Optimism and Communication∗ Ying Chen† Department of Economics, Arizona State University October 10, 2010

Abstract I examine how the communication incentive of an agent (sender) changes when the prior of the principal (receiver) about the agent’s private information becomes more optimistic (in the sense of monotone likelihood ratio dominance). I use the canonical model of strategic communication (Crawford and Sobel, 1982) under the assumption that the agent has an upward bias. The main result is that under a more optimistic prior, more information can be transmitted in equilibrium in the sense that the principal’s expected payoff is higher. Applying the result to three models, I find: (1) If the principal can choose among intermediaries to make decisions, she may benefit from choosing someone more optimistic than herself, but never benefits from choosing someone more pessimistic. (2) A more optimistic principal is more likely to gain from centralization relative to delegation. (3) In a game of two-way communication in which a privately informed principal reports to an agent first, the principal cannot credibly reveal her private signal unless it is informative enough. Keywords: communication, information transmission, cheap talk, optimism, prior, monotone likelihood ratio, intermediary, delegation, informed principal. JEL Classification: D23, D82, D83. ∗

This paper supersedes a portion of an earlier paper titled “Communication with Two-sided Asym-

metric Information.” I thank Oliver Board, Hector Chade, Alejandro Manelli, Edward Schlee and audiences at ASU, Midwest Theory Conference 2008, Canadian Economic Theory Conference 2009 and Society of Economic Design Conference 2009 for helpful suggestions and comments. † Department of Economics, Arizona State Univeristy, P.O. Box 873806, Tempe, AZ 85287-3806. Fax: 480-965-0748. Email: [email protected].

1

1

Introduction

In many situations of strategic communication, an agent (sender) who has information useful for a principal (receiver) also has an incentive to exaggerate the favorableness of his information. For example, a division manager with a desire for “empire building” (i.e., making large investments in his division) is motivated to overstate the profitability of an investment project to a CEO, and a financial analyst associated with the underwriting firm is under pressure to inflate claims about the value of a stock. As is well known, the sender’s incentive to exaggerate limits the effectiveness of his communication when his messages are costless and unverifiable (i.e., cheap talk), so only imprecise information can be conveyed to the receiver. But relatively little is known about how the effectiveness of communication is influenced by the prior belief that the receiver holds. For example, will a division manager be able to give a more nuanced report about the prospect of an investment opportunity when the CEO already deems it likely to succeed? Can an analyst better convey the value of a stock if investors have high confidence in its potential to begin with? This paper addresses these questions by comparing the amount of information that can be transmitted from a biased sender when the receiver’s prior belief becomes more optimistic. I model a more optimistic prior as a rightward shift in the sense of monotone likelihood ratio (MLR) dominance.1 As a distribution undergoes an MLR improvement, the relative likelihood of a better prospect becomes higher (in the manager-CEO example, an investment project is believed to be more likely to yield a higher profit), and it is natural to interpret it to be a more optimistic belief. In the canonical sender-receiver game introduced in the seminal paper by Crawford and Sobel (1982), I find that “more information” can be transmitted in equilibrium as the receiver’s prior becomes more optimistic. The intuition for this result lies in the mechanism in which noisy information gets transmitted in equilibrium. For a biased sender’s communication to be credible, it is necessary that his messages become increasingly noisy as the recommended action moves further in the direction of his bias. To use the manager-CEO example again, more nuanced reports can be credible when a division manager recommends relatively low 1

I define MLR dominance formally in section 2.2. For detailed discussions on MLR dominance and

comparative statics, see, for example, Milgrom (1981), Jewitt (1987), and Athey (2002).

2

investment levels, but only very coarse information can be conveyed when he recommends a high level of investment. When the receiver holds a more optimistic prior, her choice of action is higher for any given report from the sender. This results, roughly speaking, in a lower rate at which the sender’s messages become noisier. Hence the “average noise” the equilibrium messages contain is lower when the receiver is more optimistic, resulting in more information transmission in equilibrium. To describe the finding more precisely, recall that an equilibrium of a Crawford and Sobel (CS for short hereafter) game takes a partitional form, i.e., the sender partitions his type space, a bounded interval on the real line, into a finite number of steps (subintervals) and conveys to the receiver what step his type lies in. Let H be a prior that MLR dominates another prior L. Focusing on the equilibrium partition that gives the receiver the highest payoff in any given game, I find that if the receiver’s prior is H, then her expected payoff is always higher in the equilibrium partition induced under prior H than in the equilibrium partition induced under prior L. Furthermore, if the receiver’s prior is L, her expected payoff is also higher in the equilibrium partition under prior H than in the equilibrium partition under prior L, as long as H is not “too far” from L.2 So the receiver prefers the equilibrium partition under H to that under L, no matter whether her prior is L or H. In this sense, more information can be transmitted in equilibrium when the receiver is more optimistic about the sender’s type, at least locally. This parallels the finding in Crawford and Sobel (1982) that more information can be transmitted when the players’ interests are more closely aligned. I compare the two sets of comparative statics results (with respect to the prior in this paper and with respect to the sender’s bias in Crawford and Sobel) and point out their connections and differences in section 2. That optimism facilitates information transmission offers new insight into a number of questions. To illustrate, I consider three applications. Choosing an intermediary. The first application concerns the problem of a principal choosing a representative/intermediary to communicate with a privately informed agent and then make the decision on her behalf. If all potential intermediaries have the same preferences as the principal, but may differ in their prior beliefs, what intermediary should the principal choose? An application of the comparative statics result shows 2

I make it precise what it means to be “too far” in section 2.3.

3

that the principal should not use a more pessimistic intermediary to make decisions on her behalf but may benefit from using an intermediary who has a different but more optimistic belief from her own. Centralization versus delegation. The second application is about the allocation of authority in organizations, in particular, when should upper management centralize decision (with communication from a better informed lower management) and when should it delegate decision rights to lower management? I extend the analysis in Dessein (2002) of uniform prior to a more general class of Beta distributions. The main insight is to connect the value of communication and hence the advantage of centralization to the principal’s prior belief. As shown in Dessein (2002), under a uniform prior, the principal prefers centralization to delegation only when the agent’s bias is so large that communication is uninformative. By contrast, I show that a principal with a more optimistic prior may still want to centralize decision even when the agent’s bias is small enough that informative communication is possible because it is more likely for a more optimistic principal to extract useful informative from the agent. Two-way communication with an informed principal. In the third application, a principal privately observes a signal that is affiliated with the type of an agent. I address the following question: if two-way communication is possible which allows the principal to send a cheap-talk message first to the agent before the agent reports, is it possible for the principal to credibly reveal her signal? The answer, roughly, is that if the principal’s own signal is not too informative, then it is impossible for her to credibly reveal her signal because no matter what her true observation is, she would prefer the agent to believe that she has received a more favorable signal which leads her to be more optimistic about the agent’s type. My debt to the classic communication models of Crawford and Sobel (1982) and Green and Stokey (2007) is obvious. Two other closely related papers are Dessein (2002) and Che and Kartik (2009). I defer a detailed discussion of these papers to section 3 when I discuss the applications.

