Dynamic Information Revelation in Cheap Talk

Transcription

1 Dynamic Information Revelation in Cheap Talk Maxim Ivanov y McMaster University September 4 Abstract This paper studies a multi-stage version of Crawford Sobel s (98) communication game. In every period the receiver determines a test about the unknown state whose result is privately observed by the sender. After the sender sends a costless message about an outcome of the test, the receiver selects a test in the next period. After a nite number of periods of interaction, the receiver makes a decision. The paper o ers a sequence of tests that re ne sender s information step-by-step and preserve truthtelling in every period. This sequence allows the receiver to learn the state in a subinterval of the state space with an arbitrary precision and has appealing theoretical properties. It consists of simple binary tests which reveal whether the state is above a certain cuto, where the cuto s are monotonic across periods and independent from results of the previous tests. Finally, we show that the relative payo e ciency of multi-stage interaction compared to a single-stage game increases without a bound as the bias in preferences tends to zero. JEL classi cation: C7, D8, D8, D83 Keywords: Communication, Information, Cheap Talk 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 (98), hereafter CS. In general, communication between the involved parties is characterized by two features. The rst one is a con ict of interest. Di erent 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 I am grateful to Dirk Bergemann, Oliver Board, Kalyan Chatterjee, Paul Fischer, Vijay Krishna, Bart Lipman, Robert Marshall, Marek Pycia, Neil Wallace, and three anonymous referees for their valuable suggestions and discussions. Misty Ann Stone provided invaluable help with copy editing the manuscript. All mistakes are mine. y Department of Economics, McMaster University, Kenneth Taylor Hall, 8 Main Street West, Hamilton, ON, Canada L8S 4M4. mivanov@mcmaster.ca.

2 sender, since even the most knowledgeable experts may have imperfect information. However, while the di erence 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. Moreover, if players interact in multiple periods, the receiver may a ect 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 nal 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 o ers 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 nding 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 payo to the receiver than other incentive tools such as delegation or mediation with the perfectly informed sender (Ivanov, ). 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 cuto. The cuto s 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 cuto 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, 99). 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 di erence between the receiver s learning in our setup and the sequential sampling plan are the con icting 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 e ciency of decision making. We show that the constructed sequence of tests preserves truthtelling, which stems from the sender s trade-o between the set of feasible actions in each period and future informational bene ts. In particular, as the sender accumulates more precise information An answer to a natural question about the possibility of acquiring information directly by the receiver is often determined by di erent information acquisition, processing, and opportunity costs of the players. The di erence in costs forces the receiver to delegate the information acquisition problem to the sender. In delegation, the receiver delegates decision making to the sender (see, e.g., Dessein, ). In mediation, players communicate through a disinterested mediator (see Goltsman et al., 7).

3 over time, the set of feasible decisions shrinks. In order to illustrate this trade-o in the case of a positively biased sender, consider a sequence of tests with decreasing cuto s. In this case, the sender observes either a very precise signal about the state (when it is above the cuto ) which will not be updated in future periods, or a very imprecise signal (when it is below the cuto ) 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 cuto 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 cuto 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 cuto, 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 cuto, her current information is su ciently 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 uniformquadratic speci cation (with the uniform distribution of states and quadratic payo functions of players), it is worth noting that the sender reveals less information about high states in the CS equilibria. This is re ected 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 99). Also, the Department of Defense has received many accusations of manipulating test results to yield the most favorable interpretation (U.S. General Accounting O ce, 988, 99). In this light, suppose that the issue of communication is the e ciency 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 3 In short, sender s learning of information is counter to her bias. That is, the incentive of the risk-averse sender to obtain a more precise signal about the state implies that the receiver will rationally choose lower decisions. Because the sender is positively biased, she cannot gain by delaying the information that the state is high, while learning that the state is low and reporting this information would result in more precise information and the possibility of inducing lower actions in future periods. The key condition for sustaining the sender s truthtelling upon learning that the state is below the cuto is that the precision of the sender s information is su ciently low and the highest feasible action is su ciently large. This implies that there is a cuto such that precise extraction is possible if the state is above this cuto (given a su ciently large number of periods). 3

