma holds, for example, if

Eθ|θ≤θT−1>2b,

and θT is sufficiently small. Intuitively, suppose that the sender in the last communication period knows that θ is either 0 or is in (0,θT−1] and the receiver believes that the sender is truthful. If the sender observes θ=0, she has a choice between inducing the lowest rationalizable decision a=0 and a′=Eθ|θ≤θT−1. If Eθ|θ≤θT−1>2b, the sender prefers to report truthfully that θ=0 as the decision a′ is too far from the her optimal decision b. Therefore, partitioning 0,θT−1 into 0,θT and θT,θT−1, such that θT↓0 preserves the sender’s incentive to reveal her information truthfully.

4 Welfare analysis: the uniform-quadratic case

In this section, we evaluate the receiver’s ex-ante benefits in dynamic informational control versus those in one-stage communication and other organizational forms. For that purpose, we restrict attention to partitional information structures and the leading uniform-quadratic setup. ^[16] First, note that in any equilibrium of a sender–receiver communication game (i.e. the game without receiver’s commitment to actions) with quadratic preferences of players, the receiver’s ex-ante payoff is given by ^[17]

where Vary is a variance of a random variable y.

Now, we evaluate the upper limit on the receiver’s ex-ante payoff in equilibria with decreasing communication schedules when T→∞. If limT→∞maxt=1,...,T−1θt−θt−1=0, then the receiver learns θ∈θT−1,1 perfectly in the limit as T→∞. Thus, the upper limit on the receiver’s ex-ante payoff depends only on θT−1 and θT:

where

Given the constraint θT−1=θc=4b, EU¯(θT−1,θT) is maximized at θT=2b, which results in the ex-ante payoff

We can now compare the ex-ante benefits of the single-stage and multi-stage communication games. In the single-stage communication game, the incentive-compatibility constraints are

where Δθk=θk+1−θk is the length of an interval θk,θk+1 in the partitional information structure. ^[18] By (6) and θT−1=4b, there exist informative equilibria in both one-stage and multi-stage versions of the game if b≤1/4. In multi-stage communication the inequality ΔθT+ΔθT−1≥4b only must hold, while the distance θt−1−θt,t<T can be arbitrarily small. Also, since the players’ ex-ante payoffs differ by b2, all equilibria are Pareto ranked. ^[19] This results in the following theorem.

According to this theorem, if informative CS communication is feasible, i.e. b<1/4, then dynamic informational control is ex-ante payoff superior to such organizational forms as optimal delegation (with the perfectly informed sender) and communication through a disinterested mediator. This is because these mechanisms are ex-ante inferior to a single-stage informational control. ^[20] Moreover, as b→0, the ex-ante benefits of multi-stage communication relative to one-stage communication rise without a bound. This is because the average length of intervals in the optimal partition of the single-stage communication game has an order of 2b. Thus, the receiver’s ex-ante payoff has an order of −b23. In contrast, the receiver’s ex-ante payoff in truthtelling equilibria given the optimal decreasing communication schedule is characterized by the residual variance of θ conditional on θ∈0,4b multiplied by the probability of that event, which has an order of −43b3.

5 Conclusion and discussion

This paper demonstrates that through communication with an imperfectly informed sender in multiple rounds in which the receiver controls the precision of the sender’s information, the receiver can derive almost all information for a subinterval of the state space. This results in an ex-ante Pareto improvement compared to one-stage communication. Moreover, as the bias in players’ preferences decreases, the relative ex-ante performance of multi-period interaction versus single-stage game rises without a bound.

It is important to highlight several factors that influence our results. First, the performance of dynamic informational control depends on the shape of the distribution of states. At the first glance, it seems that if the distribution is concentrated near 0, for example, if the density of the distribution is decreasing, then dynamic informational control is not highly effective as the receiver only learns precise information about high states, which is unlikely. This is not true in general. For example, for b=3/14 and a decreasing communication schedule θtt=0T+1, such that θT=θc, the receiver’s ex-ante payoff EU¯ in the limit as T→∞ is approximately equal to −1/56 for the triangle distribution skewed to the left, and to the uninformative payoff −1/12 for the triangle distribution skewed to the right. This is because the value of θc depends on the shape of the distribution also. By construction, θc is the smallest interval 0,θT−1 which sustains informative communication in the subgame at the last communication period, which is a single-stage communication game with the partitional information structure [0,θT−1),θT−1. Then, if the density of the distribution is increasing and θT−1 is low, the sender strictly benefits from distorting information and inducing decision aT−1=θT−1 upon learning that θ∈[0,θT−1) since this decision is closer to the sender’s optimal decision Eθ|θ<θT−1+b than decision aT=Eθ|θ<θT−1 induced by truthful communication. As a result, θc is higher for distributions in which high states are more likely.

Second, the efficiency of dynamic informational control is affected by the communication horizon T. The perfect learning of θ∈[θc,1] requires an infinite communication horizon T. Given the receiver’s utility EUT in the game with a finite T, the relative difference between EUT and EU¯ can serve as the measure of imperfection of extracted information ε=|EU¯−EUTEU¯|. Consider, for example, the uniform-quadratic setup. For a fixed T and decreasing communication schedules θtt=0T+1 which satisfy the conditions in Lemma 2, the receiver’s ex-ante payoff is maximized for θtt=0T+1 such that θT=2b and θt=1−1−4bT−1t,t=1,...,T−1. This results in the ex-ante payoff EUT=EU¯−112(1−4b)3(T−1)2. It follows then that the communication horizon Tε, which guarantees that the loss in the ex-ante payoff does not exceed ε, increases as ε−1/2. Referring to the above example of b=314, the receiver’s ex-ante payoff EU¯ is approximately −176. However, only two periods of communication provide the ex-ante payoff −175, so that ε≃2%. ^[21]

In the analysis above we assumed that the sender’s learning of information is costless and the players do not discount the time taken by learning the state. Evaluating the efficiency of communication with costly experiments or with a discount factor seems to be an interesting avenue for future research. At the same time, the characterization of effective information schedules becomes a complicated question. For example, if the cost of a single test is positive but does not depend on the structure of the test, then multi-stage informational control can be suboptimal. This is because such a schedule updates the receiver’s information gradually, so the accumulated costs of learning may exceed the benefits of multi-stage learning and communication. In the case of a decreasing communication schedule, if the receiver in stage t+1 knows that θ is in the interval 0,θt+1 instead of 0,θt, this updates his information insignificantly. Therefore, if the cost of a single test is sufficiently high, it might be beneficial to specify a single-period information structure with two cutoffs, θt+1 andθt, instead of a two-period communication schedule with single cutoffs.