4

2

Optimism and information transmission

2.1

Benchmark model

Let us first review briefly the equilibrium characterization in the canonical senderreceiver game a la Crawford and Sobel (1982). In this game, the sender (S) privately observes the state of the world (his type) t, and then sends a message m to the receiver (R), who then chooses an action a ∈ R that affects both players’ payoffs. The sender has a twice continuously differentiable von-Neumann-Morgenstern utility function denoted by U S (a, t, b) and the receiver has a twice continuously differentiable von-Neumann-Morgenstern utility function denoted by U R (a, t). The parameter b measures the divergence of interest between the players. Assume that when b = 0, the players’ interests coincide, i.e., U S (a, t, 0) = U R (a, t). Suppose b > 0, which implies that the sender has an upward bias, i.e., he would like the receiver to believe that the state is higher than it really is.3 (To simplify notation, when it is clear that b is fixed, I omit it the the arguments and just write U S (a, t).) For each t and for i = S, R, denoting partial derivatives by subscripts in the usual way, assume i (a, t) < 0, so that U i has a unique maximum in a for U1i (a, t) = 0 for some a, and U11

each t. For each t and i = S, R, let ai (t) denote the unique solution to maxa U i (a, t). i (a, t) > 0, which implies that ai (t) is Assume U i (a, t) is supermodular in (a, t), i.e., U12 S increasing in t. Also assume U13 (a, t, b) > 0, which implies that aS (t) > aR (t) for all t

(i.e., the sender’s ideal action is always higher than the receiver’s) as b > 0. Suppose the type space T and the message space M are both equal to [0, 1]. Fix a differentiable distribution function F on [0, 1]. For 0 ≤ t0 < t00 ≤ 1, let a ¯F (t0 , t00 ) be the unique solution R t00 R to maxa t0 U (a, t)dF (t), i.e., a ¯F (t0 , t00 ) is the receiver’s optimal action when he believes that t has support on (t0 , t00 ) with distribution F . By convention, a ¯F (t, t) = aR (t). Suppose the players’ common prior is that t is distributed on [0, 1] with a differentiable distribution function F and density f . Let m (t), a mapping from T to M , be the sender’s (pure) strategy, and a (m), a mapping from M to A, be the receiver’s (pure) strategy in a Perfect Bayesian Equilibrium. 3

The case in which b = 0 is uninteresting and therefore not considered. Results analogous to those

in this paper can be derived for b < 0. In that case, a leftward shift in the prior facilitates information transmission.

5

Crawford and Sobel (1982) find that all equilibria take a simple partitional form: an equilibrium is characterized by a partition of the set of types, t(N ) = (t0 (N ), . . . , tN (N )) with 0 = t0 (N ) < t1 (N ) < . . . < tN (N ) = 1, and messages mi , i = 1, . . . , N . (The number of elements that a partition has is called its size.) The types in the same partition element send the same message, i.e., m(t) = mi for t ∈ (ti−1 , ti ]. The receiver best responds, i.e., a(mi ) = a ¯F (ti−1 , ti ). The boundary types (ti , i = 1, ..., N − 1) are indifferent between pooling with types immediately below or immediately above. So the following “arbitrage” condition holds: for all i = 1, ..., N − 1, U S (¯ aF (ti , ti+1 ), ti )) − U S (¯ aF (ti−1 , ti ), ti )) = 0,

(A)

Crawford and Sobel (1982) further make the following monotonicity assumption. For ti−1 ≤ ti ≤ ti+1 , let V (ti−1 , ti , ti+1 ) ≡ U S (¯ aF (ti , ti+1 ), ti ) − U S (¯ aF (ti−1 , ti ), ti ). A (forward) solution to (A) of length K is a sequence {t0 , . . . , tK } such that V (ti−1 , ti , ti+1 ) = 0 for i = 1, .., K − 1. Definition 1. The Monotonicity (M) Condition is satisfied if for any two solutions to (A), ˆ t and ˜ t with tˆ0 = t˜0 and tˆ1 > t˜1 , we have tˆi > t˜i for all i ≥ 2. Note that an equilibrium partition of size K satisfies (A) with t0 (K) = 0 and tK (K) = 1. Crawford and Sobel prove that if Condition (M) is satisfied, then there is exactly one equilibrium partition for each N = 1, . . . , N ∗ . The equilibrium with the highest number of partition elements (also called steps), N ∗ , is commonly referred to as the “most informative” equilibrium.4 Chen, Kartik and Sobel (2008) provide a condition, “No Incentive to Separate”, that selects the equilibrium with N ∗ steps when condition (M) holds. For the rest of the paper, I assume that (M ) holds and only consider the equilibrium with the highest number of steps.

2.2

Comparing equilibrium partitions under different priors

I now derive comparative statics of the CS equilibria with respect to the players’ prior when the priors are ranked by the monotone likelihood ratio (MLR) order. 4

This terminology is imprecise as a higher number of steps typically does not lead to a refinement

of the partition.

6

Let H (t) and L (t) be two differentiable probability distributions on [0, 1] with density functions h (t) and l (t) respectively. Assume h (t) > 0, l (t) > 0 for t ∈ [0, 1]. If

h(t) l(t)

is

increasing in t, we say that H (t) is a monotone likelihood ratio (MLR) improvement of L (t). Equivalently, H (t) MLR dominates L (t). What follows are two lemmas that are useful in the comparison of equilibrium partitions under different priors. The first lemma is a standard result in monotone comparative statics under uncertainty and the proof is therefore omitted.5 Lemma 1. a ¯H (t0 , t00 ) > a ¯L (t0 , t00 ), ∀ 0 ≤ t0 < t00 ≤ 1. This lemma says that if the receiver believes that t ∈ (t0 , t00 ), then her optimal action under belief H is higher than her optimal action under belief L. Now let tF (K) = (tFi (K))i=0,...,K with tFi (K) < tFi+1 (K) for i = 0, ..., K − 1 be a partial partition of size K satisfying the “arbitrage” condition (A) (page 6) when the players’ prior over t is F . We have the following result. (All proofs are in the appendix.) Lemma 2. Fix t and t¯. Suppose K ≥ 2. If tL (K) exists with tL0 (K) = t and tLK (K) = t¯, L H H ¯ then tH (K) exists with tH 0 (K) = t and tK (K) = t. Moreover, ti (K) > ti (K) for i = 1, 2, ..., K − 1. Lemma 2 applies to all (partial) partitions that have the same endpoints. In particL H L ular, if tL0 = tH 0 = 0 and tK = tK = 1, then t (K) is an equilibrium partition of size

K under prior L and tH (K) is an equilibrium partition of size K under prior H. So Lemma 2 implies that whenever an equilibrium partition of size K exists under prior L, it must also exist under prior H. Moreover, for equilibrium partitions of the same size, the boundary types under H are to the right of the boundary types under L. To formalize the result, let N ∗ (F ) be the maximum number of steps in an equilibrium when the players’ prior is F . We have Proposition 1, a direct implication of Lemma 2. Proposition 1. (a) The maximum size of an equilibrium partition under H is weakly higher than the maximum size of an equilibrium partition under L, i.e., N ∗ (H) ≥ N ∗ (L). (b) For a fixed equilibrium partition size, the boundary types in the equilibrium partition under prior H are to the right of those under L, pointwise. 5

For a proof, see, for example, Ormiston and Schlee (1993).

