1. Model setup

We present a stylized formal model of media skepticism induced by the spread of fake news. In our model, we view news reports as the source of information about the incumbent's characteristics on which a citizen – faced with the decision to retain or vote out the incumbent – relies. To illustrate our point that there is an incentive to spread fake news even if citizens are not directly deceived, we hold the assumption that individuals do not make biased decisions based on disinformation coming from fake news.

Our model is in line with signal-jamming models (e.g., Holmström, Reference Holmström1999; Alesina and Tabellini, Reference Alesina and Tabellini2007; Ashworth et al., Reference Ashworth, de Mesquita and Friedenberg2017) in the sense that spreading (or not spreading) fake news itself does not transmit any messages about the incumbent to the decision-maker.Footnote ² Also, our model can be seen as an application of the idea that new information helps incumbents who are behind and hurts incumbents who are ahead of the competition (Little, Reference Little2016; Ashworth et al., Reference Ashworth, de Mesquita and Friedenberg2018) to the context of spreading fake news.

1.2 Fake news and media environment

Individuals rely on news reports to estimate the incumbent's quality c more precisely. However, news producers have incentives to purposely spread fake news to mislead individuals in their favor. We assume that D can only read a limited amount of news articles, spreading more pieces of fake news reduces the likelihood that D encounters a piece of true news that is informative about the incumbent's characteristic c. Knowing this, a rational individual like D would Bayesian update her perceived quality of the incumbent upon reading news information depending on her perception about the proportion of information she reads being fake news.

Let 0 < f < 1 denote the proportion of fake news among all news articles she reads in a given media environment, which is externally given and known to D. As it is well documented in the literature of communication research, D could learn about f from sources such as her own reading of the news and personal experience, elite discussion about fake news, and fact-checking (e.g., Bachmann and Valenzuela, Reference Bachmann and Valenzuela2023; Chang, Reference Chang2021; Van Duyn and Collier, Reference Van Duyn and Collier2019). Bachmann and Valenzuela (Reference Bachmann and Valenzuela2023), in particular, find that people exposed to political fact-checks trust news less due to being aware of the prevalence of misinformation. Therefore, this probability f can be seen as an indicator of the quality of the media environment in the sense that the quality of the media environment decreases as f increases.

Formally, let x ∈ {x _f, x _t} denote a piece of news about c, where x _f and x _t indicate a piece of fake and true news, respectively. We model x _f as a completely uninformative signal about c,Footnote ³ while x _t follows

$$x_{t} = c + \epsilon,\; \ \epsilon \sim {\cal N}( 0,\; \sigma^2) ,\;$$

where $\epsilon \in {\opf R}$ denotes errors which follows the normal distribution ${\cal N}( 0,\; \sigma ^2)$. By true news, we mean any piece of news that allows D to infer an unbiased estimate of c on average (Alvarez and Franklin, Reference Alvarez and Franklin1994; Holmström, Reference Holmström1999; Edmond and Lu, Reference Edmond and Lu2021), which is captured by the zero mean of $\epsilon$. This means that D is assumed to be able to de-bias a biased piece of news as long as it contains any information about c.Footnote ⁴ The problem, however, is that individuals are not able to perfectly detect fake news. When D receives a piece of news, D knows that it is true with a probability of f, and fake with 1 − f.

2. Media skepticism

Although D is not directly deceived by fake news, spreading more pieces of fake news leads D to rely less on the media. We call this media skepticism. That is, D ignores the piece of news from the media, which prevents D from learning new information about c. Consequently, media skepticism leads D to make a decision similar to one that D would have made without new information gleaned from news reports.

Based on the above intuition, we formalize the idea of media skepticism as follows. Let c|x denote D's posterior belief about c after observing x. D does not update her belief if the piece of news is fake (i.e., x = x _f), so her posterior belief about c is the same as her prior belief (i.e., $c\vert x \sim {\cal N}( \mu ,\; \tau ^2)$). Then, the posterior expectation about c is ${\opf E}( c\vert x) = \mu$. If, on the other hand, the piece of news is true (i.e., x = x _t), then she updates her belief to $c\vert x \sim {\cal N}( \frac{\tau ^2x_{t} + \sigma ^2\mu }{\tau ^2 + \sigma ^2},\; \frac{\tau ^2\sigma ^2}{\tau ^2 + \sigma ^2})$. In this case, the posterior expectation about c is ${\opf E}( c\vert x) = \frac{\tau ^2x_{t} + \sigma ^2\mu }{\tau ^2 + \sigma ^2}$. Since x is fake news (i.e., x = x _f) with f, while it is true news (i.e., x = x _t) with 1 − f, the posterior expectation about the incumbent's characteristics is

$${\opf E}( c\vert x) = f\mu + ( 1-f) {\tau^2x_{t} + \sigma^2\mu\over \tau^2 + \sigma^2}.$$

