Dynamic Learning and Strategic Communication

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1 Dynamic Learning and Strategic Communication Maxim Ivanov y McMaster University October 014 Abstract This paper investigates a dynamic model of strategic communication between a principal and an expert with con icting preferences. In each stage, the uninformed principal can select the precision of expert s information about an unknown state without observing the information itself. We show that the principal can elicit perfect information from the expert about the state and achieve the rst-best outcome in two stages only if the expert s preference bias is not too large. If the state space is unbounded, full information revelation is possible for an arbitrarily large preference con ict. Moreover, full revelation of information is feasible in the more general frameworks than those considered in the literature, including the ones with non-quasiconcave and non-supermodular payo functions and with a privately known bias of the expert. JEL classi cation: C7, D81, D8, D83 Keywords: Communication, Information Acquisition, Cheap Talk 1 Introduction This paper focuses on the classic problem of ine cient cheap talk communication between parties with con icting interests who are asymmetrically informed and have di erent decision-making powers. It adds to the literature by investigating the following question. Consider the principal (he) who does not possess important information about the economic, political, or military consequences of his decisions. As a result, he needs to consult the I am grateful to Ricardo Alonso, Dirk Bergemann, Archishman Chakraborty, Kalyan Chatterjee, Ettore Damiano, Péter Esö, Paul Fischer, Maria Goltsman, Seungjin Han, Sergei Izmalkov, Andrei Karavaev, Vijay Krishna, Wei Li, Tymo y Mylovanov, Gregory Pavlov, Marek Pycia, Vasiliki Skreta, Joel Sobel, Dezsö Szalay, and participants of CETC 011, CEA 011, nd Game Theory Conference in Stony Brook, WZB conference on Cheap Talk and Signaling in Berlin, and a seminar at the New Economic School for valuable suggestions and discussions on di erent versions of the paper. Misty Ann Stone provided invaluable help with copy editing the manuscript. All mistakes are mine. The paper has been previously circulated under the title Dynamic Informational Control. y Department of Economics, McMaster University, 180 Main Street West, Hamilton, ON, Canada L8S 4M4. mivanov@mcmaster.ca. 1

2 expert (she), who has either more expertise in particular areas or signi cantly lower costs of acquiring and processing new information. However, the expert s preferences are biased in a sense that her optimal decisions di er from those of the principal, and, thus, never reports information truthfully (Crawford and Sobel, 198, hereafter CS). At the same time, the principal can in uence the quality of expert s private information without being able to observe its content, and then request a report from the expert about her observations. For instance, if obtaining new information requires conducting tests or experiments, the principal may select the testing procedures performed by the expert, but cannot see the outcomes of the tests. The main question of this paper is how much information can the principal elicit from the expert through communication by properly determining the expert s learning process? This paper adds to the literature in two main ways. First, it introduces a dynamic protocol of learning information by the expert which allows the principal to extract perfect information from the expert in only two rounds of cheap talk conversation. 1 Second, the paper emphasizes that fully informative communication is feasible without maintaining several fundamental assumptions of the CS model: the concavity and the supermodularity of the players payo functions. Another important feature of our learning protocol is its robustness to the assumption of the privately known bias of the expert. 3 As our work indicates, the only factors essential for e ective interaction of the principal with the imperfectly informed expert are the maximal intensity of the preference bias and the sensitivity of the expert s payo to principal s decisions for di erent states. Moreover, if the state space is unbounded, full information revelation is feasible for an arbitrarily large bias of the expert. Let us explain these ndings in a more detailed way. The failure of perfect information transmission in single-stage communication models is based on the fact that the expert can easily mimic any information arbitrarily close to the true one. To get around this di culty, we introduce a dynamic protocol for acquiring information (hereafter, a learning protocol) that (1) allows the expert to learn the state perfectly in the second period, and () sustains truthful communication in both periods. The protocol is deterministic, that is, each state is mapped into a single signal in any period. It also does not require commitment on the side of the principal who can change the quality of expert s information and his decision at any moment. The key feature of the protocol is the decomposition of communication about a continuous state into a continuum of cheap talks about binary posteriors, and then separates the true state from the irrelevant one. In particular, it includes two experiments such that the precision of the second experiment is contingent on the expert s report on the rst experiment. The rst experiment returns information about a couple of isolated points of the state space the true state and some irrelevant complement state, which is su ciently distinct from the true one but does not reveal which state is true. Hereafter, we call such a couple posterior states. The second experiment allows the expert to 1 A single-stage protocol of learning information cannot implement the rst-best outcome (Ivanov, 010a). All or some of these assumptions are used in various communication models, e.g., communication with multiple experts (Krishna and Morgan, 001a, 001b; Battaglini, 00), delegation (Dessein, 00; Alonso and Matouschek, 008; Kovác and Mylovanov, 009), mediated communication (Goltsman et al., 009), dynamic communication (Krishna and Morgan, 004; Golosov et al., 011), contracting for information (Krishna and Morgan, 008), and communication via a noisy channel (Blume et al., 007). 3 As shown by Morgan and Stocken (003) and Li and Madaràsz (008), if the expert s bias is her private information, then an outcome of communication can change drastically. In contrast, it is not the case in our setup as long as the maximum bias is below some cuto.