7

To gain some intuition for why this is true, consider the simple case of an equilibrium partition of size two. Suppose (0, tL1 , 1) is an equilibrium partition under prior L, and

a ¯L 0, tL1 and a ¯L tL1 , 1 are the receiver’s best responses. The sender of type tL1 is


¯L 0, tL1 is lower than his ideal point ¯L tL1 , 1 where a indifferent between a ¯L 0, tL1 and a
and a ¯L tL1 , 1 is higher than his ideal point. If we keep the partition but change her belief to H, then, by Lemma 1, the receiver’s best responses will shift to the right. That



¯L tL1 , 1 . Since his payoff function is single¯H tL1 , 1 > a ¯L 0, tL1 and a is, a ¯H 0, tL1 > a

peaked in a, the sender of type tL1 strictly prefers a ¯H 0, tL1 to a ¯H tL1 , 1 . Condition (M ) and the continuity of the players’ payoff functions imply that there exists a type
¯H (0, t) and a ¯H (t, 1). t ∈ tL1 , 1 such that the sender of type t is indifferent between a Letting tH 1 = t, we have an equilibrium partition of size two under prior H and the equilibrium boundary type under H is to the right of the equilibrium boundary type under L. Induction on equilibrium size shows that the result holds for partitions of larger sizes as well. It is instructive to compare these comparative statics result with respect to the players’ prior and Crawford and Sobel’s (1982) comparative statics result with respect to the sender’s bias. Crawford and Sobel find that for equilibrium partitions of the same size, the partition associated with the players’ preferences closer together (i.e., smaller b) begins with larger steps (Lemma 6) and that the maximum possible equilibrium size is nonincreasing in b (Lemma 5). So the two sets of comparative static results are parallel to each other. In what follows, I discuss how they are related. Take an equilibrium partition of size K under prior F and bias b. If we fix F but lower b, the receiver’s optimal actions associated with the steps in the original equilibrium partition remain the same but the sender’s preference changes. The indifference conditions of the boundary types no longer hold because with a smaller b, a boundary type now strictly prefers the action associated with the step immediately below to the action associated with the step immediately above. Under condition (M ), in the new equilibrium partition the boundary types must all shift to the right. Alternatively, if we fix b but change F with an MLR improvement, the sender’s preference remains the same but the receiver’s optimal actions change. With the MLR improvement of her belief, the receiver’s optimal actions associated with the steps in the original equilibrium partition all shift to the right. The indifference conditions for the

8

boundary types no longer hold because a boundary type now prefers the action associated with the step immediately below to the action associated with the step immediately above. An analogous change in the equilibrium partition follows. That is, all boundary types shift to the right in the new equilibrium partition. Does “more information” get transmitted in equilibrium when the receiver’s prior is more optimistic? If one uses the criterion that a signal6 is more informative if it is more valuable to every decision maker, then as Blackwell’s (1951) Theorem shows, a more informative signal must be a sufficient statistic for a less informative one. In the comparison of partitions, sufficiency amounts to refinement, and the equilibrium partitions that are induced under different priors typically do not satisfy this criterion.7 But as the next subsection shows, one can still establish useful results regarding the value of information contained in the equilibrium partitions induced under different priors. In particular, under certain conditions, the receiver’s payoff is higher when facing the partition induced under a more optimistic prior, no matter whether her prior is the more optimistic or the more pessimistic one. So in this sense, more information can be transmitted when the prior becomes more optimistic.

2.3

Receiver’s preference over equilibrium partitions under different priors

Fix the receiver’s belief F . Take a partial partition of size K (≥ 2), (ti )i=0,...,K . The receiver’s expected payoff on [t0 , tK ] when she faces the partition (ti )i=0,...,K is EU R = PK R ti R aF (ti−1 , ti ) , t) dF (t). i=1 ti−1 U (¯ I will first present a lemma which will be useful in comparing the receiver’s expected payoffs under different partitions. Fix the end points t0 and tK . Let (ti (x))i=0,...,K be a partition that satisfies (A) for i = 2, ..., K with tK−1 (x) = x (i.e., the last boundary type in the partition is equal to x). So the partition satisfies (A) except for (possibly) i = 1. Next, consider the receiver’s expected payoff on [t0 , tK ] when she faces the partition (ti (x))i=0,...,K as x moves to the right. 6 7

Here, a signal is generated by the sender’s message strategy. An exception is the comparison of a babbling equilibrium and a non-babbling one. The partition

in a babbling equilibrium has only one element and is therefore a strict coarsening of a partition in a non-babbling equilibrium.

9

Let y be the type that satisfies t1 (y) = t0 .

So the first step of the partition

(ti (y))i=0,...,K is degenerate: the partition has size (K − 1). Let y 0 be the type such that the partition (ti (y 0 ))i=0,...,K satisfies (A) for i = 1 as well as i = 2, ..., K. (That is, U S (¯ aF (t0 , t1 (y 0 )) , t1 (y 0 )) = U S (¯ aF (t1 (y 0 ) , t2 (y 0 )) , t1 (y 0 ))). Lemma 3. For x ∈ [y, y 0 ], the receiver’s expected payoff on [t0 , tK ] when she faces the partition (ti (x))i=0,...,K is increasing in x. Lemma 3 has important implications for the receiver’s preference over different partitions. In particular, combined with condition (M ), it implies that if we fix the players’ payoff functions and the prior and start with an equilibrium partition, then the receiver would not prefer another partition with the boundary types shifted to the left. Moreover, the receiver would prefer another partition with the boundary types shifted to the right, at least locally. Here is some intuition. Recall that for each equilibrium boundary type, the sender is indifferent between the actions induced in the steps immediately below and immediately above. Since the receiver prefers a lower action than the sender does, the receiver must prefer the action induced in the lower step to the action induced in the higher step. So, roughly speaking, if the boundary types are shifted to the left, the partition becomes even more skewed, increasing the average noise in the partition and making the receiver worse off. When the boundary types are shifted locally to the right, the partition becomes more “balanced,” reducing the average noise in the partition and making the receiver better off. From Proposition 1, we know that the boundary types of the equilibrium partition L

t (K) are to the left of the boundary types of the equilibrium partition tH (K). So we have the following result regarding the preference of the receiver with belief H. Proposition 2. The receiver with the belief H strictly prefers tH (N ∗ (H)) to tL (N ∗ (L)). The proof has two main steps. First, I fix the partition size and show that the receiver with the belief H strictly prefers the equilibrium partition under H than that under L. Second, I use Theorem 3 in Crawford and Sobel (1982), which says that for any given CS game that satisfies (M ), the receiver prefers an equilibrium with a larger size. So if N ∗ (H) > N ∗ (L), the receiver strictly prefers tH (N ∗ (H)) to tH (N ∗ (L)).

