Strategic transmission of imperfect information: why revealing evidence (without proof) is difficult

Abstract

In our cheap-talk setting, an expert privately observes multiple binary signals, her soft evidence, about a continuous state of the world and then communicates with a decision-maker. While direct transmission of evidence entails communicating the underlying signals, indirect transmission of evidence entails communicating a summary statistic of her evidence. We first establish that fully informative equilibria exist if the conflict of interest is small. Otherwise, direct transmission of evidence is impossible, as withholding one part of the soft evidence in the communication necessarily induces incentives to manipulate the report on the other part. On the contrary, indirect transmission of evidence remains partially informative for intermediate conflicts of interest. Finally, we introduce the possibility of certification. We show that, if the costs of certification are low, the expert can fully reveal her evidence regardless of the conflict of interest.

1 Introduction

Decision-makers often deal with complex issues such as whether to implement risky long-term investments or major policy reforms. Even experts will only have incomplete information on such issues and hence have to rely on evidence available to them. Evidence will in many cases be soft, consisting of measures that can be easily manipulated, e.g., unverifiable estimates of future risks (Bertomeu and Marinovic 2016). An expert can report soft evidence either directly or indirectly. Direct reporting entails communicating details on, and hence potentially revealing, each piece of evidence, e.g., estimates of various future risks. In contrast, indirect reporting entails communicating a summary of the evidence, e.g., a grade rating.

To illustrate why direct communication is important, suppose that the decision-maker receives information from multiple experts, whose soft evidence partially overlaps. Such a situation may, for instance, arise endogenously in dynamic environments such as networks. As argued by Acemoglu et al. (2014), indirect communication of a summary of the evidence then is prone to double counting of information because of the difficulty of accounting for the overlap in the experts’ evidence, and often precludes Bayesian and dynamic game theoretic analysis. Direct communication, on the other hand, allows the experts to “tag” their evidence, e.g., assign a name to each risk estimate, and thus avoid the problem of double counting (Acemoglu et al. 2014; Foerster 2019). It however requires independent reporting on each piece of information, and is hence more restrictive, but how much more is an open question.

In a simple model with one expert, we show that this downside to direct communication may be large. If the conflict of interest between parties is considerable, such that the expert cannot truthfully reveal all her soft evidence, then direct communication becomes impossible altogether: withholding one part of the evidence in the communication necessarily induces incentives to manipulate the report on the other part. Unless proof is available, the expert hence has to resort to reporting a summary of her evidence that reflects her coarse posterior belief. Our findings provide a qualification to the notion that direct communication is particularly useful in dynamic environments.

In our framework, the state of the world is unknown to both the expert and the decision-maker (DM), but it is common knowledge that the expert privately observes soft evidence—a certain number of binary signals that are correlated with the state. The expert communicates via cheap talk with the DM, who in turn takes an action. As described above, the expert may transmit her evidence either directly or indirectly. The agents’ preferences differ regarding the action to be implemented and satisfy the single-crossing property.

We first establish that a fully informative equilibrium exists if the conflict of interest is small, regardless of whether the expert reports her soft evidence directly or indirectly. Compared to the seminal cheap-talk model of Crawford and Sobel (1982, henceforth CS), the expert is less well informed in our setup, which implies coarse actions that reduce incentives to deviate even under a fully informative strategy.¹

However, if the conflict of interest is too large to reveal all evidence, then also revealing only a part of it, that is, reporting truthfully on some signals while babbling on others, is impossible. Withholding a part of the expert’s soft evidence in the communication implies that it will still affect her posterior belief but not the induced action. As we show, the induced action may be considerably further apart from her bliss point (or ideal action) than under full revelation, inducing incentives to manipulate the report on the other part of her evidence. Hence, in contrast to CS and related contributions, resorting to less informative (direct) communication tightens the expert’s IC constraints.

On the contrary, indirect transmission of her soft evidence remains possible. By reporting a summary statistic that reflects her coarse posterior belief, the expert does not completely withhold a part of her evidence. All her evidence will affect the induced action, limiting its distortion relative to the expert’s bliss point. A coarse enough summary statistic thus loosens the expert’s IC constraints as common in the literature, and communication remains partially informative for intermediate conflicts of interest.²

Comparative statics regarding the number of soft signals that the expert observes reveal that the marginal equilibrium value of information may be negative. In the “uniform-quadratic” model of CS,³ more soft evidence implies less informative equilibrium communication if it implies resorting to a coarse summary statistic instead of full revelation, because the expert then will employ only one message for each two adjacent summary statistics.

Next, we consider a well-informed expert who observes a lot of soft evidence. We establish that in equilibrium the relation between the expert’s posterior belief and the induced action of the DM is essentially the same as in CS. A well-informed expert has almost perfect information about the state, making incentives akin to those of a perfectly-informed expert. We therefore provide a robustness check for the model of CS.

Finally, we introduce the possibility for the expert to invest in costly certification, which produces hard evidence. For instance, audit risk assessment procedures may certify soft evidence on a company’s investments. We establish that, given low certification costs, a fully informative equilibrium exists regardless of the conflict of interest. We know already that the expert can reveal all her evidence without certification if the conflict is small. Otherwise, the expert can do so with certification of favorable reports. This deters the expert from persuading the DM into taking an action closer to her bliss point, since submitting favorable cheap-talk reports is not credible if the DM expects a proof for such evidence. Deviations to unfavorable reports save the costs of certification but move the DM’s action further away from the expert’s bliss point, and become more costly the larger the bias. They are hence deterred if costs of certification are low.

Our findings show that soft evidence is more prone to manipulation than one might think in light of the literature and have important implications for applications to dynamic environments such as networks. It has been argued that direct communication is particularly useful in such environments because it allows agents to tag information, thereby avoiding the problem of double counting of information (Acemoglu et al. 2014; Foerster 2019); see also the field experiment by Mobius et al. (2015) for evidence that people make use of tagging. We provide a qualification to this argument, showing that conflicts of interest being small is important. Otherwise, and unless certification is possible, agents have to resort to indirect communication to transmit information, making them prone to double count information.

There is a large literature on cheap talk that builds on CS. Our signaling framework is closely related to Morgan and Stocken (2008), who study cheap talk with multiple experts who each receive one binary signal about the state. Equilibrium communication may be more informative than in our model, because experts who are truthful in a partially informative equilibrium do not withhold any information. Notably, direct and indirect transmission coincide in this and related models on, e.g., multi-player cheap talk (Hagenbach and Koessler 2010; Galeotti et al. 2013) and communication about the impact of prosocial actions (Foerster and van der Weele 2021), as each agent observes only one soft signal. Foerster (2019) extends this framework to a dynamic game on a network and characterizes fully informative equilibria. Our current results suggest that the characterization cannot readily be extended to partially informative equilibria.

Direct communication involves reporting of a vector, which relates our paper to work on multi-dimensional communication. Chakraborty and Harbaugh (2007, 2010) find that multiple issues may improve communication. However, interaction between different dimensions according to the prior can also limit communication (Levy and Razin 2007). Battaglini (2002) and Ambrus and Takahashi (2008) show that full revelation is possible regardless of the conflict of interest if there is more than one expert, but this result may not hold with imperfect signals about the state (Ambrus and Lu 2014; Habermacher 2022). Unlike these papers, we study the communication of many soft signals about a one-dimensional state. This relates our paper to Ivanov (2010), who investigates the allocation of decision-making authority and shows that informational control, that is, restricting the precision of the expert’s information, can improve communication.⁴ In particular, cheap talk is preferred to delegation for small conflicts of interest. Argenziano et al. (2016) obtain the same result when the expert first has to acquire information, because the expert overinvests in equilibrium. Their approach to communication is related to ours in that they study indirect communication in the uniform-quadratic model of CS after the expert has acquired binary signals.

Another related literature is that on disclosure of hard evidence. Forges and Koessler (2005) and Hagenbach et al. (2014) allow players to (partially) certify their type in Bayesian games. Bertomeu and Marinovic (2016) consider a manager with both soft and hard information about her assets. Here, the credibility of cheap messages is due to some managers’ being averse to over-stating the value of their assets. Eső and Galambos (2013) introduce optional costly certification to CS’s setting and show that a larger conflict of interest may improve communication, as the expert will more heavily rely on hard evidence. Schopohl (2019) shows that a fully revealing equilibrium may exist if additionally the state space is discrete. We obtain the same result regardless of the conflict of interest with an imperfectly informed expert.

Finally, in Froeb and Kobayashi (1996)’s model of civil litigation, interested parties can incur costs to produce hard evidence—binary signals similar to our model. They show that a fully informative decision is possible even if the DM views the evidence presented as a random sample although parties in fact only present favorable evidence; see also Milgrom and Roberts (1986) and Lipman and Seppi (1995) for similar results.

The paper is organized as follows. In Sect. 2 we introduce the model and notation. Section 3 presents our main results on cheap-talk communication and comparative statics. Section 4 investigates a well-informed expert. In Sect. 5 we introduce the possibility of certification. Section 6 concludes.

2 Model and notation

An expert, E, privately observes soft evidence or information on the state of the world $\theta \in \Theta =[0,1]$ and then communicates with a decision-maker, DM, who in turn takes an action $a\in A={\mathbb {R}}$. Neither agent observes the state, but it is common knowledge that the expert observes $K\ge 1$ soft signals $s_1,s_2,\ldots ,s_K$ about $\theta$. We assume that $s_k=1$ with probability $\theta$ and $s_k=0$ with probability $1-\theta$ for all $k=1,2,\ldots ,K$, and that the signals are independent conditionally on $\theta$.⁵ High signals are thus indicative of a high state.⁶ Let $S=(s_1,s_2,\ldots ,s_K)\in {\mathcal {S}}=\{0,1\}^K$ denote the collection of the expert’s evidence and let $S_\Sigma =\sum _{k=1}^K s_k\in {\mathcal {S}}_\Sigma =\{0,1,\ldots ,K\}$ denote the summary statistic indicating the number of high signals that she has observed. Note that $S_\Sigma$ is a sufficient statistic for the state $\theta$ since the signals are independent Bernoulli-distributed random variables with mean $\theta$. The common prior over the state of the world is a distribution F on $\Theta$ with continuous and strictly positive density f.