3 distinguish between reported posterior states only. Otherwise, the outcome of the experiment is completely uninformative. Thus, the expert s information is updated in the second period if and only if she reported the truth in the previous stage, and even local distortions of information result in severe informational losses. 4 By that moment, however, the principal knows the binary distribution of posteriors, which prevents the expert from manipulating the second-stage information. If the preference con ict is not large, then the informational bene ts of learning the true state outweigh the bene ts of manipulating the rst-stage imprecise information. Also, because discrete second-period posterior beliefs sustain the fully informative equilibrium and the expert is interested in learning the state in the second period if the absolute value of the sender s bias is below some cut-o, the learning protocol is robust to the principal s knowledge of the sender s bias. The main technical innovation of our protocol is that it employs the non-convexity of posterior distributions for two purposes: rst, for acquiring and updating the information of the expert; and second, for in uencing the principal s decisions through communication. This contrasts with related works on multi-stage communication by Krishna and Morgan (004) and Golosov et al. (014). These authors establish that the overall quality of conveyed information may be improved if the expert reveals the non-convex set, which contain the state, to the principal in early stage(s) and separates it into convex subsets afterwards (given the principal s proper behavior). However, these papers consider the case of the perfectly informed expert and thus, utilize non-convex sets for in uencing the principal s beliefs only. Such a screening scheme, however, does not achieve the rst-best outcome of the principal. In contrast, using non-convex sets in the expert s rst-stage information structure and separating these sets given her truthful communication can be an incentive tool for the expert that qualitatively improves the quality of transmitted information. As a potential application of our results, consider interaction between an auto dealership, hereafter, a mechanic (expert) and a car manufacturer (principal). The mechanic repairs cars on the manufacturer warranty, whereas the manufacturer covers the costs of repairing. Before servicing a car, the mechanic gathers information about it by using the testing equipment that isolates the source of the problem and estimates its severeness. This information determines the total costs of repairing. Though both parties are concerned about repairing the car, the interests of the mechanic can be biased toward higher costs. This naturally leads to the problem of manipulating information about the severeness of the problem. Roberts (004) provides a clear example of manipulating such information due to con icting objectives of auto mechanics and the corporate headquarters. 5 In this work, we o er a potential solution to the problem of communication about car issues with high costs of repairing. In the context of our model, the car manufacturer has to impose restrictions on the tests, which estimate the problems, by selecting a particular testing equipment. Many typical tests are su ciently accurate and performed, often exclusively, 4 Ambrus and Lu (013) apply discontinuous punishment for (local) distortions of information via principal s actions in the context of communication with multiple experts. As a result, the principal can elicit almost full information from the experts. 5 In his words, Sears Roebuck in 199 sought to motivate the mechanics in its auto repair business by setting targets for the amount of work they did. The mechanics responded by telling customers that they needed steering and suspension repairs that were in fact unnecessary. Customers could not easily verify the need for the repairs themselves, and many paid for the unnecessary work. When the fraud was uncovered, Sears not only paid large nes, it lost much of the precious trust it had once enjoyed among its customers. 3

4 by the electronics, which includes aforementioned quantizers. (E.g., electronic car scanners return the code of the car s sub-system, which does not operate properly. This example is a su ciently precise approximation of a continuous state because the set of di erent outcomes is su ciently large and may exceed 100 codes for a single car model.) Thus, instead of providing a single code, the rst test has to return two codes, one of which is correct. Upon observing the outcome of this test, the mechanic reports it to the manufacturer and uses it as an input signal for the second test. The second test is conducted upon an approval from the manufacturer and allows the mechanic to distinguish between the reported values only. Finally, the last step of the procedure requires the manufacturer to audit the mechanic and verify that the reported values were identical to those used in the second test (given a large number of tests and severe punishment for misreporting, it is su cient to randomly audit a small number of tests). In the context of the above example, the principal may instead wish to acquire information without involving the expert. However, the principal s ability to obtain relevant information and gain expertise in necessary elds is often severely restricted by the large volume, diverse range, and complexity of other responsibilities. This issue has been addressed in multiple works (Radner, 1991; Krishna and Morgan, 001a; and Austen-Smith, 1994). Therefore, high costs of information acquisition and opportunity costs may force the principal to delegate information acquisition to the expert despite the existence of the preference con ict and the possibility of learning information himself. Moreover, Argenziano et al. (013) show that bene ts of multi-stage communication about expert s information which is updated over time (at some cost) can be higher compared to the case when the principal acquires information directly. Another important argument in favor of using the expert as an operator of the information acquisition technology is its decreasing returns to scale with respect to the number of performed tests. 6 In a closely related paper, Ivanov (014) investigates bene ts of a dynamic learning protocol which is selected at the beginning of the game and the posterior distribution in each period is supported on a subinterval. In each period, the expert also faces a trade-o between future informational bene ts and the set of feasible actions, however, the characters of the trade-o s in the two papers are qualitatively di erent. In the setup by Ivanov (014), the convexity of the support of posterior distributions implies that the expert can distort her information locally unconditionally on the precision of her information. As a result, even though the subset of posterior states shrinks over time, there exists a subinterval in which states cannot be separated. Outside of that subinterval, full extraction of information is feasible in the limit, i.e., as the number of periods converges without a bound. This is because the learning protocol is independent of expert s messages, so that the principal cannot reduce the number of learning periods by re ning the reported information only. In addition to the literature on multi-stage communication, our paper adds to works on communication with the imperfectly informed expert(s). In this area, Green and Stokey (007) rst demonstrated that the less informed expert can be bene cial to the 6 If a single testing procedure must be applied to multiple independent variables, then obtaining information about all variables by the central authority is costly if, e.g., there are capacity constraints on the number of tests performed. For instance, evaluating problems of cars covered by the manufacturer warranty is delegated to car dealers, who then submit reports and bills to the car producer. In this case, designing the standards on testing procedures for estimating car conditions and auditing the dealers is a less costly task for the manufacturer than testing cars directly. 4