4 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 eld tests of the weapon such that each test is performed by the military at a speci c weather temperature. This speci c temperature plays the role of the cuto, since testing the weapon at this temperature allows the military to see whether the maximum operating temperature exceeds it or not. Also, the cuto s 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 e ciency of decision making. First, Green and Stokey (7) show that the receiver s ex-ante payo is not always monotone in the quality of the sender s information. Austen-Smith (994) and Fischer and Stocken () demonstrate this result in the CS setup. Ivanov () extends this result by showing that communication with an imperfectly informed sender can be more e cient to the receiver than delegating authority to the perfectly informed sender. This paper extends the works by Fischer and Stocken () and Ivanov () by allowing the sender to learn information over time. The work closest to this paper is Ivanov (3), 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 di erences between the two papers. First, the procedure of acquiring information in Ivanov (3) 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 xed 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 a ect her incentives to communicate in our model. In contrast, this information can collapse informative communication in the setup of Ivanov (3). Our work also complements the literature on communication through multiple periods with a perfectly informed sender(s). Aumann and Hart (3) consider two-person games with discrete types and two-sided cheap talk. 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 signi cantly expanded compared to the static one. In the uniform-quadratic CS setup, Krishna and Morgan (4) investigate multi-stage communication such that the sender and the receiver communicate simultaneously in every period. Golosov et al. (3) 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 4 Also, Krishna and Morgan (4) note that simply extending the CS model to the multi-stage case, 4

5 the perfectly informed sender and thus utilize multi-stage interaction for communication only. Such an interaction, however, is not e ective enough for achieving the rst-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 rst-best outcome with any precision in a subinterval of states. 5 The paper proceeds as follows. Section presents the model. The analysis and motivating examples are provided in Section 3. Section 4 compares the receiver s ex-ante payo s in static and dynamic models of informational control. Section 5 concludes the paper. The model We consider a model of multi-stage communication in which two players, the sender and the receiver, interact during T + < periods. Players communicate about state which is constant through the game and distributed on = [; ] according to a distribution function F () with a positive and bounded density f (). The sender has access to imperfect information about, and the receiver makes a decision (or an action) a R that a ects the payo s of both players. The receiver s and sender s payo functions are U (a; ) = (a ), and V (a; ) = (a b) ; respectively, where the bias parameter b > re ects the divergence in the players interests. Actions. At the beginning of each period t = ; :::; T, the receiver determines the sender s information structure, or test I t = ff t (s t j) ; g, which consists of a family of conditional distributions of signals. 6 Then, the sender privately observes a signal s t from the signal space S, drawn from an associated distribution F t (s t j). At the end of period t, the sender sends a (possibly random) message m t from the message space M S to the receiver. Finally, in the last period T +, the receiver takes an action a. We assume that information structures are independent across periods conditional on, that is, I t does not depend on signal realizations fs g t =, messages fm g t =, or information structures fi g t = in previous periods. This restriction signi cantly decreases the receiver s possibilities of extracting information from the sender. 7 Strategies. Let I t = fi ; :::; I t g and m t = fm ; :::; m t g be the history of the receiver s and the sender s actions, respectively, at the beginning of period t. Thus, h t = fm t ; I t g determines the public history (i.e., the history available to the receiver) before selecting I t : Also, let s t = fs ; :::; s t g be the sender s private information before sending a i.e., without letting the receiver send messages to the sender does not improve communication. Because the sender knows the state before communication starts, she sends the sequence of messages that induces the most preferable action. As a result, the receiver infers the same information about the state as in the single-stage game. 5 Battaglini (4) considers a model with multi-dimensional signals and multiple imperfectly informed senders. He shows that for a large number of senders with distinct preferences and limited commitment of the receiver there is an equilibrium in which the receiver learns information with an arbitrarily precision. 6 That is, given the state, signals across periods are independent. 7 Ivanov (3) shows that if I t can depend on past messages fm g t =, then the receiver can perfectly elicit in two-period communication only. 5