Before the DM takes her action, the expert can send a cheap-talk message $m\in M= M_{\mathcal {S}}\cup M_\Sigma$ to the DM, where $M_{\mathcal {S}}={\mathcal {S}}$ and $M_\Sigma ={\mathcal {S}}_\Sigma$ without loss of generality. As we will explain in more detail below, we interpret a message $m\in M_{\mathcal {S}}$ as direct transmission of evidence and a message $m\in M_\Sigma$ as indirect transmission of evidence. A strategy for the expert is a mapping $\sigma :{\mathcal {S}}\rightarrow \Delta (M)$ and a strategy for the DM is a mapping $\rho :M\rightarrow \Delta (A)$.

The expert’s payoff function is $u_E(a,\theta ,b)$ and depends on the DM’s action a, the state $\theta$, and her bias $b>0$, which is a constant. We assume that $u_E(a,\theta ,b)$ is twice continuously differentiable and strictly concave in a, with a unique maximum for fixed $\theta$ and b. Furthermore, we impose the single-crossing conditions

$$\begin{aligned} \frac{\partial ^2 u_E(a,\theta ,b)}{\partial a\partial \theta }>0 \text { and } \frac{\partial ^2 u_E(a,\theta ,b)}{\partial a\partial b}>0\qquad \qquad \qquad (\hbox {SC}) \end{aligned}$$

SC implies that the expert would like to induce a higher action both at a higher state and with a higher bias. The DM’s payoff function is $u_{DM}(a,\theta )\equiv u_E(a,\theta ,0)$. We will sometimes consider preferences where agents suffer quadratic losses associated with a deviation of the implemented action from their bliss point, $u_E(a,\theta ,b)=-(\theta +b-a)^2$.

We study perfect Bayesian equilibria (PBE) $(\sigma ,\rho )$ of this game. Let $\mu (m)\in \Delta (\Theta )$ denote the posterior belief of the DM upon receiving message $m\in M$ derived from the expert’s strategy $\sigma$ by Bayes’ rule whenever possible. We assume without loss of generality that the DM takes an on-equilibrium-path action if she observes an off-equilibrium message, which implies that we can ignore such deviations.

3 Equilibrium analysis

We first provide formal definitions of a direct- and an indirect-transmission strategy and introduce our notion of informativeness. In Sect. 3.1, we investigate direct-transmission equilibria. We then study indirect-transmission equilibria in Sect. 3.2. Finally, we provide comparative statics on the number of soft signals K that the expert observes in Sect. 3.3.

For convenience, let $U_E(a,b|\cdot )\equiv E[u_E(a,\theta ,b) | \cdot ]$ and $U_{DM}(a | \cdot )\equiv E[u_{DM}(a,\theta ) | \cdot ]$ denote the expert’s and the DM’s (conditional) expected utilities, respectively, and let $a_E(b,\cdot )$ and $a_{DM}(\cdot )$ denote the corresponding maximizers, which are uniquely determined since preferences are strictly concave in a. Consider any strategy $\sigma$ of the expert. Upon receiving a message m such that $Pr(m|\sigma )>0$, the DM will choose

$$\begin{aligned} \rho (m)=a_{DM}( \mu (m)). \end{aligned}$$

In particular, the DM will never use mixed strategies in equilibrium. We can therefore focus the subsequent analysis on the expert.

We distinguish two types of strategies. The expert transmits her evidence directly if she reports independently on each signal; thereby, she may reveal (each piece of) her soft evidence. The expert transmits her evidence indirectly if she reports a (coarse) summary statistic of her evidence that reflects her (coarse) posterior belief. Let $(\sigma (S))_k$ denote the k-th element of the message sent by the expert after observing evidence S under the direct-transmission strategy $\sigma$, and $S_{-k}=(s_1,\ldots ,s_{k-1},s_{k+1},\ldots ,s_K)$.

Definition 1

((In)direct transmission of evidence)

• (i) The expert transmits her evidence directly under strategy $\sigma$ if for all $S\in {\mathcal {S}}$, $\sigma (S)\in \Delta (M_{\mathcal {S}})$ such that $(\sigma (S))_k$ is independent of $S_{-k}$ for all $k=1,2,\ldots ,K$.

• (ii) The expert transmits her evidence indirectly under strategy $\sigma$ if for all $S\in {\mathcal {S}}$, $\sigma (S)\in \Delta (M_\Sigma )$.

We refer to equilibria in which the expert transmits her evidence (in)directly as (in)direct-transmission equilibria. Note that direct transmission is more restrictive than indirect transmission, as it requires independent reporting on each signal. Our aim is to investigate how much more restrictive it is, and hence to understand how difficult it is to reveal soft evidence.

Our interest is in the amount of information that the expert can transmit to the DM in equilibrium. We hence focus on most informative equilibria. Let $MSE(\sigma ,K)\equiv E[(\theta -E[\theta |\mu (\sigma (S))])^2|\sigma ,K]$ denote the DM’s residual mean-squared error under the expert’s strategy $\sigma$.

Definition 2

(Informativeness)

• (i) The strategy profile $(\sigma ,\rho )$ is weakly more informative than the strategy profile $(\sigma ',\rho ')$ if $MSE(\sigma ,K)\le MSE(\sigma ',K)$.

• (ii) A most informative ((in)direct-transmission) equilibrium $(\sigma ,\rho )$ is weakly more informative than any ((in)direct-transmission) equilibrium $(\sigma ',\rho ')$.

• (iii) A fully informative equilibrium $(\sigma ,\rho )$ is weakly more informative than any strategy profile $(\sigma ',\rho ')$.

Note that the informativeness of an equilibrium is solely determined by the expert’s strategy. For convenience, we usually ignore that there may be several most (fully) informative equilibria.

3.1 Direct-transmission equilibria

We first investigate direct-transmission equilibria, in which the expert reports independently on each signal. Note that we do not restrict possible deviations of the expert, such that communication remains cheap talk. Let $\tau ^{mix}(\sigma )=\{k|(\sigma (s_k,S_{-k}))_k\ne (\sigma (1-s_k,S_{-k}))_k \text { for all } S\in {\mathcal {S}}\}$ denote the collection of evidence on which strategy $\sigma$ is (partially) informative, let $\tau (\sigma )=\{k|(\sigma (S))_k=s_k \text { for all } S\in {\mathcal {S}}\}\subseteq \tau ^{mix}(\sigma )$ denote the subset on which strategy $\sigma$ is fully informative, and let $\kappa ^{mix}(\sigma )$ and $\kappa (\sigma )$ denote the respective cardinalities of these sets. Note first that $\kappa (\sigma )\le \kappa ^{mix}(\sigma )\le K$. Second, independent reporting across signals rules out that $\sigma$ is (partially) informative on a signal $s_k$ for some but not all $S_{-k}$. We show that the most informative direct-transmission equilibrium is fully informative if the conflict of interest between the expert and the DM is small, and uninformative otherwise.

Proposition 1

The most informative direct-transmission equilibrium $(\sigma ,\rho )$ is such that

• (i) $\kappa (\sigma )=K$ if $b\le {\underline{b}}(K)\equiv \min _{k=0,1,\ldots ,K-1}{\underline{b}}(k,K)$, and

• (ii) $\kappa ^{mix}(\sigma )=0$ otherwise,

where ${\underline{b}}(k,K)>0$ solves

$$\begin{aligned} U_E(a_{DM}( S_\Sigma =k),{\underline{b}}(k,K) | S_\Sigma =k)= U_E(a_{DM}( S_\Sigma =k+1),{\underline{b}}(k,K) | S_\Sigma =k) \end{aligned}$$

(1)

for all $k=0,1,\ldots ,K-1$.

All proofs are relegated to Appendix A. We can restrict attention to upward deviations, as $b>0$. Upward deviations from a fully informative strategy $\sigma$ ($\kappa (\sigma )=K$) induce a higher action by the DM. They are not beneficial if the change in the DM’s action is large relative to the expert’s bias even in case of a small deviation, i.e., a deviation in which the expert reports one low signal as high. Equation (1) determines the threshold on the bias up to which this deviation is deterred regardless of the evidence.

