3.1. Players

There are three players in this model: a government G, a media outlet M, and a voter V. The government G and the media outlet M are either competent h or incompetent l. The type of the government (τ ^G = J) and the type of the media outlet (τ ^M = J), where J ∈ {l, h}, are each organization's private information. It is common knowledge, however, that Pr(τ ^G = h) = p and Pr(τ ^M = h) = q. The competent actors are more likely to learn the true state of the world than are the incompetent actors.

3.2. Policy choice and media report

Following the setup of Caillaud and Tirole (Reference Caillaud and Tirole2002), Nature draws the state of the world ω from a uniform distribution on the interval [0, 1], formally ω ∈ Ω ~ U(0, 1). The government chooses a policy x also from the uniform distribution. Among the infinite number of policy choices, only one unique policy is correct, meaning that only one policy matches the true state of the world. I apply this policy structure to clearly capture the event that the second mover herds its action to that of the first mover. It is noteworthy that, as in Caillaud and Tirole (Reference Caillaud and Tirole2002), the existence of an indefinite number of wrong choices makes the situation in which multiple actors receive private signals suggesting a same wrong choice to be a probability 0 event.

The government observes a signal $s^G_J$ indicating the correct policy, where J ∈ {l, h}. The signal may be noisy. When the government is competent, the probability of the signal matching the true state ${\rm Pr}(s^G_h=\omega )$ is $\overline {\gamma }$. With probability $1-\overline \gamma$, $s^G_h$ is randomly drawn from the infinitely many wrong options, and hence ${\rm Pr}(s^G_h=\omega )=0$. For an incompetent government, the probability ${\rm Pr}(s^G_l=\omega )$ is $\underline {\gamma }$, and $0 \le \underline {\gamma }\le \overline {\gamma } \le 1$.

Similar to the government's decision, the media outlet chooses its report r from the same uniform distribution. The media outlet observes x and also a private signal $s^M_J$, where J ∈ {l, h}, before reporting. The probability that the competent media outlet learns the correct policy is ${\rm Pr}(s^M_h=\omega )=\overline {\mu }$. For an incompetent media outlet, the probability ${\rm Pr}(s^M_l=\omega )$ is $\underline {\mu }$, and $0 \le \underline {\mu }\le \overline {\mu } \le 1$. The media outlet then decides what information r to report to the voter. When $x \ne s^M_J$, I call $r=s^M_J$ truthful reporting, and r = x yes-man reporting.

To simplify exposition, I denote the ex-ante expected quality of the government as $E(\gamma )=p\overline {\gamma }+ (1-p)\underline {\gamma },$ and the expected quality of the media outlet as $E(\mu )=q\overline {\mu }+ (1-q)\underline {\mu }.$ To focus on the interesting cases, I consider that the qualities of the competent media outlet, the government, and the incompetent media outlet are ordered as $\overline {\mu }\ge E(\gamma )\ge \underline {\mu }$. For numerical analyses in the comparative statics section, we only need $\overline {\mu }\ge \underline {\mu }$.

An important attribute of this model is that, policy-wise, the government and the media are unbiased. This is because I aim to show that conformist reporting is possible even when no political bias is involved. This can be understood as all members of the society share a common interest in issues such as national security or economic stability.

As yes-man behavior is most likely to be detected in this situation than in other environments such as a binary choice setting, the Caillaud and Tirole setup, combined with the most severe punishment imposed by the voter (discussed below), means that the model is analyzing the most unlikely case that the media outlet will deviate from truthful reporting. After the analysis of the general model, I discuss how the equilibria change only slightly when there is also uncertainty about the ideological position of the media outlet and when the voter is concerned about the government's honesty.

3.3. Learning, reputation, and reelection

The voter decides whether to retain the incumbent by observing the policy implemented x and learning information from the media. The voter does not observe the consequences of the policy directly. The justification is that the consequences of a policy are sometimes witnessed long after it is implemented (Besley, Reference Besley2007, 80). Instead, the voter reads the related report from the media to determine the quality of the government and decides whether to reelect the incumbent. At the same time, the media outlet builds (or ruins) its reputation via its report.

