Next, consider t=DIF, i.e., DM’s ex-ante bias is toward project 1 while S’s ex-ante bias is toward project 2. This section shows how this con‡ict of interest a¤ects S’s pandering incentive.
The analysis for each agent’s belief and DM’s strategy remains unchanged. It su¢ ces to analyze S’s strategy. On observing = 1, S sends m = 1 if he expects:
xS
S’s expected bene…t from m=1 given =1
1 (1 )
S’s expected bene…t from m=2 given =1
(8)
which is equivalent to x xDM
S+xDM given (1; DIF ) = (2; DIF ) = (i.e., full information revelation). (8) does not always hold because S is biased away from project 1.
On observing = 2, S sends m = 2 if (6) holds (i.e., S can a¤ect the selection of a project).
If (6) holds, S expects:
S’s expected bene…t from m=2 given =2
> xS (1 )
S’s expected bene…t from m=1 given =2
(9)
Lemma 2.2 Consider t=DIF. There are cuto¤ s ADIF, x xDM
S+xDM and 1+x1
DM such that:
(1) For 2 (maxfxSx+xDMDM;1+x1
DMg; 1), a full revelation equilibrium exists.
(2) For 2 ADIF;xxDM
S+xDM or 2 (1+x1DM; 1), a partial revelation equilibrium exists.
(3) Otherwise, only babbling equilibria exist.
If there are informative equilibria, their outcomes are unique. Further, if there is an informa-tive equilibrium, both agents prefer an informainforma-tive equilibrium to a babbling equilibrium.
Proof. See Appendix
For 2 (xSx+xDMDM; 1) (i.e., (8) holds with strict inequality), S is willing to fully reveal infor-mation. For 2 ADIF;xxDM
S+xDM , S partially reveals information against his ex-ante preferred project, = 2, by randomizing between two messages so that (8) holds with equality. For
< ADIF (i.e., (8) does not hold), S prefers project 2 given any . For < 1+x1
DM (i.e., (6) does not hold), DM never selects project 2 even if S fully reveals information.
Lemma 2 refers to two regions in the xSxDM-plane given t=DIF. The …rst is for xS 2 x2DM; 1 and xDM 2 (0; 1) and the second for xS2 0; x2DM and xDM 2 (0; 1). Figure 2.4 depicts Lemma 2.
Like in the previous case, both agents will bene…t from a successful project. Thus, when S’s information is precise (i.e., is high), there is full information revelation.
When S’s information is noisy (i.e., is medium), S experiences tradeo¤s given each . If his information is against his ex-ante preferred project (i.e., = 1), S still expects higher bene…t from project 2, but by recommending project 2 instead of project 1, S decreases the probability that DM carries out a project. If his information is against DM’s ex-ante preferred project (i.e.,
= 2), recommending project 1 leads DM to carry out a project with a higher probability, but S prefers project 2.
When S’s information is very noisy (i.e., is small), only babbling equilibria exist as observed in the case of t=SIM.
If the outside option is removed from the model, a full revelation equilibrium exists for a narrower range of because the probability of a project not occurring is not S’s concern any more. For example, without an outside option, given xS2 0; x2DM , a full revelation equilibrium exists for 1+x1
S and only babbling equilibria exist otherwise, where 1+x1
S > x xDM
S+xDM. S’s pandering incentive and con‡ict of interest o¤set each other and lead S to reveal information.
Therefore, the presence of the outside option can facilitate information transmission.
An interesting observation is that communication can collapse even if S is willing to reveal information. Consider xS 2 x2DM; 1 , where xxDM
S+xDM < 1+x1
DM. For > xxDM
S+xDM, S is willing to fully reveal information if he can a¤ect DM’s selection of a project. However, for < 1+x1
DM, DM ignores S’s recommendation, which discourages S to reveal information.
When there is an informative equilibrium, its welfare implication is the same as in the previous case, i.e., both agents are better o¤ in a more informative equilibrium.
2.4.3 Con‡ict of Interest and Information Transmission
This section shows that more information is revealed given t=DIF than t=SIM by comparing results in Sections 4.1 and 4.2. We also explain welfare implication of the comparison.
We focus on an informative equilibrium if it exists, and a babbling equilibrium otherwise. That is, …xing , we compare the most informative equilibria given t=DIF and t=SIM, respectively.