Now, to see why the expert cannot reveal part of her evidence if the conflict of interest exceeds this threshold, consider a partially informative pure strategy $\sigma '$ that truthfully reveals $\kappa (\sigma ')<K$ signals. Note that the evidence that the expert withholds from the DM, that is, signals $s_k$ such that $k\notin \tau (\sigma ')$, will still affect her posterior belief but not the DM’s action. We show that the expert has a profitable deviation if the realization of the evidence S is such that the signals that are not transmitted are high, i.e., $s_k=1$ for all $k\notin \tau (\sigma ')$. In this case, the expert’s report under $\sigma '$ will induce an action $\rho '(\sigma '(S))$ that is further apart from her bliss point $a_E(b, S)$ than the induced action $\rho (\sigma (S))$ under the fully informative strategy $\sigma$:

$$\begin{aligned} \rho '(\sigma '(S))=a_{DM}(s_k, k\in \tau (\sigma '))<a_{DM}(S)=\rho (\sigma (S))<a_E(b, S), \end{aligned}$$

where the inequalities follow from $E[\theta |\sigma '(S)]=E[\theta |s_k, k\in \tau (\sigma ')]<E[\theta |S]=E[\theta |\sigma (S)]$ and SC. A small deviation ${\tilde{m}}'$ from $\sigma '$, i.e., a deviation in which the expert reports one signal $s_k$ such that $s_k=0$ and $k\in \tau (\sigma ')$ as high, thus induces a lower action than a small deviation ${\tilde{m}}$ from $\sigma$, i.e., $\rho '({\tilde{m}}')<\rho ({\tilde{m}})$. If $\rho ({\tilde{m}})>a_E(b, S)$, then $\rho '({\tilde{m}}')$ is closer to the expert’s bliss point $a_E(b, S)$ than $\rho ({\tilde{m}})$; see Fig. 1 for an illustration. Thus, ${\tilde{m}}'$ is a profitable deviation from $\sigma '$, as ${\tilde{m}}$ is a profitable deviation from $\sigma$. Otherwise, if $\rho ({\tilde{m}})<a_E(b, S)$,⁷ then ${\tilde{m}}'$ is a profitable deviation from $\sigma '$, as $\rho '(\sigma '(S))<\rho '({\tilde{m}}')<\rho ({\tilde{m}})<a_E(b, S)$. Thus, if $\sigma$ is not part of an equilibrium, then neither is $\sigma '$.⁸

Fig. 1

Deviations ${\tilde{m}}$ and ${\tilde{m}}'$ from a fully informative strategy $\sigma$ and a partially informative strategy $\sigma '$, respectively, with corresponding best replies $\rho$ and $\rho '$ under direct transmission for $b>{\underline{b}}(K)$, such that $(\sigma ,\rho )$ is not an equilibrium

In other words, resorting to less informative direct communication tightens the expert’s IC constraints, implying that if full revelation of soft evidence is precluded, then so is partial revelation. The following example provides the exact threshold on the bias for the uniform-quadratic case.

Example 1

Suppose that $F={\mathcal {U}}(0,1)$ and $u_E(a,\theta ,b)=-(\theta +b-a)^2$. The most informative direct-transmission equilibrium $(\sigma ,\rho )$ is such that $\kappa (\sigma )=K$ if $b\le {\underline{b}}(K)=\frac{1}{2(K+2)}$, and $\kappa ^{mix}(\sigma )=0$ otherwise.

We spare a discussion of equilibria other than most informative equilibria and only note that such equilibria generically do not exist due to the tightening of the expert’s IC constraints under a partially informative strategy discussed above.

3.2 Indirect-transmission equilibria

Next, we investigate indirect-transmission equilibria, in which the expert reports a summary statistic of her evidence. To characterize these equilibria, we generalize the notion of a consecutive partition, allowing partition elements to slightly overlap. Two sets P and Q are pairwise incomparable (with respect to inclusion) if $P\nsubseteq Q$ and $Q\nsubseteq P$.

Definition 3

(Consecutive (overlapping) partition)

We say that ${\mathcal {P}}=\{P_0,P_1,\ldots ,P_{\kappa -1}\}$ is a consecutive (overlapping) partition of ${\mathcal {S}}_\Sigma$ if $P_0,P_1,\ldots ,P_{\kappa -1}$ are pairwise incomparable subsets of ${\mathcal {S}}_\Sigma$ such that $\cup _{k=0}^{\kappa -1}P_k={\mathcal {S}}_\Sigma$ and $\max P_{k-1}<(\le )\min P_{k}$ for all $k=1,2,\ldots ,\kappa -1$.

We show that indirect-transmission equilibria are characterized by a consecutive overlapping partition of the set of possible summary statistics, which indicate the number of high signals that the expert has observed and reflect her posterior belief. The expert reports the partition element that her true summary statistic belongs to—a (coarse) summary statistic of her evidence.

Proposition 2

An indirect-transmission strategy profile $(\sigma ,\rho )$ is a PBE if and only if there exists a consecutive overlapping partition ${\mathcal {P}}=\{P_0,P_1,\ldots ,P_{\kappa -1}\}$ of ${\mathcal {S}}_\Sigma$ such that

• (i) there exist distinct $m_0,m_1,\ldots ,m_{\kappa -1}\in M_\Sigma$ such that $m_k\in \hbox {supp}(\sigma (S))$ if and only if $S_\Sigma \in P_k$ for all $k=0,1,\ldots ,\kappa -1$,

• (ii) $\rho$ satisfies $\rho (m_k)=a_{DM}(\mu (m_k))$ for all $k=0,1,\ldots ,\kappa -1$, and

• (iii)
  $$\begin{aligned} U_E(\rho (m_{k-1}),b | S_\Sigma =\max P_{k-1})&\ge U_E(\rho (m_{k}),b | S_\Sigma =\max P_{k-1}) \text { and }\\ U_E(\rho (m_{k}),b | S_\Sigma =\min P_{k})&\ge U_E(\rho (m_{k-1}),b | S_\Sigma =\min P_{k}) \end{aligned}$$

  for all $k=1,2,\ldots ,\kappa -1$.

Proposition 2 adapts the main result of CS to our setting. (We investigate the relation between our model and CS in Sect. 4.) It shows that any indirect-transmission equilibrium is such that the expert reports the partition element that her true summary statistic belongs to (condition (i)). In turn, the DM best replies to the expert’s message (condition (ii)). The reported information needs to be “coarse enough” to deter the expert from deviating to another message (condition (iii)). Note first that pure indirect-transmission equilibria are characterized by a consecutive, non-overlapping, partition. Second, condition (iii) generalizes Lemma 2 of Argenziano et al. (2016) to overlapping partitions (and hence mixed strategies) and more general preferences. Finally, note that we do not restrict the expert’s strategy under indirect communication, such that Proposition 2 characterizes all equilibria of the game.

Building on Proposition 2, we show that the condition for a fully informative equilibrium to exist coincides with the corresponding condition for direct-transmission equilibria (Proposition 1). However, with indirect transmission, partially informative communication remains possible in case a fully informative equilibrium does not exist.

Proposition 3

The most informative indirect-transmission equilibrium $(\sigma ,\rho )$ is characterized by a consecutive overlapping partition ${\mathcal {P}}=\{P_0,P_1,\ldots ,P_{\kappa -1}\}$ of ${\mathcal {S}}_\Sigma$ such that

• (i) $\kappa =K+1$ if $b\le {\underline{b}}(K)$,

• (ii) $2\le \kappa <K+1$ if ${\underline{b}}(K)< b\le {\overline{b}}(K)\equiv \max _{k=0,1,\ldots ,K-1}{\overline{b}}(k,K)$, and

• (iii) $\kappa =1$ otherwise,

where ${\overline{b}}(k,K)>0$ solves

$$\begin{aligned} U_E(a_{DM}(S_\Sigma \le k),{\overline{b}}(k,K) | S_\Sigma =k)=U_E(a_{DM}(S_\Sigma >k),{\overline{b}}(k,K) | S_\Sigma =k) \end{aligned}$$

(2)

for all $k=0,1,\ldots ,K-1$.

Under a fully informative strategy profile ($\kappa =K+1$), a small upward deviation (reporting one more high signal than one actually has got) induces the same change in the DM’s action as under direct transmission. Thus, the condition for a fully informative equilibrium to exist is the same for both types of strategies.

Now, if this condition is not satisfied, partial transmission of evidence remains possible. The expert can simply report a less precise summary statistic of her evidence by resorting to a coarse partition. Thereby, she does not completely withhold a part of her evidence contrary to direct transmission. All her evidence will affect the induced action, which limits the distortion of the action relative to the expert’s bliss point. A coarse partition that shifts the DM’s best replies to different messages far enough apart will hence loosen the expert’s IC constraints as common in the literature, and partially informative communication remains possible for intermediate conflicts of interest. See Fig. 2 for an illustration.

Fig. 2

Partially informative indirect-transmission strategy $\sigma$, characterized by the partition ${\mathcal {P}}=\{P_0,P_1\}=\{\{0,1\},\{2,3,4\}\}$ such that $\sigma (S)(m_k)=1$ if $S_\Sigma \in P_k$ for all $k=0,1$, and the corresponding best reply $\rho$ for $K=4$

Equation (2) determines the threshold on the bias up to which a partition with two elements constitutes an equilibrium. The following example illustrates this result. Let $\lfloor x \rfloor$ denote the greatest integer less or equal to x.

Example 2

Suppose that $F={\mathcal {U}}(0,1)$ and $u_E(a,\theta ,b)=-(\theta +b-a)^2$. The most informative indirect-transmission equilibrium $(\sigma ,\rho )$ is characterized by a consecutive partition ${\mathcal {P}}=\{P_0,P_1,\ldots ,P_{\kappa -1}\}$ of ${\mathcal {S}}_\Sigma$ such that $\kappa =K+1$ if $b\le {\underline{b}}(K)=\frac{1}{2(K+2)}$, $2\le \kappa \le \lfloor K/2+1 \rfloor$ if ${\underline{b}}(K)< b\le {\overline{b}}(K)=\frac{K+1}{4(K+2)}$, and $\kappa =1$ otherwise. Notably, if the most informative equilibrium is partially informative, then the expert employs at most about half of the messages that she would employ under a fully informative strategy, as each partition element except the first contains at least two adjacent summary statistics.

3.3 Comparative statics and the value of information

We finish this section with comparative statics on the number of soft signals K that the expert observes. Let $(\sigma _K,\rho _K)$ denote the most informative equilibrium when the expert observes K signals. We can assume without loss of generality that $(\sigma _K,\rho _K)$ is an indirect-transmission equilibrium.

Recall that a fully informative equilibrium exists if the change in the DM’s action upon a small deviation is large relative to the expert’s bias. Now, as the expert becomes better informed, the change in the action upon such a deviation becomes smaller, making a deviation more beneficial. In the limit, most informative equilibria are (at most) partially informative:

Corollary 1

The threshold ${\underline{b}}(K)>0$ converges to zero as $K\rightarrow \infty$.