This paper also considers the case that, with probability ρ, information from other exogenous sources, such as foreign media reports or natural disasters, reveal a perfect signal of the true state s _V = ω to the voter. In other words, the voter could potentially, with probability ρ, receive perfect ex-post feedback about the state of the world. Formally, Pr(s _V = ω) = ρ and ${\rm Pr}(s_{\rm V}=\emptyset )=1-\rho$. With the policy decision x, the media report r, and the exogenous information s _V, the voter updates her beliefs about the type of the government and the media outlet, and makes her reelection decision.Footnote ³ The voter chooses the new government G′ either by retaining the incumbent or by a random draw from the candidate pool. In the latter case, I suppose Pr(τ ^G′ = h) = p. Also, I assume that if indifferent, the voter retains the incumbent.

It is assumed that the voter applies the most severe punishment to the media outlet once she finds it does not report truthfully. That is, once the media outlet is caught cheating, the voter believes it is incompetent, or formally, Pr(τ ^M = h|s _V = ω ≠ (r = x)) = 0. Relaxing this assumption only disincentivizes the media outlet to report truthfully.

3.4. The payoffs

The model incorporates the idea of reputational payoff used in Kartik and Van Weelden's (Reference Kartik and Van Weelden2019) analysis of cheap talk in elections. The payoff for the media outlet U _M is a strictly increasing mapping of its reputation for competence. Formally, I denote $U_M =\Pi ({\hat q})$, where ${\rm Pr}(\tau ^M=h \vert x, r, s_{\rm V})={\hat q}$, and $\Pi : {\hat q} \rightarrow {\opf R}_+$ can be any strictly increasing function. The media outlet enjoys utility from an increase in its reputation. For example, higher reputation could lead to an expansion of readership or an increase in advertising revenue.

Similarly, I define the payoff of the government as $U_G =\Delta ({\hat p})$, where ${\rm Pr}(\tau ^G=h \vert x, r, s_{\rm V})={\hat p}$. $\Delta : {\hat p} \rightarrow {\opf R}_+$ can be any strictly increasing function. The government's increasing utility from reputation comes from the rising probability of winning reelection, increasing political donations due to popularity, etc.

The voter wants to know the quality of the government to facilitate her political selection. The payoff for the voter is $U_{\rm V} ={\tf="PS8B4B"\char"31}_{\{\tau ^{G'}=h\}}$, which means the voter enjoys a utility normalized to one when she selects a competent government G′.

3.5. Timing

The timing of this game is as follows and can be summarized as in Figure 1:

1. 1. Nature chooses the types of actors τ ^G = J and τ ^M = J, J ∈ {l, h} and the state of the world ω

2. 2. G observes $s^G_J$ and implements a policy x

3. 3. M observes $s^M_J$ and x, and reports r

4. 4. With probability ρ, V learns the true state via other information sources

5. 5. V forms the posterior belief about the types of G (${\hat p}$) and M (${\hat q}$) and selects G′

6. 6. The payoffs are realized, and the game ends.

Figure 1. Timeline.

3.6. Equilibrium concept

The equilibrium concept applied is pure-strategy perfect Bayesian equilibrium, and weakly dominated strategies are excluded. Similar to Ottaviani and Sørensen (Reference Ottaviani and Sørensen2006), the main goal of the analysis is to check whether there could exist truthful equilibrium in which both types of media outlet report truthfully. Formally, truthful reporting equilibrium means the media outlet reports the information it learnt truthfully to the voter, that is, $r= s^M_J$. When a truthful reporting equilibrium does not exist, I characterize a yes-man equilibrium, which is a pooling equilibrium in which both types of the media outlet report an endorsement of the government's decision independent of the private information, formally r = x. Discussing the yes-man equilibrium is reasonable because media outlets are not likely to challenge the government without very good reason. A good relationship with officials is usually critical for future news reporting (Eraslan and Özertürk, Reference Eraslan and Özertürk2017). Hence, intuitively, we can think that if a truthful reporting equilibrium fails to exist, the media outlet will automatically switch to the yes-man reporting strategy. With a truthful reporting equilibrium, the voter updates her belief about the quality of the government based on the media's report. With a yes-man equilibrium, the voter does not update her belief. Off the yes-man equilibrium path, it is required that the voter should not believe a media outlet to be more likely to be competent (${\hat q} \gt q$) upon observing r ≠ x; otherwise, the outlet will always report r ≠ x even when ${s^M_J}=x$.