6 message m t. Denote by h t = fm t ; I t ; s t g = fh t ; I t ; s t g the sender s (private) history. A behavioral strategy of the receiver in communication periods (communication schedule) I t : Ht! I ; t = ; :::; T, is a sequence of functions which maps the set of public histories Ht into the space of information structures I. In the last period, the receiver s strategy (hereafter, decision rule) a : HT +! R maps H T + into the decision space R.8 The behavioral strategy of the sender t E : H t! M; t = ; :::; T, is the sequence of functions, which maps the set of sender s histories into the space of probability distributions on the message set M. Thus, for any, a strategy pro le = f t (h t )g T t= ; fi t (h t )g T t= ; a h T + determines a stochastic process Q () for a triple q t = (I t ; s t ; m t ), which takes values in I S M, and a random variable a R, which represents the decision. This completely describes what happens if the players follow. The continuation of the process Q () following a history h t, denoted Q (jh t ), is also a stochastic process which describes the future behavior of the triple following h t. Similarly, a random variable a (jh t ) describes the behavior of the decision following h t. Beliefs. The receiver s belief system t : Ht! ; t = ; :::; T +, is the sequence of mappings which determines the probability distribution over for each public history h t. We call the belief system = f t (h t )g T + t= consistent if it is derived from the player s strategies on the basis of Bayes rule where applicable. Equilibrium. Formally, a communication schedule fi t (h t )g T t= is a part of the receiver s strategy and hence is determined endogenously. However, we investigate the bene ts of a particular class of fi t (h t )g T t=, such that each I t (h t ) is a partition of into a nite number of intervals. Because the receiver in our model does not optimize over all communication schedules, we provide a de nition of an equilibrium for an arbitrary fi t (h t )g T t=. In particular, given a communication schedule fi t (h t )g T t=, a perfect Bayesian equilibrium (hereafter, an equilibrium) is a pair of a strategy pro le and the belief system such that: () is consistent with ; () given fi t (h t )g T t= ; a h T + and ht, t (h t ) maximizes the sender s interim payo in period t = ; :::; T : V (jh t ) = E EQ (jh t)v (a (jh t ) ; ) ; and (3) given T + h T + ; f t (h t )g T t=, and any history h t, a h T + maximizes the receiver s payo E U (a; ) j T + h T + in period T +. 3 Dynamic informational control Fischer and Stocken () show that the receiver can increase his ex-ante payo by restricting the quality of the sender s information in the CS setup, in particular, by partitioning into intervals f k g N k= = f[ k; k+ ]g N k=, such that the sender observes k 3. 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, 8 Because U (a; ) is strictly concave in a, the receiver never mixes over actions. N CS 9 Any equilibrium CS partition CS k = CS k= k ; CS k ; :::; N who are indi erent between adjacent decisions a CS k 6 N CS is determined by cuto types CS k ; k = and a CS k+. However, if the sender observes k=

7 which increases the receiver s ex-ante payo. In this section, we investigate the bene ts of multi-stage updating of the sender s information. Before focusing on the main setup, we consider a modi ed model in which the receiver commits to a communication schedule. 3. Commitment to the communication schedule Any equilibrium in the original game can be replicated in the modi ed game in which the receiver commits to a communication schedule fi t (h t )g T t= from the beginning but does not commit to an action. The following result shows that the converse is also true. All proofs are collected in the Appendix. Lemma Any equilibrium in the game with commitment to a communication schedule fi t (h t )g T t= is outcome equivalent to some equilibrium in the game without commitment. The intuition for this result is most easily gained from the fact that in any cheap-talk game (i.e., unconditionally on T and fi t (h t )g T t= ) there is the babbling equilibrium in which the sender does not reveal any useful information and the receiver ignores the sender s messages. In the multi-period game, the sender can punish the receiver with a babbling equilibrium in the continuation subgame. If the receiver follows the equilibrium fit (h t )g T t=, the sender continues to update the receiver s beliefs about the state. Otherwise, no additional information will be provided. Because information cannot hurt the receiver, his equilibrium behavior is optimal. Because the lack of commitment to the communication schedule does not a ect the outcome of communication, without loss of generality, we can restrict attention to the setup with the receiver s commitment to the information structures. This preserves all of the results about the e ciency of communication but simpli es the analysis, since we do not need to be concerned about the sender s reaction to out-of-equilibrium information structures. 3. Truthtelling equilibria Our main focus is truthtelling equilibria in which the sender truthfully reports her information in each period, i.e., m t (h t ) = s t ; 8h t ; t = ; :::; T. A communication schedule fi t (h t )g T t= is incentive-compatible if there is a truthtelling equilibrium with such a schedule. First, consider a game with single-stage communication, that is T =. De ne a function w (z; x) =! x z! z ; z x ; where! = E j ; is a posterior mean of conditional on ;. (For =, we put! =.) Since f () is bounded, w (z; x) is continuous in (z; x). Note only CS k 3, she strictly prefer the associated decision a CS k, since all CS k strictly prefer a CS k. That is, the partitional information structure replaces k CS by mean types! k = E j CS k. Because of that, the receiver can specify a ner partition than the equilibrium CS partition without violating the sender s incentives to communicate truthfully. The sender s behavior is reminiscent of the grim-punishment strategy in the classical prisoners dilemma game. That is, the sender cooperates by sending informative messages as long as the receiver updates her information according to some scenario. In contrast, any deviation by the receiver results in permanent punishment in terms of the quality of received information. 7