It follows immediately from Corollary 1 that there exists a non-negative integer ${\underline{K}}(b)\equiv \max \{\max \{K\ge 1\ |\ {\underline{b}}(K)\ge b\},0\}$ such that a fully informative equilibrium exists if the expert is informed about at most ${\underline{K}}(b)$ signals. Hence, while the expert becomes better informed as K increases, it is unclear whether the same holds for the DM in equilibrium.

Definition 4

(Marginal equilibrium value of information)

The marginal equilibrium value of information at K signals is $MSE (\sigma _K,K)-MSE(\sigma _{K+1},K+1)$.

To keep the analysis tractable, we now consider a uniformly distributed state of the world, $F={\mathcal {U}}(0,1)$, and preferences where agents suffer quadratic losses, $u_E(a,\theta ,b)=-(\theta +b-a)^2$. Note that in this case ${\underline{K}}(b)=\max \{\left\lfloor \frac{1}{2b}-2\right\rfloor ,0\}$. We show that more soft evidence implies less informative equilibrium communication if it implies resorting to a coarse summary statistic.

Proposition 4

Suppose that $F={\mathcal {U}}(0,1)$ and $u_E(a,\theta ,b)=-(\theta +b-a)^2$. If $b\le 1/8$, such that ${\underline{K}}(b)\ge 2$, then the marginal equilibrium value of information is positive at $K<{\underline{K}}(b)$ signals, and negative at $K={\underline{K}}(b)$ signals.

The marginal equilibrium value of information is positive as long as the expert could still perfectly report her evidence if she had observed an additional signal. However, it turns negative at $K={\underline{K}}(b)$ signals, where the expert can still perfectly report her evidence, while she could not do so any more if she had observed an additional signal. In this case, the additional signal would almost halve the number of messages induced in equilibrium, as the expert then would employ one message for each two adjacent summary statistics (see also Example 2). As a result, the equilibrium would be less informative. Figure 3 illustrates this result and the extent to which frictions may impair communication.

Fig. 3

Residual mean-squared errors in the most informative equilibrium $(\sigma _K,\rho _K)$ for $b=1/10$, such that ${\underline{K}}(b)=3$

4 Well-informed experts and robustness of CS

We investigate a well-informed expert who observes a lot of soft evidence. Just like in Sect. 3.3, let $(\sigma _K,\rho _K)$ denote the most informative equilibrium when the expert observes K signals. Note that a well-informed expert will generally employ coarse summary statistics of her evidence in the communication with the DM, and let $\kappa _K$ denote the number of messages induced in $(\sigma _K,\rho _K)$.

We show that the relation between the expert’s posterior belief and the induced action of the DM is essentially the same as in CS.⁹

Proposition 5

There exists a positive integer ${\overline{K}}$ and a partition ${{\bar{\theta }}}_0=0<{{\bar{\theta }}}_1<\cdots <{{\bar{\theta }}}_{{{\bar{\kappa }}}}=1$ of $\Theta$, where ${{\bar{\theta }}}_k$ solves

$$\begin{aligned} U_E\big (a_{DM}(\theta \in ({{\bar{\theta }}}_{k-1},{{\bar{\theta }}}_{k})),b | \theta ={{\bar{\theta }}}_{k}\big )=U_E\big (a_{DM}(\theta \in ({{\bar{\theta }}}_{k},{{\bar{\theta }}}_{k+1})),b | \theta ={{\bar{\theta }}}_{k}\big ) \end{aligned}$$

(3)

for all $k=1,2,\ldots ,{{\bar{\kappa }}}-1$, such that the most informative equilibrium $(\sigma _K,\rho _K)$ is characterized by a consecutive overlapping partition ${\mathcal {P}}^K=\{P_0^K,P_1^K,\ldots ,P_{\kappa _K-1}^K\}$ of ${\mathcal {S}}_\Sigma$ such that

• (i) $\kappa _K={{\bar{\kappa }}}$ if $K\ge {\overline{K}}$, and

• (ii)
  $$\begin{aligned} \lim _{K\rightarrow \infty }E[\theta | S_\Sigma =\max P_{k-1}^K]=\lim _{K\rightarrow \infty }E[\theta | S_\Sigma =\min P_{k}^K]={{\bar{\theta }}}_{k} \end{aligned}$$

  (4)

  for all $k=1,2,\ldots ,{{\bar{\kappa }}}-1$.

First, Proposition 5 shows that a well-informed expert employs a certain number of messages ${{\bar{\kappa }}}$ in the most informative equilibrium. Second, the partition that the expert employs in equilibrium “converges”—in the sense of (4)—to a certain partition of the state space as she becomes perfectly informed. This partition corresponds to the partition that CS obtain in their main result (Theorem 1) for the most informative equilibrium, as Eq. (3) coincides with their condition (A). Thus, a well-informed expert behaves analogously to the perfectly-informed expert in CS: she employs the same number of messages and the relation between her posterior belief and the induced action of the DM is essentially the same. This exercise therefore provides a robustness check for the model of CS.

The following example illustrates this result. Let $\lceil x \rceil$ denote the smallest integer greater or equal to x.

Example 3

Suppose that $F={\mathcal {U}}(0,1)$ and $u_E(a,\theta ,b)=-(\theta +b-a)^2$. There exists a positive integer ${\overline{K}}$ such that the most informative equilibrium $(\sigma _K,\rho _K)$ is such that

$$\kappa _{K} = \bar{\kappa } = \left\lceil {\frac{1}{2}\sqrt {1 + \frac{2}{b}} - \frac{1}{2}} \right\rceil {\text{ if }}K \ge \bar{K}$$

The partition that the expert employs in $(\sigma _K,\rho _K)$ converges to the partition ${{\bar{\theta }}}_0=0<{{\bar{\theta }}}_1<\cdots <{{\bar{\theta }}}_{{{\bar{\kappa }}}}=1$ of $\Theta$, where

$$\begin{aligned} {{\bar{\theta }}}_k=\frac{1}{{{\bar{\kappa }}}}+2b(2k-{{\bar{\kappa }}}-1) \text { for all } k=1,2,\ldots ,{{\bar{\kappa }}}-1. \end{aligned}$$

Note that this partition corresponds to the partition that CS obtain in their uniform-quadratic model for the most informative equilibrium. Figure 4 illustrates the case $b=1/10$ and $K=50$, where $\kappa _{50}={{\bar{\kappa }}}=2$.

Fig. 4

Most informative equilibrium $(\sigma _{50},\rho _{50})$ for $b=1/10$ and $K=50$ in Example 3

5 Certification

Finally, we introduce the possibility of costly certification, which produces hard evidence. Let $M_{\mathcal {S}}^v(S)=\prod _{k=1}^K\{0,1,v(s_k)\}$, with $1<v(0)<v(1)$, denote the set of available messages given evidence $S\in {\mathcal {S}}$ under direct transmission, and let $\overline{M_{\mathcal {S}}^v}= \cup _{S\in {\mathcal {S}}} M_{\mathcal {S}}^v(S)$. The set of available messages under indirect transmission remains $M_\Sigma ={\mathcal {S}}_\Sigma$.¹⁰ Under direct transmission, the expert now may transmit the k-th soft signal $s_k$ either via cheap talk, $m_k\in \{0,1\}$, or subject it to certification, $m_k=v(s_k)$. Message $m_k=v(s_k)$ produces hard evidence of the k-th signal’s being $s_k$. Strategies extend to $\overline{M_{\mathcal {S}}^v}$ in the obvious way.

The preferences of the expert are given by $u_E^v(a,\theta ,b,m)= u_E(a,\theta ,b)-c\cdot \nu (m)$, where $u_E(a,\theta ,b)$ is as before and

$$\begin{aligned} \nu (m)=\left\{ \begin{array}{cl} |\{k|m_k\notin \{0,1\}\}|, &{} \text{ if } m\in \overline{M_{\mathcal {S}}^v}\\ 0, &{} \text{ if } m\in M_\Sigma \end{array}\right. \end{aligned}$$

denotes the number of signals that the expert certifies with message m, such that $c>0$ measures the costs per certified soft signal. The preferences of the DM remain unchanged, $u_{DM}(a,\theta )= u_E(a,\theta ,0)$.

To focus the analysis on the issue of certification, we restrict attention to direct-transmission equilibria. We show that a fully informative equilibrium exists even when the conflict of interest is potentially large, as long as costs of certification are low enough. In this case, the expert provides proof only for favorable reports. Let $\tau ^v_+(\sigma )=\{k | (\sigma (S))_k=v(s_k) \text { if } s_k=1 \text { and } (\sigma (S))_k\in \{0,1\} \text { if } s_k=0\}$ denote the collection of soft signals that are certified if high but not if low under strategy $\sigma$, let $\kappa ^v_+(\sigma )$ denote its cardinality, and note that $\kappa (\sigma )+\kappa ^v_+(\sigma )\le \kappa ^{mix}(\sigma )\le K$.

Proposition 6

The most informative direct-transmission equilibrium with certification $(\sigma ,\rho )$ is such that

• (i) $\kappa (\sigma )=K$ if $b\le {\underline{b}}(K)$,

• (ii) $\kappa ^v_+(\sigma )=K$ if $c\le {\overline{c}}(b,K)$, and

• (iii) $0\le \kappa ^{v}_+(\sigma )=\kappa ^{mix}(\sigma )<K$ otherwise,

where ${\overline{c}}(b,K)>0$ is given by

$$\begin{aligned} \min _{k=1,2,\ldots ,K} \min _{d=1,2,\ldots ,k}\frac{U_E(a_{DM}(S_\Sigma =k),b | S_\Sigma =k)-U_E(a_{DM}(S_\Sigma =k-d),b | S_\Sigma =k)}{d}. \end{aligned}$$

The first part of Proposition 6 follows directly from Proposition 1. The second part establishes that, if costs of certification are low, certifying each soft signal if high but not if low constitutes a fully informative equilibrium. First, certification of high signals deters the expert from persuading the DM into taking an action closer to her bliss point, since submitting favorable (high) cheap-talk reports is not credible if the DM expects certification of such evidence. Second, deviations to unfavorable (low) cheap-talk reports save the costs of certification but move the DM’s action further away from the expert’s bliss point, and are therefore deterred if the costs of certification are low. Note that the equilibria in part (i) and part (ii) may coexist.

