Social Media and Democracy

Before we proceed to the formal model we provide an illustrative example. We begin with Condorcet’s jury theorem, proceed to the basic logic of Bayesian persuasion, and then show how social-media persuasion can overturn the conclusion of the theorem.

Our general model, of which the above example is a special case, shows how in more encompassing circumstances the designer is able to sway the vote towards his desired outcome. As in the example, in some cases, the preferred outcome of the designer is chosen with probability one. In other cases, the designer is unable to exert any influence at all. The ability of the designer to influence outcomes depends both on the extent to which users are exogenously informed and on the heterogeneity in the accuracy levels of users’ exogenous information.

We begin with a setting in which some voters are uninformed, whereas others are exogenously informed with the same accuracy. We show that, as long as this accuracy is not too high, the social media platform can completely sway the outcome of the vote. Interestingly, we also show that, in this case, the designer can sway the outcome of the vote with an unbiased signal. We also show that, in some cases, the designer can only sway the outcome with a signal biased towards B—that is, one that sends message “B is better” more often than the message “A is better”. Finally, we show that when users’ exogenous signal accuracy is high, the designer can no longer affect the outcome.

Next, we extend the model to a scenario in which there are two kinds of voters corresponding to two levels of exogenous-signal accuracy. Here we show that if both accuracies are sufficiently low, the designer can determine the outcome of the vote with probability one; if both accuracies are sufficiently high, the designer cannot exert any influence; and if accuracies are in an intermediate range, then whether or not the designer has influence depends on both the specific accuracies of information and the fractions of users of each kind. We show that, in this last case, there is a nonmonotonicity wherein the designer can determine the outcome if most users are of one kind and if most users are of the other kind, but not when there is a substantial fraction of each kind. There is also a nonmonotonicity wherein the designer can determine the outcome of the vote when the two accuracies are sufficiently close or sufficiently far, but not in between. Finally, we provide conditions under which the designer can control the outcome of the vote only with a signal that is biased towards A and only with a signal that is biased towards B.

Suppose a voter’s belief that the state is $\theta_{A}$ is equal to $1/2$. Then any additional signal $s:\Theta\mapsto\{\mathfrak{a},\mathfrak{b}\}$ induces a pair of posteriors $(\alpha,\beta)$, where $\alpha=\mathrm{P}\left[\theta_{A}|\mathfrak{a}\right]$ and $\beta=\mathrm{P}\left[\theta_{A}|\mathfrak{b}\right]$. The signal is uninformative if $\alpha=\beta=0.5$. If the signal is informative we can assume without loss of generality that $0\leq\beta<\frac{1}{2}<\alpha\leq 1$, and in this case there exists a (unique) signal that induces these posteriors. Thus, we can identify any informative signal with the pair of posterior distributions it induces when starting with prior $\frac{1}{2}$.⁷⁷7Representing signals as distributions over posteriors is standard in Bayesian persuasion (Kamenica, 2019). Representing them as posteriors starting with a prior of $\frac{1}{2}$ is common in the literature on social learning (see, e.g., Acemoglu et al., 2011; Arieli et al., 2020, 2021). For any $\alpha<\frac{1}{2}<\beta$, the exact mapping can be derived as follows: First, since the expected prior is equal to the posterior, we have

$\displaystyle\alpha p_{\mathfrak{a}}$

$\displaystyle+\beta(1-p_{\mathfrak{a}})=\frac{1}{2}~{}~{}~{}~{}\Rightarrow~{}~{}~{}~{}p_{\mathfrak{a}}=\frac{\frac{1}{2}-\beta}{\alpha-\beta},~{}~{}~{}~{}p_{\mathfrak{b}}=\frac{\alpha-\frac{1}{2}}{\alpha-\beta},$

where $p_{\mathfrak{a}}=\mathrm{P}\left[\mathfrak{a}\right]$ and $p_{\mathfrak{b}}=\mathrm{P}\left[\mathfrak{b}\right]=1-p_{a}$. Next, we have that

$$\mathrm{P}\left[s(\theta_{A})=\mathfrak{a}\right]=\mathrm{P}\left[\mathfrak{a}|\theta_{A}\right]=\frac{\mathrm{P}\left[\theta_{A}|\mathfrak{a}\right]p_{\mathfrak{a}}}{\mathrm{P}\left[\theta_{A}\right]}=\frac{\alpha p_{\mathfrak{a}}}{1/2}=\frac{\alpha-2\alpha\beta}{\alpha-\beta}~{}~{}\Rightarrow~{}~{}\mathrm{P}\left[\mathfrak{b}|\theta_{A}\right]=\frac{2\alpha\beta-\beta}{\alpha-\beta}.$$

