A Good Listener and a Bad Listener

Transcription

1 A Good Listener and a Bad Listener Hiromasa Ogawa This version:march 2016 First draft:september 2013 Abstract This paper investigates how a listener s sensitivity, which represents the extent to which a listener incorporates others advice, affect information transmission. We develop a multi-sender cheap talk model in which one decision maker (DM) consults two experts in making a decision. We find that the value of sensitivity is dependent on the degree of conflict among the experts over the DM s decision. Specifically, high sensitivity improves the quality of communication when the conflict is not serious but hurts it otherwise. Furthermore, we demonstrate that the DM can improve her payoff by delegating the decision right to an agent who has the desired sensitivity. The degree of the DM s sensitivity may depend on her estimation of the importance of information possessed by the DM and the experts as well as the level of the DM s self-confidence in her skill to accurately interpret messages from the experts. Keywords : Cheap talk, Sensitivity toward opinions, Self-confidence, Bilateral Communication JEL classifications:d03,d83 1 Introduction When we advise others, we often take into account how the listener values our opinions. If a listener considers that a speaker has valuable information, she is sensitive to the advice in the sense that she gives more importance to the advice and is likely to incorporate it into the decisions. For example, consider a CEO of a manufacturing firm who needs to decide the quality of a new product (e.g., low-end, high-end, or between models). While the CEO knows the needs of consumers in the domestic market, she may have less information regarding the needs of consumers in overseas markets. To maximize the total sales of the firm, the CEO has to consult the sales managers in those markets regarding consumers preference for products. However, if sales managers prefer that the product is sold mostly in their own markets, they may exaggerate I am grateful to Akihiko Matsui for his helpful comments and advice. I also would like to thank Kohei Kawamura and seminar participants at Contract Theory Workshop East, Contract Theory Workshop summer camp, and Kwansei Gakuin University for their helpful comments. This research was supported by a grant from the Japan Society for the Promotion of Science (JSPS). National Graduate Institute for Policy Studies, Roppongi, Minato-ku, Tokyo, Japan ( hogawa@grips.ac.jp). 1

2 the needs in their respective markets (e.g., they may say our consumers need a much higher-end product ) even if the extent of the needs is not so extreme, because they know that the CEO will consider their opinions as well as those of the CEO and other managers whose opinions may conflict with that of their own. If the CEO attaches much importance to sales in overseas markets, she may decide the quality of the product giving a lot of weight on managers opinions. How does a listener(ceo) s sensitivity affect the amount of information provided by speakers(managers)? Do high sensitivity help elicit more information from speakers and improve the payoff of the listener? This paper addresses these questions by developing a model of strategic information transmission between decision maker (DM, she) and two experts (he). The DM s objective is to make a decision d that maximizes the weighted sum of three functions π 1 (d, θ 1 ), π 2 (d, θ 2 ), π L (d, θ L ) (interpreted as the total sales of the firm), but expert i s aim is to maximize only π i (d, θ i ) (interpreted as the sales in each overseas market). The DM and experts have non-overlapped private information, that is, the DM has information only on θ L, and each expert i has information only on θ i. The DM and experts can communicate via cheap talk before a decision is made. If the weight on π L is small in the DM s objective, then the DM is sensitive to the experts opinions in making a decision. Even truthful communication is not strategy-proof because the DM does not fully incorporate each expert s opinion into the decision. Only partially informative communication can be achieved as Crawford and Sobel (1982), which partitions the type space into some intervals so that the expert merely reveals that the type belongs to a certain interva. Somewhat counter-intuitively, high sensitivity may hurt the quality of communication with the experts. In fact, if there is only one expert, the DM can elicit more information as her sensitivity increases. If the DM has high sensitivity, the expert will be more willing to reveal information because such a DM will not persist with her opinions but would carefully consider and incorporate the expert s opinion. However, if there is more than one expert, the negative side of high sensitivity arises. Although a highly sensitive DM may incorporate more of one expert s opinion, she may also be easily influenced by another expert s opinion. If experts believe that their rivals hold opposite opinions, they may exaggerate their information to change the DM s mind. To clarify the intuition, we provide a brief sketch regarding how the DM s sensitivity affect the incentive of experts to provide information. When experts communicate their opinions, they consider the two following factors. First, how will the DM incorporate their opinions into the decision? If the DM gives substantial weight to the experts opinions, the incentive to misrepresent information becomes weak and experts are inclined to provide more information. Second, what is the collective opinion (which is the expected weighted 2

3 sum of the opinions of the DM and the other expert)? If the expert believes that the DM and the other expert hold opinions that are similar to his, then that expert does not need to misrepresent information. However, if the expert believes that they have different opinions and that the collective opinion differs from his ideal decision, there is a larger incentive to exaggerate information to change the DM s mind. Such a conflict of opinions over the ideal decision results in less informative communication. Thus, the sensitivity of a DM may exacerbate misrepresentation by the experts when their opinions are likely to be in conflict. These results suggest that it can be desirable to delegate the decision right to an agent who is either more or less sensitive than the DM. We examine the case in which the DM can delegate the decision right to an agent who may have decision policies that are different from those of the DM. We call the agent who is more sensitive toward the experts opinions than the DM a good listener and the agent who is less sensitive a bad listener. Delegation directly and strategically affects the DM s payoff. The direct effect is a negative effect caused by applying a distorted decision policy that leads to a not-optimal decision in the ex-post sense, and the strategic effect can be positive by the improving the quality of communication. We demonstrate that the DM always improves the payoff by delegation to a good (bad) listener if conflict between the experts is serious (not serious) as long as the distortion of the agent s sensitivity is slight, We address two issues in Discussion section. The first concerns the value of high sensitivity when there is some bias on the side of the DM. We show that the value of high sensitivity is more likely to be high when the DM has biased opinions. This is because the DM s extreme opinions provide an incentive to experts to exaggerate their information. Such a DM can elicit more information by giving up her own extreme opinion. Another form of distortion in sensitivity arise from the DM s favoritism, where the DM shows different sensitivities toward the experts. We demonstrate that the DM s payoff improves if she has high sensitivity toward the expert whose information is distributed in the widest range. The second issue relates to the DM s confidence in her skill to accurately interpret the advice from the experts. We argue that the DM who is overconfident in her own skill and irrationally underestimates the possibility of misinterpretation incorporates more of the experts opinions into the decision, whereas the underconfident DM who irrationally overestimates the possibility of misinterpretation incorporates less of the experts opinions. We demonstrate that the distortion in DM s sensitivity can arise from the irrational self-confidence in the interpretation skill. We further examine a bilateral communication setting, in which the DM can send a one-time costless message to the experts on her opinion before the experts send messages. We find that the DM s communication strategy can take only two forms, binary strategy (with which the DM induces only two different posteriors) and babbling strategy, despite the DM s type space being a continuum and more than three 3