The last part establishes that, if partial transmission of evidence remains possible when neither the bias is small nor the costs of certification are low, then each transmitted signal is certified if high but not if low. Proposition 1 implies that no evidence can be transmitted without certification in this case, and costly certification implies that the expert would never certify unfavorable (low) reports.

It follows from the proof of Proposition 6 that the upper bound on the costs of certification in part (ii), ${\overline{c}}(b,K)$, is strictly increasing in b. Due to strict concavity of preferences, deviations to unfavorable cheap-talk reports that move the DM’s action further away from the expert’s bliss point become more costly for the expert the larger the bias. Thus, there exists a fully informative equilibrium regardless of the conflict of interest if (and only if) the costs of certification do not exceed ${\overline{c}}(K)\equiv {\overline{c}}({\underline{b}}(K),K)>0$: the expert can transmit all evidence without certification if the conflict is small (Proposition 6 (i)), and she can transmit it with certification of high signals otherwise (Proposition 6 (ii).

Corollary 2

There exists a fully informative direct-transmission equilibrium with certification $(\sigma ,\rho )$ for all $b>0$ if and only if $c\le {\overline{c}}(K)$.

Example 4

Suppose that $F={\mathcal {U}}(0,1)$ and $u_E(a,\theta ,b)=-(\theta +b-a)^2$. Then there exists a fully informative direct-transmission equilibrium with certification $(\sigma ,\rho )$ for all $b>0$ if and only if $c\le {\overline{c}}(K)=(K+2)^{-2}$.

6 Concluding remarks

We have investigated communication between an expert who privately observes soft evidence and a DM. In situations of considerable conflict of interest, direct communication of soft evidence is impossible, as withholding one part of the evidence in the communication necessarily induces incentives to manipulate the report on the other part. The expert therefore has to resort to coarse summary statistics to transmit any information. However, if the expert can provide proof for her evidence at low costs, then she can reveal all evidence regardless of the conflict of interest. In particular, she provides proof only for favorable reports. These results show that revealing evidence is difficult and prone to manipulation unless proof is provided (for favorable reports). They further provide a qualification to the notion that direct communication is particularly useful in dynamic environments (Acemoglu et al. 2014; Foerster 2019), showing that conflicts of interest being small is important.

We finish with a brief discussion of our modelling choices. First, we assume binary soft signals that are independent conditionally on the state of the world, but our results are qualitatively robust to signals that can take more than two values and to correlations between the signals (conditionally on the state). In both cases, the DM would react less to small deviations, regardless of whether the expert transmits her evidence directly or indirectly.¹¹ Hence, deviations from fully informative communication would be more beneficial for the expert, implying a lower upper bound on the bias for a fully informative equilibrium to exist.

Second, we know from Aumann and Hart (2003) and Krishna and Morgan (2004) that multiple rounds of cheap talk may improve information transmission. One may hence ask whether sequential instead of simultaneous communication of the soft signals may improve communication in our setting. Suppose that the expert communicates one signal at each stage and that the next stage is reached with some probability, and consider truthful reporting. In the last stage, which is reached with positive probability, a deviation by the expert yields the same change in the DM’s reply than a small deviation on one signal in the simultaneous game, such that sequential communication does not improve information transmission. A similar reasoning holds for partially informative strategies.

Third, instead of one expert who observes K signals, we could consider K experts with identical preferences who each observe one signal. In a pure equilibrium, each expert then either reveals her information, that is, communicates truthfully, or not. Since the information of the experts who do not reveal their information does not affect the beliefs of the experts who do, partially informative communication remains possible if fully informative communication is precluded. In particular, there exists an equilibrium in which $K'\le K$ experts communicate truthfully if and only if $b\le {\underline{b}}(K')$, i.e., if and only if fully informative communication is possible in our model with $K'$ signals. Note that for the uniform-quadratic model of CS, this result also follows from Corollary 1 in Galeotti et al. (2013).

Appendix A: Proofs

Proof of Proposition 1. Let $(\sigma ,\rho )$ denote a strategy profile under which the expert transmits her evidence directly such that $\kappa (\sigma )=K$ and the DM best responds to $\sigma$, i.e., $\rho (m)=a_{DM}( \mu (m))$ for all m such that $Pr(m|\sigma )>0$. SC implies that

$$\begin{aligned} a_E(b, S)>a_{DM}( S)=\rho (\sigma (S)) \text { for all } S\in {\mathcal {S}}, \end{aligned}$$

(5)

i.e., it is sufficient to consider deviations on low signals. Therefore, consider $S_\Sigma <K$ and a deviation to message ${\tilde{m}}^d$ such that $\sum _{k=1}^K {\tilde{m}}_k^d=S_\Sigma +d$ for some $1\le d\le K-S_\Sigma$. The expert has no incentives to deviate if and only if

$$\begin{aligned} U_E(\rho (\sigma (S)),b | S)\ge U_E(\rho ({\tilde{m}}^d),b | S) \text { for all } 1\le d\le K-S_\Sigma . \end{aligned}$$

By (5) and SC, this requires $\rho ({\tilde{m}}^d)>a_E(b, S)>\rho (\sigma (S))$ for all $1\le d\le K-S_\Sigma$. Hence, a deviation on one signal, ${\tilde{m}}^1$, is most profitable, and strict concavity implies that there exists ${\underline{b}}(S_\Sigma ,K)>0$ such that

$$\begin{aligned}&U_E(\rho (\sigma (S)),{\underline{b}}(S_\Sigma ,K) | S)= U_E(\rho ({\tilde{m}}^1),{\underline{b}}(S_\Sigma ,K) | S)\\ \Leftrightarrow&U_E(a_{DM}( \sigma (S)),{\underline{b}}(S_\Sigma ,K) | S_\Sigma )= U_E(a_{DM}( {\tilde{m}}^1),{\underline{b}}(S_\Sigma ,K) | S_\Sigma ). \end{aligned}$$

Thus, by SC, a deviation is not profitable regardless of S if and only if $b\le {\underline{b}}(K)\equiv \min _{k=0,1,\ldots ,K-1}{\underline{b}}(k,K)$, which finishes the first part.

Next, suppose that $b>{\underline{b}}(K)$. First, let $(\sigma ',\rho ')$ denote a strategy profile under which the expert transmits her evidence directly such that $0<\kappa (\sigma ')=\kappa ^{mix}(\sigma ')<K$ and the DM best responds to $\sigma '$. (Note that this case does not exist if $K=1$.) Consider $S\in {\mathcal {S}}$ such that $K-\kappa (\sigma ')\le S_\Sigma <K$ and $\sum _{k\in \tau (\sigma ')}s_k=S_\Sigma -\left( K-\kappa (\sigma ')\right)$. Moreover, consider the deviation ${\tilde{m}}'$ such that $\sum _{k\in \tau (\sigma ')}{\tilde{m}}_k'=S_\Sigma -\left( K-\kappa (\sigma ')\right) +1$. We show by case distinction that $(\sigma ',\rho ')$ is not an equilibrium.

• Case 1. $\rho '({\tilde{m}}')> a_E(b,S)$. Note that

  $$\begin{aligned}&f(\theta |\sigma '(S))>f(\theta |\sigma (S))\\&\quad \Leftrightarrow \frac{\left( {\begin{array}{c}\kappa (\sigma ')\\ S_\Sigma -\left( K-\kappa (\sigma ')\right) \end{array}}\right) \theta ^{S_\Sigma -\left( K-\kappa (\sigma ')\right) }(1-\theta )^{K-S_\Sigma }f(\theta )}{\int _0^1\left( {\begin{array}{c}\kappa (\sigma ')\\ S_\Sigma -\left( K-\kappa (\sigma ')\right) \end{array}}\right) \theta ^{S_\Sigma -\left( K-\kappa (\sigma ')\right) }(1-\theta )^{K-S_\Sigma }f(\theta )d\theta }\\&\quad>\frac{\left( {\begin{array}{c}K\\ {S_\Sigma }\end{array}}\right) \theta ^{S_\Sigma }(1-\theta )^{K-{S_\Sigma }}f(\theta )}{\int _0^1\left( {\begin{array}{c}K\\ {S_\Sigma }\end{array}}\right) \theta ^{S_\Sigma }(1-\theta )^{K-{S_\Sigma }}f(\theta )d\theta }\\&\quad \Leftrightarrow \underbrace{\left( \frac{{\int _0^1\theta ^{S_\Sigma }(1-\theta )^{K-{S_\Sigma }}f(\theta )d\theta }}{\int _0^1\theta ^{S_\Sigma -\left( K-\kappa (\sigma ')\right) }(1-\theta )^{K-{S_\Sigma }}f(\theta )d\theta } \right) ^{1/\left( K-\kappa (\sigma ')\right) }}_{\in (0,1)}>\theta . \end{aligned}$$

  Hence, $F(\theta |\sigma '(S))>F(\theta |\sigma (S))$ for all $\theta \in (0,1)$, which yields

  $$\begin{aligned} \begin{aligned}&E[\theta |\sigma (S)]-E[\theta |\sigma '(S)]\\&\quad =\int _0^1 \theta (f(\theta |\sigma (S))-f(\theta |\sigma '(S))) d\theta \\&\quad =\left[ \theta (F(\theta |\sigma (S))-F(\theta |\sigma '(S)))\right] _0^1-\int _0^1 (F(\theta |\sigma (S))-F(\theta |\sigma '(S))) d\theta \\&\quad =\int _0^1 (F(\theta |\sigma '(S))-F(\theta |\sigma (S))) d\theta \\&\quad >0. \end{aligned} \end{aligned}$$