Similarly,

$$\mathrm{P}\left[s(\theta_{B})=\mathfrak{a}\right]=\mathrm{P}\left[\mathfrak{a}|\theta_{B}\right]=\frac{\mathrm{P}\left[\theta_{B}|\mathfrak{a}\right]p_{\mathfrak{a}}}{1/2}=\frac{1-2\beta-\alpha+2\alpha\beta}{\alpha-\beta}~{}~{}\Rightarrow~{}~{}\mathrm{P}\left[\mathfrak{b}|\theta_{B}\right]=\frac{2\alpha+\beta-1-2\alpha\beta}{\alpha-\beta}.$$

Furthermore, this analysis implies that the bias of the signal $(\alpha,\beta)$ is:

$$\mathrm{bias}(\alpha,\beta)=\mathrm{P}\left[s(\theta_{B})=\mathfrak{a}\right]-\mathrm{P}\left[s(\theta_{A})=\mathfrak{b}\right]=\frac{1-\beta-\alpha}{\alpha-\beta}.$$

(1)

Next, we have the following lemma, which allows us to focus in our analysis on a small set of $2\times 3$ signals, such that, if the election is manipulable, one of these signals must be optimal (and any other optimal signal has weakly higher values of $\alpha$ and $\beta$).

The intuition for Lemma 1 is that for any signal, slightly decreasing either $\alpha$ or $\beta$ increases the probability of the $A$-favorable realization $\mathfrak{a}$, and that if $\alpha\notin\{q_{\ell},q_{h}\}$ and $\beta\notin\{0,1-q_{h},1-q_{\ell}\}$, then this slight decrease has no effect on the behavior of any voter conditional on her signals. See Appendix A for the proof. Table 1 summarizes the key properties of these $2\times 3$ optimal signals (which are derived from straightforward calculations). Note that the share of voters who vote for policy $A$ in state $\theta_{A}$ is always larger than in state $\theta_{B}$, and, thus, a signal is optimal if and only if it induces at least half of the agents to vote for policy $A$ in state $\theta_{B}$.

In this section we focus on the case in which the low-accuracy signal is uninformative (i.e., $q_{\ell}=\frac{1}{2}$). That is, we assume that a share $\lambda$ of the population is uninformed, and the remaining agents obtain each a private signal with accuracy $q_{h}$. An interesting special case, discussed towards the end of the section, is that of homogeneous populations, in which all voters have the same signal accuracy (formally, $\lambda=0$).

Our first observation is that if the signal accuracy $q_{h}$ is below some threshold, denoted by $q_{NI}(\lambda)$, then policy $A$ is always chosen, even without any manipulation by the designer.⁸⁸8 If one changes the tie-breaking rule to favor policy $B$, then under the same condition on $\lambda$, the designer can perfectly manipulate the election’s outcome by sending the signal $(\frac{1}{2}+\varepsilon,1-q_{h})$ for a sufficiently small $0<\varepsilon\ll 1$. This signal is almost-uninformative in the sense that with an arbitrarily high probability of $1-O(\varepsilon)$ the agents get the realization $\mathfrak{a}$, which only slightly increases the likelihood of state $\theta_{A}$, and which is sufficient to make the uninformed voters to strictly prefer policy $A$.

It is straightforward to verify that $q_{NI}(\lambda)$ is increasing in $\lambda$, that $q_{NI}(25\%)=\frac{2}{3}$, and that $q_{NI}(50\%)=1$. This is illustrated by the solid (red) curve in Figure 1. In what follows we focus on the more interesting case in which $q_{h}$ is above this threshold—namely, the case in which policy $B$ is chosen in state $\theta_{B}$ unless the designer manipulates the outcome.