10

Combined with the first step of the proof, this implies that the receiver strictly prefers tH (N ∗ (H)) to tL (N ∗ (L)). Proposition 2 says that the receiver with prior H is better off in the equilibrium partition of the largest size induced under H than in the equilibrium partition of the largest size induced under L. What about the receiver with prior L? Does she prefer the equilibrium partition induced under H as well? We have already argued that the receiver benefits when the boundary types in an equilibrium partition shift to the right locally. Indeed, as long as the the boundary types are not shifted “too far” to the right, the receiver is better off. (Intuitively, if the boundary types are shifted too far to the right, then the value of information contained in the partition under H becomes too low for the receiver with belief L as she places a lot of weigh on low states.) Proposition 3 below makes it precise what “too far” means. Basically, if at the boundary between two steps the receiver still prefers the action induced in the lower step to the action induced in the higher step, then she benefits from a shift of the boundary types to the right.
H

H
H H Proposition 3. If U R a ¯L tH , ti ≥ U R a ¯L tH for i = 1, ..., N ∗ (H), i−1 , ti i , ti+1 , ti then the receiver with belief L strictly prefers the equilibrium partition tH (N ∗ (H)) to the equilibrium partition tL (N ∗ (L)). The proof also has two main steps. The first step shows that for a fixed size, the receiver with the belief L strictly prefers the equilibrium partition under H than that under L if the conditions in the proposition are satisfied. The second step shows that the argument can be extended even if N ∗ (H) > N ∗ (L). To summarize, if the conditions in Proposition 3 are satisfied,8 then the receiver has a higher expected payoff in the equilibrium partition of the largest size induced under H than in the equilibrium partition of the largest size induced under L, no matter whether her prior is H or L. So the value of information transmitted in equilibrium under prior H is higher and in this sense, more information can be transmitted when the prior becomes more optimistic. Crawford and Sobel (1982) have a related result on the receiver’s preference over equilibrium partitions. Their Theorem 4 says that for a given size, the receiver prefers the equilibrium associated with more similar preferences (i.e., a smaller b). Again, it 8

These conditions are sufficient, but not necessary.

11

is instructive to compare their results with the results derived here. As shown in CS, fixed the prior F , when b gets smaller, the equilibrium boundary types shift to the right. Observe that this shift is never “too far” to the right to benefit the receiver, that is, the conditions on the receiver’s payoffs given in Proposition 3 are always satisfied. To see this, note that the indifference condition of the boundary type ti requires that S > 0, U S (¯ aF (ti−1 , ti ) , ti , b) = U S (¯ aF (ti , ti+1 ) , ti , b). Since U R (a, t) = U S (a, t, 0) and U13

it follows that U R (¯ aF (ti−1 , ti ) , ti ) > U R (¯ aF (ti , ti+1 ) , ti ) for any b > 0.

3

Applications

I illustrate the usefulness of the comparative statics results derived in the previous section with the following three applications.

3.1

Choosing an intermediary

Instead of communicating with an informed agent and then making the decision herself, a principal sometimes may choose a representative (intermediary) to communicate with an agent and make decisions on her behalf. For example, diplomatic envoys are given the right to negotiate with a foreign country and make certain decisions on behalf of their government. Similarly, executives sometimes delegate decisions to external consultants.9 What is the advantage of using an intermediary and what characteristics of an intermediary make him attractive to the principal? Dessein (2002) examines intermediaries who have different preferences from the principal. The main insight of his analysis is that if an intermediary’s preference is closer to the agent’s, giving authority to the intermediary may facilitate information transmission from the agent, which in turn benefits the principal. Another possibility, not discussed in Dessein (2002), is that intermediaries may have different beliefs from the principal, 9

Note that the intermediary discussed here is different from the non-strategic mediator in Goltsman,

Horner, Pavlov and Squintani (2009) and the strategic intermediaries in Li (2010), Ambrus, Azevedo and Kamada (2009) and Ivanov (2010). Here, the principal gives the communication and decision right to the intermediary whereas in the other papers, the mediator or intermediary only passes information from the agent to the principal and does not make any payoff-relevant decision himself. Moreover, the papers on strategic intermediaries focus on how difference in preference may impede or facilitate information transmission whereas the focus here is on the difference in beliefs.

12

and this is the case to be analyzed here. To highlight the effect of prior beliefs on the choice of an intermediary, assume that the intermediary’ preference is the same as the principal’s. Giving the communication and decision making right to an intermediary has two effects: (1) Since the agent now faces someone with a different prior, his communication incentives are different,10 which may result in either informational gain or loss, as will be seen more precisely in the discussion that follows. (2) Given the information conveyed by the agent, the intermediary’s optimal choice of action is generally different from that of the principal’s because of their differences in beliefs. Since this always results in a loss for the principal, the use of an intermediary makes a principal better off only when there is sufficient informational gain. Suppose the principal’s prior is F and the intermediary’s prior is G. If F MLR dominates G, i.e., the intermediary is more pessimistic, then by Proposition 2, the principal strictly prefers the equilibrium partition of the largest size if she communicates directly with the agent (tF (N ∗ (F ))) to the equilibrium partition of the largest size if the intermediary communicates with the agent (tG (N ∗ (G))). So using the intermediary results in informational loss and therefore the principal does not benefit from giving the decision making right to a more pessimistic intermediary. Now suppose the intermediary’s prior G MLR dominates the principal’s prior F , i.e., the intermediary is more optimistic. In this case, as shown in Proposition 3, the principal may prefer the partition tG (N ∗ (G)) to tF (N ∗ (F )). This informational gain could be sufficiently high that the principal ultimately is made better off by giving authority to a more optimistic intermediary. The following example illustrates this point. Example 1. Suppose F (t) = t, G (t) = 21 t + 12 t2 , the principal’s (and also the intermediary’s) utility function is − (a − t)2 and the agent’s utility function is − (a − t − 0.2)2 . Since

g(t) f (t)

= 21 + t is increasing in t, the intermediary with the prior G is more optimistic

than the principal with the uniform prior F . If the agent communicates directly with the principal, then the equilibrium partition of the largest size is (0, 0.1, 1) and the principal’s expected payoff is −0.061. If the intermediary makes the decision, then the equilibrium partition of the largest size in the communication game between the agent and the intermediary is (0, 0.158, 1) and the principal’s expected payoff is −0.031. Hence the principal is better off giving the authority 10

The intermediary’s belief is assumed to be known to the agent.

13

to the intermediary. More generally, we have the following proposition. Proposition 4. The principal never benefits from choosing a more pessimistic intermediary, but may benefit from choosing a more optimistic intermediary. Che and Kartik (2009) show that to motivate an advisor to acquire information, it is optimal for a decision maker to choose someone with a different prior than her own. Proposition 4 complements their result, but there are a number of important differences. First, the decision maker chooses which advisor to communicate with in Che and Kartik (2009) whereas the principal chooses which representative to give the communication and decision-making right to here. Second, the information that an advisor possesses is endogenous in Che and Kartik (2009) whereas the agent’s information is exogenously given here. Most importantly, Che and Kartik (2009) focus on the incentives created by the differences in opinion for information acquisition, but the focus here is on the advantage of optimism (a particular kind of difference in opinion) for information transmission.

