Cheap Talk when the Receiver Has Uncertain
Information Sources
∗Junichiro Ishida†and Takashi Shimizu‡ February 24, 2017
Abstract
In this paper we analyze a cheap talk model with a partially informed receiver. In clear contrast to the previous literature, we find that there is a case where the receiver’s prior knowledge enhances the amount of information conveyed via cheap talk. The point of depar- ture is our explicit focus on the “dual role” of the sender’s message in our context: when the receiver’s prior belief is subject to higher-order uncertainty, the sender’s message provides information not only about the true state but also about the reliability of the receiver’s pri- vate information. Building on this result, we argue that whether information acquisition and communication are complements or substitutes depends crucially on the extent of uncertainty regarding the information source.
Keywords: Cheap talk, Strategic communication, Informed receiver, Higher-order uncer- tainty.
JEL Classification Number: D23; D82.
∗This paper was previously circulated under the title “Can More Information Facilitate Communication?” We thank Shinsuke Kambe and Dan Sasaki for helpful comments on earlier versions of the manuscript. The first author acknowledges financial support from JSPS KAKENHI Grant-in-Aid for Scientific Research (S) 15H05728, (A) 20245031, and (C) 24530196 as well as the program of the Joint Usage/Research Center for Behavioral Economics at ISER, Osaka University. The second author acknowledges financial support from JSPS KAKENHI Grant-in-Aid (C) 26380252 and the Kansai University Domestic Researcher, 2014. Of course, any remaining errors are our own.
†Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki-shi, Osaka 567-0047 JAPAN (E-mail: jishida@iser.osaka-u.ac.jp)
‡Graduate School of Economics, Kobe University, 2-1 Rokkodai-cho Nada-ku, Kobe, 657-8501 JAPAN (E-mail: shimizu@econ.kobe-u.ac.jp)
1 Introduction
Since the pioneering work of Crawford and Sobel [5] (hereafter, CS), much attention has been paid to strategic aspects of communication between players with misaligned preferences. However, the original setup of CS focuses exclusively on the case where a perfectly informed sender (he) sends a message to a receiver (she) who has no information of her own, thereby leaving some potentially intriguing questions unanswered. One question which seems particularly relevant is how the nature of communication alters when the receiver has access to additional information sources on top of the messages received from the sender. Adding private information to the receiver’s side is a natural extension of the existing literature and poses an intriguing question in itself, as it is a priori not clear whether the quality of communication improves or deteriorates when the receiver has more precise (yet imperfect) information. The extension also sheds light on whether information acquisition and communication are complements or substitutes, which yields more practical implications when how much information to collect on her own is the receiver’s endogenous choice.1
Several recent papers explore this issue, either in a setup with a (partially) informed receiver (Chen [3, 4], Lai [11], Moreno de Barreda [6]) or with multiple senders (Austen-Smith [2], Morgan and Stocken [14], Galeotti et al. [7], Kawamura [9, 10]). An emerging consensus in the literature is that the receiver can extract less information via cheap-talk communication as she becomes more informed, i.e., information acquisition and communication are substitutes.2 The underlying force is fairly simple once we carefully dissect why any information can be conveyed in a cheap- talk environment. When the receiver is endowed with her own private information, the receiver naturally becomes less sensitive to whatever the sender insists. In order to sway the receiver’s decision in his favor, he must then exaggerate more, which would inevitably deteriorate the quality of communication.
1For expositional clarity, we say that information acquisition and communication are complements (substitutes) when more precise prior knowledge of the receiver facilitates (obstructs) communication.
2An exception is Moreno de Barreda [6] who shows that information acquisition and communication can be complements when the receiver is sufficiently risk averse. As we will detail later, the underlying mechanism of this paper is totally different as it has nothing to do with the receiver’s risk aversion. Also, in settings that are somewhat different from CS, some previous works show that the receiver can extract more information as she becomes more informed (Seidmann [16], Watson [17], and Olszewski [15]). Seidmann [16] and Watson [17] consider an environment in which the sender’s payoff function does not satisfy the single-crossing property. They present examples in which full information revelation can be attained when the receiver is endowed with her own private information whereas it is impossible otherwise. Olszewski [15] shows that the sender’s concern to be perceived as honest, combined with the receiver’s private information, induces truth-telling as a unique equilibrium.
Does this mean that the impact of the receiver’s prior knowledge on the quality of communi- cation is invariably negative? In this paper, we ask whether there is any plausible circumstance in which the receiver’s prior knowledge facilitates communication, and if so, identify a crucial link that has been overlooked in the literature in order to provide clearer insight into the role of prior knowledge in strategic communication. To address this issue, we analyze an extension of a canonical cheap-talk model in which the receiver obtains her own private information from an unknown information source. The key element of our model is that the receiver is subject to what is called “higher-order uncertainty,” which stems from the fact that the receiver does not know where the information originates from. More precisely, the receiver draws a signal from one of two possible information sources before communicating with the sender. She knows that one source is more reliable than the other, in that it generates a more precise signal, but does not know which source any given signal is actually drawn from.3 As a consequence, the receiver faces uncertainty not only about the true state of nature but also about the reliability (variance) of the information source.
The point of departure from the existing literature is our explicit focus on the “dual role” of the sender’s message in this context: when the receiver faces higher-order uncertainty as described above, the sender’s message provides information about the reliability of the receiver’s private information as well as about the true state. The latter aspect, which has been largely neglected in the literature, is the driving force of our model. To see how this works, suppose that the receiver knows that her private information is less reliable than the sender’s.4 If the sender’s message is consonant with what she privately knows, she thinks that her private information is more likely to be correct and, consequently, places more weight on it. As the receiver relies more on her own information, her action necessarily becomes less sensitive to the sender’s message. If the message is not consonant with her private information, on the other hand, she loses her confidence in her private information and her action becomes more sensitive to the message. Since the sender’s message is more likely to be consonant with the receiver’s private information when the sender truthfully reveals his information, the receiver reacts less to the truthful message and more to the misrepresented one. As we will clarify in more detail later, this asymmetric response of the receiver is essential in disciplining the sender to be more truthful and hence facilitating communication between them.