4.1. Ideal case: when truthful reporting exists

The voter wants the media outlet to be an effective watchdog. The ideal world for the voter is one in which only a competent media outlet exists and it always truthfully reports its information. Thus, when the media report endorses the government's policy, the voter can infer that the government is more likely to be of high competence, and when the media report conflicts with the government's choice, then the voter knows the government is more likely to be of low quality. The voter can therefore decide whether to reelect the incumbent following the media report.

Formally, this first-best situation means

$$ {\rm Pr}(\tau^G = h \vert r = x)\ge p\ge {\rm Pr}(\tau^G = h \vert r \ne x).$$

Suppose Pr(τ ^M = h) = 1 and the media outlet reports information truthfully. By Bayes' rule,

(1)$$ {\rm Pr}(\tau^G = h \vert r = x) = {\displaystyle{\,p\overline{\gamma \mu}} \over {\,p\overline{\gamma \mu} + (1 - p)\underline{\gamma \mu}}} = {\displaystyle{\,p\overline{\gamma}} \over {\,p\overline{\gamma} + (1 - p)\underline{\gamma}}}, $$

and Pr(τ ^G = h|r = x) is always greater than p when $\overline {\gamma } \gt \underline {\gamma }$.

Similarly, we have

(2)$$ {\rm Pr}(\tau^G = h \vert r \ne x) = {\displaystyle{\,p(1 - \overline{\gamma \mu})} \over {\,p(1-\overline{\gamma \mu}) + (1 - p)(1 - \underline{\gamma} {\overline \mu})}}, $$

and p is always greater than Pr(τ ^G = h|r ≠ x) when $\overline {\gamma } \gt \underline {\gamma }$.

As such, we know that a media report is a meaningful signal of the government's competence if the media outletFootnote ⁴ reports information truthfully (instead of reporting as a yes man), but in the following sections, I show that because of the media outlet's reputational concerns, the voter is not likely to learn the type of the government from the media report.

4.2. No other source of information

Following the practice of Gentzkow and Shapiro (Reference Gentzkow and Shapiro2006), the analysis begins with a case with a single media outlet. In this case, there are no other sources of information available for the voter, so she relies solely on one media outlet's report to learn the quality of the government. In a democratic context, this happens in a country with a media monopoly, such as Brazil (see Amaral and Guimarães, Reference Amaral and Guimarães1994), or with a very homogeneous media market evolved after a series of mergers (as discussed in Anderson and McLaren, Reference Anderson and McLaren2012).

I solve this game using backward induction and checking if there is a profitable deviation for the media outlet from the truthful reporting equilibrium. The voter updates her beliefs and makes her electoral decision based on the media report when the media outlet reports truthfully, instead of sending a babbling signal. It is obvious that once the media outlet receives a signal $s^M_J= x$, it will transmit the signal truthfully. Whereas, if the media outlet receives a signal $s^M_J\ne x$, the media outlet needs to decide to report either r = x or r ≠ x. If the voter believes the media outlet reports truthfully, by Bayes' rule, the voter's posterior belief about the media outlet's competence when r ≠ x is:

(3)$$ {\rm Pr}(\tau^M=h \vert r \ne x)={\displaystyle{q(1-\overline{\mu}E(\gamma))}\over{q(1-\overline{\mu}E(\gamma))+(1-q)(1-\underline{\mu}E(\gamma))}}. $$

If r = x, again supposing that the voter believes the media outlet reports truthfully, the posterior is:

(4)$$ {\rm Pr}(\tau^M=h \vert r=x)={\displaystyle{qE(\gamma)\overline{\mu}} \over {qE(\gamma)\overline{\mu}+(1-q)E(\gamma)\underline{\mu}}}={\displaystyle{q\overline{\mu}} \over {q\overline{\mu}+(1-q)\underline{\mu}}}. $$

By comparing the two posteriors, we can see that, except when the media outlet does not have to signal its type by its report,Footnote ⁵ endorsing the government (r = x) leads to strictly higher reputation for the media outlet. Thus, when receiving a signal that is different from the government's decision, the media outlet will deviate from truthful reporting. The mechanism driving this result is simple: Being a yes man yields higher posterior reputation for a media outlet if the voter believes the report is truthful. Thus, the truthful reporting equilibrium does not exist.

When it is in yes-man equilibrium, because the report is a babbling signal and the voter does not update beliefs with it, the posterior is Pr(τ ^M = h|r = x) = Pr(τ ^M = h|r ≠ x) = q. Thus, there is no profitable deviation for the media to report r ≠ x instead of r = x. As noted earlier, it is intuitive to understand that the media outlet will not issue a report challenging the government's decision for no obvious reason.