3.2

Centralization versus delegation in organizations

When to delegate decisions to lower management and when to centralize authority is an important problem in organizations. The central trade off that the upper management faces is that delegation makes better use of the lower management’s local knowledge but typically results in distortions because of the different objectives (“biases”) that the lower management may have. Whether centralized decision making (with communication from the informed agent) dominates delegation depends on how much information can be conveyed from the agent to the principal under centralization, which in turn depends on the principal’s prior. Dessein (2002) provides a detailed characterization for uniform distributions and finds that under a uniform prior, delegation is better than centralization whenever the agent’s bias is small enough that informative communication is feasible. This implies that whenever the principal chooses centralization over delegation, there is no information conveyed by the agent’s report and the principal makes the decision based only on her prior. As one would expect, this result relies on the uniform assumption and Dessein 14

(2002) provides more general results for symmetric priors, but it is still unclear what happens with asymmetric priors. Although a full characterization is hard to obtain, I take a step forward and provide some new insight into the value of centralization relative to delegation for a family of Beta distributions which includes the uniform distribution as a special case. Suppose the principal’s prior Fα is a Beta distribution on [0, 1] with Fα (x) = xα and fα (x) = αxα−1 . If α = 1, then the distribution is uniform. Moreover, Fα MLR dominates Fα0 if and only if α > α0 . Suppose the players’s utility function take the commonly used quadratic form, i.e., the principal’s utility function is − (a − t)2 and the agent’s utility function is − (a − t − b)2 . If the principal delegates the decision to the agent, then the agent chooses the action a = t + b and the principal’s payoff is −b2 . If the principal keeps the authority and makes a decision based on her prior  Fα , then, given  the quadratic-loss utility funcα tion, her expected payoff is equal to − (1+α)2 (α+2) , the negative of the variance of t. Moreover, if the principal keeps her authority but communicates with the agent, then an informative equilibrium exists when the agent’s bias is sufficiently small. In
α this case, when b < 12 α+1 . Since the principal’s payoff in an informative equilibrium is higher than a decision based on only her prior, it immediately  if she makes 
α follows that if − (1+α)α2 (α+2) ≥ −b2 and b < 12 α+1 , then informative communication is feasible under centralization and it dominates delegation. When are these condi
α tions satisfied? In the case of uniform distribution (α = 1), we have 21 1+α = 14 
2 1 1 and − (1+α)α2 (α+2) = − 12 . Since − 12 < − 41 , delegation dominates centralization whenever informative communication is possible, consistent with the finding in Dessein q √
α (2002). However, if (1+α)α2 (α+2) < 21 1+α , i.e., α > 5 − 1, there exists a range of biases such that informative communication is better than delegation. Note that for q
α α > 1, (1+α)α2 (α+2) is decreasing in α and 21 1+α is increasing in α. Since α indicates a more optimistic prior, this shows that a more optimistic principal is more likely to centralize authority as it is more likely for her to gain from communication.

3.3

Two-way communication with an informed principal

If the principal privately observes a signal s which is correlated with the agent’s type t, then the principal’s belief about t depends on the realization of the signal s. For 15

simplicity, I will focus on the case in which the signal s has two realizations: sH and sL (sH > sL ). Let H denote the principal’s belief on t when observing s = sH and L denote her belief on t when observing s = sL . Suppose the conditional probability distribution p (s|t) satisfies the monotone likelihood ratio property, i.e., t and s are affiliated. Then H MLR dominates L. (Note that the observation sH is more “favorable” news than sL in the sense of Milgrom, 1981.) There are numerous applications in which the principal has private information which is correlated with the agent’s type.11 For example, managers ask their subordinates to evaluate workers to help with compensation and promotional decisions, but they may have their own assessment of workers from occasional interaction with them. In a setting like this, the principal often has an opportunity to communicate to the agent first, before the agent reports. For example, a manager can discuss a worker’s performance with the worker’s supervisor before the supervisor submits his evaluation. In this game of twoway cheap-talk communication,12 is it possible for the principal to credibly reveal her signal to the agent in the first stage of communication? To see how the comparative statics results from the previous section helps us answer this question, note that if the principal reveals her signal truthfully in the first stage, then she no longer has any private information in the second stage. The continuation game is a Crawford and Sobel game with appropriately updated beliefs. Specifically, if the principal reveals her signal to be sH , the continuation game is a CS game with common prior H (t) and if she reveals her signal to be sL , the continuation game is a CS game with common prior L (t). (The joint distribution between the state and the principal’s signal is assumed to be common knowledge.) Proposition 2 in the previous section shows that the principal with belief H prefers the equilibrium partition of the largest size under H to the equilibrium partition of the 11

Only a few papers have modeled privately informed receivers in the literature. These include

Seidmann (1990), Watson (1996), Olszewski (2004), Harris and Raviv (2005) and Lai (2008), but these models have different assumptions on information structure and address different questions from my paper. 12 To be more precise, consider the following extensive-form game. At the beginning of the game, the agent privately observes t and the principal privately observes s. Then two-way communication takes place sequentially: in stage 1, the principal sends a message to the agent; in stage 2, the agent sends a message back to the principal. Then the principal chooses an action.

16

largest size under L. Therefore the principal who has observed sH has no incentive to pretend to have observed sL . Moreover, Proposition 3 gives sufficient conditions under which the principal with belief L also prefers the equilibrium partition of the largest size under H to the equilibrium partition of the largest size under L. Under these conditions, the principal with the signal sL has the incentive to pretend that she has observed sH . So in the augmented game of two-way cheap-talk communication, we have the following result regarding the failure of truthful revelation by the principal. Proposition 5. If the conditions in Proposition 3 are met, then no equilibrium exists in which the principal reveals her signal truthfully to the agent.13 Proposition 5 says that truthful revelation of the principal’s signal fails if the boundary types in the equilibrium partition are not shifted “too far” to the right when the belief changes from L to H. Whether the boundary types are shifted too far to the right depends on how different the distributions L and H are, which in turn depends on the informativeness of the principal’s signal. To illustrates this idea, let us consider an example in which the principal’s signal belongs to the following class of information structure. The conditional probability of signal s is given by: p (sH |t) = πq (t) + (1 − π) c and p (sL |t) = π (1 − q (t)) + (1 − π) (1 − c), i.e., the information structure is a mixture between an informative and an uninformative experiment. With probability π, the experiment is successful and the signal is informative (the probability of observing sH or sL depends on t); with probability (1 − π), the experiment fails and the signal is uninformative (the probability of observing sH or sL does not depend on t). The principal knows the probability that the experiment is successful, but does not observe whether the experiment has succeeded or not.14 Assume q (t) is increasing in t, which implies that for a fixed π, the posterior of the principal after observing sH MLR dominates her posterior after observing sL . Moreover, a higher π means that the signal s is more informative (in the sense of Blackwell). So π is an informativeness parameter of the principal’s signal. One useful characteristic of this class 13

I focus on the equilibrium of the largest size in the continuation game after the principal reveals

her signal, which is uniquely selected by the “no incentive to separate” criterion. 14 This assumption on information structure is analogous to Ottaviani and Sorensen (2006). It is different from the success-enhancing models (e.g. Green and Stokey 2007) that assume whether the experiment has succeeded or failed is observed.