4 messages being available. Furthermore, we find that although there may be an infinite number of binary equilibrium, all of them can be ranked in terms of the DM s payoff and the amount of information provided by the experts and that the DM s payoff is best in babbling equilibrium than in any other equilibrium. The results imply that the DM should intentionally obfuscate her own opinion before she listens to the experts. 2 Related Literature This paper make some contributions to the literature on cheap talk game. Crawford and Sobel(1982) is a seminal work in this field. They consider the situation in which only a sender can observe the true state, but only a receiver can make a decision that affects both utilities. They show that there is a Perfect Bayesian equilibrium such that state spaces are divided by finite numbers of partitions, and the sender reveals only the partition in which the true state is as long as the parties never have the same preference in the decision. Some researchers have examined the multi-sender situations where senders private information is correlated, such as Gilligan and Krehbiel (1987), Krishna (2001), and Battaglini (2002). Following the terminology used in the literature, this paper considers a situation in which multiple senders exist and they have independent private information. The closest model to ours is the individually biasedagents model in Kawamura (2011), who considers the case of a DM gathering information via cheap talk about members preferences for the level of public goods provision that affects the welfare of all members. The situation considered by Hori(2009) and Mcgee and Yang (2013) is similar to ours. They consider the case of a DM gathering information from two agents who have non-overlapped information on the true state. The information gathering problem in centralized organization as studied by Alonso et al.(2008) and Rantakari (2008), is also close to our model. However, this paper differs from theirs as we examine the effect of a receiver s sensitivity to messages from senders on the precision of the messages in a multi-sender situation. Some researchers have examined the effect of participants self-confidence about the precision of the sender s signal on the quality of communication and welfare in one-receiver and one-sender information transmission games. Admati and Pfleiderer (2004) argue that the sender s overconfidence in his observation skill may improve the quality of communication and the receiver s welfare. Kawamura (2015) also shows that slight overconfidence on the part of the sender can always increase information transmission and the receiver s welfare whenever the sender has a different preference from the receiver on the decision, whereas underconfidence does not. We consider a situation in which distortion in the receiver s sensitivity can arise from her positive/negative self confidence in her interpretation skill. We derive the clear welfare result that the receiver s overconfidence is likely to be negative as conflict among senders over the receiver s decision 4

5 intensifies in the multi-sender setting. The analysis in a bilateral communication setting contributes to the issue of whether disclosing conflict before communication is beneficial. Li and Madarasz (2008) consider the information transmission game and show that mandatory disclosure about the extent of conflict between the sender and receiver is not beneficial. While this paper shares a similar result that revealing information about differences in preferences among the players before communication hurts the quality of communication, no informative message can be transmitted via cheap talk in their model, whereas informative communication is feasible even via cheap talk in our model. We believe that this paper has a few contributions to the literature on leadership, by reinterpreting a DM as a leader and experts as followers. 1 Many previous studies on leadership have investigated the role of leaders in eliciting followers efforts (e.g.,rotemberg and Saloner (1993, 2000) and Hermalin (1998)). More recently, studies have focused on the role of leadership in communication in coordination games. Dewan and Myatt (2008) describe a situation in which followers actions need to be adapted to the environment and coordinated with the actions of other followers. Only leaders can observe the signals on the environment and transmit these signals to the followers. They emphasize the importance of not only skillful observation but also speaking skill in efficient coordination. Brunnermeier et.al (2012) argue that leader resoluteness, which is a very similar concept to less sensitivity, helps mitigate coordination difficulties arising from a time-consistency problem. Although our model does not address coordination issues, the implication of this paper are similar to those model. A novel contribution of our paper is to indicate that a leader s traits may help mitigate communication difficulties arising from conflicting interest among followers. Many researchers have also examined leader overconfidence. Gervais and Goldstein (2007), Van Den Steen (2005), Vidal and Mollar (2007), and Brunnermeier et.al (2012) emphasize the positive sides of a leader s overconfidence, and Goel and Thakor (2008) address why a leader might be overconfident. Our paper indicates that a leader s overconfidence in the interpretation skill can positively work to improve the efficiency of information gathering via cheap talk communication. 1 An example of a good listener is Konosuke Matsushita, the founder of Panasonic. It is well known that he incorporated followers opinions into his own decision giving them a lot of weight and did not persist with his own opinion, saying Listening is the first priority for managers. An example of a bad listener is Steve Jobs, the former charismatic CEO of Apple, who tended to persist with his own opinion rather than incorporate the opinions of others, saying Don t let the noise of others opinions drown out your own inner voice. 5