3In this context, we say that the receiver is more informed when her signal is more likely to come from the more informative source.
4As in most cheap talk models, we assume that the sender knows the true state with precision.
As the main, and possibly testable, implication of our analysis, we contend that uncer- tainty regarding the information source plays a crucial role in determining whether information acquisition and communication are complements or substitutes. To fix this idea, consider an environment where the receiver has access to different information sources, each of which is rep- resented by a signal distribution, and can solicit advice from one of them before communicating with an outside expert. When the receiver knows precisely where her information comes from, the situation is as if she draws a signal from a known distribution. This is the structure that is implicitly assumed in essentially all of the previous works where the receiver faces uncertainty only about the state of nature. In contrast, the nature of uncertainty changes to some extent when the receiver obtains information from an anonymous source, e.g., via the Internet or word- of-mouth communication, and is unsure of its exact origin, in which case it is more appropriate to specify that the receiver draws a signal without knowing which distribution generates that particular signal. Our analysis suggests that this seemingly subtle change in the structure of information can yield a substantial strategic impact on the nature of communication and can in fact overturn the relationship between information acquisition and communication that has been robustly found in the previous literature.
As already mentioned, there are now a few works which examine the role of the receiver’s private information by extending the canonical model of CS in various ways.5 Lai [12] assumes that the receiver’s private information consists of two components where the first is a threshold level and the second is a signal indicating whether the realized state is higher or lower than the threshold, and shows that the existence of the receiver’s private information lowers the efficiency of communication due to the information effect. Similarly, Chen [3] assumes that the receiver obtains a binary signal regarding the state and presents an example in which the quality of communication becomes deteriorates as the receiver becomes more informed.6 Chen [4] considers a situation in which the sender and the receiver can access to a public signal regarding the state, and shows that an increase in the quality of the public signal leads to less efficient communication. Moreno de Barreda [6] is most closely related to our paper, and the relationship with this work will be discussed in detail in Section 4.1.
Similar conclusions are also obtained in models with multiple senders. Austen-Smith [2]
5In a discrete-state model, Ishida and Shimizu [8] also show that the receiver’s prior information becomes an impediment to efficient communication.
6Note, however, that the main focus of her analysis is to show the existence of non-monotone equilibria and investigate whether the receiver can truthfully report to the sender about her signal.
considers a cheap-talk model with two senders, where each sender draws a signal from a binomial distribution while the state is distributed uniformly. He then shows that the sender’s incentive for truth-telling becomes more stringent in the case with two senders than in the case with a single sender. Mogan and Stocken [14] extends this setting to analyze a situation where a policy maker solicits information from constituents, each of whom privately observes a signal, and find that there is an upper bound of the number of constituents who reveal their signals truthfully.7 Kawamura [10] considers a model of social surveys where a social planner solicits information from a random sample of individuals with heterogeneous preferences in order to estimate the distribution of individual preferences. He shows that the true distribution cannot be identified because the respondents have incentives to exaggerate their preferences when the sample size grows large, and moreover that the optimal sample size is bounded when the prior belief is relatively weak.8
Mandler [13] considers a voting model with an information structure similar to ours. In his model, there are two candidates and two signals, where each voter observes one of the signals with probability q and the other with probability 1 − q. As a departure from the classical jury model, it is assumed that q itself is a random draw from a distribution, conditional on which candidate is correct, and shows that information fails to aggregate when voters are subject to this type of uncertainty. The information structure of our model shares one important aspect with this setting: the receiver, like voters in Mandler [13], faces multiple distributions and is unsure of which distribution a particular signal is drawn from.9 In a broader perspective, our model can be seen as yet another attempt to discern implications of higher-order uncertainty in strategic environments.
The paper is organized as follows. Section 2 outlines the model which is an extension of CS. Section 3 characterizes equilibria of the model and discusses the relationship with the existing literature. Section 4 discusses the relationship with the literature and provides some extensions.
7To analyze a setup with many senders (many-to-one communication), Mogan and Stocken [14] assume that signals are distributed according to a Beta distribution. Galeotti et al. [7] also extends this setting to a network (many-to-many communication) and obtain a similar conclusion in the case of private communication.
8With a similar reasoning, Kawamura [9] shows that the most informative equilibrium converges to binary communication as the number of respondents increases because binary messages do not allow exaggeration.
9Aside from this, however, our information structure differs substantially from Mandler [13] due to the difference in objectives: strategic information transmission in our model and information aggregation through strategic voting in Mandler [13]. The most notable difference is that in our model, there is another player – the sender – who observes the true state, so that the receiver can learn from the sender’s message not only about the true state but also about which distribution the observed signal is more likely to come from. This draws clear contrast to Mandler [13] where all voters face the same (though uncertain) signal distribution.
Finally, section 5 offers concluding remarks. All the proofs are relegated to Appendix A.
2 The Model
We extend the canonical model of CS by endowing the receiver with her own source of private information. There are two players, the sender (male) and the receiver (female), and the model goes as follows.
1. The state of nature t ∈ [0, 1] is drawn from a distribution F with its corresponding density f where f (t) > 0 for any t ∈ (0, 1).
2. The sender observes the realized state and sends a message m ∈ [0, 1] to the receiver. 3. The receiver observes a private signal r ∈ [0, 1], where it reflects the true state with
probability q while it is randomly drawn from F with probability 1 − q. 4. Upon observing m and r, the receiver chooses an action a ∈ [0, 1].
For most part, we consider quadratic preferences in order to make our points in the most emphatic manner: as shown in Moreno de Barreda [6], the quality of communication deteriorates as the receiver becomes more informed in the uniform-quadratic specification. The receiver’s payoff is hence given by
UR(t, a) = −(t − a)2, whereas the sender’s payoff is
US(t, a) = −(t + b − a)2. We call b the bias and assume b ∈ (0, 0.5).