5 principal. Fischer and Stocken (001) show this result for the CS model. Ivanov (010a) extends this result by demonstrating that communication with the imperfectly informed expert can bene t the principal more than optimally delegating authority to the perfectly informed expert. Besides cheap-talk communication, the situations in which the uninformed party(s) can in uence the quality of private information acquired by the informed agent(s) have attracted attention in various areas. Applications include auctions (Bergemann and Pesendorfer, 007; Board, 009), monopoly (Lewis and Sappington, 1994; Johnson and Myatt, 006), and markets of di erentiated products (Damiano and Li, 007; Ivanov, 014). 7 Our work also complements a few papers on (almost) full information extraction in communication games. These include cheap talk with multiple experts investigated by Krishna and Morgan (001a, 001b), Battaglini (00), Esö and Fong (008), and Ambrus and Lu (013). Kartik et al. (007) consider a setting where the expert incurs lying costs. The main distinction of our paper is the endogenous and dynamic quality of expert s information, whereas the information structure of the expert(s) in the aforementioned works is exogenously given at the beginning of the game. The rest of the paper is structured as follows. Section presents the formal model. Section 3 highlights an illustrative example. The general analysis of the model is performed in Section 4. Section 5 concludes the paper. The model Consider a two-stage communication game with two players, an expert and a principal. The state is distributed on the interval = [0; 1] according to a prior distribution F () with a positive and continuous density f (). The expert can privately observe some information about, whereas the principal makes a decision a R that a ects the payo s of both players. The payo functions are of the form U (a; ; b i ) ; i fe; P g where the inherent bias parameter b i R re ects the divergence in players interests. The principal s bias is normalized to be 0, whereas the expert s bias is b 0. We consider a more general class of players preferences than those in the CS setting. In particular, the function U (a; ; b) is continuous in (a; ; b) and has a unique ideal decision a (; b) = arg max a U (a; ; b), which is continuous in and b. Hereafter, we write U (a; ) = U (a; ; 0) and V (a; ; b) = U (a; ; b) as the principal s and the expert s payo functions, respectively. We also write a p () = a (; 0) and y (; b) = a (; b) as the ideal decisions of the principal and the expert, respectively. Denote the set of the principal s ideal decisions A = fa p () j g. We call a decision a rationalizable if a A. Also, we assume there exists s (0; 1) and a continuously di erentiable bijective function ' : [0; s]! [s; 1], such that ' 0 (s) > 0 and a p () 6= a p (' ()) ; [0; s]. This condition re ects two facts. First, the subset [0; 1) of can be split into two-point sets f; 0 g, such that < s 0 and the principal s ideal decisions for any pair of states and 0 are distinct. Second, all pairs of states f; 0 g with distinct principal s ideal decisions are not arbitrarily close. Hereafter, we refer to ' () as the complement state function. 8 7 A recent paper by Klein and Mylovanov (011) investigates the con ict between the expert s reputational concerns about appearing competent and the incentives to report information truthfully in the dynamic environment in which the expert can accumulate information about her competence over time. 8 Compared to the CS setup, our model does not require the supermodularity of players payo functions 5

6 Actions. At the beginning of period t = 1;, the principal selects a publicly observable expert s information structure I t = ff t (s t j)g S, where the signal space S is compact. That is, an information structure determines conditional distributions of the signal for all in each period. Denote I = S the space of all distributions of signals conditional on the state. Then, the expert privately observes a signal s t S drawn from an associated distribution F t (s t j). At the end of period t, the expert sends a message m t M to the principal. Finally, upon receiving messages fm 1 ; m g, the principal makes a decision a R. 9 Strategies. We call a pair of rst-period and second-period information structures a learning protocol fi 1 ; I (m 1 ; I 1 )g, where I : M I! I is a function of the rst-period principal s history fm 1 ; I 1 g. Note the principal does not commit to the second-stage information structure I (m 1 ; I 1 ) from the beginning. Instead, he takes into account his history fm 1 ; I 1 g while selecting I in the second stage. A behavioral strategy of the principal consists of the learning protocol and a decision rule a : M I! R, which is a function of the second-period principal s history fm 1 ; m ; I 1 ; I g. The behavioral strategy of the expert is a pair of functions f 1 ; g, where 1 : I S! M and : I S M! M, which map expert s private histories h 1 = fi 1 ; s 1 g and h = fh 1 ; I ; s ; m 1 g in the rst and the second periods, respectively, into the spaces of probability distributions on measurable message sets M in these periods. Beliefs. The principal s belief is a pair = f 1 (m 1 ; I 1 ) ; (m 1 ; m ; I 1 ; I )g, where 1 : M I! and : M M I! determine the principal s beliefs in end of the rst and the second periods, respectively. The belief is consistent if it is derived from the players strategies on the basis of Bayes rule where applicable. 10 Equilibrium. A perfect Bayesian equilibrium (hereafter, equilibrium) consists of the learning protocol fi 1; I (m 1 ; I 1 )g, the decision rule a (:), the belief (:) and the expert s strategy f 1 (:) ; (:)g, such that (:) is consistent with the players strategies and the strategies are optimal given the principal s beliefs and any sender s history at each moment. The formal de nition of an equilibrium is given in the Appendix. 3 Example: the CS uniform-quadratic case We start with an illustrative example, which outlines the two-stage learning protocol under which the expert truthfully reveals the state upon perfectly learning it in the second stage. This example highlights the key ideas that we use below in the more general economic environments than the standard CS settings. For our example, we employ the uniform-quadratic setup in which the prior distribution of is uniform on = [0; 1] and the preferences are quadratic: 11 U (a; ; b i ) = (a b i ) ; i fe; P g : U a (a; ; b i ) > 0 or equivalently, the monotonicity of ideal decisions a (; b i ) in, and the single-sided bias y (; b) 6= a () for all b > 0 and. 9 For simplicity, we restrict attention to pure strategies of the principal as functions of messages fm 1 ; m g. 10 For all experts s messages m t = M; t = 1;, we de ne the principal s belief in such a way that he interprets them as some m 0 t M. 11 The uniform-quadratic setup is known for the tractability in various modi cations of the basic CS model. See, for example, Blume et al. (007), Goltsman et al. (007), Krishna and Morgan (001a, 001b, 004, 008), Melumad and Shibano (1991), Ottaviani and Squintani (006), and Esö and Szalay (010). 6