Notice that the posterior expectation of the incumbent's characteristic (i.e., ${\opf E}( c\vert x)$) is still a random variable, which follows the following normal distribution

$${\opf E}( c\vert x) \sim {\cal N}\left(\mu,\; {( 1-f) ^2\tau^4\sigma^2\over ( \tau^2 + \sigma^2) ^2}\right).$$

The posterior expectation of D about the incumbent's characteristic is influenced by the quality of the media environment f, which is captured by the fact that ${\opf E}( c\vert x)$ is a random variable with the variance $ \frac{(1-f)^2\tau^4\sigma^2}{(\tau^2+\sigma^2)^2}$. By inversing this variance term, we can measure the degree of D's media skepticism as a function of f, denoted by S(f), as follows:

$$S( f) = {( \tau^2 + \sigma^2) ^2\over ( 1-f) ^2\tau^4\sigma^2}.$$

To see the rationale behind this, first, note that S(f) is increasing in f. So, the greater f is, the stronger D's media skepticism S(f) is. Next, observe that D's posterior expectation of the incumbent's characteristic would not deviate much from D's initial expectation (i.e., μ) when S(f) is large. That is, the piece of news from the media environment does not crucially change D's expectation about the incumbent's characteristic when D's media skepticism is strong. By contrast, D's posterior expectation of the incumbent's characteristic becomes more likely to deviate from D's initial expectation as S(f) becomes smaller (i.e., as D's media skepticism becomes weaker). That is, the piece of news from the media environment meaningfully affects D to form a posterior expectation quite different from D's initial expectation.

Figure 1 graphically demonstrates the above intuition. As D's media skepticism intensifies (i.e., as S(f) increases), the peak of the distribution of ${\opf E}( c\vert x)$ becomes taller, and the tails become lighter as depicted in Fig. 1. In other words, weak media skepticism of D leads to a mean-preserving spread of D's posterior expectation about the incumbent's characteristic, while strong media skepticism of D leads to a mean-preserving contraction of it. This implies that D less seriously considers the piece of news (i.e., x) as D becomes more skeptical of the media environment, which results in D's posterior expectation of the incumbent's characteristic being more densely distributed around the initial expectation (i.e., μ).

Figure 1. The distributions of D's posterior expectation of the incumbent's characteristics (i.e., ${\opf E}( c\vert x) \sim {\cal N}( \mu ,\; {1}/{S( f) }$) where 0 < f _s < f _m < f _l < 1 and S(f _s) < S(f _m) < S(f _l).

In conclusion, we present Proposition 1 as a summary of the effect of spreading fake news on D's decision-making mediated by its effect on D's media skepticism.

3. The incentive to spread fake news

In our model, spreading fake news cannot directly deceive the decision-maker in expectation, which is captured by $ {\opf E}( c\vert x) \sim {\cal N}\left( \mu ,\; \frac{( 1-f) ^2\tau ^4\sigma ^2}{( \tau ^2 + \sigma ^2) ^2}\right)$. Nevertheless, is there still an incentive to spread fake news? Our answer to this question is yes because spreading fake news increases the decision-maker's media skepticism, which, in turn, can impact her/his decisions. Specifically, media skepticism leads the decision-maker's updated expectation of the incumbent's characteristics based on news reports to be similar to the decision-maker's initial expectation. Then, it is rational for those who benefit from maintaining the decision-maker's initial expectation of the incumbent's characteristics to spread fake news.