6 3 Model We consider one DM (she) and two experts (he) indexed by i = 1, 2. The DM and experts have their preferences regarding the DM s decision. Specifically, we assume that expert i s payoff (e.g., the sales in each overseas market) is given by π i (d, θ i ) = (d θ i b i ) 2, where d R denotes the DM s decision (e.g., the quality of a new product), and θ i + b i denotes expert i s ideal decision (e.g., the consumers needs in the oversea market managed by manager i). θ i denotes expert i s private information and b i R is public information. θ i is uniformly distributed in [ s i, s i ] where s i R + and the distribution is public knowledge. We assume that b 1 = b 2 = b > 0, that is, the expected experts ideal decision is symmetrical around zero. 2 Then, b represents the extent of ex-ante conflict between the experts on the decision. Each expert s objective is to maximize only his own payoff. 3 The DM s payoff (e.g., the total sales of the firm) is given by Π = α L (d θ L ) 2 + α i π i (d, θ i ). θ L denotes the DM s private information regarding the ideal decision (e.g., the consumers needs in the domestic market). θ L follows a density function f L and a cumulative function F L, and let the mean and the variance be µ L and σl 2 respectively and the distribution is public knowledge. α L represents the importance of the DM s information (e.g., the importance of sales in the domestic market) and α i represents the importance of expert i s information (e.g., the importance of sales in each overseas market) on the DM s payoff. i=1,2 The experts can communicate their own information before the DM makes a decision. Each expert sends a one-time costless message r i [ s i, s i ] to the DM. 4 We suppose that the DM can not commit any mechanism and monetary transfer contingent on the messages, that is, any communication is cheap talk. We denote a posterior belief after communication as m i E[θ i r i ] for i = 1, 2. Finally, to make our model tractable, we make the following assumption. Assumption 1. (i) b min{ s1 2, s2 2 }, and (ii) µ L min{ s1 2, s2 2 }. In words, (i) assumes that the extent of the conflict between the experts is not too large, and (ii) assumes that the expected value of the DM s information is not largely biased. 2 We study an asymmetric case in Discussion section. 3 An unaligned incentive is typical in organizations. In organizations, it is undesirable to fully align a member s incentives from the viewpoint of preventing a free-rider problem, even if a misalignment in their incentives creates communication difficulties. Athey and Roberts (2001), Dessein et al.(2010), and Friebel and Raith (2010) address this issue. 4 In the later section, we extend the model to allow bilateral communication in which the DM can also send a one-time, costless, and publicly-observable message to the experts before the experts send messages. 6

7 The game proceeds as follows: 1. The DM and the experts privately observe θ L and θ i for i = 1, The experts send their messages to the DM (they are not necessarily truthful). 3. The DM decides d. 4 Decision making 4.1 The DM s decision The problem is solved backwards. Note that the equilibrium decision is represented as a liner combination such as d = z L θ L + z 1 (m 1 + b 1 ) + z 2 (m 2 + b 2 ) with z L + z 1 + z 2 1. We refer to z i as the degree of the DM s sensitivity toward expert i s opinion and denote a vector z = (z L, z 1, z 2 ) as the DM s decision policy. From the first order condition, the DM s optimal decision is given by d(θ L, r 1, r 2 ) = α L α L + α 1 + α 2 θ L + α 1 α L + α 1 + α 2 (m 1 + b 1 ) + α 2 α L + α 1 + α 2 (m 2 + b 2 ). (1) We denote the optimal decision policy as z = (z L, z 1, z 2) = ( 4.2 Communication strategy α L α L +α 1+α 2, α 1 α L +α 1+α 2, α 2 α L +α 1+α 2 ). Before we consider the experts communication strategy, we show that the experts have incentives to misrepresent the information they possess. Following Alonso et al.(2008), we first suppose that expert 1 can credibly misrepresent his information, that is, he can arbitrarily choose the DM s posterior belief about the information. For a given decision policy of the DM, the optimal posterior for expert 1, denoted by m 1, satisfies E[d m 1 = m 1] = θ 1 + b 1, or equivalently, Define functions B 1 as and q 1 as m 1 = θ 1 + b 1 z L µ L z 2 E[m 2 + b 2 ] z 1 b 1. B 1 (θ 1, z) m 1 θ 1 = (1 z 1)(θ 1 + b 1 ) z L µ L z 2 E[m 2 + b 2 ] z 1 q 1 (z) z Lµ L +z 2 E[m 2 +b 2 ] 1 z 1 b 1. B 1 (θ 1, z) represents the difference between what expert 1 wants the DM to believe and his true information. The first term of q 1 (z) is interpreted as an expected collective opinion of the other participants from 1 s 7