17

of information structure is that as we vary the informativeness parameter, the resulting posteriors of the principal are ordered by MLR dominance. In particular, suppose s˜ is ˜ (t) denote the principal’s posterior a signal with informativeness parameter π ˜ and let H ˜ (t) denote her posterior after observing s˜ = sL . If after observing that s˜ = sH and L ˜ (t) and L ˜ (t) MLR dominates L (t) (intuitively, π > π ˜ , then H (t) MLR dominates H the posteriors induced by the more informative signal s are more spread out than the posteriors induced by the less informative signal s˜) and the comparative statics from section 2 therefore apply.15 The following example illustrates that the principal can credibly reveal her signal only when it is sufficiently informative. Example 2. For the informative experiment, assume q (t) = 0 if t ∈ [0, 0.45), q (t) = 1 4

+ 12 t if t ∈ [0.45, 0.55] and q (t) = 1 if t ∈ (0.55, 1]. For the uninformative experiment,

assume s = sL and s = sH with equal probability. To summarize, the conditional
probabilities are p (sH |t) = (1 − π) 21 if t ∈ [0, 0.45), p (sH |t) = π 41 + 12 t + (1 − π) 12 if t ∈ [0.45, 0.55] and p (sH |t) = π+ (1 − π) 21 if t ∈ (0.55, 1]. Also, assume the principal’s utility function is − (a − t)2 , the agent’s utility function is − (a − t − 0.08)2 , and the common prior on t is uniform on [0, 1]. Consider the case in which π = 35 . The equilibrium partition under L has at most size two: tL = (0, 0.21, 1); the equilibrium partition under H has at most size three:
H
H , ti ≥ ¯L tH tH = (0, 0.014, 0.38, 1). Straightforward calculation shows that U R a i−1 , ti

H H UR a ¯L tH for i = 1, 2. Hence the principal cannot credibly reveal s to the i , ti+1 , ti agent. For any less informative signal s˜ with π ˜ < 53 , the principal cannot credibly reveal her signal either. As the principal’s signal gets more informative, she may be able to reveal her signal credibly in equilibrium. This is most easily seen in the limit as π goes to 1. In this case, the support of L becomes [0, 0.55] and the support of H becomes [0.45, 1]. The most informative equilibrium partition under L is tL = (0, 0.103, 0.55) and the most informative equilibrium partition under H is tH = (0.45, 0.586, 1). Since 0.586 > 0.55, the information contained in partition tH has no value to the principal with observation sL and therefore the principal with signal sL has no incentive to pretend that her signal is sH . Hence there exists an equilibrium in which the principal reveals her signal credibly in the first stage of communication. 15

A proof is provided in the appendix.

18

4

Conclusion

This paper contributes to the understanding of strategic communication by linking the amount of information transmitted in equilibrium to the principal’s prior over the state of the world. The main finding is that optimism (in the sense of monotone likelihood ratio dominance) facilitates information transmission. Three applications are considered, demonstrating the usefulness of the result in various settings. In the choice of intermediaries, a principal may benefit from choosing a more optimistic intermediary, but should not let someone more pessimistic than herself make decisions on her behalf. A more optimistic principal is more likely to centralize decisions rather than delegate to an informed agent because it is more likely for a more optimistic principal to elicit useful information from communication. Finally, in a model of two-way communication between a privately informed principal and an agent, unless her signal is highly informative, the principal fails to credibly reveal it to the agent because she wants to appear to be more optimistic than her private signal suggests.

Appendix Proof of Lemma 2: By induction on K. Suppose K = 2 and tL (K) exists. Condition (A) requires that U S (¯ aL (tL0 , tL1 ), tL1 )) =
S U S (¯ aL (tL1 , tL2 ), tL1 )) where a ¯L (tL0 , tL1 ) < aS tL1 < a ¯L (tL1 , tL2 ). Since U11 (a, t) < 0, and a ¯H (tLi−1 , tLi ) > a ¯L (tLi−1 , tLi ) for i = 1, 2 by Lemma 1, it follows that U S (¯ aH (tL0 , tL1 ), tL1 )) > S U S (¯ aH (tL1 , tL2 ), tL1 )). Since U11 (a, t) < 0, U S is continuous and a ¯H (t0 , t00 ) is contin
uous and increasing in both arguments, there must exist a t ∈ tL1 , tL2 such that

U S (¯ aH (tL0 , tL1 ), tL1 )) = U S (¯ aH (tL1 , t), tL1 )). Condition (M ) then implies that there exists a L H 0 t0 > tL1 such that U S (¯ aH (tL0 , t0 ), t0 )) = U S (¯ aH (t0 , tL2 ), t0 )). Let tH 0 = t0 = t, t1 = t and L H H L ¯ tH 2 = t2 = t. The partition that satisfies (A) under H, t (K), exists and t1 > t1 .

Suppose the claim holds for all i = 2, .., K − 1. Suppose tL (K) exists. Then
tLi (K) i=0,K−1 is a partial partition of size (K − 1) satisfying (A). By the induction hypothesis, there exists a partial partition of size (K − 1) satisfying (A) under distribution 19


L L ˆH ˆH ˆH H, tˆH i i=0,K−1 , with t0 = t, tK−1 = tK−1 (K) and ti > ti for all i = 1, ..., K − 2. So



H L L H H L L ˆ ˆ ˆ a ¯H tˆH , t > a ¯ t , t , and by Lemma 1, a ¯ t , t > a ¯ t , t H H L K−2 K−1 K−2 K−1 K−2 K−1 K−2 K−1 .

S L L L S L L L H H Since U (¯ aL (tK−2 , tK−1 ), tK−1 )) = U (¯ aL (tK−1 , tK ), tK−1 )), a ¯H tˆK−2 , tˆK−1 > a ¯L tLK−2 , tLK−1 ,
H S ˆH ˆ (a, t) < 0, it follows that U S (¯ aH tˆH a ¯H (tLK−1 , tLK ) > a ¯L (tLK−1 , tLK ) and U11 K−2 , tK−1 , tK−1 )) >
L L S ˆH ˆH U S (¯ aH (tLK−1 , tLK ), tˆH aH (tˆH K−1 )). So there exists a t ∈ tK−1 , tK such that U (¯ K−2 , tK−1 ), tK−1 )) = ˆH U S (¯ aH (tˆH K−1 , t), tK−1 )). Condition (M ) implies that there exists a partition of size K H H ˆH ¯ satisfying (A) under H, tH (K) such that tH 0 = t, ti > ti for i = 1, ..., K − 1 and tK = t. L L H L ˆH Since tˆH i > ti for all i = 1, ..., K − 2 and tK−1 = tK−1 , it follows that ti (K) > ti (K)

for i = 1, ..., K − 1. Proof of Lemma 3: The argument is similar to that in the proof of Theorem 3 in Crawford and Sobel (1982). P R ti (x) R aF (ti−1 (x) , ti (x)) , t) dF (t). Since t0 (x) and Note that EU R (x) = K i=1 ti−1 (x) U (¯ tK (x) are fixed and a ¯F (ti−1 (x) , ti (x)) is the receiver’s optimal action on [ti−1 , ti ], the envelope theorem implies that K−1 dEU R (x) X dti (x) R = f (ti (x)) (U (¯ aF (ti−1 (x) , ti (x)) , ti (x)) dx dx i=1

−U R (¯ aF (ti (x) , ti+1 (x)) , ti (x)) . Condition (M ) implies that

dti (x) dx

> 0 for all i = 1, ..., K − 1. Also, since (ti (x)) satis-

fies (A) for i = 2, ..., K, U S (¯ aF (ti−1 (x) , ti (x)) , ti (x)) = U S (¯ aF (ti (x) , ti+1 (x)) , ti (x)) for i = 2, ..., K−1. Hence U R (¯ aF (ti−1 (x) , ti (x)) , ti (x))−U R (¯ aF (ti (x) , ti+1 (x)) , ti (x)) > 0 for i = 2, ..., K − 1. Also, (M ) implies that for x ∈ (y, y 0 ), U R (¯ aF (t0 , t1 (x)) , t1 (x)) > U R (¯ aF (t1 (x) , t2 (x)) , t1 (x)). Hence

dEU R (x) dx

> 0.