8 that! q s is the receiver s decision if the sender truthfully reports that [q; s]. Thus, w (z; x) determines the distance between posterior means obtained by splitting an interval [; x] into two subintervals, [; 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 F [j [; x]] and the sender s information structure given by the partition f[; z] ; [z; x]g. Suppose that b satis es w (; ) = E [] > b: () Intuitively, condition () guarantees that there is informative communication in the singlestage game. Consider a (degenerate) information structure that discloses only whether is or not, and suppose that the receiver treats the sender s messages as truthful. In this case, the sender has a choice between decisions! = and! = E []. Condition () states that the sender prefers to induce! by truthful reporting upon learning that [; ), since this decision is closer to the sender s optimal interim decision! + b. Also, the sender prefers to induce! after learning =. By the continuity of (z; ) in z, there is an informative equilibrium with the partitional information structure f[; z] ; [z; ]g, where z is close to. Consider c (; ) de ned as c = max fxj w (x; x) = bg = max fxjx E [j x] = bg ; () which exists because function w (x; x) = x E [j x] is continuous in x and w (; ) = < b < E [] = w (; ). The interpretation of c is as follows. (For simplicity, let c be a unique solution to w (x; x) = b.) Suppose that both players believe that [; x], where < x <. Then, c is the smallest interval [; x] such that there exists informative single-stage communication with the partitional information structure f[; x); fxgg. In other words, c is a measure of the lowest sender s uncertainty about relative to her bias, such that the sender is willing to disclose whether = x or < x. Approximately full information revelation in [ c ; ]. We restrict the analysis to a particular class of communication schedules such that the sender s information structure in each period t = ; :::; T is the two-interval partition f t ; t g = f[; t ] ; [ t ; ]g for any history h t. That is, the sender s signal s t f t ; t g is such that s t ; ; t = ; :::; T. Equivalently, a communication schedule in this class can be determined by the sequence of cuto s f t g T + t=, where we put = and T + =. Consider a decreasing communication schedule, i.e., a schedule with a strictly decreasing f t g T + t=. Such a communication schedule is characterized by two important properties which can be clearly seen if T is large and cuto s t are distributed uniformly over, i.e., the distance between any adjacent cuto s t t > is small. The rst property re ects 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 [; t ], the quality of her information is low because shrinking the set of possible states from t = [; t ] to t \ t = [; t ] updates the sender s information insigni cantly. However, if the sender There may exist multiple solutions to equation w (x; x) = b. Consider b = =5 and f () = =; [; =] [ [=5; ], and f () = ; (=; =5). Then, the solutions to w (x; x) = b are x = :8; x = :, and x 3 = :8: 8

9 observes the higher interval [ t ; ], she infers that is in the set t \ t = [ t ; t ]. This substantially updates her previous information. The second property determines the sender s future informational bene ts. 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. Suppose that is distributed uniformly on [; ] and let b = 3=4. Consider the two-period communication schedule f ; g = f[; 6=7] ; [6=7; ]g ; f ; g = f[; 3=7] ; [3=7; ]g and the decision rule a (m ; m ) = a = 3=4; a (m ; m ) = a = 9=4; a (m ; :) = a = 3=4 depicted in Fig.. t = t =??? a a a Figure : Two-stage communication Suppose that the sender observes the higher subinterval in the rst period. Hence, her information is not updated in the second period. Then, the truthful message m induces the decision a unconditionally on the sender s message in the second round. In contrast, reporting m allows the sender to induce two decisions, a and a. Because these decisions are strictly inferior to a given, the sender reveals her information truthfully. If the sender observes, her information will be updated in the second period. Then, distorting information by sending message m induces the decision a unconditionally on the future messages. This deprives the sender of all future informational bene ts because new information does not a ect the receiver s decision. As a result, the sender s interim payo V (a j ) = =7 is lower than that of 3=49 in the case of truthful reporting in both periods. 3 Now, suppose that the sender observes the subinterval in the second period after observing in the rst period. She infers that \ = [3=7; 6=7]. In this period, however, only decisions a and a are feasible, where a is strictly preferable to a. 4 That is, the sender induces a by truthful communication. If the sender observes in the second round, she deduces that [; 3=7]. Given this information, a and a results in the same interim payo 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 payo is approximately =75, which This is because V aj is increasing in a for a < a, and max fa ; a g < a < a, where a = arg max a V aj = 8=7. R 3 E[V a (m ; :) ; )j R = V (a ; ) d + V (a ; ) d = 3 49 > 7. 4 Since a < a < a, where a = 6=7 is the sender s optimal decision given updated information. 9