8 viewpoint weighted by z, and it is straightforward to show B 1 (q 1 (z), z) = 0. Intuitively, if θ 1 = q 1 (z), expert 1 s ideal decision is identical to the expected collective opinion, then he has no incentive to misrepresent his information. However, he has an incentive to exaggerate his information whenever θ 1 q 1 (z). Expert 1 induces a higher posterior belief than his true information when θ 1 is higher than q 1 (z) and a lower posterior belief when θ 1 is lower than q 1 (z). Moreover, B 1 (θ 1, z) increases as θ 1 is further away from q 1 (z). Thus, the greater the difference between expert 1 s ideal decision and the expected collective opinion, the stronger the incentive for misrepresentation. We define functions B 2 as and q 2 as B 2 (θ 2, z) m 2 θ 2 = (1 z 2)(θ 2 + b 2 ) z L m L z 1 E[m 1 + b 1 ] z 2 q 2 (z) z Lm L +z 1E[m 1+b 1] 1 z 2 b 2. Expert 2 also has no incentive to reveal his information truthfully and has an incentive to exaggerate his information whenever θ 2 q 2 (z). The incentive for misrepresentation becomes larger as the difference between θ 2 and q 2 (z) increases like it does for expert 1. While a truth-telling equilibrium does not exist, partially informative communication may still be achieved as shown by Crawford and Sobel (1982). It is achieved by partitioning the type space so that any message r i reveals only that θ i belongs to some interval. Divide expert i s type space into N i intervals and name cutoff points from left as a ij for j = 0,..., N i, which satisfies boundary conditions a i0 = s i and a ini = s i and order constraints a ij < a ij+1. In equilibrium, expert i sends a randomized message that is drawn from the uniform distribution supported on [a ij 1, a ij ) if θ i [a ij 1, a ij ). If the receipt message is in [a ij 1, a ij ), the DM s posterior belief is given by m ij = aij 1+aij 2. On each cutoff point, expert i is indifferent between reporting that θ i belongs to either one of the two intervals around that cutoff point. That is, any cutoff a ij for j = 1,...N i 1 must satisfy the following indifferent condition, E[π i θ i = a ij, m i = m ij ] = E[π i θ i = a ij, m i = m ij+1 ]. (2) Solving and arranging (2), we obtain the second order difference equation as follows; for j = 1,..., N i 1, a ij+1 a ij = a ij a ij 1 + 4B i (a ij, z). (3) Figure 1 illustrates the cutoffs in an equilibrium; here we put s 1 = 1, z L = z 1 = z 2 = 1/3, and b = 1/3. From the second order difference equation (3), we can see how B i (a ij, z) determines the size of each interval. 8

9 At any cutoff a ij such that a ij < q i (z) = 1/2, the size of the interval a ij+1 a ij is smaller than the size of the preceding intervals a ij a ij 1 by 4 B i (a ij, z), and the changes in the sizes of intervals decrease as j increases. The change in the size of the intervals becomes quite small when a ij is near q i (z). In turn, at any cutoff a ij such that a ij > q i (z), the size of the interval a ij+1 a ij is larger than the size of the preceding intervals a ij a ij 1 by 4 B i (a ij, z), and the changes in the sizes of intervals increase as j increases. a 10 a 11 a 1N1 1 a 1N q 1 (z) θ 1 Figure 1: Communication strategy There is no upper bound for the number of equilibrium cutoffs except for the case where q i (z) is extremely high or low. The following lemma identifies a sufficient condition to ensure that N i is not limited. Lemma 1. For i = 1, 2, the upper bound of N i does not exist if q i (z) [ s i, s i ]. The proof is in Appendix. In words, the limit disappears when the expert has an information that is identical to the expected collective opinion from 1 s viewpoint with strict positive probability. Intuitively, if q i (z) [ s i, s i ], we can find the equilibrium where an infinite number of intervals of negligibly small size exist around q i (z). We remark that Assumption 1 ensures that q i (z) [ s i, s i ] for any z. To summarize the results so far, we present the following proposition. Proposition 1. Suppose Assumption 1 holds. Given the DM s decision policy z, for i = 1, 2 for any positive integer N i there exists at least one equilibrium such that 1. the DM s decision is given by (1), 2. expert i sends the randomized message r i, which is drawn from the uniform distribution supported on [a ij 1, a ij ) if θ i [a ij 1, a ij ) for j = 1,...N i 1 and on [a ini 1, a ini ] if θ i [a ini 1, a ini ], 3. the DM makes her belief m i as aij 1+aij 2 if r i is in [a ij 1, a ij ) for j = 1,...N i 1 and a in i 1+a ini 2 if the receipt message r i is in [a ini 1, a ini ], and 4. for j = 1,..., N i 1, a ij follows (3), and a i0 = s i and a ini = s i. The proof is in Appendix. 9

10 Note that when N i is large enough any cutoffs can be approximately represented by the following explicit form; for the j-th cutoff from the left edge, and for the j-th cutoff from the right edge, ( ) a ij = 1 x(z i ) s j i x(z i ) q j i (z), (4) ( ) a ini j = 1 x(z i) s j i x(z i) q j i (z). (5) ) ( ) 2 4 (2 + where x(z i ) = 4zi 2 4 z i 2 > 1. The derivations are in Appendix. Then, j-th cutoff from the left (resp. right) edge is represented as an internally divided point between the left (resp. right) edge and q i (z) ( ) j ( j. 1 in the ratio x(z i) : 1 1 x(z i)) 4.3 Quality of communication A residual variance E[(θ i m i ) 2 ] indicates how the information that expert i provides is precise. If the posterior belief of the DM about i s information is close to (resp. far from) his true one, it becomes small (resp. large). By applying the law of iterated expectation, we obtain E[(θ i m i ) 2 ] = E[θi 2] E[m2 i ]. We refer to E[m 2 i ] as the quality of communication with expert i, because E[θ2 i ] is independent of the equilibrium profile and the residual variance decrease as E[m 2 i ] increases. The quality of communication with expert i increases as N i increases, that is, the more the intervals, the more precise is the communication. Lemma 2. E[m 2 i ] is increasing in N i. Proof is in Appendix. A high quality of communication also improves the DM s payoff as we see in Welfare Implication section. Therefore, the following section focuses on the equilibrium with an infinite partitioned communication strategy, in which the DM s payoff is maximized within any partitioned communication strategy. In the infinitely partitioned equilibrium, the residual variance is given simply as lim E[(θ i m i ) 2 ] = 1 z ( ) i s 2 i N i 4 z i 3 + q i(z) 2 and the quality of communication with i is given by lim N E[m2 i ] = 1 s 2 i 1 z i q i (z) 2. (6) i 4 z i 4 z i We denote lim Ni E[m 2 i ] by M i(z i, q i (z)). If Assumption 1 holds M i (z i, q i (z)) is well-defined. Two key parameters, the DM s sensitivity to expert i s opinion z i and i s expected collective opinion q i (z), characterize the quality of communication with expert i. From (6), we can observe the following property. 10