  (6)

  Thus, SC implies that

  $$\begin{aligned} \rho '(\sigma '(S))<\rho (\sigma (S))<a_E(b,S). \end{aligned}$$

  (7)

  Next, recall that ${\tilde{m}}^1$ is a deviation from $\sigma$ such that $\sum _{k=1}^K {\tilde{m}}_k^1=S_\Sigma +1$. Analogously to (6), we obtain $E[\theta |\mu ({\tilde{m}}^1)]-E[\theta |\mu '({\tilde{m}}')]>0$. Thus, SC implies that $\rho ({\tilde{m}}^1)>\rho '({\tilde{m}}')> a_E(b,S)$. Finally, (7) and $b>{\underline{b}}(K)$ imply that

  $$\begin{aligned} U_E(\rho '(\sigma '(S)),b | S)<U_E(\rho (\sigma (S)),b | S)<U_E(\rho ({\tilde{m}}^1),b | S)<U_E(\rho '({\tilde{m}}'),b | S). \end{aligned}$$

• Case 2. $\rho '({\tilde{m}}')\le a_E(b,S)$. As $E[\theta |\sigma '(S)]<E[\theta |\mu '({\tilde{m}}')]$, SC implies that $\rho '(\sigma '(S))<\rho '({\tilde{m}}')$ and hence $U_E(\rho '(\sigma '(S)),b | S)<U_E(\rho '({\tilde{m}}'),b | S)$.

In both cases, ${\tilde{m}}'$ yields a higher payoff than $\sigma '(S)$, which establishes that $(\sigma ',\rho ')$ is not an equilibrium.

Second, let $(\sigma '',\rho '')$ denote a strategy profile under which the expert transmits her evidence directly such that $0\le \kappa (\sigma '')<\kappa ^{mix}(\sigma '')\le K$ and the DM best responds to $\sigma ''$. Consider $S\in {\mathcal {S}}$ such that $S_\Sigma \ge K-\kappa (\sigma '')$ and $\sum _{k\in \tau (\sigma '')}s_k=S_\Sigma -\left( K-\kappa (\sigma '')\right)$. Then $s_k=1$ for all $k\in \tau ^{mix}(\sigma '')\backslash \tau (\sigma '')$ implies that, for any $m\in \hbox {supp}(\sigma ''(S))$,

$$\begin{aligned} E[\theta |\mu ''(m)]<E[\theta |(s_k)_{k\in \tau ^{mix}(\sigma '')}]\le E[\theta |S]=E[\theta |\sigma (S)], \end{aligned}$$

where the last inequality follows analogously to (6). Hence, SC implies that

$$\begin{aligned} \rho ''(\mu ''(m))<\rho (\sigma (S))<a_E(b,S). \end{aligned}$$

Thus, strict concavity of preferences yields

$$\begin{aligned} U_E(\rho ''(\mu ''(m)),b | S)\ne U_E(\rho ''(\mu ''(m')),b | S) \end{aligned}$$

for any $m,m'\in \hbox {supp}(\sigma ''(S))$ such that $\rho ''(\mu ''(m))\ne \rho ''(\mu ''(m'))$, i.e., $(\sigma '',\rho '')$ is not an equilibrium. $\square$

Proof of Proposition 2. Let $(\sigma ,\rho )$ be a strategy profile under which the expert transmits her evidence indirectly and suppose that $1\le \kappa \le K+1$ distinct actions are induced by the expert’s strategy $\sigma$, i.e., without loss of generality, $\rho (0)<\rho (1)<\cdots <\rho (\kappa -1)$. Hence, we can restrict attention to messages $m=0,1,\ldots ,\kappa -1$. Let $P_m\equiv \{k| m\in \hbox {supp}(\sigma (S)) \text { if } S_\Sigma =k\}$ for $m=0,1,\ldots ,\kappa -1$.

The expert has no incentives to deviate from her strategy $\sigma$ if and only if

$$\begin{aligned} \hbox {supp}(\sigma (S))\subseteq \hbox {argmax}_{m\in \{0,1,\ldots ,\kappa -1\}}U_E(\rho (m),b | S) \text { for all } S\in {\mathcal {S}}, \end{aligned}$$

which is equivalent to

$$\begin{aligned} m\in \hbox {argmax}_{m'\in \{0,1,\ldots ,\kappa -1\}}U_E(\rho (m'),b | S) \text { if } S_\Sigma \in P_m \text { for } m=0,1,\ldots ,\kappa -1. \end{aligned}$$

(8)

SC and $\rho (0)<\rho (1)<\cdots <\rho (\kappa -1)$ imply that the sets $P_m$ form a consecutive overlapping partition of ${\mathcal {S}}_\Sigma$, i.e., $\max P_{m-1}\le \min P_{m}$ for all $m=1,2,\ldots ,\kappa -1$. Furthermore, SC implies that (8) is equivalent to

$$\begin{aligned} U_E(\rho (m),b | S) \ge U_E(\rho (m+1),b | S) \end{aligned}$$

(9)

for $m=0,1,\ldots ,\kappa -2$ and $S_\Sigma \in P_m$, and

$$\begin{aligned} U_E(\rho (m),b | S) \ge U_E(\rho (m-1),b | S) \end{aligned}$$

(10)

for $m=1,2,\ldots ,\kappa -1$ and $S_\Sigma \in P_m$. Fix $m=1,2,\ldots ,\kappa -1$. By SC, (9) and (10) are equivalent to

$$\begin{aligned} U_E(\rho (m-1),b | S_\Sigma =\max P_{m-1})&\ge U_E(\rho (m),b | S_\Sigma =\max P_{m-1}) \text { and}\\ U_E(\rho (m),b | S_\Sigma =\min P_{m})&\ge U_E(\rho (m-1),b | S_\Sigma =\min P_{m}). \end{aligned}$$

In turn, the DM has no incentives to deviate if her strategy $\rho$ satisfies

$$\begin{aligned} \rho (m)=a_{DM}( \mu (m)) \text { for all } m=0,1,\ldots ,\kappa -1. \end{aligned}$$

(11)

Since the overlapping Partition ${\mathcal {P}}=\{P_0,P_1,\ldots ,P_{\kappa -1}\}$ is consecutive, SC implies that the best replies (11) satisfy the assumed order. $\square$

Proof of Proposition 3. Consider a strategy profile $(\sigma ,\rho )$ under which the expert transmits her evidence indirectly and that is characterized by the consecutive overlapping partition ${\mathcal {P}}=\{P_0,P_1,\ldots ,P_K\}$ of ${\mathcal {S}}_\Sigma$, with $P_k=\{k\}$ and $\sigma (S)(k)=1$ if $S_\Sigma =k$ for all $k=0,1,\ldots ,K$. The second condition in Proposition 2 yields

$$\begin{aligned} \rho (k)=a_{DM}( S_\Sigma \in P_k)=a_{DM}( S_\Sigma =k) \text { for } k=0,1,\ldots ,K. \end{aligned}$$

Fix any $k=1,2,\ldots ,K$. The last condition in Proposition 2 simplifies to

$$\begin{aligned} \begin{aligned} U_E(\rho (k-1),b | S_\Sigma =k-1)&\ge U_E(\rho (k),b | S_\Sigma =k-1) \text { and }\\ U_E(\rho (k),b | S_\Sigma =k)&\ge U_E(\rho (k-1),b | S_\Sigma =k). \end{aligned} \end{aligned}$$

(12)

Analogously to the proof of Proposition 1, SC implies that it is sufficient to consider a deviation on one low signal, which corresponds to a deviation from message $m=k-1$ to ${\tilde{m}}=k$. Therefore, (12) is equivalent to

$$\begin{aligned} U_E(\rho (k-1),b | S_\Sigma =k-1)\ge U_E(\rho (k),b | S_\Sigma =k-1). \end{aligned}$$

(13)

Next, consider a strategy profile $(\sigma ',\rho ')$ under which the expert transmits her evidence directly such that $\kappa (\sigma ')=K$ and the DM best responds to $\sigma '$. Moreover, consider $S\in {\mathcal {S}}$ such that $S_\Sigma =k-1$ and a deviation ${\tilde{m}}^1$ such that $\sum _{k=1}^K {\tilde{m}}_k^1=S_\Sigma +1$. Then (13) is equivalent to $U_E(\rho (\sigma '(S)),b | S)\ge U_E(\rho ({\tilde{m}}^1),b | S)$. Applying the same steps as in the proof of Proposition 1 establishes the first claim.