10 exceeds that of control. 5 =48 in the most informative equilibrium in the single-stage informational The following theorem generalizes the above example and is the main result of the paper. Theorem For a decreasing communication schedule f t g T + t=, such that T c, there is a truthtelling equilibrium in the game with f t g T + t=. An implication of this result is that if there are no exogenous restrictions on the communication horizon T, the receiver can learn [ c ; ] with an arbitrary precision. Corollary For any " >, there is T < and a decreasing f t g T + t= such that: T = c ; max ( t t ) < ", and there is a truthtelling equilibrium in the game with f t g T + t=;:::;t t=. It is worthy to note that dynamic informational control demonstrates some similarity to the model of communication investigated by Krishna and Morgan (4). 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 bene ts, which are a ected by the sender s current message. However, the mechanisms of information extraction in the two models are di erent. In Krishna and Morgan s model, the uncertainty stems from the random outcome of simultaneous communication in the rst 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 a ects the sender s incentives in the rst period. In addition, in the case of success, the sender reveals more information in the second round for su ciently 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 simpli es 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. Partial information extraction in [; c ]. Condition () states that if the sender learns that < c at t = T, so she can induce two actions,! T + = E [j c ] and! T = E [j [ c ; T ]], then she prefers to induce! T + by truthfully reporting the lower interval. However, if T is close to c and so is! T, splitting the interval [; c ] into two subintervals, [; z] and [z; c ], by specifying T = c and T = z can violate the sender s incentives to report truthfully about the lower interval. This is because the action! c = E [j [z; c ]] is feasible in the last communication period and is below c. Hence, condition () seems to be too restrictive for informative communication about [; c ]. Nevertheless, this logic is imperfect, and the receiver can extract partial information about 5 Note that the receiver cannot replicate the equilibrium outcome in the single-stage game via specifying the partition with cuto s f3=7; 6=7g and reacting to sender s messages as if they were truthful. The reason is that if [3=7; 6=7], the sender can pro tably deviate from inducing a.

11 even if it is below c. In fact, if < z, then partitioning the interval [; c ] updates the sender s information in the last communication period, which decreases her posterior valuation of from! T + to! z = E [j z] <! T +. Hence, the sender s optimal action decreases from! T + + b to! z + b. This suppresses the sender s incentive to overstate information. We use this observation in order to show the following result. Lemma For a decreasing f t g T + t= such that T c and w ( T ; T ) b, there is a truthtelling equilibrium in the game with f t g T + t=. According to Lemma, given a decreasing (T ) period communication schedule f t g T t= such that T c and w (z; T ) b for some z (; T ), the receiver can extract partial information in [; T ] by specifying the decreasing T period communication o T + schedule n^t which replicates the original one in periods t = ; :::; T and is such t= that ^ T = z. Also, the condition w ( T ; T ) b in the Lemma holds, for example, if E [j T ] > b; and T is su ciently small. Intuitively, suppose that the sender in the last communication period knows that is either or is in (; T ] and the receiver believes that the sender is truthful. If the sender observes =, she has a choice between inducing the lowest rationalizable decision a = and a = E [j T ]. If E [j T ] > b, the sender prefers to report truthfully that = as the decision a is too far from the her optimal decision b. Therefore, partitioning [; T ] into [; T ] and [ T ; T ], such that T # 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 bene ts 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. 6 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 payo is given by 7 EU = Var [a] Var [] ; (3) where Var[y] is a variance of a random variable y. Now, we evaluate the upper limit on the receiver s ex-ante payo in equilibria with decreasing communication schedules when T!. If lim max j t t j =, then the T!t=;:::;T 6 The uniform-quadratic setup is widely used in the literature on strategic communication. See, for example, Blume et al. (7), Gilligan and Krehbiel (989, 987), Goltsman et al. (7), Krishna and Morgan (8, 4, ), Melumad and Shibano (99), Ottaviani and Squintani (6). 7 See, for example, Goltsman et al. (9).