11 Lemma 3. (i) Mi z i (z i, q) q=qi (z) > 0. (ii) Mi q i (z i, 0) = 0 and 2 M i (z i, q) < 0. Figure 2 depicts a change in cutoffs as z 1 increases, for fixed q 1. In figure 2, we put s 1 = 1 and z 1 = 1/2 and fix q 1 = 1/2. Because for any j 1 x(z i) j q 2 i increases as z 1 increases, according to (4) and (5), any cutoffs to the left (resp. right) of q 1 shift toward left (resp. right) edge. Thus, the equilibrium partitions on both sides become fine and the quality of communication is improved as z 1 increases, for a given fixed q 1. Compared with the case in depicted in Figure 1, a 11 shifts to from and a 1N1 1 shifts to from 0.551, and the quality of communication is improved to 1/4 from 5/22. a 10 a 11 a 1N1 1 a 1N q 1 θ 1 Figure 2: Communication strategy with z 1 = 1/2, given fixed q 1 Lemma 3 (ii) means that M i is single peaked at q i = 0. Figure 3 depicts a change in cutoffs as q 1 (> 0) increases, for a given fixed z 1. In Figure 3, we put s 1 = 1 and q 1 = 0.75 and fix z 1 = 1/3. According to (4) and (5), any cutoffs shift toward the right as q 1 increases, then the equilibrium partitions to the right of q 1 become fine but ones to the left of q 1 become coarse. Compared with the case in depicted in Figure 1, a 11 shifts to from and a 1N1 1 shifts to from 0.551, and the quality of communication declines to 15/88 from 5/22. Note that the residual variance becomes large as the sizes of larger intervals increase even if the sizes of smaller intervals decrease. In contrast, the residual variance becomes small when the sizes of larger intervals decrease even if the sizes of smaller intervals increase. Then, the residual variance is minimized when the equilibrium partitions are symmetric around zero, that is, when q 1 = 0. a 10 a 11 a 1N1 1 a 1N q θ 1 Figure 3: Communication strategy with q 1 = 3/4, given fixed z 1 5 The DM s sensitivity and the quality of communication Here we examine how the DM s sensitivity affect the quality of communication. To make the intuition of the results clear, we put α 1 = α 2 = α > 0 and µ L = 0 in the following sections unless otherwise noted. 11

12 We consider how a decrease in α L affect the quality of communication with expert i. The marginal effect can be represented as follows; M i α L (z i, q i (z)) = M i z i (z i, q) q=qi (z) Note that M i z i (z i, q) q=qi (z) > 0 from Lemma 3 (i). ( z ) i + M ( i (z i, q i ) q ) i (z). (7) α L q i α L Because z i α L > 0, the first term of (7) is always positive. The first term of (7) is interpreted as the positive side of a highly sensitive listener, which is coming from the notion that expert i s incentive to misrepresent information becomes weaker according to the DM s willingness to incorporate i s opinions into the decision. On the other hand, the second term of (7) represents the negative side of a highly sensitive listener. To see this, we focus on the quality of communication with expert 1. Because we obtain q1 α L (z) > 0. In the same manner, since we obtain q 2 α L (z) < 0. Note that, for i = 1, 2, q 1 (z) = z Lµ L + z 2 E[m 2 + b 2 ] 1 z 1 b 1 (8) = α L + 2α b > 0, (9) α L + α q 2 (z) = z Lµ L + z 1 E[m 1 + b 1 ] 1 z 2 b 2 (10) = α L + 2α b < 0, (11) α L + α M i q i (z i, q i ) > 0 if q i < 0 and M i q i (z i, q i ) < 0 if q i > 0 from Lemma 3 (ii). These imply that the second term of (7) is always negative. The result above can be interpreted as follows. While a highly sensitive listener gives expert 1 s opinion significant weight in the decision, she also gives expert 2 s opinion a significant weight. This implies that from expert 1 s view point, the voice of the rival becomes more influential, and the collective opinion becomes biased toward the rival s ideal decision. If expert 1 believes that the rival has a greatly different ideal decision from his, expert 1 s incentive to exaggerate his information becomes strong to change the DM s mind against the rival s voice. Even though the negative side of high sensitivity may exist, it can be negligible when b is quite small. When b is small, q i is hard to change with regard to α L. In words, if the rival likely has a similar opinion to him, each expert pays less attention to the rival s voice being influential. Thus, we arrive at the next proposition, which identifies the situation where high sensitivity improves the quality of communication with the experts. Proposition 2. Suppose Assumption 1 (i) and µ L = 0 holds. M i is increased as α L decreases if and only if b 2 < (α L + α) 2 (α L + 2α)(5α L + 8α) s2 i. (12) 12

13 In Appendix, we provide the proof and confirm the threshold of b can exist within the area restricted by Assumption 1. Proposition 2 claims that when the conflict between experts is not serious, a highly sensitive listener is more likely to improve the quality of communication because the negative side of a highly sensitive listener is quite small and only the positive side arises. However, if the conflict is serious, the negative side is no longer negligible and a less sensitive listener can elicit more information than a highly sensitive one. Note that a sensitive listener always improves the quality of communication in the case of one DM and one expert. To treat that case in the same framework, we assume that α 1 = α and α 2 = 0. If α 2 = 0, expert 2 s information does not matter for the DM s payoff and expert 2 s opinion is never incorporated into the decision. Then, the problem under the the assumption is identical to one in a situation with the DM and only expert 1. In this case, q 1 (z) = b 1 from (8) and it is independent of α L. This implies that the negative side of a good listener disappears in this case, though the positive side exists. The next proposition immediately follows. Proposition 3. M 1 is always increased as α L decreases in the one expert case. 6 Welfare implication We allow for the DM being able to delegate the decision right to an agent. We assume that the agent can observe θ L but values the importance of the information in a different way from that of the DM. Specifically, we assume that the agent believes the importance of her own information to be higher or lower than α L. The importance of θ L that the agent believes is denoted as ᾱ L. As in the example of new product development illustrated in Introduction, such a heterogeneity of the belief may arise from a bias in compensation contract for the CEO. If the shareholders of a firm can offer a compensation contract to the CEO contingent on the sales in each market, they can arbitrarily design the values of ᾱ L. The agent decides d to maximize Π = ᾱ L (d θ L ) 2 + α i π i. We say that the agent is a good listener when ᾱ L < α L and a bad listener when ᾱ L > α L. If the DM is a good listener, z i > z i i=1,2 for i = 1, 2, that is, a good listener shows a highly sensitive attitude toward both the experts opinions. On the other hand, if the DM is a bad listener, z i < z i shows a low sensitive attitude toward both the experts opinions. for i = 1, 2, that is, a bad listener We consider the effect of a decrease in ᾱ L on the DM s payoff. The ex-ante expected payoff is represented 13