The government's action is very straightforward in this model. When the government is making the policy decision, the only information on hand is the signal $s^G_J$. Thus, irrespective of the media outlet's reporting strategy, the probability that the decision and the report match ($r= s^G_J$) is always higher than the probability that they don't match ($r \ne s^G_J$).Footnote ⁶ Assuming a truthful reporting equilibrium, when observing $r= s^G_J$, the voter believes the government is more likely to be competent. Therefore, it is a dominant strategy for the government to set the policy $x= s^G_J$, and this is true in all cases discussed in this paper.

The yes-man equilibrium is the unique equilibrium in this case. It is robust to Cho and Kreps's Intuitive Criterion because the competent media outlet cannot do better by reporting r ≠ x. On the other hand, any separating equilibrium—for example, only the competent media outlet reports truthfully—cannot survive the Intuitive Criterion. In this situation, the incompetent media outlet has an incentive to always report r ≠ x to be viewed as competent. This paper focuses on studying pure-strategy equilibrium. Indeed, there are many mixed-strategy equilibria in which the media outlet randomizes its reporting independent of its private signal, but they are weakly dominated by the yes-man equilibrium, because the media outlet will not gain higher reputation by randomizing its reporting than by reporting as a yes-man.

The following example can demonstrate the mechanism clearly: Suppose the government and the media outlet received different signals, and $\{\overline {\mu }=0.6, \underline {\mu }=0.4, E(\gamma )=0.6, q=0.6\}$. Then, if the voter believes the news report is truthful, the posterior about the media outlet's competence is 0.558. If, instead of reporting r ≠ x, the media outlet reports r = x, the posterior increases to 0.692. Thus, it is a profitable deviation for the media outlet. In this case, the media outlet never has an incentive to report truthfully. The media outlet is a yes man instead of a watchdog, and the voter cannot infer the quality of the government from the media report. Following the assumed tie-break rule, the voter retains the incumbent because she does not find the incumbent and challenger discernibly different from one another (${\hat p}=p$).

4.3. When the voter may learn from other sources

Now I analyze the general case of the model where, with some probability ρ ∈ (0, 1), the voter learns the true state from other information sources.

When ρ = 1: To find the conditions for the media outlet to report truthfully when ρ ∈ (0, 1), we need to find the posteriors attainable with different actions when ρ = 0 and ρ = 1. We already have the posteriors for ρ = 0, so now we need to find the posteriors for ρ = 1. The detailed calculations are given in the online Appendix. The analysis shows that when the voter can perfectly learn the true state ex post, the competent media outlet will always report truthfully because the probability for the competent media outlet to be correct is higher than that for the government ($\overline {\mu } \gt E(\gamma )$). However, the incompetent media outlet could still have an incentive to report untruthfully even when yes-man behavior is very likely to be detected.Footnote ⁷ To have a truthful reporting equilibrium, we need both types of media outlets to report truthfully. Thus, a truthful reporting equilibrium is not guaranteed despite the fact that the voter can perfectly learn the true state ex post. If the truthful reporting equilibrium is not guaranteed even in this extreme case,Footnote ⁸ we can expect that the truthful reporting equilibrium is less likely to exist in the general case.

When ρ ∈ (0, 1): With the posteriors in different situations when ρ = 0 and ρ = 1, we can analyze the general case and find the conditions for the existence of the truthful reporting equilibrium. The detailed calculations are given in the online Appendix.

The condition for the competent media to report truthfully is:

(5)$$ \rho \ge {{\displaystyle{{\displaystyle{\overline{\mu}} \over {E(\mu)}}-{\displaystyle{1-\overline{\mu} E(\gamma)} \over {T}}}} \over {(1+\overline{\mu}-E(\gamma)){\displaystyle{\overline{\mu}} \over {E(\mu)}}+(1-\overline{\mu}) {\displaystyle{1-\overline{\mu}} \over {1-E(\mu)}}-{\displaystyle{1-\overline{\mu}E(\gamma)} \over {T}}}}\equiv \rho_h, $$

where $T=q(1-\overline {\mu }E(\gamma ))+(1-q)(1-\underline {\mu }E(\gamma ))$.Footnote ⁹