In order to state our result, we define $\overline{q}:[0,1)\Rightarrow[0.5,1]$ as follows:

$$\overline{q}(\lambda)=\frac{-\lambda+\sqrt{\lambda^{2}+2-2\lambda}}{2-2\lambda}.$$

It is straightforward to verify the following, illustrated by the dashed (blue) line in Figure 1:

Our first result shows that the designer can manipulate the election’s outcome iff $q_{h}\leq\overline{q}(\lambda)$, and that manipulation (when possible) can be implemented by an unbiased signal. Moreover, for values of precision in the interval $(\frac{2}{3},\overline{q}(\lambda)$ (values above the horizontal dotted (black) line in Figure 1), all optimal signals have non-positive biases.

An interesting feature of the optimal signals is that they have non-positive bias (bullets 2 and 3 of Proposition 1). Observe that this is the opposite direction of bias relative to the standard application of Bayesian persuasion (Kamenica and Gentzkow, 2011), in which the bias is positive (i.e., in favor of $A$, the designer-optimal outcome). The intuition underlying the contrast between the standard application and our setup is the following. A voter who is informed and obtains signal $a$ is prepared to vote for $A$, whereas a voter who is informed and obtains signal $b$ is prepared to vote for $B$. The designer would like to convince the latter to also vote for $A$, while at the same time not causing the former to switch. The way to do this is to choose a signal for which realization $\mathfrak{a}$ is stronger than realization $\mathfrak{b}$, so that voters with realizations $(b,\mathfrak{a})$ switch to $A$ but voters with realizations $(a,\mathfrak{b})$ do not switch to $B$. Of course, since $\mathfrak{a}$ is stronger than $\mathfrak{b}$, it must be the case that $\mathfrak{b}$ is realized with higher probability, or, in other words, that the signal has non-positive bias. We now prove Proposition 1.

An interesting special case is a homogeneous population in which all agents have the same signal quality (i.e., $\lambda=0$). Applying Proposition 1 yields the following characterization of election manipulation in homogeneous populations.

In this section we return to the general binary model in which the low-accuracy signal is informative ($q_{\ell}>1/2$). We prove two results: first, a characterization of conditions under which the designer can manipulate the election; and second, an identification of conditions under which optimal signals have positive or negative biases.

In order to state our first result we define $\underline{\lambda}:(\frac{1}{2},\frac{\sqrt{2}}{2})]\rightarrow[0,1]$ as follows: $\underline{\lambda}(q_{h})=\frac{0.5-q_{h}^{2}}{q_{h}\left(1-q_{h}\right)}.$ We will show in the proof of Proposition 2 below that $\underline{\lambda}(q_{h})$ is the highest share of $\lambda$ for which the signal $(q_{h},1-q_{h})$ induces a majority of the voters to vote for $A$ in state $\theta_{B}$ for all values of $q_{\ell}\leq q_{h}$. Observe that:

Our results are illustrated in Figure 2. Bullet (3) of Proposition 2 implies an interesting non-monotonicity in $\lambda$: When $q_{h}$ is in the interval $\left(\frac{2}{3},\frac{\sqrt{2}}{2}\right)$, then the elections are manipulable if either the share of agents with low accuracy is sufficiently small (namely, $\lambda\leq\underline{\lambda}$) or if it is sufficiently large, but the elections cannot be manipulated if this share is intermediate. The intuition for this non-monotonicity is as follows. Our designer is constrained by the fact that he must choose the same signal distribution to all agents. When $\lambda$ is either very low or very high, the designer can tune this distribution to the large majority of agents by sending an unbiased signal with the “right” level of accuracy (i.e., a level that induces most voters to vote $A$ if either of their signals is in favor of $A$). By contrast, when $\lambda$ is close to $50\%$, the designer has a trade-off, and must send a “versatile” signal that can handle both groups of agents (and is not optimally “tailored” to either group), and this limits his ability to manipulate the elections.

Moreover, one can show that the ability to manipulate the elections is also non-monotonic in $q_{\ell}$. Specifically, for each $q_{h}$, the interval of $\lambda$-s for which the elections cannot be manipulated is non-monotone in $q_{\ell}$: the interval is small (and it might even disappear) when $q_{\ell}$ is either small (close to $50\%$) or large (close to $q_{h}$), while it is largest when $q_{\ell}$ is intermediate. The intuition for the non-monotonicity in $q_{\ell}$ is as follows. When $q_{\ell}$ is low, it is easy to manipulate the low-accuracy agents. The designer can send a positively biased signal that with high probability sends a “weak-pro-$A$” realization, which manipulates most low-accuracy agents (and a sufficient number of high-accuracy agents). When $q_{\ell}$ is close to $q_{h}$, the designer can exploit the relative homogeneity of the population by sending the negatively biased signal $(q_{h},1-q_{\ell})$, which induces both types of agents to vote for $A$ if either of their signals is “pro-$A$”. In contrast, when $q_{\ell}$ has an intermediate value the designer has neither of these two benefits, and thus his ability to manipulate the elections is limited.