7 s 1 () 6 1 s (; m 1 ) s m t 1 m 1 t ; m 1 0 = m 0 1 6= m Figure 1: Information structures in two periods Because b P = 0 and b E = b, the ideal decisions of the principal and the expert are a p () = and y (; b) = + b, respectively. Suppose that b b = 1, where b is the largest bias, which 4 sustains informative communication in the CS model. De ne the rst-period information structure as follows (see Fig. 1). Each pair of state < 1 and the complement state 0 = ' () = + 1 map into the same signal s =. Such a partitioning of into two-point sets generates a binary posterior distribution of conditional on s with probabilities Pr f = sjsg = Pr = s + 1js = 1; s < 1. The second-stage information structure updates the expert s reported information only. In particular, given message m 1 < 1, the expert can separate state m 1 from the complement state ' (m 1 ). For = fm 1 ; ' (m 1 )g, the information structure returns an uninformative signal s 0 = [0; 1). Thus, the expert learns the true state in the second stage if and only if she truthfully reports her information in the rst stage. Otherwise, reporting m 1 6= s does not update the expert s information. Upon receiving message m fm 1 ; ' (m 1 )g, the principal interprets it as truthful and makes a decision a (m ) = m. According to such a learning protocol, the expert in the rst period faces a trade-o between informational bene ts and exibility over actions. On the one hand, she can induce any action y in the set of rationalizable decisions A = [0; 1] by sending either m 1 = y or 1 m 1 = y in the rst stage depending on whether y is below or above 1, respectively, and then sending m = y in the second stage. In this case, however, the decisions are based on the expert s interim information only. On the other hand, truthfully reporting m 1 = s allows the expert to learn the state perfectly, but shrinks the set of feasible actions to two actions. These are the principal s best-responses to posterior realizations s and ' (s), a p (s) = s and a p s + 1 = s + 1, respectively. From the expert s prospective, distorting information in the rst period and inducing decision y A results in the payo E [V (y; ; b) js] = 1 V (y; s; b) + 1 V (y; ' (s) ; b) = 1 (y s 1 b) y s 1 b ; 7

8 which is maximized at y 1 (s; b) = min s b; 1. Since b 1 and s < 1, it follows that 4 4 y 1 (s; b) = s b < 1. Inducing the optimal interim decision y 4 1 (s; b) provides the payo to the expert: E [V (y 1 (s; b) ; ) js] = 1 4 (' (s) s) = 1 16 ; s < 1 : Truthful reporting in both stages results in the payo : E [V (a p () ; ; b) js] = 1 V (a p (s) ; s; b) + 1 V (a p (' (s)) ; ' (s) ; b) = 1 (s s b) 1 (' (s) ' (s) b) = b ; s < 1 : Finally, the second-stage incentive compatibility constraints must prevent the positively biased expert from distorting information in period two. That is, inducing a p () = upon learning < 1 must be more bene cial than inducing a p (' ()) = ' () > a p (), or V (a p (s) ; s; b) V (a p (' (s)) ; s; b) ; s < 1 : (1) These constraints are satis ed if a p (' ()) a p () = ' (s) s = 1 b. Thus, if b 1 4, then the inequalities (1) and the learning incentive constraints E [V (a p () ; ; b) js] = b 1 16 = E [V (y 1 (s; b) ; ) js] ; imply that the expert prefers to convey her information truthfully in both periods and, hence, reveals the true state in the second stage. Since the principal achieves his rst-best outcome, he does not have pro table deviations in any period. However, we need to specify the players reaction to the principal s deviations from the prescribed information structures. In this case, the expert sends an uninformative message, or babbles, until the end of the game, whereas the principal ignores all expert s messages after his deviation. Because there is a babbling equilibrium given any information structure, such out-of-equilibrium behavior is optimal for both players. Intuitively, the e ciency of the constructed learning protocol is driven by a combination of two factors. First, the quality of the expert s information in the second period, and, therefore, the bene ts of updating the information are contingent on her rst-period report. If the expert conveys her imprecise information truthfully, the second-period information structure allows her to perfectly learn the state. Second, truthtelling in the rst period reveals the particular binary distribution of posteriors to the principal, who can use it in order to (partially) verify the expert s report in the next period. This signi cantly restricts the expert s possibilities of manipulating her information in round two, because the principal makes only those decisions that are consistent with received information in both stages. Together, these factors imply that the expert faces a trade-o between the future informational bene ts received in the case of truthful reporting and exibility over available actions, which can be induced by properly distorting her interim information. If the preference bias is not large, the former e ect outweighs the latter one. In addition, a smaller set of feasible actions in round two helps sustain truthful communication in that stage. In 8