14 as E[Π] = α L E[(d(θ L, r 1, r 2 ) θ L ) 2 ] αe[(d(θ L, r 1, r 2 ) θ i b i ) 2 ] i=1,2 = (α L (1 z L ) 2 + 2αz 2 L)σ 2 L α s2 1 3 αs2 2 3 z 1(z 1 (α L + 2α) 2α)E[m 2 1] z 2 (z 2 (α L + 2α) 2α)E[m 2 2] α L ((1 z L )µ L z 1 b 1 z 2 b 2 ) 2 α((1 z 1 )b 1 z L µ L z 2 b 2 ) 2 α((1 z 2 )b 2 z L µ L z 1 b 1 ) 2.(13) Then the marginal effect of a decrease in ᾱ L on the payoff is represented by E[Π] ᾱ L = E[Π] M1 =M ᾱ 1 (z 1,q 1 (z)),m 2 =M 2 (z 2,q 2 (z)) + E[Π] ( M ) 1 (z 1, q 1 (z)) + E[Π] ( M ) 2 (z 2, q 2 (z)). (14) L M 1 ᾱ L M 2 ᾱ L The first term of (14) represents the direct effect of a decrease in ᾱ L on the DM s payoff. The second and third terms of (14) represent the strategic effect of a decrease in ᾱ L on the quality of communication. 5 We can further separate the strategic effect into two. The first effect is how ᾱ affects the quality of communication, and the second effect is how the quality of communication affects the payoff. If a distortion of ᾱ L from α L is slight, we can neglect any direct effect by applying the envelope theorem. It is straightforward to show that E[Π] M i > 0 at ᾱ L = α L, that is, the DM s payoff is improved as the quality of communication is improved. Then, using proposition 2, we may identify the situation where delegating the decision right to a good listener can increase the DM s payoff. Proposition 4. Suppose Assumption 1 (i) and µ L = 0 holds. The DM can increase her payoff by delegating the decision right to a good listener if and only if b 2 < (α L + α) 2 2(α L + 2α)(5α L + 8α) (s2 1 + s 2 2). Otherwise, the DM can increase the payoff by delegating the decision right to a bad listener. Proposition 4 claims that a good listener can achieve a better performance if the conflict between the experts is not serious, and otherwise a bad listener can achieve better performance. 7 Discussion 7.1 A Biased opinion of the DM The value of high sensitivity is dependent on the DM s opinion. Roughly speaking, if the expert believes that the DM has a more different opinion than the rival, increasing the DM s sensitivity to the experts opinions 5 If communication is not strategic, because the strategic effect should be zero, it immediately follows that the optimal decision policy should be z. 14

15 make the collective opinion close to the expert s ideal decision and the negative side of high sensitivity disappears. To demonstrate this, we assume µ L 0 and focus on the quality of communication with expert 1. Note that M 1 (z 1, q 1 ) q 1 ( q ) 1(z) α L = 2α(µ L b)(α L µ L + (α L + 2α)b) (α L + α) 2. (15) (4α + 7α) From (15), if µ L α L+2α α L b then the negative side does not exist. Intuitively, if the DM has an extremely biased opinion in a negative direction, the collective opinion is also extremely biased in a negative direction, and q 1 (z) goes toward zero as the voice of expert 2 becomes more influential, because expert 2 s ideal decision is biased in a positive direction. Nor does the negative side exist if b µ L. Intuitively, if the bias of expert 2 s ideal decision is more moderate (in the sense that it closer to zero) than that of the DM s opinion, the collective opinion goes toward zero as the voice of expert 2 becomes more influential. The negative side of high sensitivity arises if the DM s opinion is moderate, that is, α L+2α α L b < µ L < b. Since q1(z) α L > 0 and q 1 (z) > 0 and Lemma 3 (ii) suggests M1(z1,q1) q 1, the negative side exists. The seriousness of the negative side is dependent on µ L. Note that µ L ( M1 (z 1, q 1 ) q 1 ( q )) 1(z) 4α(α L µ L + αb) = α L (α L + α) 2 (4α L + 7α). (16) From (16), we derive that the negative side is the most serious when µ L = α α L b. We can show that the negative side on the quality of communication with expert 2 disappears if µ L b or α L+α α L b µ L, it exists if b < µ L < α L+α α L b, and it is most serious when µ L = α α L b in the same manner. If the bias of the DM s opinion is moderate, a decrease in α L heterogeneously affects the quality of communication with each expert. As µ L increases, the negative side on the quality of communication with expert 1 becomes small but that with expert 2 becomes large. However, the negative side of high sensitivity on the quality of communication with the both experts is totally weak or diminished when the DM s opinion is biased. Proposition 5. Suppose Assumption 1 holds. High sensitivity is more likely to improve the quality of communication with both experts when the DM has a biased opinion. Proof in Appendix. 7.2 Heterogeneous sensitivity In this subsection, we consider a delegation case where the agent shows heterogeneous sensitivity toward each expert s opinion. Our concern here is to examine whether heterogeneous sensitivity is attractive, and 15