This treatment seems reasonable because Lemmas 1 and 2 showed that an informative equilibrium makes DM and S better o¤ than a babbling equilibrium if it exists.7
We interpret the probability that S reveals his signal as a measure of informativeness for several reasons. First, we focus on information received by DM. Second, we will show that regardless of t, DM’s ex-ante expected payo¤ is larger as S reveals his information with the larger probability.8 Regardless of t, when S’s information is precise (i.e., is high), both agents agree to select
7We follow CS’s approach that focuses on the most informative equilibrium, an equilibrium including the maxi-mum number of steps, in conducting comparative statics analysis. In CS, both agents’ex-ante expected payo¤s are increasing in the number of steps.
A di¤erent rationale for selecting cheap talk equilibria could be an equilibrium re…nement such as NITS (Chen, Kartik and Sobel 2008) or neologism-proofness (Farrell 1993). Unfortunately, these notions do not isolate an equilibrium in this model, even though they may do in CS.
8Our measure of informativeness is di¤erent from standard measures such as Blackwell informativeness.
a project which is more likely to succeed. Hence, a full revelation equilibrium exists. When S’s information is noisy (i.e., is low), DM ignores S’s recommendation in selecting a project, and/or S is unwilling to reveal information against his ex-ante preferred project. Hence, only babbling equilibria exist.
However, S’s information is in the middle range, there is a di¤erence between t=SIM than t=DIF. Consider t=SIM. Given = 2 (i.e., S’s information is against both agents’ ex-ante pre-ferred project), S’s pandering incentive as well as his ex-ante bias tempts him to hide this infor-mation. On the other hand, consider t=DIF. Given = 1 (i.e., S’s information is against his ex-ante preferred project), S’s ex-ante bias tempts him to hide this information, but his pandering incentive encourages S to reveal information. Given = 2 (i.e., S’s information is against DM’s preferred project), S’s pandering incentive tempts him to hide this information, but his ex-ante bias encourages S to reveal information.
Thus, S is more tempted to hide information given t =SIM than given t =DIF. Con‡icts of interest can facilitate communication.9
Proposition 2.1 Given any xS and xD, at least as much information is revealed in the most informative equilibrium given t=DIF than given t=SIM.
Proof. See Appendix.
Proposition 3 follows from Lemmas 1 and 2, which identify four key regions in the xSxDM -plane. The …rst is for xS 2 (1+x2DM; 1) and xDM 2 (0; 1), the second for xS 2 (x2DM;1+x2DM) and xDM 2 (0; 1), the third for xS 2 2x3DM xDM; x2DM and xDM 2 (0; 1), and the last for xS 2 0; 2x3DM xDM and xDM 2 (0; 1). In any region, S reveals the same or strictly more information given t=DIF than given t=SIM. Figure 2.5 depicts Proposition 3.
The interesting comparison is for 2 ASIM;x xD
S+xD in the last region. In this case, a partial revelation equilibrium exists given t=SIM as well as t=DIF, i.e., S’s strategy is (1; SIM ) = 1 and (2; SIM ) 2 (0; 1) given t=SIM and (1; DIF ) 2 (0; 1) and (2; DIF ) = 1 given t=DIF. The result is (2; SIM ) < (1; DIF ), i.e., S reveals more information given t=DIF than t=SIM.
9The result is not driven by the stochastic nature of DM’s cost or the number of projects. We investigated a model where DM’s cost is predetermined and known to every agent. We also studied a model where there are more than two projects. The result remains unchanged in each model such that there is more information to be revealed by S given t =DIF than t=SIM. The description, results and proofs for both models are available on request.
We examine the welfare implication of Proposition 3. Suppose a decision maker (DM) can select a speaker (S) from a pool of speakers with di¤erent types (i.e., di¤erent levels of con‡ict of interest compared to DM). Which type (t ) of S will bene…t DM? The answer is that DM can bene…t relying on S who has larger con‡ict of interest with DM.
Proposition 2.2 For any xDM and xS, there are cuto¤ s A1 and A2, where 12 < A1 < A2 < 1, such that:
(1) DM prefers t=DIF to t=SIM for 2 (A1; A2).
(2) DM is indi¤ erent between t=DIF and t=SIM otherwise.
Proof. See the Appendix.
Regardless of t, DM is better o¤ as he collects more information. However, the same result may not hold for S. Alignment of ex-ante biases complements information for S. In a full information revelation equilibrium, S is better o¤ given t=SIM than given t=DIF. Moreover, S may be better o¤ in a less informative equilibrium given t=SIM than in a more informative equilibrium given t=DIF.
We have also considered an extension of the model where S puts costly e¤ort into acquiring information. Speci…cally, before observing a signal, S can put e¤ort into improving precision of his signal. The result is that S puts more e¤ort into acquiring information given t =DIF than t=SIM.10