The only difference from the original setup of CS is that we allow for the possibility that the receiver observes a possibly informative signal of the state of nature. The signal is either perfectly informative (with probability q) or noisy (with the remaining probability), but the receiver cannot tell whether any given signal is informative or not – the type of noise structure which may be referred to as “replacement noise” (Ambrus and Lu [1]).10 What is critical in this
10In contrast, we assume that the sender has perfect information about the realized state. This is mainly for expositional clarity, as the main logic of the model, especially the confirmation effect, survives even when the sender’s information is noisy in the same way as the receiver’s.
specification is the way noise is introduced into the receiver’s signal: in the current setting, the receiver faces uncertainty not only about the state of nature but also about the accuracy of the signal. The current specification is a way to capture the gist of this “higher-order uncertainty” in the simplest way, which is essential in giving rise to the dual role of the sender’s message as we detail below. We interpret q as the accuracy of the signal where the model is equivalent to the original CS model when q = 0 and say that the agent is more informed as q increases. Within this framework, we ask: can more information (on the receiver’s side) facilitate communication?
3 Analysis
3.1 The equilibrium concept
Throughout the analysis, we focus on the class of monotone partition equilibria, which is a subset of perfect Bayesian equilibria, defined as below.11
Definition 1 Let µt denote the type t sender’s strategy (i.e., a probability distribution over the message space [0, 1]). Letting I denote some index set, a monotone partition strategy (MPS) is the sender’s strategy where there exists a partition of [0, 1], {Tn}n∈I, such that
• Tn is a non-empty interval for any n ∈ I,
• µt= µt′ for any n ∈ I and any t, t′∈ Tn, and
• Supp µt∩ Supp µt′ = ∅ for any distinctive n, n′ ∈ I and any t ∈ Tn, t′∈ Tn′.
A monotone partition equilibrium (MPE) is a perfect Bayesian equilibrium where the sender’s strategy is an MPS.
We can show that any MPE consists of finite intervals. Proposition 1 For any MPE, I is a finite set.
11The need for this focus arises from a special feature of cheap-talk models with an informed receiver. When the receiver is endowed with some information of her own, the sender typically induces a lottery over actions rather than a single action. This feature produces some complicated equilibria that never exist in the original CS model: for example, Chen [3] shows that there exist a non-monotone equilibrium in a version of cheap-talk models with an informed receiver. We rule out this possibility given the question we set out to solve, although it is certainly intriguing as a theoretical possibility. Note that CS shows that in the case of q = 0, there exists only monotone partition equilibria.
Based on this proposition, we denote an MPE partition by {Tn}Nn=1where N is some natural number and sup Tn= inf Tn+1 for n = 1, . . . , N − 1. We call Tn the nth interval. Alternatively, we occasionally identify an MPS with partition {Tn}Nn=1 with a vector t = (t0, . . . , tN) defined as
tn:=
(0 for n = 0,
sup Tn for n = 1, . . . , N.
We call each tn a threshold. Note that 0 = t0 < t1 < · · · < tN = 1. Define τn:=
Z tn
tn−1
f (t)dt, βn:= Rtn
tn−1tf (t)dt
Rtn
tn−1f (t)dt
.
3.2 The receiver’s problem
The receiver observes her own signal r and the sender’s message m before she chooses her action. The receiver’s equilibrium strategy is pure and denoted by α(m, r). Furthermore, the actions induced on any equilibrium path are expressed as follows:
α(m, r) =
( q
q+(1−q)τnr +
(1−q)τn
q+(1−q)τnβn if ∃n ∀t ∈ Tn µt(m) > 0 and r ∈ Tn, βn if ∃n ∀t ∈ Tn µt(m) > 0 and r /∈ Tn.
Note that the receiver uses her own information only when the sender’s message falls into the same interval. On the other hand, the receiver sees her private information as a noise and disregards it altogether if the message does not agree with her private signal, given that the sender plays the equilibrium (truth-telling) strategy. This is the critical feature of the current model: when the message is “close” to the receiver’s signal, she places more confidence in her signal; when it is “further away,” she relies less on it and more on the sender’s message. As we will see later, this asymmetric response, which we broadly refer to as the confirmation effect for expositional purposes, is what disciplines the sender to be more truthful.
3.3 The equilibrium conditions
Given the receiver’s strategy, we can now identify the equilibrium conditions by checking the sender’s incentives. In particular, what we need to see is that given some partition t and the true state t, the sender has no incentive to deviate by sending a “nearby” message. Define
∆(t; tn−1, tn, tn+1) := ErUS(t, α(mn+1, r)) | t − ErUS(t, α(mn, r)) | t
= − Z 1
0
[t + b − α(mn+1, r)]2P (dr | t) + Z 1
0
[t + b − α(mn, r)]2P (dr | t)
where mnis a message sent (with positive probability) when the true state lies in the nth interval.
∆(t; tn−1, tn, tn+1) is the difference in the sender’s payoff between sending mn+1 and sending mn for a given state t.
In the original CS model, the length of each interval must satisfy certain conditions in equilib- rium: for instance, in the linear-quadratic specification, each interval must be exactly 4b longer than the last. Similar conditions also characterize the equilibria of our model. Clearly, it is nec- essary to have ∆(t; tn−1, tn, tn+1) ≤ 0 for t ∈ (tn−1, tn) and ∆(t; tn−1, tn, tn+1) ≥ 0 t ∈ (tn, tn+1). In particular, these conditions must be satisfied around a threshold, i.e.,
limt↓tn
∆(t; tn−1, tn, tn+1) ≥ 0 ≥ lim
t↑tn
∆(t; tn−1, tn, tn+1).
With some computation, these local incentive conditions are represented by the following two functions:
G(tn−1, tn, tn+1; q) :=
− (βn+1− βn)(βn+1+ βn− 2tn− 2b)
− q
2(1 − q)
(q + (1 − q)τn+1)2 Z
r∈Tn+1
(r − βn+1)2f (r)dr + q
2(1 − q)
(q + (1 − q)τn)2 Z
r∈Tn
(r − βn)2f (r)dr
− q
2(q + 2(1 − q)τn)(tn− βn)2 (q + (1 − q)τn)2 −
2q2(tn− βn) q + (1 − q)τnb,
G(tn−1, tn, tn+1; q) :=
− (βn+1− βn)(βn+1+ βn− 2tn− 2b)
− q
2(1 − q)
(q + (1 − q)τn+1)2 Z
r∈Tn+1
(r − βn+1)2f (r)dr + q
2(1 − q)
(q + (1 − q)τn)2 Z
r∈Tn
(r − βn)2f (r)dr +q
2(q + 2(1 − q)τn+1)(tn− βn+1)2
(q + (1 − q)τn+1)2 +
2q2(tn− βn)
q + (1 − q)τn+1
b.