Proof of Proposition 2. First, I establish the following lemma. Lemma 4. For a fixed number of steps K ≥ 2, the receiver with the belief H strictly prefers the equilibrium partition tH (K) to the equilibrium partition tL (K). Proof. By induction on K.
H

H
Suppose K = 2. Since U R a ¯H 0, tH > UR a ¯H tH and tL1 < tH 1 , t1 1 , 1 , t1 1 , the claim is true as immediately implied by Lemma 3 when t0 = 0 and tK = 1. Suppose the claim holds for steps i = 2, ..., K − 1. In what follows I show that it holds for i = K. 20


Consider equilibrium partitions tL (K) = tL0 = 0, tL1 , ..., tLK = 1 under prior L and
H H tH (K) = tH = 0, t , ..., t = 1 under prior H. Lemma 2 implies that there exists a 0 1 K
H H H L ˆH partition ˆt (K) = tˆ0 = 0, tˆ1 , ..., tˆH K = 1 such that t1 = t1 and the condition (A) holds L ˆH for all tˆH i (i = 2, ..., K − 1) under distribution H. Moreover, ti > ti for all i = 2, ..., K −

1. By the induction hypothesis, the receiver with belief H must strictly prefer partition ˆtH (K) to tL (K). All we need to show is that the receiver with belief H prefers partition
H


ˆ ˆH ˆH tH (K) to ˆ tH (K). By (M ), U R a ¯H 0, tˆH ≥ UR a ¯H tˆH 1 , t1 1 , t2 , t1 . Lemma 3 implies that the receiver indeed prefers tH (K) to ˆtH (K). In Lemma 4, the number of steps is fixed, but as Lemma 2 shows, N ∗ (H) ≥ N ∗ (L), i.e., the equilibrium partition of the largest size under H may have more steps than the equilibrium partition of the largest size under L. To show that the receiver with the belief H strictly prefers tH (N ∗ (H)) to tL (N ∗ (L)) even when N ∗ (H) > N ∗ (L), recall Theorem 3 in Crawford and Sobel (1982), which says that when the payoff functions and the prior are fixed, the receiver prefers an equilibrium with a higher number of steps. So the receiver with belief H prefers tH (N ∗ (H)) to tH (N ) for any N < N ∗ (H). Combined with Lemma 4, the receiver with the belief H strictly prefers tH (N ∗ (H)) to tL (N ∗ (L)). Proof of Proposition 3: First, I establish the following lemma. Lemma 5. Fix the receiver’s prior F . Take two partial partitions of the same size K ≥ 2, t = (ti )i=1,...,K and ˆ t =(tˆi )i=1,...,K . Suppose t0 = tˆ0 , tK = tˆK and tˆi > ti for all

i = 1, ..., K − 1. If U R (¯ aF (ti−1 , ti ) , ti ) ≥ U R (¯ aF (ti , ti+1 ) , ti ) and U R a ¯F tˆi−1 , tˆi , tˆi ≥

UR a ¯F tˆi , tˆi+1 , tˆi , then the receiver strictly prefers the partition ˆ t to t. Proof. By induction on K. Step 1. Suppose K = 2. Lemma 3 implies that the claim is true. Step 2. Suppose K ≥ 3 and the claim holds for all i = 2, ..., K − 1. Consider the partitions (ti )i=0,...,K and (t0 , t1 , ..., tˆK−1 , tK ). There are two possibilities.



(i) Suppose U R a ¯F tK−2 , tˆK−1 , tˆK−1 ≥ U R a ¯F tˆK−1 , tK , tˆK−1 . Then by step 1,
the receiver prefers the partial partition tK−2 , tˆK−1 , tK to (tK−2 , tK−1 , tK ). It follows that the receiver prefers (t0 , ..., tK−2 , tˆK−1 , tK ) to (ti )i=0,...,K . Now compare the partitions (t0 , t1 , ..., tˆK−1 ) and (tˆi )i=0,...,K−1 . Since tˆi ≥ ti , by the induction hypothesis, the receiver

21

prefers (tˆi )i=0,...,K−1 to (t0 , t1 , ..., tˆK−1 ). It follows that the receiver prefers (tˆi )i=0,...,K−1 to (ti )i=0,...,K .



(ii) Suppose U R a ¯F tK−2 , tˆK−1 , tˆK−1 < U R a ¯F tˆK−1 , tK , tˆK−1 . Find t0K−2 such



¯F tˆK−1 , tK , tˆK−1 . Note that t0K−2 > tK−2 . If ¯F t0K−2 , tˆK−1 , tˆK−1 = U R a that U R a




UR a ¯F 0, t0K−2 , t0K−2 < U R a ¯F t0K−2 , tˆK−1 , t0K−2 , then find t0K−3 ∈ 0, t0K−2 such



that U R a ¯F t0K−3 , t0K−2 , t0K−2 = U R a ¯F t0K−2 , tˆK−1 , t0K−2 . Otherwise, let t0K−3 = 0.



¯F t0K−i , t0K−i+1 , t0K−i , ¯F 0, t0K−i , t0K−i < U R a Similarly, for i ≥ 3, if t0K−i > 0 and U R a




find t0K−i−1 ∈ 0, t0K−i such that U R a ¯F t0K−i−1 , t0K−i , t0K−i = U R a ¯F t0K−i , t0K−i+1 , t0K−i . Otherwise let t0K−i−1 = 0. Note that by (M ), t00 = t0 = 0. Let i∗ = min{i : t0K−i−1 ≤ tK−i−1 }. So t0K−i∗ −1 ≤ tK−i∗ −1 and t0K−i∗ > tK−i∗ . Since t0K−i∗ > tK−i∗ , we can find t00K−i∗ +1 , ..., t00K−2 , t00K−1 (with t00K−i > t00K−i−1 ) such that





UR a ¯F t00K−2 , t00K−1 , t00K−1 = U R a ¯F t00K−1 , tK , t00K−1 , U R a ¯F t00K−i−1 , t00K−i , t00K−i =



UR a ¯F t00K−i , t00K−i+1 , t00K−i for 1 < i ≤ i∗ − 2 and U R a ¯F tK−i∗ , t00K−i∗ +1 , t00K−i∗ +1 =

¯F t00K−i∗ +1 , t00K−i∗ +2 , t00K−i∗ +1 . UR a



Since U R a ¯F t0K−i∗ −1 , t0K−i∗ , t0K−i∗ = U R a ¯F t0K−i∗ , t0K−i∗ +1 , t0K−i∗ and t0K−i∗ −1 ≤



tK−i∗ −1 , it follows that U R a ¯F tK−i∗ −1 , t0K−i∗ , t0K−i∗ ≥ U R a ¯F t0K−i∗ , t0K−i∗ +1 , t0K−i∗ . Also, since t0K−i∗ > tK−i∗ , (M ) implies that t0K−i > t00K−i for i = 2, 3, ..., i∗ − 1 and tˆK−1 > t00K−1 . So by Lemma 3, the receiver prefers the partition tK−i∗ −1 , t0K−i∗ , t0K−i∗ +1 , ..., t0K−2 , tˆK−1 , tK
to the partition tK−i∗ −1 , tK−i∗ , t00K−i∗ +1 , ..., t00K−2 , t00K−1 , tK .