The condition parallel to (5) for the incompetent media is:

(6)$$ \rho \ge {\displaystyle{\displaystyle{\overline{\mu} \over {E(\mu)}}-{\displaystyle{1-\overline{\mu}E(\gamma)} \over {T}}} \over {(1+\underline{\mu}-E(\gamma)){\displaystyle{\overline{\mu}} \over {E(\mu)}}+(1-\underline{\mu}){\displaystyle{1-\overline{\mu}} \over {1-E(\mu)}}-{\displaystyle{1-\overline{\mu}E(\gamma)} \over {T}}}}\equiv \rho_l. $$

Given $\overline {\mu }\ge \underline {\mu }$, we have ρ _l ≥ ρ _h. As expected, an incompetent media outlet is less likely to report truthfully than a competent one. Hence, ρ _l is the binding condition for the truthful reporting equilibrium. A numerical example is when $\{E(\gamma )=0.5, \overline {\mu }=0.6, \underline {\mu }=0.2, q=0.5 \}$, we have ρ _h = 0.6 and ρ _l ≈ 0.882. Hence, the truthful reporting equilibrium exists if and only if ρ ≥ ρ _l ≈ 0.882. This means that unless the probability of receiving information from other sources is very high, a truthful reporting equilibrium does not exist.

By fixing $\{\overline {\mu }=0.6, \underline {\mu }=0.2, q=0.5\}$, the conditions for the two types of media outlets to report truthfully ρ _h and ρ _l can be graphically shown as in Figure 2.Footnote ¹⁰ The area between the horizontal line ρ = 1 and ρ _l is where the truthful reporting equilibrium exists. If the government is expected to be reasonably competent, that is E(γ) ≥ 0.5, the truthful reporting equilibrium only exists in a very restricted area, and it requires a very high probability for the voter to learn from other sources to sustain this equilibrium. Figure 2 also demonstrates that even when the quality of the government is expected to be very low, the truthful reporting equilibrium can exist only with a sufficient probability for the voter to learn from other information sources.

Figure 2. Conditions for truthful reporting $(\{\overline {\mu }=0.6, \underline {\mu }=0.2, q=0.5 \})$.

4.4. Comparative statics

Based on Proposition 2, I perform three numerical comparative statics exercises.Footnote ¹¹ First of all, I set the quality of the competent media outlet $\overline {\mu }$ as a variable and fix the quality of the incompetent outlet $\underline {\mu }$ and the probability to have the competent outlet q. Figure 3 shows that, counter intuitively, when the quality of the competent media outlet increases, the truthful reporting equilibrium becomes less likely to occur.

Figure 3. Comparative statics 1 $({\rm fix }\ \underline {\mu }= 0.2, q= 0.5)$.

Two forces drive this surprising result. First, as the quality gap between the two types of media outlets widens, if the true state is not revealed ex post but the voter believes the media report is truthful, as before, the media outlet has a higher posterior when its report matches the policy decision and a lower posterior when its report does not match the policy decision. However, now the difference between these two posteriors becomes larger. Thus, matching the report to the policy leads to higher return from reputation. Second, if ex post the report is verified to be correct, the media outlet is more likely to be competent, and because the difference in posteriors of being correct and being incorrect increases when the quality gap widens, being correct becomes a strong signal of competence and leads to higher reputational returns. The incompetent outlet becomes more willing to take the risk of being caught cheating by reporting as a yes man.

For the second set of comparative statics, I fix the quality of the competent media outlet $\overline {\mu }$ and the probability to have the competent media outlet q, and vary the quality of the incompetent outlet $\underline {\mu }$. From Figure 4, we know that when $\underline {\mu }$ increases, the required ρ _l to support the truthful reporting equilibrium decreases. The result again indicates that the likelihood of the truthful reporting equilibrium decreases as the difference between the types of media outlets increases. When the two types are of similar quality, such as the case where $\overline {\mu }=0.6$ and $\underline {\mu }=0.5$, a truthful reporting equilibrium is more likely to exist. Even in this situation, though, the quality of the government cannot be too high, and the probability of learning from other sources of information cannot be too low. The driving force for this result is that deviating from truthful reporting is more rewarding for a media outlet with a lower competence $\underline {\mu }$. This is because when the incompetent media outlet is less likely to receive a correct signal, it is more willing to pander to the government policyto increase the probability the media report is correct.