Lemma 1
(i) limt↑tn∆(t; tn−1, tn, tn+1) ≤ 0 ⇔ G(tn−1, tn, tn+1) ≤ 0. (ii) limt↓tn∆(t; tn−1, tn, tn+1) ≥ 0 ⇔ G(tn−1, tn, tn+1) ≥ 0.
The two local incentive conditions are obviously necessary for any MPE but not sufficient. To constitute an MPE, we also need to have the sender follow the equilibrium strategy not
just locally (at each threshold) but also globally, which imposes an additional condition. More precisly, the following statement identifies a set of sufficient conditions for an MPE with N intervals.
Proposition 2 A partition t constitutes an MPE with N intervals if for n = 1, ..., N − 1, (i) G(tn−1, tn, tn+1) ≥ 0 ≥ G(tn−1, tn, tn+1),
(ii) limt↑tn ∂∆∂t = 2(βn+1− βn) −2q
2(q+2(1−q)τ n)
(q+(1−q)τn)2 (tn− βn) − 2q2
q+(1−q)τnb ≥ 0.
Note that condition (ii) is not particularly stringent as it is necessarily satisfied for many distributions, including the uniform distribution (see Proposition 5). In the case where the state is distributed uniformly over [0, 1], therefore, G(tn−1, tn, tn+1) ≥ 0 ≥ G(tn−1, tn, tn+1) is both necessary and sufficient for an MPE with N intervals.
The sender’s incentives are nicely summarized by the two functions G and G that can be used to illuminate the role of the receiver’s information in the current setting. Take G for instance. The first line in the righthand side of G is a term that also appears in CS and characterizes the equilibrium with q = 0. Aside from this, an increase in q above zero yields two additional effects as we have discussed. The second line stems from the stochastic nature of the receiver’s action which is closely related to what Moreno de Barreda [6] refers to as the risk effect. The third line captures the novel feature of our model, which we label as the confirmation effect. Note that the confirmation effect (the third line) is unambiguously negative. While the sign of the risk effect (the second line) is ambiguous, it is verified that
τn(tn− βn)2 >
Z
r∈Tn
(r − βn)2f (r)dr. It then follows from this that
G(tn−1, tn, tn+1) < −(βn+1− βn)(βn+1+ βn− 2tn− 2b). By the same token, we can also show that
G(tn−1, tn, tn+1) > −(βn+1− βn)(βn+1+ βn− 2tn− 2b).
This argument indicates that the local conditions are less stringent than that of CS, which may allow us to construct a more informative equilibrium.
A simple example may help develop the intuition behind this result. Suppose that the state t is distributed uniformly over [0, 1], and consider an equilibrium with two intervals, {[0, t1], (t1, 1]}. When q = 0, it is straightforward to compute α(m1, r) = t21 and α(m2, r) = t1+12 regardless of r. At t = t1, the sender must be indifferent between the two messages, i.e., his bliss point must be at the midpoint of α(m1, r) and α(m2, r). Let b′ denote the bias which satisfies
t1+ b′= 1 2
t1 2 +
t1+ 1 2
!
⇔ b′= 1 − 2t1
4 ,
where t1 < 12 by definition. We now let q increase from zero and see how that changes the sender’s incentives. To this end, it suffices to focus on how the incentive of the boundary type t = t1 is affected by a change in q. There are two cases to consider, depending on whether the receiver observes a correct signal or not.
The risk effect emerges when the receiver’s information is a noise, in which case she (mis- takenly) uses her private signal with some positive probability. This happens when the sender’s message happens to agree with the signal. From the sender’s point of view, this induces a lottery over actions and introduces irrelevant noise, thereby reducing the sender’s expected payoff. While this can happen regardless of whether the sender reveals truthfully m = m1 or not (m = m2), it is more likely when the sender misreports since t1 < 12. As such, the risk effect tends to discipline the sender to be more truthful although it is typically not strong.
In contrast, the confirmation effect emerges when the receiver’s information reflects the true state. If the sender reveals truthfully, i.e., m = m1, the receiver’s information is consonant with the sender’s message, and the receiver combines the two pieces of evidence to determine her action. The resultant action is hence necessarily gravitated towards the true state t1 and away from what the message indicates, i.e., t1/2. If the sender chose to deviate and sent m = m2, on the other hand, the receiver’s reaction would totally be different, now that the sender’s message is dissonant with the receiver’s signal. Under the presumption that the sender plays the equilibrium strategy, the receiver must think that her signal is a noise and places zero weight in Bayesian updating: the resultant action hence stays at (t1 + 1)/2 regardless of q. This is the essence of what we call the confirmation effect which works to discipline the sender to be more truthful.
With these two effects at work, t = t1 is no longer on the border. It is now strictly better for the sender with t = t1 to send m = m1, meaning that the threshold can be “pushed further to the right.” Even with the same number of intervals, we can thus construct a more informative
equilibrium by having more equally divided intervals. We will formalize this result in the next subsection.
3.4 Can more information facilitate communication?
We are now ready to address our main question of how a change in q affects the quality of communication. We first establish that an increase in q per se enhances the payoffs of both players. Define Vi(t; q), i = S, R, as player i’s ex ante expected payoff for a given partition t. We can then obtain the following result.
Proposition 3 ∂V∂qi > 0 for i = S, R.
This result is fairly intuitive as it simply states that the players benefit from having access to more information. What is more interesting is whether an increase in q can afford a more efficient way of communication, or a more efficient configuration of the equilibrium partition. To this end, it is convenient to associate each MPE explicitly with the information accuracy q and denote it by an MPE-q. Given this, we make the following statement which is derived directly from Proposition 2.