9 fact, the expert can induce only the principal s best responses to either true or complement states. If the latter option is unfavorable, the expert prefers to tell the truth. In short, the learning protocol gives a lot of freedom over decisions to the expert, because the principal can be easily manipulated by the expert s messages. However, the value of such freedom is low if the expert is not well informed about the true situation. 1 Technically, the learning protocol decomposes communication about a continuous variable into the continuum of cheap-talk conversations about discrete posteriors. This decomposition plays a dual role. First, it introduces substantial uncertainty in the rst-period expert s information and, thus, motivates her to learn more information in the second round. Second, the discretization of the state space prevents the expert s from manipulating her information by claiming a di erent state in the second period. An important feature of the introduced protocol is its sensitivity to reported information. For implementation, however, this feature is less critical than it may seem at rst glance. The high sensitivity is dictated by the extreme e ciency of the learning protocol within the very limited time of interaction. If the principal does not aim to achieve the rst-best outcome, he can use simpler information structures that are less sensitive to conveyed information while preserving su cient e ciency. This is because the main principle of the learning protocol is equally applicable to arbitrary non-convex subsets as to single points. In the more general learning protocol, the rst-stage information structure allows the expert to observe the nite family of non-convex measurable subsets Q k = k j that contain the state. In the second period, the expert can identify the individual subset k j Q k containing the state upon truthtelling in the previous stage. Consider, for example, the bias b = 1=5 and the simplest non-trivial learning protocol with the posterior sets f 1 1; 1 g = 0; 4 1 ; 1 ; 3 4 and f 1; g = 1 ; 1 4 ; 3 ; It results in an ex-ante payo to the principal of, 4 19 which is far better than all known non- rst-best incentive schemes. 14 In general, for quadratic preferences and an arbitrary distribution, a learning protocol that achieves the principal s ex-ante losses " relative to the rst-best payo must contain approximately (1") 1= sub-intervals k j in total In this light, it is important to note that the e ciency of interaction relies on the expert s being unable to get outside information. Otherwise, the expert could run the rst test to nd out the two possible states and then privately eliminate one of them with an outside investigation. Having then discovered the true state, he could mislead the principal afterwards. 13 By the simplest non-trivial learning protocol, we imply the protocol with the smallest number of Q k and k j, in which the expert acquires private information in rst period (that is, it contains at least two families Q k ) and can improve the obtained information in the second period (i.e., each Q k contains at least two subsets). 14 To the best of our knowledge, these are contracting for information with full commitment (Krishna and Morgan, 008) in the case of the perfectly informed expert and static informational control with commitment (Ivanov, 010a) in the case of the imperfectly informed expert. They provide the ex-ante payo s to the 1 principal approximately 38 and 1 98, respectively. 15 The principal s ex-ante payo in the most informative equilibrium is equal to the average residual variance across sub-intervals k j. Suppose that the total number of sub-intervals in the learning protocol is N and all sub-intervals are of the equal length = 1 N. Since the density function can be approximated by the piecewise constant density on each sub-interval for large N, this gives EU ' 1 = 1 1N. Setting EU = ", where " > 0 is su ciently small, determines the necessary number of sub-intervals. N k j=1 9

10 4 Dynamic information extraction In this section, we extend the example above to general prior distributions of the state and players payo functions. In particular, we construct learning protocols that sustain the fully informative equilibria in which the expert truthfully discloses the state upon learning it in the second stage. 4.1 Learning protocols Consider the rst-period information structure which generates the expert s signal s 1 () = if < s or = 1; ' 1 () if s < 1: () Because the rst-period information structure maps state < s and a complement state 0 = ' () s into the same signal, the expert cannot distinguish between states and ' () upon observing a signal s =. Thus, the information structure () generates the family of the rst-stage posterior distributions ff (js) ; s [0; s); fs; ' (s)gg and the degenerate distribution at = 1, such that each F (js) supported on fs; ' (s)g with the probabilities: 16 f (s) p s = Pr f = sjsg = f (s) + f (' (s)) ' 0 (s), and p c s = Pr f = ' (s) jsg = 1 p s, if s [0; s): Hereafter, we call the support fs; ' (s)g of a posterior distribution F (js) the posterior set, and each pair of states s and s 0 = ' (s) posterior states. Given message m 1 [0; s), the expert s second-stage information structure generates a signal, which separates posterior states m 1 and ' (m 1 ), but keeps other states unseparated.: if fm1 ; ' (m 1 )g and m 1 [0; s); s (; m 1 ) = s 0 = [0; 1) otherwise. (3) The learning protocol () and (3) is characterized by a few key properties that in uence the expert s incentives to convey information. First, the binary set of posteriors makes the expert s information in the rst stage su ciently imprecise. Because the principal s ideal decisions for states s and ' (s) are di erent, he would be is interested in learning the state upon knowing that fs; ' (s)g. If the expert s bias is not large, her ideal decisions are close to those of the principal. Therefore, the expert is also interested in learning the true state in the second stage. 17 At the same time, the principal wants to limit the set of posteriors 16 The derivation of the formulas for p s and p c s is in the Appendix. 17 The e ect of the binary rst-stage information structure on expert s uncertainty is clearly seen in the case of the risk-averse expert, i.e., if V (a; ; b) is concave in a for all (; b). For a xed interval [ 1 ; ], consider all distributions of a random variable X with the support L [ 1 ; ] and a given mean value E [X] ( 1 ; ). E[X] In this class, the binary distribution on f 1 ; g with the probabilities p 1 = Pr fx = 1 g = 1 and p = 1 p 1, respectively, is dominated by any other distribution by the second-order stochastic dominance. (The proof follows from (3.A.8) in Shaked and Shanthikumar (007).) That is, the expert is the most uncertain about X if it takes binary values, 1 or. Applying this intuition to our model means that the 10