12 receiver learns [ T ; ] perfectly in the limit as T!. Thus, the upper limit on the receiver s ex-ante payo depends only on T and T : where EU ( T ; T ) = lim EU T = T!; max t=;:::;t jt t j!; = 3 T EU T = TX Z =T d ( T T ) 3 ; (4) TX Z = d: Given the constraint T = c = 4b, EU ( T ; T ) is maximized at T = b, which results in the ex-ante payo EU = 4 3 b3 : (5) We can now compare the ex-ante bene ts of the single-stage and multi-stage communication games. In the single-stage communication game, the incentive-compatibility constraints are k+ + k 4b; 8k; (6) where k = k+ k is the length of an interval [ k ; k+ ] in the partitional information structure. 8 By (6) and T = 4b, there exist informative equilibria in both one-stage and multi-stage versions of the game if b =4. In multi-stage communication the inequality T + T 4b only must hold, while the distance t t ; t < T can be arbitrarily small. Also, since the players ex-ante payo s di er by b, all equilibria are Pareto ranked. 9 This results in the following theorem. Theorem For b < 4 and any equilibrium ex-ante payo EU in the single-stage communication game, there is an equilibrium in the multi-stage communication game such that EU T > EU. Also, EU EU! as b!. According to this theorem, if informative CS communication is feasible, i.e., b < 4, then dynamic informational control is ex-ante payo 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. Moreover, as b!, the ex-ante bene ts of multi-stage 8 Given the message that k, the receiver s best response to the sender s truthtelling strategy is a k =! k = E [j k ] = k+ k+. Upon observing k, the sender prefers a k to a a k+ > a k if a k++a k! k + b = a k + b. This implies a k+ a k = k++ k+ k + k+ b and k+ k = k+ + k 4b. Also, because V (aj k ) is decreasing in a for all a <! k + b, the sender never prefers a < a k to a k. 9 For quadratic preferences, CS show that EV = EU b, where EV and EU are the sender s and the receiver s ex-ante payo s, respectively. This relationship holds in any equilibrium of our model as well. See Ivanov (). Goltsman et al. (9) show that the optimal delegation is an optimal arbitration mechanism, i.e., the communication protocol in which players communicate via the neutral arbiter who can

13 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 b game has an order of b. Thus, the receiver s ex-ante payo has an order of. In 3 contrast, the receiver s ex-ante payo in truthtelling equilibria given the optimal decreasing communication schedule is characterized by the residual variance of conditional on 4 [; 4b] multiplied by the probability of that event, which has an order of 3 b3. 5 Conclusion and Discussion We have demonstrated that through communication with an imperfectly informed sender in multiple rounds in which the receiver controls the precision of the sender s information in every round without learning its content, 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 divergence in players preferences decreases, the relative ex-ante performance of multi-period interaction versus single-stage game rises without a bound. However, the performance of dynamic informational control depends on a few factors. First, it depends on the shape of the distribution of states. At the rst glance, it seems that if the distribution is concentrated near, for example, if the density of the distribution is decreasing, then dynamic informational control is not highly e ective as the receiver only learns precise information about high states, which is unlikely. This is not true in general. For example, for b = 3 and a decreasing communication schedule f 4 tg T + t=, such that T = c, the receiver s ex-ante payo EU in the limit as T! is approximately equal to for the triangle distribution skewed to the left, and to the uninformative payo 56 for the triangle distribution skewed to the right. This is because the value of c depends on the shape of the distribution also. By construction, c is the smallest interval [; T ] which sustains informative communication in the subgame at the last communication period, which is a single-stage communication game with the partitional information structure f[; T ); f T gg. Then, if the density of the distribution is increasing and T is low, the sender strictly bene ts from distorting information and inducing decision a T = T upon learning that [; T ) since this decision is closer to the sender s optimal decision E [j < T ] + b than decision a T = E [j < T ] induced by truthful communication. As a result, c is higher for distributions in which high states are more likely. Second, the e ciency of dynamic informational control is a ected by the communication horizon T. The perfect learning of [ c ; ] requires an in nite communication horizon T. Given the receiver s utility EU T in the game with a nite T, the relative di erence between EU T and EU can serve as the measure of imperfection of extracted information " = EU EU T. Consider, for example, the uniform-quadratic setup. For a xed T and decreasing EU communication schedules f t g T + t= which satisfy the conditions in Lemma, the receiver s exante payo is maximized for f t g T + t= such that T = b and t = 4b t; t = ; :::; T. This T results in the ex-ante payo EU T ( 4b) = EU 3. It follows then that the communication (T ) enforce his recommendations. Because mediation is a special case of arbitration, it is ex-ante payo inferior to the optimal delegation. 3