Definition 2 We say that:
(i) A partition t is more efficient than t′ at q if Vi(t, q) > Vi(t′, q) for i = S, R.
(ii) More information facilitates communication at (q, q′), q > q′, if for any MPE-q′ with partition t′, there exists an MPE-q with partition t which is more efficient than t′ at q′. Like CS, for any given q, the model generates a plethora of equilibrium with different numbers of intervals. In addition, since there is a payoff discontinuity at each threshold, our model admits a continuum of equilibria even with a fixed number of intervals. To evaluate the impact of a change in q, therefore, we focus on the most efficient equilibrium for each q. In words, we say that more information facilitates communication at (q, q′) if the most efficient equilibrium under q is more efficient than that under q′.
Proposition 4 For any MPE-0 with t and any q > 0, t also constitutes an MPE-q. Furthermore, more information facilitates communication at (q, 0) for any q > 0.
While we show the above result by comparing among the MPEs with the same number of intervals, we can also easily construct an example where an increase in q results in an increase in the maximum number of intervals. For instance, suppose that b = 0.25, in which case the unique equilibrium is babbling when q = 0. This conclusion does not hold, however, when q > 0. Since G(0, 0, 1) > 0 and G(0, 0, 1) < 0, for a sufficiently small t1 > 0, (0, t1, 1) also satisfies G(0, t1, 1) > 0 and G(0, t1, 1) < 0 due to the continuity of G and G. In other words, there always exists an MPE-q, q > 0, with two intervals when b = 0.25.
Proposition 4 establishes that more information facilitates communication at (q, 0) for any q > 0, but does not show that more information monotonically facilitates communication. To examine whether this condition holds for any arbitrary pair (q, q′) for q > q′ > 0, we now assume that the state is distributed uniformly over [0, 1]. By assuming the uniform distribution for t, we can provide more detailed equilibrium conditions.
Proposition 5 Suppose that the state is distributed uniformly over [0, 1]. Then, for any τn∈ (0, 1) and q ∈ [0, 1),
(i) there exists τ (τn, q) and τ (τn, q) such that G(tn−1, tn, tn+1; q) ≥ 0 ≥ G(tn−1, tn, tn+1; q) if and only if τ (τn, q) ≥ τn+1≥ τ (τn, q);
(ii) G(tn−1, tn, tn+ τ (τn, q); q) = 0 and G(tn−1, tn, tn+ τ (τn, q); q) = 0; (iii) τ (τn, q) ≥ τn+ 4b ≥ τ (τn, q) > 2b where
q = 0 ⇔ τ (τn, q) = τn+ 4b = τ (τn, q), q > 0 ⇔ τ (τn, q) > τn+ 4b > τ (τn, q).
Moreover, G(tn−1, tn, tn+1; q) ≥ 0 ≥ G(tn−1, tn, tn+1; q) for n = 1, . . . , N − 1 is both necessary and sufficient for an MPE with N intervals.
Even with this simplification, however, it is still difficult to fully characterize when more information facilitates communication for an arbitrary pair (q, q′). The reason for this is that it becomes increasingly difficult to identify the most efficient MPE-q (an MPE with the most efficient partition) when q is large. More precisely, if τn < τn+1 for n = 1, . . . N − 1, the most efficient equilibrium requires τn+1= τ (τn, q) for n = 1, . . . , N − 1 (Lemma 6 in the Appendix), in which case it suffices to focus on τ . Unfortunately, in our setting, τn < τn+1 is violated in
some MSE, thereby precluding us from extending Proposition 4 for an arbitrary pair of (q, q′). See the following example:
Example 1 Suppose that t is uniformly distributed over [0, 1]. When b < 18 and q is sufficiently large, there exists an MPE with two intervals where τ1 ≥ τ2. This is verified from the fact that G(0, 0.5, 1) < 0 holds for a sufficiently large q.
One can verify, however, that a partition of this kind never constitutes an MPE whenever q is sufficiently small (Lemma 7 in the Appendix). Moreover, one can also verify that τ is strictly decreasing in q whenever q is sufficiently small (Lemma 8 in the Appendix). From these facts, we conclude that more information facilitates communication whenever we focus on the situation where the accuracy is sufficiently low.
Proposition 6 Suppose that the state t is uniformly distributed over [0, 1]. There exists q > 0 such that for any q ∈ (0, q] and q′ < q, more information facilitates communication at (q, q′).
As a final note on this matter, we would like to present some counterexamples to show that the effect of the receiver’s prior knowledge does not monotonically improve the quality of communication. Figure 1 focuses on MPEs which admit two intervals (while assuming the uniformed distribution for t), and depicts the length of the first interval of the most efficient MPE for different values of b. Since this length is below 0.5, it also directly represents the efficiency level. The figure shows that more information may result in a less efficient partition when q is sufficiently close to one and the bias is relatively large (b = 0.225 and b = 0.25). The reason why more information may make communication coarser when q is large pertains to the risk effect. To see this, note that the risk effect relies on the fact that the right-hand interval is always longer than the left-hand one, which means that the risk effect becomes weaker as the equilibrium becomes more efficient. In addition, since the risk effect arises when the receiver’s signal is a noise, it becomes less relevant as q increases.
It is important to note, however, that our focus is generally on the case where q is relatively small and b is moderate because, otherwise, there would be little point in consulting experts in the first place. We can thus argue that in a class of situations where communication is relevant (a small q) and beneficial (a moderate b), the effect of the receiver’s prior knowledge on the quality of communication is largely positive, if the underlying information structure has a feature that gives rise to the dual role of the sender’s message.
4 Discussion
4.1 Relationship to the existing literature
Recently, there are some attempts to explore the role of the receiver’s private information by examining variants of CS where the receiver is also (partially) informed about the state of nature. The underlying mechanism of those models is most succinctly illustrated by Moreno de Barreda [6]. Her analysis identifies two effects, labeled as the information and risk effects, which inherently arise when the receiver is endowed with her own information in the original setup of CS:
• Information effect: The receiver’s action necessarily becomes less sensitive to the sender’s message when she has her own private information. To sway the receiver in his favor, the sender must exaggerate more, which impedes information transmission.