Figure 4. Comparative statics 2 $({ \rm fix }\ \overline {\mu }= 0.6, q= 0.5)$.

For the final set of comparative statics, I fix the qualities of the two types of media outlets $\overline {\mu }$ and $\underline {\mu }$, and allow the probability of having the competent media outlet q to vary. Figure 5 shows that when the media outlet is more likely to be competent, the likelihood of truthful reporting increases. Therefore, although the previous results show that homogeneity of the types of media outlet leads to truthful reporting, ceteris paribus, it is more likely for the voter to receive a truthful report when the probability of having a competent media outlet is higher.

Figure 5. Comparative statics 3 $({\rm fix} \ \overline {\mu }= 0.6, \underline {\mu }= 0.2)$.

It is quite intuitive to think that critical reporting is less likely to occur when the quality of the incompetent outlet declines, but it is surprising that when the quality of the competent outlet is further improved, the probability of critical reporting also decreases. As government subsidy is usually considered as a promising means of improving the quality of news reporting (see Sunstein, Reference Sunstein2017), the first set of comparative statics serves as a warning. Those rewards and supports for competent news outlets, aiming at further improving their quality, may turn out to make critical reporting even more unlikely. Improving the quality of incompetent news outlets, as the second set of comparative statics shows, could be a more effective way to encourage critical reporting.

7.1. Two media outlets

First, consider the case where the voter can learn the information from two media outlets M ₁ and M ₂. Suppose that the voter believes the two media outlets play the truthful reporting equilibrium, and the true state is not revealed by other sources (ρ = 0). The posteriors given different actions and the probabilities for these different situations to occur are summarized in Table 1. Throughout this section, to simplify the notation regarding the probability of occurrence of different situations, I use μ (without overline or underline) to denote that it is either of two cases: $\underline {\mu }$ when the media outlet M ₁ is incompetent and $\overline {\mu }$ when M ₁ is competent.

Table 1. Posteriors given different actions: two media outlets (ρ = 0)

Note: Denote $\displaystyle{{q(1-E(\gamma)E(\mu)-E(\gamma)\overline{\mu}-E(\mu)\overline{\mu}+2E(\gamma)E(\mu)\overline{\mu})} \over {q(1-E(\gamma)E(\mu)-E(\gamma)\overline{\mu}-E(\mu)\overline{\mu}+2E(\gamma)E(\mu)\overline{\mu})+(1-q)(1-E(\gamma)E(\mu)-E(\gamma)\underline{\mu}-E(\mu)\underline{\mu}+2E(\gamma)E(\mu)\underline{\mu})}}$ as Ψ₂.

When deviating from truthful reporting, the posterior will be ${q\overline {\mu }}/({q\overline {\mu }+(1-q)\underline {\mu }})$, which is greater or equal to both Ψ₂ and ${q(1-\overline {\mu })}/({q(1-\overline {\mu })+(1-q)(1-\underline {\mu })})$ (unless q = 0, q = 1 or $\overline {\mu }=\underline {\mu }$). Thus, the media outlet has an incentive to deviate from the truthful reporting equilibrium, and there is no truthful reporting equilibrium.

Now, consider the situation where the voter can learn the true state of the world from other sources of information with certainty (ρ = 1). Table 2 summarizes the posteriors given different actions and the probabilities for these different situations to occur when the voter believes the media outlets play the truthful reporting equilibrium.

Table 2. Posteriors given different actions: two media outlets (ρ = 1)

If a media outlet deviates from reporting truthfully when the voter will observe the true state with certainty, its posterior reputation is

$$ E(\gamma)(1-\mu) {\displaystyle{q\overline{\mu}} \over {q\overline{\mu}+(1-q)\underline{\mu}}}.$$

As in the baseline case, the competent media outlet has no incentive to deviate from truthful reporting, but the incompetent type will only report truthfully if the following condition is satisfied:

$$ (1-\underline{\mu}){\displaystyle{q(1-\overline{\mu})} \over {q(1-\overline{\mu})+(1-q)(1-\underline{\mu})}} \ge [E(\gamma)[1-\underline{\mu}+E(\mu)\underline{\mu}]-\underline{\mu}]{\displaystyle{q\overline{\mu}} \over {q\overline{\mu}+(1-q)\underline{\mu}}}.$$

This condition does not always hold. For example, when $\{q= 0.2, \overline {\mu }= 0.8, \underline {\mu }= 0.2\}$, the condition is only satisfied when E(γ) ≤ 0.34. Hence we know that ρ = 1, again, does not guarantee the existence of a truthful reporting equilibrium.