Second, consider a strategy profile $(\sigma ,\rho )$ under which the expert transmits her evidence indirectly and that is characterized by a consecutive overlapping partition ${\mathcal {P}}=\{P_0,P_1,\ldots ,P_{\kappa -1}\}$ of ${\mathcal {S}}_\Sigma$ such that $\kappa \ge 2$, $\sigma (S)\in \Delta (\{k|S_\Sigma \in P_k\})$ for all $S\in {\mathcal {S}}$ and $\rho (k)=a_{DM}(\mu (k))$ for all $k=0,1,\ldots ,\kappa -1$, i.e., Proposition 2 (i) and (ii) hold. Then $(\sigma ,\rho )$ is an equilibrium if and only if Proposition 2 (iii) holds, i.e., if and only if

$$\begin{aligned} \begin{aligned} \min \Big \{&U_E(\rho (k-1),b | S_\Sigma =\max P_{k-1})-U_E(\rho (k),b | S_\Sigma =\max P_{k-1}),\\&U_E(\rho (k),b | S_\Sigma =\min P_{k})-U_E(\rho (k-1),b | S_\Sigma =\min P_{k}) \Big \}\ge 0 \end{aligned} \end{aligned}$$

(14)

for all $k=1,2,\ldots ,\kappa -1$. Since, by SC, merging two adjacent partition elements eases the constraints (14), $(\sigma ,\rho )$ is an equilibrium for some $\kappa \ge 2$ if and only if

$$\begin{aligned}&\max _{\kappa \ge 2}\max _{{\mathcal {P}}=(P_0,P_1,\ldots ,P_{\kappa -1})}\min _{k=1,2,\ldots ,\kappa -1}\min \nonumber \\&\Big \{U_E(\rho (k-1),b | S_\Sigma =\max P_{k-1})-U_E(\rho (k),b | S_\Sigma =\max P_{k-1}),\nonumber \\&U_E(\rho (k),b | S_\Sigma =\min P_{k})-U_E(\rho (k-1),b | S_\Sigma =\min P_{k}) \Big \}\ge 0\nonumber \\ \Leftrightarrow&\max _{{\mathcal {P}}=(P_0,P_1)}\min \Big \{U_E(\rho (0),b | S_\Sigma =\max P_0)-U_E(\rho (1),b | S_\Sigma =\max P_0),\nonumber \\&U_E(\rho (1),b | S_\Sigma =\min P_{1})-U_E(\rho (0),b | S_\Sigma =\min P_{1}) \Big \}\ge 0\nonumber \\ \Leftrightarrow&\max _{{\mathcal {P}}=(P_0,P_1)}\left( U_E(\rho (0),b | S_\Sigma =\max P_0)-U_E(\rho (1),b | S_\Sigma =\max P_0)\right) \ge 0. \end{aligned}$$

(15)

By SC, there exists ${\overline{b}}(K)>{\underline{b}}(K)$ such that (15) holds if and only if $b\le {\overline{b}}(K)$. $\square$

Proof of Corollary 1. Recall from Proposition 1 that ${\underline{b}}(K)\equiv \min _{k=0,1,\ldots ,K-1}{\underline{b}}(k,K)$, where ${\underline{b}}(k,K)>0$ solves (1). Hence, SC implies that

$$\begin{aligned} a_{DM}(S_\Sigma =k+1)>a_E({\underline{b}}(k,K),S_\Sigma =k)>a_{DM}(S_\Sigma =k) \end{aligned}$$

(16)

for all $k=0,1,\ldots ,K-1$. Note that

$$\begin{aligned} \min _{k=0,1,\ldots ,K-1}E[\theta |S_\Sigma =k+1]-E[\theta |S_\Sigma =k]\rightarrow 0 \text { as } K\rightarrow \infty . \end{aligned}$$

Hence, SC implies that $a_{DM}(S_\Sigma =k^*(K)+1)-a_{DM}(S_\Sigma =k^*(K))\rightarrow 0 \text { as } K\rightarrow \infty$ for $k^*(K)\in \hbox {argmin}_{k=0,1,\ldots ,K-1}E[\theta |S_\Sigma =k+1]-E[\theta |S_\Sigma =k]$. Together with (16), we obtain ${\underline{b}}(K)\le {\underline{b}}(k^*(K),K)\rightarrow 0 \text { as } K\rightarrow \infty$. $\square$

Proof of Proposition 4. First note that if $K\le {\underline{K}}(b)$, then $(\sigma _K,\rho _K)$ is fully informative with

$$\begin{aligned} MSE (\sigma _K,K)=&\sum _{k=0}^K \left( {\begin{array}{c}K\\ k\end{array}}\right) \int _0^1\left( \theta -\frac{k+1}{K+2}\right) ^2\theta ^k(1-\theta )^{K-k}d\theta =&\frac{1}{6(K+2)}. \end{aligned}$$

(17)

Hence, $K<{\underline{K}}(b)$ implies that both $(\sigma _K,\rho _K)$ and $(\sigma _{K+1},\rho _{K+1})$ are fully informative with $MSE (\sigma _K,K)-MSE(\sigma _{K+1},K+1)>0$, which proves the first claim. Next, consider $K={\underline{K}}(b)$. Then $(\sigma _{K+1},\rho _{K+1})$ is only partially informative. We proceed by case distinction. We know from Example 2 that, if K is even, $\kappa \le \lfloor (K+1)/2+1 \rfloor =K/2+1$ in $(\sigma _{K+1},\rho _{K+1})$. In particular, $\kappa =(K+2)/2$ corresponds to the partition ${\mathcal {P}}=\{P_0,P_1,\ldots ,P_{K/2}\}=\{\{0,1\},\{2,3\},\ldots ,\{K,K+1\}\}$ of ${\mathcal {S}}_\Sigma$. Therefore, we obtain

$$\begin{aligned} MSE (\sigma _{K+1},K+1)&\ge\sum _{k=0}^{K/2}\sum _{l=2k}^{2k+1}\left( {\begin{array}{c}K+1\\ l\end{array}}\right) \int _0^1(\theta -\underbrace{E[\theta |S_\Sigma \in \{2k,2k+1\}]}_{=\frac{4k+3}{2(K+3)}})^2\\&\quad \cdot \theta ^l(1-\theta )^{K+1-l}d\theta \\ &=\frac{2K+9}{12(K+3)^2}, \end{aligned}$$

which is larger than (17) and thus proves the claim. Similarly, if K is odd, then $\kappa \le \lfloor (K+1)/2+1 \rfloor =(K+1)/2+1$ in $(\sigma _{K+1},\rho _{K+1})$. In particular, $\kappa =(K+1)/2+1$ corresponds to the partition ${\mathcal {P}}=\{P_0,P_1,\ldots ,P_{(K+1)/2}\}=\{\{0\},\{1,2\},\{3,4\},\ldots ,\{K,K+1\}\}$ of ${\mathcal {S}}_\Sigma$. Therefore, we obtain

$$\begin{aligned} MSE (\sigma _{K+1},K+1)&\ge\int _0^1(\theta -\underbrace{E[\theta |S_\Sigma =0]}_{=\frac{1}{K+3}})^2(1-\theta )^{K+1}d\theta +\sum _{k=1}^{(K+1)/2}\sum _{l=2k-1}^{2k}\left( {\begin{array}{c}K+1\\ l\end{array}}\right) \nonumber \\&\quad \cdot \int _0^1(\theta -\underbrace{E[\theta |S_\Sigma \in \{2k-1,2k\}]}_{=\frac{4k+1}{2(K+3)}})^2\theta ^l(1-\theta )^{K+1-l}d\theta \nonumber \\ &=\frac{(2K+3)(K+5)}{12(K+2)(K+3)^2}. \end{aligned}$$

(18)

Hence,

$$\begin{aligned} MSE (\sigma _{K+1},K+1)>MSE (\sigma _K,K)\Leftarrow \frac{(2K+3)(K+5)}{12(K+2)(K+3)^2}>\frac{1}{6(K+2)}\Leftrightarrow K>3. \end{aligned}$$

It is left to prove the claim for $K={\underline{K}}(b)=3$. Proposition 2 (iii) implies that $\kappa =\lfloor (K+1)/2+1 \rfloor =3$ in $(\sigma _4,\rho _4)$, which corresponds to the partition ${\mathcal {P}}=\{P_0,P_1,P_2\}=\{\{0\},\{1,2\},\{3,4\}\}$, only if

$$\begin{aligned}&\max _{S:S_\Sigma \in P_1} U_E(\rho (m_1),b | S)\ge \max _{S:S_\Sigma \in P_1} U_E(\rho (m_2),b | S)\nonumber \\&\quad \wedge \min _{S:S_\Sigma \in P_2} U_E(\rho (m_2),b | S) \ge \min _{S:S_\Sigma \in P_2} U_E(\rho (m_1),b | S)\nonumber \\&\quad \Leftrightarrow \max _{S:S_\Sigma \in P_1}E[\theta | S]+b-\rho (m_1)\le \rho (m_2)-\left( \max _{S:S_\Sigma \in P_1}E[\theta | S]+b\right) \nonumber \\&\quad \wedge \rho (m_2)-\left( \min _{S:S_\Sigma \in P_2}E[\theta | S]+b\right) \le \min _{S:S_\Sigma \in P_2}E[\theta | S]+b-\rho (m_1)\nonumber \\&\quad \Leftrightarrow \underbrace{\frac{\rho (m_1)+\rho (m_2)}{2}}_{=7/12}-\underbrace{\min _{S:S_\Sigma \in P_2}E[\theta | S]}_{=2/3} \le b\le \frac{\rho (m_1) + \rho (m_2)}{2}-\underbrace{\max _{S:S_\Sigma \in P_1}E[\theta | S]}_{=1/2}\nonumber \\ \Leftrightarrow&b\le 1/12. \end{aligned}$$

(19)

However, by definition of ${\underline{K}}(b)$, $1/12<b\le 1/10$ if ${\underline{K}}(b)=3$, which is a contradiction to (19). Therefore, $\kappa < 3$ in $(\sigma _4,\rho _4)$, such that the DM’s residual mean-squared error is strictly larger than (18). $\square$

Proof of Proposition 5. Consider the sequence of most informative indirect-transmission equilibria $( \sigma _K,\rho _K)_{K\ge 1}$. By construction and Proposition 2, $(\sigma _K,\rho _K)$ is such that ${\mathcal {P}}^K=\{P_0^K,P_1^K,\ldots ,P_{\kappa _K-1}^K\}$ minimizes the residual mean-squared error conditional on

$$\begin{aligned} \begin{aligned} U_E(\rho _K(k-1),b | S_\Sigma =\max P_{k-1}^K)&\ge U_E(\rho _K(k),b | S_\Sigma =\max P_{k-1}^K) \text { and }\\ U_E(\rho _K(k),b | S_\Sigma =\min P_{k}^K)&\ge U_E(\rho _K(k-1),b | S_\Sigma =\min P_{k}^K) \end{aligned} \end{aligned}$$