¯F tˆi , tˆi+1 , tˆi , we have tˆK−i > t0K−i for i = ¯F tˆi−1 , tˆi , tˆi ≥ U R a Since U R a 2, ..., i∗ . Also, since tˆi > ti , by the induction hypothesis, the receiver prefers the partition
(tˆi )i=0,...,K−1 to t0 , ..., tK−i∗ −1 , t0K−i∗ , t0K−i∗ +1 , ..., , t0K−2 , tˆK−1 . Hence the receiver prefers
the partition (tˆi )i=0,...,K to t0 , ..., tK−i∗ −1 , t0K−i∗ , t0K−i∗ +1 , ..., , t0K−2 , tˆK−1 , tK . Moreover, since U R (¯ aF (ti−1 , ti ) , ti ) ≥ U R (¯ aF (ti , ti+1 ) , ti ), we have t00K−i > tK−i for i = 1, ..., i∗ − 1. So by the induction hypothesis, the receiver prefers the parti
tion tK−i∗ , t00K−i∗ +1 , ..., t00K−2 , t00K−1 , tK to the partition (ti )i=K−i∗ ,...,K . Hence the receiver
prefers the partition t0 , ..., tK−i∗ −1 , tK−i∗ , t00K−i∗ +1 , ..., t00K−2 , t00K−1 , tK to (ti )i=0,...,K . It follows that the receiver prefers the partition (tˆi )i=0,...,K to the partition (ti )i=0,...,K .



Equilibrium condition implies that U R a ¯L tLi−1 , tLi , tLi > U R a ¯L tLi , tLi+1 , tLi always holds. Since the boundary types under H are to the right of the boundary types under L, Lemma 5 immediately implies that the receiver with belief L prefers the equilibrium partition under H to the equilibrium partition under L of the same size, if

22





H

H
∗ H H ¯L tH , ti ≥ U R a ¯L tH UR a i , ti+1 , ti , i.e., Proposition 3 holds if N (H) = i−1 , ti N ∗ (L). L Now suppose N ∗ (H) > N ∗ (L). Condition (M ) imply that tH N ∗ (H)−i > tN ∗ (L)−i for

i = 1, ..., N ∗ (L)−1. Note that the partition tH (N ∗ (H)) has (N ∗ (H) − N ∗ (L)) more elH ements than the partition tL (N ∗ (L)) does. By inserting the elements tH 1 , ..., tN ∗ (H)−N ∗ (L)

into the partition tL (N ∗ (L)), one can construct a new partition that has size N ∗ (H) : ˆt. Note that this partition is finer and more informative (in the Blackwell sense) ˆ than tL (N ∗ (L)) and therefore is preferable to the receiver. Also, since tH i > ti for



H H H ¯L tH ≥ UR a , tH ¯L tH i = N ∗ (H) − N ∗ (L) , ..., N ∗ (H) − 1 and U R a i , ti+1 , ti , i i−1 , ti by Lemma 5, the receiver with belief L prefers the partition tH (N ∗ (H)) to the partition ˆt and hence to the partition tL (N ∗ (L)). ˜ and L ˜ MLR dominates L. Since Proof that if π > π ˜ , then H MLR dominates H p (sH |t) = πq (t)+(1 − π) c and p (sL |t) = π (1 − q (t))+(1 R − π) (1 − c), we have the con 1 ditional density g (t|sH , π) = f (t) (πq (t) + (1 − π) c) / 0 πq (x) f (x) dx + (1 − π) c . ˜ (t), it suffices to show that g(t|sH ,π) > g(t00 |sH ,π) To show that H (t) MLR dominates H 0

for t > t . Note that f (t0 )(πq(t0 )+(1−π)c)/(

R1

g(t|sH ,π) g(t|sH ,˜ π)

=

f (t)(πq(t)+(1−π)c)/( f (t)(˜ π q(t)+(1−˜ π )c)/(

R1 0 R1 0

g(t|sH ,˜ π) g(t |sH ,˜ π) πq(x)f (x)dt+(1−π)c) g(t0 |sH ,π) and g(t0 |sH ,˜π) = π ˜ q(x)f (x)dt+(1−˜ π )c)

πq(x)f (x)dt+(1−π)c)

(πq(t)+(1−π)c) 0 R > and therefore it suffices to show that (˜ π q(t)+(1−˜ π )c) f (t0 )(˜ π q(t0 )+(1−˜ π )c)/( 01 π ˜ q(x)f (x)dt+(1−˜ π )c) 0 (πq(t )+(1−π)c) . Since π > π ˜ , t > t0 and q (·) is increasing, the inequality holds. A similar (˜ π q(t0 )+(1−˜ π )c)

˜ (t). argument shows that L (t) MLR dominates L

References [1] Athey, S (2002): “Monotone Comparative Statics under Uncertainty.” Quarterly Journal of Economics,Vol. 117, No. 1, 187-223. [2] Ambrus, A, E. Azevedo and Y. Kamada (2009): “Hierarchical Cheap Talk.” Working Paper, Harvard University. [3] Blackwell, D (1951): “The Comparison of Experiments.” in Proceedings, Second Berkeley Symposium on Mathematical Statistics and Probability, 93-102. 23

[4] Che, Y-K and N. Kartik (2009): “Opinions as Incentives.” Journal of Political Economy, Vo. 117, No. 5, 815-860. [5] Chen, Y., N. Kartik and J. Sobel (2008): “Selecting Cheap-talk Equilibria.” Econometrica, Vol. 76, No.1, 117-136. [6] Crawford, V. and J. Sobel (1982): “Strategic Information Transmission.” Econometrica, Vol. 50, No. 6, 1431-1451. [7] Dessein W. (2002): “Authority and Communication in Organizations.” Review of Economic Studies, 69, 811-838. [8] Goltsman, M., J. Horner, G. Pavlov and F. Squintani (2009): “Mediation, Arbitration and Negotiation.” Journal of Economic Theory, 144, 1397-1420. [9] Green, J. and N. Stokey (2007): “A Two-person Game of Information Transmission.” Journal of Economic Theory, 135, 90-104. [10] Harris M. and A. Raviv (2005): “Allocation of Decision-making Authority.” Review of Finance, 9(3) 353-383. [11] Ivanov, M. (2010): “Communication via a Strategic Mediator.” Journal of Economic Theory, 145, 869-884. [12] Jewitt I. (1987): “Risk Aversion and the Choice between Risky Prospects: the Preservation of Comparative Statics Results.” Review of Economic Studies, 54, 7385. [13] Lai, E. (2008): “Expert Advice for Amateurs.” Working Paper, University of Pittsburgh. [14] Li, W. (2010): “Peddling Influence through Intermediaries.” American Economic Review, 100 (3), 1136-1162. [15] Milgrom, P. (1981): “Good News and Bad News: Representation Theorems and Applications.” Bell Journal of Economics, Vol. 12, No 2, 380-391. [16] Olszewski, W. (2004): “Informal Communication.” Journal of Economic Theory, 117, 180-200. 24

[17] Ormiston, M. and E. Schlee (1993): “Comparative Statics Under Uncertainty for a Class of Economic Agents.” Journal of Economic Theory, 61, 412-422. [18] Ottaviani, M and P. Sorensen (2006): “Professional Advice.” Journal of Economic Theory, 126, 120-142. [19] Seidmann, D. (1990): “Effective Cheap Talk with Conflicting Interests.” Journal of Economic Theory, 50, 445-458. [20] Watson, J. (1996): “Information Transmission When the Informed Party is Confused.” Games and Economic Behavior, 12, 143-161.

25