16 if so, when it is attractive. We suppose the agent s objective is given by Π = α L (d θ L ) 2 + ᾱ i π(d, θ i ). where ᾱ 1 = α 1 + ɛ and ᾱ 2 = α 2 ɛ with ɛ R. Then, the agent s decision policy, denoted by z ɛ, is given by z ɛ L = i=1,2 α L α L + 2α, zɛ 1 = α + ɛ α L + 2α, and zɛ 2 = α ɛ α L + 2α. Because z ɛ 1 is increasing in ɛ, the agent becomes more sensitive to expert 1 s opinion and gives it a significant weight in the decision as ɛ increases. Similarly, because z ɛ 2 is decreasing in ɛ, the agent becomes less sensitive to expert 2 s opinion giving a small weight in the decision as ɛ increases. We can interpret ɛ as the degree of favoritism toward expert 1. As ɛ increases, the quality of communication with expert 1 always improves, but that with expert 2 always declines. Note that q 1 (z ɛ ) = α L + 2α 2ɛ α L + α ɛ b > 0 q 2 (z ɛ ) = α L + 2α + 2ɛ α L + α + ɛ b < 0, and it is straightforward to show that q1 ɛ (zɛ ) < 0 and q2 ɛ (zɛ ) < 0. Then, from Lemma 3, we obtain that M 1 ɛ (zɛ 1, q 1 (z ɛ )) > 0 and M 2 ɛ (zɛ 2, q 2 (z ɛ )) < 0. This suggests that heterogeneity in the sensitivity has a tradeoff effect on the quality of communication with each expert. The following proposition identifies a situation where the heterogeneity improves the DM s payoff. Proposition 6. Suppose Assumption 1 and µ L = 0 holds. Some small ɛ improves the DM s payoff if s 1 > s 2. Proposition 6 claims that the agent should be sensitive toward the expert whose information is distributed in the widest range. To make the intuition clear, take an extreme example where s 2 is near zero. Although the information of both experts is equally important for the DM s payoff, the DM does not need to extract information from expert 2 because almost accurate information can be extrapolated without communication. In such a case, it is effective to be sensitive toward expert 1 s opinion to extract more information from him, even if the quality of communication with expert 2 declines. In Figures 4 and 5, we compare the equilibrium communication strategy under decision policy z with that under decision policy z ɛ when s 1 > s 2. Figure 4 illustrates the partitions of type spaces in the equilibrium under z. The listener can know which grid the experts private information lies in. Under z, expert 1 s type space is divided by relatively coarser partitions than expert 2 s type space. Figure 5 illustrates the partitions of type spaces in the equilibrium under z ɛ with ɛ > 0. As ɛ increases, the size of the partitions in expert 1 s 16

17 type space becomes finer and that of the partitions in expert 2 s type space becomes coarser. In this case, the grids become more balanced, whereupon the sum of residual variances E[(θ 1 m 1 ) 2 ] + E[(θ 2 m 2 ) 2 ] decreases and the listener can infer the ideal decisions of both experts more precisely on average. s 2 θ 2 s 2 θ 2 s 1 s 1 θ 1 s 1 s 1 θ 1 q 2 (z ) q 2 (z ɛ ) s 2 q 1 (z ) s 2 q 1 (z ɛ ) Figure 4: The partitions of type spaces under decision policy z. Figure 5: The partitions of type spaces under decision policy z ɛ with ɛ > Self-confidence and sensitivity In this subsection, we attempt to explain that a distortion in the DM s sensitivity can arise from the DM s confidence in her interpretation skill. We introduce two assumptions regarding her skill level in interpreting experts messages accurately. First, the DM misinterprets an expert s message with probability λ and unconsciously forms a wrong posterior which is independent of the message. Formally, when expert i sends message r i, the DM receives it with probability 1 λ but receives a message r i which is independent of r i and randomly drawn from [ 1, 1] with probability λ and she is unaware of misinterpreting it. 6 Second, the DM may have a wrong belief in her own skill. Let λ be her belief of the probability of misinterpretation. We say that the DM is overconfident if λ < λ and underconfident if λ > λ. We denote the posterior belief with the belief of misinterpretation probability λ as m λ i = E[θ i λ, r i ]. It can be represented as m λ i = (1 λ)e[θ i r i ] + λe[e[θ i r i ]] = (1 λ)m i. The DM s decision d λ(θ L, r 1, r 2 ) is represented as d λ(θ L, r 1, r 2 ) = α L α L + 2α θ (1 λ)α L + α L + 2α m (1 λ)α 1 + α L + 2α m 2. (17) 6 Here we employ the same framework as that for communication noise, which was introduced by Blume et al. (2007). 17