• Risk effect: When the receiver has her own private information, the sender’s message induces a lottery over actions, rather than a single action. Since each interval is always longer than the previous one in the canonical setup of CS, the sender faces more uncertainty when he announces a larger message. This diminishes the sender’s incentive to exaggerate, which facilitates information transmission.
The overall informativeness of communication is determined by the tradeoff between these two forces. An emerging consensus in the literature is that the information effect typically dominates the risk effect, suggesting that the receiver can extract less information via communication as she becomes more informed. In particular, Moreno de Barreda [6] shows that the risk effect can never dominate the information effect when (a version of) the uniform-quadratic specification of CS – the setup most commonly used in applications – is considered.12
The key departure from the existing literature is that the information effect is replaced by the confirmation effect in our model with its role totally reversed; with the positive confirmation effect overriding the negative information effect, the receiver can now extract more information from the sender. To illustrate this point, consider the following signal distribution:
s = (1 − σ)t + σε,
12To be more precise, Moreno de Barreda [6] obtains this result when the noise distribution is also uniform. Also, as can easily be expected, the risk effect may dominate the information effect as the sender becomes more risk averse. This is in fact what is found in Moreno de Barreda [6] who shows through a numerical example that the risk effect can dominate the information effect when the sender is sufficiently risk averse.
where ε ∈ [0, 1] is drawn from some distribution F and σ ∈ [0, 1] is a measure of the “accuracy” of the signal which itself is also unknown to the receiver. In the most general form, one can specify that σ is a random variable drawn from some distribution G. In this model, therefore, the receiver may not know how accurately the signal reflects the true state.
Our setup is a special case of this general model where the unknown random variable σ has a two-point distribution where
P (σ = 0) = q, P (σ = 1) = 1 − q.
That is, the signal contains no noise and perfectly reveals the true state with probability q while it is a pure noise which provides no information with the remaining probability. For practical purposes, this can be interpreted as a situation where there are two possible information sources, but the receiver does not know which source her observed signal comes from. As emphasized earlier, due to this higher-order uncertainty, the sender’s message is not only a signal of the true state, but also a signal of the reliability of the information source, which gives rise to the confirmation effect.
Given this formulation, the difference from the existing literature, in particular Morreno de Barreda [6], should now be clearer. The existing models also focus on a special case of the general model which corresponds to the case where σ is a known constant which lies between 0 and 1, i.e., for some σ0∈ (0, 1),
P (σ = σ0) = 1.
Note that this setup can generate the risk effect because the signal contains some noise. Given the degenerate distribution of σ, however, the role of the sender’s message is essentially the same as in the canonical setup of CS where it can only convey information about the true state. There is hence no confirmation effect in this setup.
It is also worth emphasizing that although both the risk and confirmation effects tend to improve the quality of communication, there is a fundamental difference between them: the risk effect works through its direct impact on the sender’s payoff whereas the confirmation effect works through its impact on the receiver’s behavior. Because of this, the way the risk effect facilitates communication depends crucially on the minute details of the underlying model. First, the risk effect dominates the information effect only when the sender is sufficiently risk averse. In contrast, the reasoning which builds on the confirmation effect is independent of the sender’s risk
preferences and therefore holds in a wide range of circumstances including the uniform-quadratic specification. Second, the way the risk effect facilitates communication also depends on a special feature of CS that each interval is longer than the previous one. Although the presence of the risk effect is inherent in models with an informed receiver, it is not necessarily clear whether it improves the quality of communication in a more general sense.
4.2 Linear preferences
Throughout the analysis, we have restricted our attention to a model with quadratic prefer- ences where the utility loss is quadratic in the distance from the bliss point. We focus on this specification because it is perhaps the most commonly used setup in applications. The focus on the quadratic specification also facilitates comparison with previous works, many of which build on this specification. There is a drawback, however, as the quadratic specification necessarily entails the risk effect that happens to work in the same direction as the confirmation effect and somewhat obscures the extent of the confirmation effect – the novel feature of our model – as a consequence.
A way to isolate the confirmation effect is to consider a model with linear preferences where the utility loss is proportional to the distance from the bliss point. Now suppose that the sender’s payoff takes the following form:
US(t, a) = |t + b − a|. It is then verified that the integrand is positive, and we have
Z
r∈Tn
tn+ b − βn− q q + (1 − q)τn
(r − βn)
f (r)dr
= Z
t∈Tn
(tn+ b − βn) − q
q + (1 − q)τn(r − βn)
f (r)dr
= τn(tn+ b − βn)
Suppose tn+ b − βn+1≤ 0.13 If q is small enough to satisfy q ≤ τn+1(βn+1− tn− b)
τn+1(βn+1− tn) + (1 − τn+1)b
,
13This condition is satisfied by any CS partition.
the integrand is negative and we have Z
r∈Tn+1
tn+ b − βn+1− q
q + (1 − q)τn+1(r − βn+1)
f (r)dr
= Z
t∈Tn+1
−(tn+1+ b − βn+1) +
q
q + (1 − q)τn(r − βn+1)
f (r)dr
= −τn+1(tn+ b − βn+1).
As above, the local incentive condition for the sender to send mn instead of mn+1 at the threshold can be written as H ≤ 0 where
H := −(βn+1+ βn− 2tn− 2b) − q
2
q + (1 − q)τn
(tn− βn).
Note that the risk effect disappears and the confirmation effect makes the condition less stringent, possibly allowing for a more informative partition. Similarly, the local incentive condition for the sender to send mn+1 instead of mn at the threshold can be written as H ≥ 0 where
H := −(βn+1+ βn− 2tn− 2b) + q
2
q + (1 − q)τn+1(βn+1− tn).
Again, the risk effect disappears and the confirmation effect makes the condition less stringent, possibly allowing for a more informative partition. As we show in Appendix B, H ≥ 0 ≥ H is not only necessary but also sufficient (for any CS partition) when q is sufficiently small.