11 for each signal s. This is necessary in order to restrict the expert s possibilities of mimicking another posterior after perfectly learning the state in round two. Also, the distance between the posterior states ' (s) s a ects the expert s incentives due to two e ects. First, a smaller distance reduces the value of learning in the second stage. This is because it increases the expert s interim payo from inducing her optimal decision while the interim payo from learning the state and inducing an ideal principal s decision does not change signi cantly. Second, it provides incentive to the expert to distort information in the second stage upon learning. For example, the positively biased expert may bene t from inducing a p (' ()) if it is closer to her ideal decision than a p (). However, the principal is limited in maximizing the distance ' (s) s for all s by choosing a di erent function ' (s). This is because for any partition of [0; 1) consisting of the two-point subsets, the minimum distance between pairs of points is bounded by Information extraction in bounded state space We show now how the constructed learning protocol allows the principal to elicit full information from the expert. Suppose that the expert acquires information according to the information structures () (3) and sends the sequence of messages fm 1 ; m g. If m 1 [0; s) and m fm 1 ; ' (m 1 )g, the principal interprets them as truthful and implements the decision a p (m ). If m 1 = 1, the principal makes a decision a p (1) unconditionally on m. 18 Let Y 1 (s; b) be the set of maximizers of the expert s payo given rst-stage information: Y 1 (s; b) = arg max aa p s V (a; s; b) + (1 p s ) V (a; ' (s) ; b) ; s [0; s): That is, each y 1 (s; b) Y 1 (s; b) is an optimal interim decision based on the rst-stage information only. The expert can induce y = y 1 (s; b) by sending the proper messages in both periods. 19 Because her information will not be updated in the second stage, inducing y 1 (s; b) results in the payo : E [V (y 1 (s; b) ; ; b) js] = max aa p sv (a; s; b) + (1 p s ) V (a; ' (s) ; b) (4) = p s V (y 1 (s; b) ; s; b) + (1 p s ) V (y 1 (s; b) ; ' (s) ; b) ; s [0; s): If the expert reveals her information truthfully by sending m 1 = s, her interim information will be updated in the second stage, but the set of feasible actions shrinks to fa p (s) ; a p (' (s))g. In this case, reporting the truth in both periods results in the payo : E [V (a p () ; ; b) js] = p s V (a p (s) ; s; b) + (1 p s ) V (a p (' (s)) ; ' (s) ; b) : (5) rst-stage information structure, which generates a binary distribution of posterior values of for each signal, maximizes the expert s uncertainty about the state. This increases her bene ts from perfectly learning the state in the second stage and, thus, forces her to communicate truthfully in the rst period. 18 If the expert reports m 1 = [0; s)[f1g, the principal interprets it as some m 0 1 [0; s) and induces a P (m 0 1) for any m. If m 1 [0; s) [ f1g, but m = fm 1 ; ' (m 1 )g, the principal induces a P (m 1 ) for any m. 19 The messages fm 1 ; m g that induce y = y 1 (s; b) can be constructed as follows. Denote A 1 = fa p () j [0; s)g and A = fa p () j [s; 1)g. Then, de ne m 1 = m [0; s) \ ap 1 (y) if y A 1, m 1 = ' 1 (m ) ; m [s; 1) \ ap 1 (y) if y A, and m 1 = m = 1 if y = a p (1). 11