18 and the decision policy is given by z λ = ( α L α L +2α, (1 λ)α α L +2α, (1 λ)α α L +2α ). We remark that the decision policy is dependent on the DM s belief on her skill, not on the skill itself. Since the DM believes that she has not formed a correct assessment of one expert s message with probability λ, the possibility of misinterpretation is taken into account optimally by discounting the weight on the experts opinions in decision making. As a result, the overconfident (resp. underconfident) DM puts an excessively large (resp. small) weight on experts opinion and a small (resp. large) weight on her own information. We can then discuss the effect of self-confidence on the quality of communication in almost the same manner as in the earlier sections and derive a similar claim to Proposition 2 and 4 without qualitative differences as follows. Proposition 7. Suppose Assumption 1 (i) and µ L = 0 holds. The overconfident DM improves the quality of communication with i if and only if b 2 < α L + (1 λ)α (α L + 2α)(5α L + (8 + 2λ)α) s2 i. Furthermore, the DM s slight overconfidence increases her payoff if and only if Proof in Appendix. b 2 < 8 Bilateral communication α L + (1 λ)α (α L + 2α)(5α L + (8 + 2λ)α) (s2 1 + s 2 2). In this section, we consider a bilateral-communication situation whereby the DM can send a one-time costless message concerning her own information to the experts. We suppose that the DM s message is publicly observable and that the experts send messages after observing the DM s message. 7 We denote the DM s message as r L R and posterior beliefs about the DM s information after communication as m L E[θ L r L ]. While the DM s message does not affect the decision policy, it can affect the amount of information the experts provide to the DM. Given the DM s message r L and decision policy z, the experts communication strategy is derived by the following indifferent condition, E[π i θ i = a ij, m i = m ij, r L ] = E[π i θ i = a ij, m i = m ij+1, r L ]. (18) Solving and arranging (18), we obtain the second order difference equation that is same as (3), except that B i and q i is dependent on m L as follows, B 1 (θ 1, z) = (1 z 1)(θ 1 +b 1 ) z L m L z 2 E[m 2 +b 2 ] z 1 B 2 (θ 1, z) = (1 z 2)(θ 2 +b 2 ) z L m L z 1 E[m 1 +b 1 ] z 2, 7 Even if the DM s message is not public but private, the results can hold without qualitative differences. 18

19 and q 1 (z) = z Lm L +z 2E[m 2+b 2] 1 z 1 b 1 q 2 (z) = z Lm L +z 1 E[m 1 +b 1 ] 1 z 2 b 2. Then, the collective opinion can be dependent on the DM s message in the context of bilateral communication. As we considered the experts communication strategy in unilateral communication, likewise we first consider the DM s incentive to misrepresent her information. Suppose that the DM can credibly misrepresent her information, that is, she can arbitrarily choose the experts posterior beliefs of θ L. The optimal posterior for the DM, denoted by m L, maximizes the DM s objective for given θ L: m L = argmax ml E[Π θ L, m L ]. Because the DM s message affect the DM s payoff only through changes in M i, from (13) it can be represented as m L = argmax ml P (M 1, M 2 ) z 1 (z 1 (α L + 2α) 2α)M 1 + z 2 (z 2 (α L + 2α) 2α)M 2. (19) The first order condition gives that m L = 0, then the DM has an incentive to misrepresent her information unless θ L = 0. Furthermore, the DM s objective is single peaked and symmetric around zero with regard to m L. This fact gives that the number of posteriors induced in equilibrium is two at most and the absolute values of the posteriors are the same. We refer to an equilibrium where the DM induces two different posteriors as binary equilibrium and an equilibrium where the DM induce only one posterior (that is, she sends an uninformative message) as babbling equilibrium. The following proposition holds. Proposition 8. In bilateral communication, the equilibrium takes only two forms; babbling equilibrium or binary equilibrium. Proof in Appendix. Only in binary equilibrium, the DM sends an informative message to the experts. Posteriors induced in binary equilibrium are not uniquely determined. For example, θ L is a random variable drawn from a uniform distribution on [ 1, 1]. Consider the following communication strategy. If θ L < 0, the DM sends message A with probability p and B with probability 1 p, and if θ L 0, she sends message B with probability p and A with probability 1 p, where 1/2 < p 1. Given this strategy, the experts posterior after observing each message is given by E[θ L r L = A] = p/2 + (1 p)/2 = (2p 1)/2 and E[θ L r L = B] = (1 p)/2 + p/2 = (2p 1)/2 respectively. The two posteriors are indifferent for 19

20 the DM then she does not have an incentive to deviate from such a communication strategy. Then, there exist a continuum of binary equilibrium where the DM induces two different posteriors characterized by p (1/2, 1]. 8 Although a continuum of binary equilibrium may exist as we see above, all equilibrium can be ranked in terms of the DM s payoff. We define k-binary equilibrium as the equilibrium where the induced absolute values of the posteriors equal k. To make a countably infinite partition strategy of the experts feasible, we modify the Assumption 1 as follows. Assumption 2. (i) b min{ s1 2, s2 2 }, and (ii) the range of distribution of θ L is within [ s 2, s 2 ], where s = min{s 1, s 2 }. Assumption 2 ensures that k min{ s1 2, s2 2 } in any k-binary equilibrium and the condition of Lemma 1 is satisfied. Then, the following proposition holds. Proposition 9. Suppose Assumption 2 hold. The quality of communication and the DM s payoff in babbling equilibrium are the highest among those in any binary equilibrium. The quality of communication and the DM s payoff in k-binary equilibrium are higher than those in k -binary equilibrium for any k > k. Proof in Appendix. The first claim follows from the fact that k > µ L ] has to hold, and the second claim follows from the single peakedness of P (M 1, M 2 ) with regard to m L. Furthermore, the following claim immediately follows from Proposition 5. Corollary 1. High sensitivity is more likely to be valuable in any binary equilibrium than the babbling equilibrium. High sensitivity is more likely to be valuable in k -binary equilibrium than k-binary equilibrium for any k < k. 9 Concluding remarks This paper examines the effect of a listener s sensitivity toward experts opinions on the amount of information provided by the experts. Developing a multi-sender cheap talk model between one DM and two experts, we show that a highly sensitive DM is likely to facilitate communication with experts when conflict between the experts is not serious. Otherwise, such a DM actually may hurt the quality of communication with the experts, whereas a less sensitive DM is likely to facilitate communication. The DM can improve her payoff by delegating the decision right to the agent who has a slightly distorted sensitivity from the DM. We 8 If we focus only on a binary partition strategy, unique binary equilibrium exist. 20