The analysis here illuminates the impact of the sender’s risk preferences on the equilibrium outcome and its efficiency. In Figure 2, we compute the lengths of the first interval of the most efficient MPE with two interval under linear and quadratic preferences when the state is uniformly distributed and compare them as a function of q. To understand this, it is important to note that the sender in this context faces two types of uncertainty. First, the sender does not know what signal the receiver observes. This is obviously the source of the risk effect and thus tends to facilitate communication as the sender becomes more risk averse. Second, in addition to that, the sender does not know whether the receiver’s signal reflects the true state or not, which we refer to as the higher-order uncertainty throughout the analysis (though mainly from the receiver’s point of view). The presence of this higher-order uncertainty weakens the positive impact of the two effects as the sender becomes more risk averse, thereby counteracting the risk effect which moves in the opposite direction. Due to this tradeoff, the overall effect of risk aversion on the quality of communication is indeed ambiguous. As can be seen from this argument, since the effect of the higher-order uncertainty becomes weaker as q decreases, the sender’s risk aversion tends to facilitate communication when q is sufficiently small (but positive).
4.3 The finite-state model and convergence
Our baseline model has a natural finite-state counterpart where the state space is a finite set. The analysis of a finite-state model is often useful as it can better clarify the key underlying mechanism due to its simple and intuitive structure. In this section, we provide a finite-state model whose limit converges to the continuous-state baseline model. Throughout the analysis here, we restrict attention to the case where the state is uniformly distributed.
Formally, each finite-state model is identified with a natural number L such that
• The state space is TL=0,L1, . . . ,L−1L , 1 ,
• The probability function of r conditional on t, pL(r|t), is defined as
pL(r|t) =
(q +L+11−q if r = t,
1−q
L+1 otherwise.
The receiver chooses an action from a continuous interval [0, 1] with the same payoff functions as in the baseline model. It is a natural finite-state counterpart of the baseline model where the accuracy of the signal converges to q in the limit L → ∞. Let tLℓ = Lℓ for ℓ = 0, . . . , L. In this setup, we can analogously define a finite-state version of a monotone partition equilibrium (MPE): an MPE is characterized by a finite partition of TL, {TnL}n=1,...,N, such that for any i < j, ti ∈ TiL, and tj ∈ TjL, ti < tj.
Example 2 Suppose that L = 3, so that the set of possible states is given by T3 = {0, 1/3, 2/3, 1}. In any finite-state model, there exists a fully separating equilibrium if the bias is sufficiently small. In the case with L = 3, it is verified that a fully separating equilibrium exists if and only if 16 ≥ b. For the sake of the argument, we exclude this possibility by assuming b > 16.
In this case a candidate for the most efficient equilibrium is the equilibria with two intervals of equal length, i.e., there exists two distinctive messages, say m1 and m2, such that
µ(t) =
(m1 if t = 0, 1/3, m2 if t = 2/3, 1.
Let β(t|m, r) the receiver’s posterior probability over t when she receives a message m and her
own private signal r, then it is written as
β(t|m, r) =
1+3q
2(1+q) if m = µ(t) and r = t,
1−q
2(1+q) if m = µ(t) , r 6= t, and m = µ(r),
1
2 if m = µ(t) , s 6= t, and m 6= µ(r), 0 if m = m1 or m2, but m 6= µ(t). Then, the receiver’s best response α is determined as
α(m1, 0) = 1 − q
6(1 + q), α(m1, 1/3) =
1 + 3q
6(1 + q), α(m1, 2/3) = 1
6, α(m1, 1) = 1 6, α(m2, 0) = 5
6, α(m2, 1/3) = 5
6, α(m2, 2/3) =
5 + 3q
6(1 + q), α(m2, 1) =
5 + 7q 6(1 + q). Using these, we derive the equilibrium condition as
b ≤ 2 + 4q + 3q2 6(1 + q)(2 + 2q − q2).
Moreover, we can show that the upper bound of b (the RHS) is strictly larger than 16 whenever q > 0 and it is strictly increasing in q. The upper bound of b is illustrated in Figure 3.
Finally, to motivate the use of finite-state models, we show that our baseline model is the limit of a sequence of finite-state models as L → ∞. More precisely, we establish that almost any MPE of the original model can be approximated by an MPE of a finite-state model whenever L is sufficiently large.
Proposition 7 Consider an original model with the uniform state distribution. Given any MPE with {Tn}n=1,...,N of the original model, which satisfies the equilibrium conditions with strict inequalities, i.e., G(tn−1, t, tn+1) > 0 > G(tn−1, t, tn+1) for n = 1, . . . , N − 1, then there exists Λ such that for any finite model with L > Λ there exists an MPE with {TnL}n=1,...,N such that TnL⊂ Tnfor any n.
5 Conclusion
In this paper we analyze a cheap talk model with a partially informed receiver. In clear contrast to the previous literature, we find that the receiver’s prior knowledge enhances the amount of information conveyed via cheap talk messages. This contrasting result is mainly due to the confirmation effect which stems from the uncertainty about the reliability of the information
source. When the receiver faces the higher-order uncertainty, the sender’s message provides information not only about the true state of nature but also about the reliability of the receiver’s information. The dual role of messages gives rise to the asymmetric response by the receiver, where she reacts less to truthful messages and less to misrepresented ones, which is essential in disciplining the sender to be more truthful.
Appendix A: the proofs
Proof of Proposition 1:
The proof closely follows that of Proposition 1 of Moreno de Barreda [6] and directly stems from the following lemma.
Lemma 2 The width of an interval Ti in Difinition 1 is longer than or equal to 2b unless inf Ti= 0.
Proof:
Suppose to the contrary that there exists an interval Ti such that 0 < inf Ti ≤ sup Ti< inf Ti+ 2b.
We divide the situation into the following two cases: {inf Ti} /∈ Ti or {inf Ti} ∈ Ti.