12 Since the expert can induce only a p (s) or a p (' (s)) upon reporting s < s, her second-stage incentive-compatibility constraints are given by: V (a p (s) ; s; b) V (a p (' (s)) ; s; b) ; s [0; s), and (6) V (a p (' (s)) ; ' (s) ; b) V (a p (s) ; ' (s) ; b) ; s [0; s): (7) Also, the learning incentive constraints in the rst stage require: E [V (a p () ; ; b) js] E [V (y 1 (s; b) ; ; b) js] ; s [0; s): (8) Finally, we model the out-of-equilibrium behavior of the players in the case of principal s deviation to a di erent information structure at any period as in the example above. Then, fully informative communication is sustainable if (6) (8) hold. The following theorem demonstrates that the trade-o between the informational bene ts and the exibility over available actions is in favor of the former option if the bias in preferences is not large. All proofs are collected in the Appendix. Theorem 1 There exists b such that if b b, there is a fully informative equilibrium with the learning protocol determined by () and (3). The proof is constructive and follows from the structure of the learning protocol. If the players interests are close enough, the ideal decisions of the players are su ciently close for each state. Since the expert s information in the rst period is imprecise, she is more interested in learning the state perfectly at the cost of inducing the principal s ideal decision than inducing an optimal interim decision. Because the expert can learn the state only by revealing her rst-stage information, the principal knows the binary distribution of posteriors at the beginning of the second period. For this distribution, truthtelling communication is sustainable if the bias is not large. A few comments are necessary. First, it is clear why communication in our model does not rely on the crucial assumptions about U (a; ; b i ) in the CS setup: the supermodularity condition Ua 00 > 0 and the (quasi)-concavity of U with respect to a. These are the local properties of the payo functions that determine the expert s reaction to changes in the state and the principal s reaction to expert s messages. As a result, they determine the incentives of the perfectly informed expert to manipulate her information locally by mimicking a slightly di erent state. However, the learning protocol in our setup precludes the expert from claiming a slightly di erent type, and hence, makes these assumptions unnecessary. Second, dynamic learning contrasts with mediated communication introduced by Myerson (1991) and studied by Blume et al. (007), Goltsman et al. (009), and Ivanov (010b). In mediated communication, the expert faces the message-contingent lottery over induced actions chosen by the mediator, who privately requests information from the expert and gives recommendations to the decision maker. In our model, the expert faces the message-contingent lottery over her own posterior types generated by the information structure in period one. The higher e ectiveness of a lottery over expert s types relative a lottery over principal s decisions is driven by two factors. First, in order to screen a continuum of types, the mediated protocol has to be delicate in punishing the expert for lying. Suppose that the mediator aims to punish the expert s type for placing some spin on 1

13 information and claiming type 0 close to by inducing some unfavorable distribution over actions. By the continuity of the expert s payo function, such a lottery also punishes type 0 for telling the truth and motivates this type to distort information. In contrast, the lottery over posterior types is able to separate the intensity of expert s punishment for distorting information across rst-stage types (signals). As a result, each expert s type can be punished for slight distortions of information without damaging the incentives of arbitrarily close types to convey their information. Sending a message s 0 di erent from the observed signal s in the rst period does not provide new information to the particular type s only. These informational losses, however, do not a ect type s 0, who can still bene t from truthtelling. Also, mediated communication critically relies on the risk-aversion of the expert. The mediator explores this property the randomization over recommended actions, which depends on the expert s reported type. A side e ect of this randomization is that each state maps into several actions, which cannot be the ideal principal s decisions together. In dynamic learning, the expert in the rst stage is interested in acquiring more information unconditionally on being risk-averse. That is, instead of imposing the (quasi)-concavity condition on the sender s payo function, we only assume that the sender s ideal decisions for posterior states are unique and distinct. Also, there is no need to randomize over actions after the expert learns in the second period. By that time, the principal knows the distribution of posterior types, which are distant enough to separate themselves without additional incentives. 0 Finally, note that if the expert s bias is her private information, but any value of b is below b, Theorem 1 still holds. In contrast, if b is su ciently large, then the fully informative equilibrium might not exist for any function ' () in the learning protocol given by () and (3). In order to demonstrate this, assume that a p (' (s)) > a p (s) for some s and and y (s; b) > sup A for a su ciently large b. Then, it follows that V (a p (' (s)) ; s; b ) = V (y(s; b ); s; b ) > V (a p (s) ; s; b ) = V (y (s; 0) ; s; b ) for some b. Thus, the expert strictly prefers to induce action a p (' (s)) in the second stage upon learning that the state is s if b b Comparative statics In the light of Theorem 1, an important question to examine is whether full information extraction is monotone in the magnitude of the expert s bias. In other words, what are the 0 Dynamic learning is also di erent from multi-stage communication investigated by Krishna and Morgan (004). Though active participation of the principal in the communication process in both models a ects expert s incentives to reveal more information by entailing some uncertainty for her, the nature of uncertainty in these models is completely di erent. In Krishna and Morgan s model, the expert is imperfectly informed about future decisions of the principal because of a random binary outcome of the simultaneous dialog in the rst round. In particular, if the outcome of the dialog is success, the expert may update her information in the next stage. Thus, even though the expert might not be allowed to improve her report later, the uncertainty about future interaction a ects the expert s incentives in the rst stage. Moreover, in the case of success, the expert actually reports more precise information in the second round. In our model, uncertainty arises directly from expert s imprecise information about the true realization of the state. The principal may in uence this uncertainty by amending the second-stage information structure, which depends on the rst-period report of the expert. 1 In the uniform-quadratic example above, the distance js s 0 j 1 for s0 = 1. Hence, if b > 1 4, then the second-stage incentive-compatibility constraints (1) are violated for some s. In addition, E [V (y 1 (s; b) ; ) js] 1 16 > E [V (a p () ; ; b) js] = b means that the expert s learning sup s;'(s):[fs;'(s)g= incentive constraints (8) are also violated. 13