(20)

for all $k\in \{1,2,\ldots ,\kappa _K-1\}$. Take any $\kappa \ge 1$ such that $\kappa _K\ge \kappa$ for all K large enough and fix $k\in \{1,2,\ldots ,\kappa -1\}$. Then

$$\begin{aligned} {{\bar{\theta }}}_{k}\equiv \lim _{K\rightarrow \infty }E[\theta | S_\Sigma =\max P_{k-1}^K]=\lim _{K\rightarrow \infty }E[\theta | S_\Sigma =\min P_{k}^K] \end{aligned}$$

is well defined. Next, (20) yields

$$\begin{aligned}&\lim _{K\rightarrow \infty }U_E(\rho _K(k-1),b | S_\Sigma =\max P_{k-1}^K)\ge \lim _{K\rightarrow \infty } U_E(\rho _K(k),b | S_\Sigma =\max P_{k-1}^K)\ \wedge \nonumber \\&\lim _{K\rightarrow \infty } U_E(\rho _K(k),b | S_\Sigma =\min P_{k}^K) \ge \lim _{K\rightarrow \infty } U_E(\rho _K(k-1),b | S_\Sigma =\min P_{k}^K)\nonumber \\ \Leftrightarrow&U_E\big (\lim _{K\rightarrow \infty }\rho _K(k-1),b | \theta ={{\bar{\theta }}}_{k}\big )=U_E\big (\lim _{K\rightarrow \infty }\rho _K(k),b | \theta ={{\bar{\theta }}}_{k}\big ), \end{aligned}$$

(21)

where

$$\begin{aligned} \lim _{K\rightarrow \infty } \rho _K(k)=\lim _{K\rightarrow \infty } a_{DM}(\mu (k))=a_{DM}(\theta \in ({{\bar{\theta }}}_{k},{{\bar{\theta }}}_{k+1})). \end{aligned}$$

Note that it is left to define ${{\bar{\theta }}}_0$ and ${{\bar{\theta }}}_\kappa$. SC and Theorem 1 of CS imply that there exists a positive integer ${{\bar{\kappa }}}$ such that

$$\begin{aligned} {{\bar{\theta }}}_0=0<{{\bar{\theta }}}_1<\cdots<{{\bar{\theta }}}_{\kappa -1}<{{\bar{\theta }}}_\kappa =1, \end{aligned}$$

(22)

where ${{\bar{\theta }}}_{k}$ solves (21) for all $k\in \{1,2,\ldots ,\kappa -1\}$, if and only if $1\le \kappa \le {{\bar{\kappa }}}$. In particular, (22) constitutes a partition of $\Theta$.

By construction, there exists a positive integer ${\overline{K}}$ such that $(\sigma _K,\rho _K)$ is characterized by the partition ${\mathcal {P}}^K=\{P_0^K,P_1^K,\ldots ,P_{{{\bar{\kappa }}}-1}^K\}$ such that (20) holds for all $k\in \{1,2,\ldots ,{{\bar{\kappa }}}-1\}$ if $K\ge {\overline{K}}$. $\square$

Proof of Proposition 6.

• (i) Follows directly from Proposition 1.

• (ii) Let $(\sigma ,\rho )$ be a strategy profile under which the expert transmits her evidence directly such that $\kappa ^v_+(\sigma )=K$, i.e., $(\sigma (S))_k=v(s_k)$ if $s_k=1$ and $(\sigma (S))_k=0$ otherwise for all $k=1,2,\ldots ,K$. Furthermore, the DM best responds to $\sigma$, i.e., $\rho (m)=a_{DM}(\mu (m))$. We assume that the DM treats an off-equilibrium message ${{\tilde{m}}}\in \overline{M_{\mathcal {S}}^v}$ such that ${{\tilde{m}}}_k=1$ for some $k\in \tau ^v_+(\sigma )$ as she treats the message $m=(1-{{\tilde{m}}}_k,{{\tilde{m}}}_{-k})$, i.e., we can ignore such deviations. Since a deviation on a low signal to certification is costly and would not change the DM’s belief, we only need to consider deviations on high signals. Therefore, consider $S_\Sigma \ge 1$ and a deviation ${{\tilde{m}}}^d$ such that $\nu ({{\tilde{m}}}^d)=S_\Sigma -d<S_\Sigma =\nu (\sigma (S))$ for some $1\le d\le S_\Sigma$. The expert has no incentives to do this deviation if and only if

  $$\begin{aligned}&E[u_E^v(\rho (\sigma (S)),\theta ,b,\sigma (S)) | S]\ge E[u_E^v(\rho ({{\tilde{m}}}^d),\theta ,b,{{\tilde{m}}}^d) | S]\nonumber \\&\quad \Leftrightarrow U_E(\rho (\sigma (S)),b | S)-c\cdot \nu (\sigma (S))\ge U_E(\rho ({{\tilde{m}}}^d),b | S)-c\cdot \nu ({{\tilde{m}}}^d)\nonumber \\&\quad \Leftrightarrow U_E(\rho (\sigma (S)),b | S)-U_E(\rho ({{\tilde{m}}}^d),b | S)\ge c\cdot d. \end{aligned}$$

  (23)

  SC implies that $\rho ({{\tilde{m}}}^{d+1})<\rho ({{\tilde{m}}}^d)<\rho (\sigma (S))<a_E(b,S)$ for any $1\le d\le S_\Sigma -1$, i.e., the left-hand-side of (23) is strictly positive and strictly increasing in d. Hence, inequality (23) holds for all $S\in {\mathcal {S}}$ such that $S_\Sigma \ge 1$ and $1\le d\le S_\Sigma$ if and only if

  $$\begin{aligned} c\le \min _{S\in {\mathcal {S}}: S_\Sigma \ge 1} \min _{1\le d\le S_\Sigma }\frac{U_E(\rho (\sigma (S)),b | S)-U_E(\rho ({{\tilde{m}}}^d),b | S)}{d}\equiv {\overline{c}}(b,K), \end{aligned}$$

  which is strictly positive, and thus finishes the second part.

• (iii) Suppose that $b>{\underline{b}}(K)$ and $c>{\overline{c}}(b,K)$. Further suppose that $(\sigma ,\rho )$ is a most informative equilibrium such that $0<\kappa ^{mix}(\sigma )<K$. Proposition 1 implies that no signal is at least partially transmitted without (any) certification, i.e.,

  $$\begin{aligned} v(s_k)\in \hbox {supp}((\sigma (S))_k) \text { if } s_k=0 \text { or if } s_k=1 \text { for all } k\in \tau ^{mix}(\sigma ). \end{aligned}$$

  Suppose that there exists $k\in \tau ^{mix}(\sigma )$ such that $v(s_k)\in \hbox {supp}((\sigma (S))_k)$ if $s_k=0$. The proof of Proposition 1 implies that there exists $S\in {\mathcal {S}}$ such that $s_k=0$ and a deviation ${{\tilde{m}}}$ such that ${{\tilde{m}}}_l=(\sigma (S))_l$ for all $l\ne k$ and ${{\tilde{m}}}_k\in \{0,1\}$ that weakly increases the expert’s payoff from the DM’s action independent of the belief the DM attaches to this deviation.¹² This deviation is beneficial since it also strictly reduces the costs of certification, a contradiction. Therefore,

  $$\begin{aligned} v(s_k)\in \hbox {supp}((\sigma (S))_k) \text { if and only if } s_k=1 \text { for all } k\in \tau ^{mix}(\sigma ). \end{aligned}$$

  Suppose without loss of generality that there exists $k\in \tau ^{mix}(\sigma )$ such that $\hbox {supp}((\sigma (1,S_{-k}))_k)=\{0,v(1)\}$. It follows from the proof of Proposition 1 that $\rho (m)=\rho (m')$ for all $S\in {\mathcal {S}}$ such that $s_k=0$, $m\in \hbox {supp}(\sigma (S))$ and $m'=(0,m_{-k})$, as otherwise $\rho (m')<\rho (m)$, such that m would be a profitable deviation. This implies that $\hbox {supp}((\sigma (0,S_{-k}))_k)=\{0\}$. Consider the strategy profile $(\sigma ',\rho ')$ such that $(\sigma '(S))_l=(\sigma (S))_l$ for all $l\ne k$, $(\sigma '(1,S_{-k}))_k=v(1)$ and $(\sigma '(0,S_{-k}))_k=0$ for all $S\in {\mathcal {S}}$. Furthermore, consider $S\in {\mathcal {S}}$ such that $s_k=1$, $m\in \hbox {supp}(\sigma (S))$ such that $m_k=v(1)$ and a deviation ${{\tilde{m}}}=(0,m_{-k})$. Note that SC and $\hbox {supp}((\sigma (1,S_{-k}))_k)=\{0,v(1)\}$ imply $\rho '({{\tilde{m}}})<\rho ({{\tilde{m}}})<\rho '(m)=\rho (m)$ and $a_E(b,S)>\rho ({{\tilde{m}}})$. Hence,

  $$\begin{aligned} 0=U_E(\rho (m),b | S)-c-U_E(\rho ({{\tilde{m}}}),b | S)&=U_E(\rho '(m),b | S)-c-U_E(\rho ({{\tilde{m}}}),b | S)\\&<U_E(\rho '(m),b | S)-c-U_E(\rho '({{\tilde{m}}}),b | S), \end{aligned}$$

  i.e., deviation ${{\tilde{m}}}$ is strictly not beneficial under $(\sigma ',\rho ')$, which implies that we have constructed an equilibrium that is more informative than $(\sigma ,\rho )$, a contradiction. Therefore,

  $$\begin{aligned} (\sigma (S))_k=v(s_k) \text { if } s_k=1 \text { and } (\sigma (S))_k\in \{0,1\} \text { if } s_k=0 \text { for all } k\in \tau ^{mix}(\sigma ), \end{aligned}$$

  i.e., $\kappa ^{v}_+(\sigma )=\kappa ^{mix}(\sigma )$. $\square$