We first consider the case of {inf Ti} /∈ Ti. In this case the point inf Ti belongs to another interval, say Tj, and
inf Tj ≤ sup Tj = inf Ti < sup Ti
holds. Pick any two distinctive messages mi and mj such that µt(mi) > 0 iff t ∈ Ti, µt(mj) > 0 iff t ∈ Tj. Then,
sup Ti > α(mi, r) > inf Ti≥ α(mj, r)
holds for any r ∈ [0, 1]. This implies that
|inf Ti+ b − α(mi, r)| < b ≤ |inf Ti+ b − α(mj, r)|
holds for any r ∈ [0, 1]. However, this eliminates the sender’s incentive to send a message mj at t = inf Ti. This is a contradiction.
Next, we consider the case of {inf Ti} ∈ Ti. In this case there exists an interval Tj such that inf Tj < sup Tj = inf Ti ≤ sup Tj.
Pick any two distinctive messages mi and mj such that µt(mi) > 0 iff t ∈ Ti, µt(mj) > 0 iff t ∈ Tj. Then,
sup Ti ≥ α(mi, r) ≥ inf Ti> α(mj, r) holds for any r ∈ [0, 1]. This implies that
|inf Ti+ b − α(mi, r)| < b < |inf Ti+ b − α(mj, r)|
holds for any r ∈ [0, 1]. Then, there exists a sufficiently small ǫ > 0 such that {inf Ti− ǫ} ∈ Tj
and
|inf Ti− ǫ + b − α(mi, r)| < |inf Ti− ǫ + b − α(mj, r)|
holds. However, this eliminates the sender’s incentive to send a message mj at t = inf Ti− ǫ. This is a contradiction.
Proof of Lemma 1:
For t ∈ Tn, ∆ can be written as
∆(t; tn−1, tn, tn+1)
= −q(t + b − βn+1)2
− (1 − q) Z
r∈Tn+1
t + b − βn+1− q
q + (1 − q)τn+1(r − βn+1)
2
f (r)dr
− (1 − q) Z
r /∈Tn+1
(t + b − βn+1)2f (r)dr
+ q
t + b − βn− q
q + (1 − q)τn(t − βn)
2
+ (1 − q) Z
r∈Tn
t + b − βn− q q + (1 − q)τn
(r − βn)
2
f (r)dr + (1 − q)
Z
r /∈Tn
(t + b − βn)2f (r)dr. Since
(1 − q) Z
r∈Tn+1
t + b − βn+1− q
q + (1 − q)τn+1
(r − βn+1)
2
f (r)dr
= (1 − q) Z
r∈Tn+1
(t + b − βn+1)2+ q
2(1 − q)
(q + (1 − q)τn+1)2 Z
r∈Tn+1
(r − βn+1)2f (r)dr,
(1 − q) Z
r∈Tn
t + b − βn− q
q + (1 − q)τn(r − βn)
2
f (r)dr
= (1 − q) Z
r∈Tn
(t + b − βn)2− q
2(1 − q)
(q + (1 − q)τn)2 Z
r∈Tn
(r − βn)2f (r)dr,
q
t + b − βn− q
q + (1 − q)τn(t − βn)
2
= q(t + b − βn)2−q
2(q + 2(1 − q)τn) (t − βn)2 (q + (1 − q)τn)2 −
2q2(t − βn) q + (1 − q)τnb, we can show that
∆(t; tn−1, tn, tn+1)
= −(βn+1− βn)(βn+1+ βn− 2t − 2b)
− q
2(1 − q)
(q + (1 − q)τn+1)2 Z
r∈Tn+1
(r − βn+1)2f (r)dr + q
2(1 − q)
(q + (1 − q)τn)2 Z
r∈Tn
(r − βn)2f (r)dr
− q
2(q + 2(1 − q)τn) (t − βn)2 (q + (1 − q)τn)2 −
2q2(t − βn)
q + (1 − q)τnb. (1)
Taking the limit t ↑ tn then yields G(tn−1, tn, tn+1). Similarly, for t ∈ Tn+1, ∆ can be written as
∆(t; tn−1, tn, tn+1)
= −q
t + b − βn+1− q
q + (1 − q)τn+1(t − βn+1)
2
− (1 − q) Z
r∈Tn+1
t + b − βn+1− q
q + (1 − q)τn+1(r − βn+1)
2
f (r)dr
− (1 − q) Z
r /∈Tn+1
(t + b − βn+1)2f (r)dr + q(t + b − βn)2
+ (1 − q) Z
r∈Tn
t + b − βn− q
q + (1 − q)τn(r − βn)
2
f (r)dr + (1 − q)
Z
r /∈Tn
(t + b − βn)2f (r)dr. As above, this is further reduced to
∆(t; tn−1, tn, tn+1)
= −(βn+1− βn)(βn+1+ βn− 2t − 2b)
− q
2(1 − q)
(q + (1 − q)τn+1)2 Z
r∈Tn+1
(r − βn+1)2f (r)dr + q
2(1 − q)
(q + (1 − q)τn)2 Z
r∈Tn
(r − βn)2f (r)dr + q
2(q + 2(1 − q)τn+1) (t − βn+1)2 (q + (1 − q)τn+1)2 +
2q2(t − βn)
q + (1 − q)τn+1b. (2)
Taking the limit t ↓ tn then yields G(tn−1, tn, tn+1).
Proof of Proposition 2:
The local incentive condition is necessary for an MPE but not sufficient. To show the existence of an MPE, we need to show that the sender has an incentive to follow the equilibrium strategy for all t.
First, consider t ∈ Tn. If limt↑tn ∂∆∂t ≥ 0, then ∂∆∂t ≥ 0 holds and the sender has no incentive to deviate for any t ∈ Tnsince ∂∂t2∆2 ≤ 0. Taking derivative of (1) with respect to t yields
∂∆
∂t = 2(βn+1− βn) −
2q2(q + 2(1 − q)τn)
(q + (1 − q)τn)2 (t − βn) −
2q2 q + (1 − q)τnb,
![Figure 2: Efficiency in a two-interval equilibrium with different payoff functions [solid: quadratic preferences, dashed: linear preferences, b = 0.1]](https://thumb-ap.123doks.com/thumbv2/123deta/5735242.24313/43.892.289.649.693.982/efficiency-interval-equilibrium-different-functions-quadratic-preferences-preferences.webp)