14 horizon T ", which guarantees that the loss in the ex-ante payo does not exceed ", increases as " =. Referring to the above example of b = 3, the receiver s ex-ante payo EU is 4 approximately. However, only two periods of communication provide the ex-ante payo 76, so that " ' %. 75 Finally, 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 e ciency 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 e ective 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 bene ts of multi-stage learning and communication. In the case of a decreasing communication schedule, if the receiver in stage t + knows that is in the interval [; t+ ] instead of [; t ], this updates his information insigni cantly. Therefore, if the cost of a single test is su ciently high, it might be bene cial to specify a single-period information structure with two cuto s, t+ and t, instead of a two-period communication schedule with single cuto s. Appendix In this section we provide proofs of the results. ~ Proof of Lemma. Denote ~ = E t (h t ) T t= ; ~a h T + an equilibrium pro le of the game n with commitment to the information structures ~It h o T t. Note that for any history t= h T +, the optimal decision rule is ~a h T + = E jh T +. In the game without commitment, de ne the following trigger function n d h t ; I t T + if I t = ~Ii h o t t ; i= = n o min i : I i 6= ~Ii h t t otherwise, i= n which indicates the rst period in which the receiver deviates from ~It h o T t t=. Consider the sender s strategy ~ t E E (h t ) = t (h t ) if d h t ; I t = T + and m t otherwise, (7) which implies that the sender follows the strategy ~ t E (h t ) T as long as the receiver follows n t= ~It h o T t. In contrast, any deviation by the receiver results in completely uninformative t= messages m t in the current and all future periods. The receiver s best response to this strategy is a h h i T + = ~a h d(h T ;IT ) = E jh d(h T ;IT ) : Also, T " is quite sensitive to the value of b and the order of b 3=. For example, for " =, it follows that T " = if b = and T " = 5 if b = : 4

15 That is, the receiver makes a decision on the basis of information up to the period of deviation n from ~It h o T t only and ignores all other messages. t= n Consider the strategy pro le = E t (h t ) T t= ; ~It h o T t ; a t= h T +, and suppose that It the receiver deviates from it by choosing P = h T t t= ; a h T +, where I 6= I ~ h for some h. It follows that d h t ; I t = < T + ; t = ; :::; T, so that t E (h t ) = m t ; t = ; :::; T. Since the receiver does not receive useful information in periods t = ; :::; T, his interim payo is bounded from above by U P P ; E jh E U P ~a h ; jh : However, U P P ; E jh E U P ~a h n ; jh, because following ~It h o T t t= results in a history h T +, which includes h. Therefore, the receiver may choose a h T + = ~a h and replicate the interim payo E U P ~a h ; jh. Also, the sender s strategy (7) is the best response to the receiver s strategy. First, for all h t such that the receiver follows I ~ t h t, the sender cannot bene cially deviate from ~ E t (h t ), since it is optimal in the game with commitment to the communication schedule. Finally, if h t involves deviation from I ~ t h t, then sending uninformative messages m T t t= is also optimal, since a h T + does not depends on them. Proof of Theorem. De ne t = [ t ; t ] ; t = ; :::; T +. Given the sender s truthtelling strategy fm (h )g T = =fs g T =, the receiver s consistent posterior beliefs are such that is distributed on according to distribution F [j ], where min f : m = = g if fm tg T t= 6= t T t=, and T + if fm t g T t= = ft gt t= : For such beliefs, the receiver s best response is a h T + =! = E [j ] : (8) We show now that the sender cannot strictly bene t by distorting information at any stage conditional on any history. Equilibrium path. Given the state and the receiver s best response (8), consider the sender s incentives in any period conditional on truthful reporting in previous periods. Let t = min f : s = g be rst stage in which the sender observes the higher interval t. For a decreasing communication schedule, we have + ; 8. Thus, s = ; t, and the sender s information is not updated in periods > t. Also, we have fs g t = = f gt =. Hence, the sender infers that t, so that her optimal interim decision is! t + b. Truthtelling in the previous periods implies fm g t = = f gt =. Then, message m t = t induces the decision a t =! t unconditionally on messages fm g T +. In contrast, if m t = t, then the set of feasible decisions in the future periods is fa g T + + = f! g T + +. Because! < a t <! t + b; > t, this means V (a j t ) < V (a t j t ) ; > t. Hence, the sender is strictly worse o by sending m t = t. The receiver s posterior beliefs are consistent for both equilibrium and o -equilibrium messages of the sender. If the sender sends, say, a sequence fm ; m g, which implies [ ; ] \ [; ] =?, the receiver interprets these messages as [ ; ]. 5