To formally demonstrate the above rationale, we introduce an application of the model analyzed above in which D (the representative voter in this context) determines the winner of a quality competition between an incumbent and a challenger. As mentioned above, D forms her belief after observing the piece of news (i.e., x) and decides whether to retain or replace the incumbent. D retains the incumbent if and only if D's expectation of the incumbent's quality ${\opf E}( c\vert x)$ is greater than 0. Using the fact that ${\opf E}( c\vert x) \sim {\cal N}\left( \mu ,\;\frac{( 1-f) ^2\tau ^4\sigma ^2}{( \tau ^2 + \sigma ^2) ^2}\right)$, we can write the probability that the incumbent wins the competition as the probability that ${\opf E}( c\vert x)$ is greater than 0,

$$Pr\left[{\opf E}( c\vert x) > 0\right] = 1-\Phi\left({-\mu( \tau^2 + \sigma^2) \over ( 1-f) \tau^2\sigma}\right),\;$$

where Φ denotes the CDF of the standard normal distribution.

Knowing this, pro-incumbent news producers have the incentive to maximize the probability that the incumbent wins the competition (i.e., $Pr[ {\opf E}( c\vert x) > 0]$), while anti-incumbent producers want to minimize the probability. Notice that the above probability is influenced by the value of f. It is innocuous to assume that spreading more pieces of fake news increases the probability of encountering fake news (i.e., f), which intensifies D's media skepticism (i.e., S(f) increases). Then, influencing D's media skepticism by adding pieces of fake news to the media environment (i.e., increasing f) enables news producers to pursue their goals. Building on this, we present our second proposition, which has a similar flavor to Little (Reference Little2016) and Ashworth et al. (Reference Ashworth, de Mesquita and Friedenberg2018).

The rationale behind Proposition 2 is as follows. Spreading more pieces of fake news would increase the probability that the incumbent wins the competition when D's initial expectation about the incumbent's characteristic is greater than 0 (i.e., μ > 0). By contrast, the probability decreases as more fake news is proliferated when D's initial expectation is less than 0 (i.e., μ < 0). Figure 2 visualizes the rationale behind this. Each curve in Fig. 2 represents the probability that the incumbent wins the competition with different degrees of media skepticism. We can see that the probability curve in Fig. 2 shifts from the red line to the dashed line to the blue line as D's media skepticism becomes stronger.

Figure 2. The probability that the incumbent wins the competition (i.e., $Pr[ {\opf E}( c\vert x) > 0]$), where 0 < f _s < f _m < f _l < 1, and S(f _s) < S(f _m) < S(f _l). The horizontal axis represents potential locations of 0.

Consider the case in which D's initial expectation about the incumbent's quality is greater than 0 (i.e., μ > 0), which is captured by setting a = 0 in Fig. 2. Then, as D's media skepticism becomes stronger, the probability that the incumbent wins the competition (i.e., $Pr[ {\opf E}( c\vert x) > a = 0]$) increases from a ₁ to a ₂ to a ₃ in Fig. 2. This is why a pro-incumbent news producer has the incentive to spread fake news (i.e., increase f) to lead D to be more skeptical. Substantively, maintaining D's initial expectation would be the best situation for a pro-incumbent producer because, given μ > 0, D would retain the incumbent. If it is not possible, the pro-incumbent producer will still try to keep D's posterior expectation about the incumbent's quality as close to D's initial expectation as possible. In other words, the pro-incumbent producer wants to hinder D's ability to update her/his opinion of the incumbent based on news reports. To do so, it is necessary to minimize the impact of the information extracted from news reports on D's updating, which can be done by spreading fake news.

Consider the case in which D's initial expectation of the incumbent's quality is less than 0 (i.e., μ < 0), which is captured in Fig. 2 by setting b = 0. As we can see in Fig. 2, the probability that the incumbent wins the competition decreases from b ₁ to b ₂ to b ₃ as D's media skepticism becomes stronger. Therefore, an anti-incumbent producer now has an incentive to spread fake news to strengthen D's media skepticism. D's initial expectation is favorable to the challenger, so the anti-incumbent producer wants to obstruct D's updating through news reports to keep D's posterior expectation about the incumbent's quality similar to the initial expectation.

In conclusion, the incentive to spread fake news is persistent because the dissemination of fake news can cause media skepticism. For example, Llwellyn et al. (Reference Llwellyn, Cram, Favero and Hill2019) found that the bot/troll network that disseminated fake news about Brexit did not seem to aim to gain any particular outcome. This lack of a particular direction corroborates that inducing media skepticism – and not directly influencing public opinion – is what motivates the spread of fake news.