The general case we are interested in is a convex combination of the situations in terms of ρ, summarized in the two tables. That is, the voter may sometimes learn the true state from the perfect feedback, but not always. The analysis shows that, as in the baseline case, an incompetent media outlet does not always follow the truthful reporting strategy. Essentially, when there are two media outlets, they play the same signaling game simultaneously, so that one report cannot discredit the other since the voter cannot learn the state of the world from other sources. The decision made by the government then serves as a focal point for the media outlets to tend toward. The truthful reporting equilibrium exists only when the voter can learn the true state of the world from other sources with a sufficiently high probability.Footnote ¹³

7.2. Three or more media outlets

To analyze cases with three or more media outlets, I use a general approach and categorize n media outlets into three groups: M ₁, M _i, and M _j. M _i and M _j partition the n − 1 media outlets other than M ₁ into two groups, each of which consists of at least one media outlet. Each media outlet within a group does not necessarily receive the same signal.

Again, the analysis should focus on the situation where M ₁ receives a signal different from the government's decision x and compare posteriors from r ≠ x and r = x in different scenarios, presuming every other actor follows the truthful reporting strategy. The different scenarios, their probabilities of occurrence, and the corresponding posterior beliefs, are summarized in Tables 3 and 4. Table 3 is for the situation ρ = 0, and Table 4 is for ρ = 1.

Table 3. Posteriors given different actions: more than two media outlets (ρ = 0)

Note: Denote Σ₁ ≡ [1 − E(γ)][(n − 1)E(μ)(1 − E(μ))ⁿ⁻²]μ; Σ₂ ≡ [1 − E(γ)][1 − (1 − E(μ))ⁿ⁻¹ − (n − 1)E(μ)(1 − E(μ))ⁿ⁻²]μ; Σ₃ ≡ [1 − E(γ)][1 − (1 − E(μ))ⁿ⁻¹ − (n − 1)E(μ)(1 − E(μ))ⁿ⁻²](1 − μ); Φ_n ≡ [1 − E(γ)][1 − E(μ)]ⁿ⁻¹(1 − μ) + E(γ)[1 − E(μ)]ⁿ⁻¹(1 − μ) + (n − 1)E(μ)[1 − E(γ)](1 − E(μ))ⁿ⁻²(1 − μ) + [1 − E(γ)][1 − E(μ)]ⁿ⁻¹μ; $\displaystyle{\Psi _n\equiv {{q\Phi _n^h} \over {q\Phi _n^h+(1-q)\Phi _n^l}}}$.

Table 4. Posteriors given different actions: more than two media outlets (ρ = 1)

The difference between this case and those previously analyzed is that if some M _i and M _j receive the correct signal and report truthfully, yes-man behavior will be detected. Hence, supposing all other media outlets follow the truthful reporting strategy, even when ρ = 0, yes-man reporting is harmful (to the media outlet) if: (1) M ₁ actually received a correct signal but deviates from truthful reporting (because it is possible to match reports with one or more other news outlets on a non-x option, and such a report is perfect evidence of being correct) or (2) there are at least two media outlets that receive the correct signal (because the true state is revealed, and the yes-man behavior is detected).

When the number of media outlets becomes large, it is certain that at least some media outlets will receive the correct signal, but truthful reporting equilibrium is still not guaranteed. The analysis shows that when the media market is in perfect competition (n → ∞), the role of exogenous information sources (ρ) can be replaced by the existence of a large number of media outlets because it is certain that at least two media outlets will receive the correct signal. Thus, supposing that the media outlets report truthfully, the voter is going to learn the true state for sure ex post. Indeed, when n → ∞, the case with ρ = 0 is equivalent to the case with ρ = 1. Even so, an incompetent media outlet still has an incentive to be a yes man whenever $[E(\gamma )-\underline {\mu }]\overline {\mu }-(1-\overline {\mu })E(\mu ) \ge 0$ (e.g., when $\{E(\gamma )=0.5, \overline {\mu }=0.8, \underline {\mu }=0.2, q=0.5\}$). Thus, a truthful reporting equilibrium is still not guaranteed and the baseline result is robust in this extreme case.