14 conditions on the primitives, i.e., the players payo functions and the distribution of states, which sustain the full information extraction if and only if b is below some cut-o? To address this issue and extend the results to the case of the unbounded, we impose the standard CS conditions on the players preferences through this subsection: A1: U (a; ; b) is twice-di erentiable and satis es Uaa 00 < 0; Ua > 0, and Uab > 0. Due to these conditions, a (; b) is strictly increasing in and b. Through this subsection, we also suppose that U 0 b (a (; b) ; ; b) 0. Consider a di erentiable and strictly increasing function ' : [0; 1=]! [1=; 1] and the di erence between the expert s interim payo s in the cases of truthful communication and inducing an arbitrary decision y R: V (y; s; b) = E [V (a p () ; ; b) js] E [V (y; ; b) js] = p s V (a p (s) ; s; b) + (1 p s ) V (a p (' (s)) ; ' (s) ; b) p s V (y; s; b) (1 p s ) V (y; ' (s) ; b) : Suppose that y 1 (s; b) R is the expert s optimal (possibly unfeasible) interim decision: y1 (s; b) = arg max ar = arg max ar E [V (a; ; b) js] p s V (a; s; b) + (1 p s ) V (a; ' (s) ; b) ; s [0; 1=): Then, if the di erence in the payo s in the cases of truthful communication and inducing the (non-feasible) decision y 1 (s; b) is not increasing in b, then the fully informative equilibrium exists if and only if the bias is below some cut-o. Lemma 1 If V 0 b (y = y 1 (s; b) ; s; b) 0; s < 1=, then there is a fully informative equilibrium with the learning protocol determined by () and (3) if and only if b b. The main reason for using y1 (s; b) instead of y 1 (s; b) in Lemma 1 is that it is not necessary to check if the boundary condition y 1 (s; b) a p (1) is binding for all s and b. As a result, verifying the sign of Vb 0 (y = y 1 (s; b) ; s; b) is a signi cantly less complicated exercise than checking it for Vb 0 (y = y 1 (s; b) ; s; b). The following corollary is an implication of Lemma 1 for a class of preferences broadly used in cheap-talk and mechanism design literature. 3 Corollary 1 If V (a; ; b) = V (a (b; ) ; ), where (0; ) = 0; b 0 0, and 00 b 0, then there is a fully informative equilibrium with the learning protocol determined by () and (3) if and only if b b. 4.4 Information extraction in unbounded state space If the state space is unbounded, the expert s rst-stage information structure can be partitioned into posterior sets, such that the distance between any posterior states and That is, there are no private bene ts to the expert for simply being more biased. 3 Besides the quadratic payo function with a constant bias, this class includes the generalized quadratic functions V (a; ; b) = (a y (; b)) used by Alonso and Matouschek (008) and Kovác and Mylovanov (009), and the symmetric functions V (a; ; b) = V (ja ( + b)j) exploited by Dessein (00). Krishna and Morgan (004) use a special case of the symmetric functions, V (a; ; b) = ja ( + b)j. 14

15 0 = ' () is arbitrarily large. This might suppress the expert s incentive to distort her information because of two factors. First, a large distance between the posterior states implies that the optimal decision y 1 in the case of observing the signal s and deviating from truthtelling in period one would di er signi cantly from at least one of the ideal decisions y (s; b) and y (' (s) ; b). As a result, the expected losses from distorting information should increase. Second, upon communicating truthfully in the rst stage and observing s < s 0 = ' (s) in the next period, inducing a p (s 0 ) results in high losses of the expert if s 0 is su ciently distant from s. These observations suggest that the principal can potentially extract all information from the expert for an arbitrary magnitude of the expert s bias. The intuition above, however, misses an essential detail. The value of updated information in the second stage is high if the posterior states are quite distinct and approximately equally likely. For a xed state, if the distance between the posterior states and 0 > increases, then the density f ( 0 ) eventually converges to 0 and the posterior probability of converges to 1. Thus, the expert infers that the posterior realization = s is more likely than 0 = ' (s). This decreases her informational bene ts in the second stage and, as a result, the incentives to truthfully convey information in the rst stage. These arguments imply that truthtelling can be sustained only if the magnitudes of f () and f ( 0 ) do not di er signi cantly. In order to generalize the logic above, consider the distribution of states with a positive density f () > 0 on = R +. In addition to the assumptions on the payo functions maintained for the case of bounded, suppose that the set of rationalizable decisions of U (a; ) is unbounded, A = [a p (0) ; 1). Let y1 (; 0 ; b) be the expert s optimal action given that the state is either or 0 with probabilities p = p 0 = 1=: y 1 (; 0 ; b) = arg max aa V (a; ; b) + V (a; 0 ; b) : (9) Suppose V (a; ; b) is strictly pseudo-concave in a and satis es the following conditions. A: For any b, there exists (b) > 0, such that jy (; b) a p ()j (b) ; 8. A3: For any b and > 0, there exists (b; ) > 0, such that V (y (; b) ; ; b) V y (; b) (b; ) ; ; b, for all : A4: For any b and > 0, there exists d (b; ) > 0, such that y (; b) + < y 1 (; + d (b; ) ; b) < y ( + d (b; ) ; b) ; 8. Condition (A) states that there is a uniform bound on the expert s bias. Condition (A3) requires that the expert is not in nitely more sensitive to the actions below her ideal decision than to the actions above and vice versa. 4 Finally, (A4) establishes that (1) moving the posterior states 0 and su ciently apart from each other guarantees that the expert s ideal decisions will be su ciently apart as well, and () the optimal decision given information that the true state is either or 0 is not too close to the ideal decisions for either realization. These conditions do not impose strong restrictions on the shape of the expert s payo function, and imposing only (A) might be su cient for satisfying (A3) and (A4). 5 4 Ambrus and Lu (013) use this condition in the context of communication with multiple experts. 5 Consider the symmetric payo function V (a; ; b) = V (ja (b; )j), where V 0 (x) 0; V 00 (x) < 0; x 0. Then, a p () = ; y (; b) = + (b; ), and y1 (; + d; b) = + d + (b;)+(b;+d). For a xed b, (A) requires (b; ) (b) ; 8. This implies V (y (; b) ; ; b) = V () V = V y (; b) ; ; b if 15