16 If s t = t, the sender infers that t. Given fm g t = = f gt =, message m t = t induces action a t =! t unconditionally on fm g T +. This results in the interim payo to the sender: V a t j t = E V (!t ; ) j t h i h i = E (! t b) j t = E (! t ) j t h i = E (! t v t + v t ) j t + be! t j t b = (! t v t ) (! t v t ) E v t j t h E ( v t ) j t h i = (! t v t ) E ( v t ) j t + b (! t v t ) b h i = (! t v t ) (! t v t b) E ( v t ) j t b ; + be! t j t i + be! t j t where v t = E j t = E [j t ] < t <! t. By! t > t c and (), we have! t v t b > t E [j t ] b. This leads to (! t v t ) (! t v t b) > and V a t j t h i E ( v t ) j t b : b b Now, consider the sender s interim payo from sending fm g T = fs g T : h i E V a fs g T ; j t = Z TX V (! + ; ) df j t = + X T Z (! + b) df j t + = Z = Z TX (! + ) df j t X T + b (! + ) df j t + + TX (! + ) df j t + b Z t Z ( v t ) df j t b = b h i E ( v t ) j t b ; where the inequality follows from Z TX (! + ) df j t = + X T Z ((! + v t ) ( v t )) df j t + = Z t Z ( v t ) df j t X T + (! + v t ) ( v t ) df j t + TX P + j t (!+ v t ) = Z t = ( v t ) df j t X T + P + j t X T (!+ v t ) P + j t (!+ v t ) Z t ( v t ) df j t X T + P + j t (!+ v t ) Z t ( v t ) df j t ; 6

17 where P + j t = F ( ) F ( + ). Therefore, F ( t) h E V a fs g T i h i ; j t = E ( v t ) j t b V a t j t ; so the sender cannot bene cially deviate from reporting truthfully at any t = ; :::; T conditional on truthtelling in all previous stages. O -equilibrium path. Consider the sender s incentives at stage t = ; :::; T given that she was not truthful in previous stage(s), i.e., h t is such that fm g t = 6= fs g t =. If fm g t = 6= f gt =, then the receiver takes an action!, where = min f : m = g < t. Because the receiver s decision does not depend on fm g T, the sender cannot bene t by distorting information at t. Now, suppose that fm g t = = f gt =. Since fs g t = 6= fm g t =, it follows that s i = i, where i = min f : s = g < t, and fs g T =i = f gt =i. Hence, the sender s information at t is given by i and is not updated afterwards. Since fm g t = = f gt =, then message m t = t induces action a t =! t unconditionally on fm g T +. If m t = t, then the set of feasible actions is fa g T + + = f! g T + +. Because a <! t <! t + b; > t, we have V (a j t ) < V (a t j t ) ; > t. Hence, the sender is strictly worse o by sending m t = t. Thus, she cannot bene cially deviate from truthtelling at any t conditional on misreporting at previous stages. Proof of Lemma. Given [; T ] and truthful reporting in periods t = ; :::; T, consider the subgame at the beginning of the last communication period t = T. This sub-game is a game with single-period communication, the prior distribution F [j [; T ]], and the sender s information structure given by the two-interval partition f[; T ] ; [ T ; T ]g. Then, the receiver s best responses to the sender s truthful reporting of signals s t = T and s t = T are! T + =! T = E [j T ] and! T =! T T = E [j T T ], respectively. Because the sender s incentive-compatibility constraint w ( T ; T ) =! T T! T =! T! T + b holds by conditions of the Lemma, then Lemma in Ivanov () implies that there is a truthtelling equilibrium in this game. Because T c, then following the same lines as those in Theorem shows that (8) and truthful reporting is periods t = ; :::; T are the receiver s and sender s equilibrium strategies in the original game. Proof of Theorem. Consider a single-stage communication game with the partitional information structure given by strictly increasing cuto s f g K =, where = and T =, and focus on equilibria which maximize the receiver s ex-ante payo. By Lemma in Ivanov (), we can restrict attention to incentive-compatible information structures and truthtelling equilibria under these structures. In these equilibria, a = E [j [ ; ]] = + ; = ; :::; K, and the sender s incentive-compatibility constrains are + = + + 4b; = ; :::; K ; (9) where = ; = ; :::; K. If b < =4, Lemma 3 in Ivanov () implies that informative communication is feasible, i.e., there is a partitional information structure with K intervals which satis es (9). The ex-ante payo to the receiver in the truthtelling equilibrium under this structure is KX Z KX EU = (a ) d = 3 : = = 7