Recall that in the special case of $q_{\ell}=0.5$ (partially informed agents), manipulation, when possible, could always be implemented by an unbiased signal (and in some cases only by signals with non-positive biases). In what follows, we show that this is no longer the case in the general setup (with $q_{\ell}>0.5$). Specifically, we characterize both setups in which manipulation can only be implemented by positively biased signals (in which the signal $\mathfrak{a}$ is sent more often than $\mathfrak{b}$), and setups in which manipulation can only be implemented by negatively biased signals. Specifically, Proposition 3 below shows that the direction of the bias is uniquely determined (for an interior interval of $\lambda$-s) in the following setups:

1. 1.

   All optimal signals must have a positive bias if $q_{h}>\frac{2}{3}$ and $q_{\ell}$ is sufficiently low.

2. 2.

   All optimal signal must have a negative bias if $q_{h}\in(\frac{2}{3},\frac{\sqrt{2}}{2})$ and $q_{\ell}$ is sufficiently high.

Observe that in the second case, the manipulation is implemented by signals with the opposite direction of bias relative to the standard applications of Bayesian persuasion (see, e.g., Kamenica, 2019). The intuition for this is that the optimal negatively biased signal $(q_{h},1-q_{\ell})$ exploits the (modest) heterogeneity of the population: the relatively infrequent signal $\mathfrak{a}$ is sufficiently informative to induce both types of agents to vote $A$ regardless of their exogenous signal, while the more frequently used signal $\mathfrak{b}$ is sufficiently weakly-informative, such that it does not change the voting of any agent who got exogenous signal in favor of policy $A$.

In this section we extend our model to a general distribution over signal accuracies (and not just binary). Specifically, we consider here a variant of our model in which the exogenous signal’s accuracy is continuous, and is distributed according to density $f$ with a support ${\textrm{supp}}(f)$ in $[0.5,1]$. We note that the closely related extension of the model to a setup with a finite (but greater than 2) number of signal accuracies is analogous (and is omitted for brevity).

Our result here is an extension of Proposition 2 to this general setup, showing that the election is manipulable if all agents have accuracies of at most $\frac{2}{3}$, and that it is non-manipulable if all agents have accuracies of at least $\frac{\sqrt{2}}{2}$. Formally,

In our main analysis, we required the designer to design one signal for all voters. What if the designer can differentiate between voters based on their signal accuracies, and can thus design different signals for different voters?

Suppose that the designer can distinguish between the two types of voters, $q_{\ell}$ and $q_{h}$. Furthermore, suppose she can commit to two different signals, $s_{h}$ and $s_{\ell}$, where the former targets $q_{h}$ voters and the latter targets $q_{\ell}$ voters. Observe first that Proposition 2 applies: if $q_{\ell}<q_{h}\leq 2/3$ then the $(q_{h},0)$ signal is optimal regardless of the type of voter, and so the designer can manipulate the election, whereas if $q_{h}>q_{\ell}>\sqrt{2}/2$ then the designer cannot manipulate the election. The proof is the same as that of the proposition.

However, targeted signals differ in the intermediate case. Note that for either $i\in\{\ell,h\}$, the signal that leads to the maximal fraction of $q_{i}$ voters to vote for $A$ is one of $(q_{i},0)$ or $(q_{i},1-q_{i})$, by (the proof of) Lemma 1. Both signals lead to a majority of voters for $A$ in state $\theta_{A}$. In state $\theta_{B}$, however, $(q_{i},0)$ leads to $\mathrm{P}\left[A|\theta_{B}\right]=\frac{1-q_{i}}{q_{i}}$, whereas $(q_{i},1-q_{i})$ leads to $\mathrm{P}\left[A|\theta_{B}\right]=1-q_{i}^{2}$ (see Table 1). For $q_{i}\in[1/2,\frac{\sqrt{5}-1}{2})$ the former is larger, whereas for $q_{i}\in(\frac{\sqrt{5}-1}{2},1)$ the latter is larger. Overall, the designer can manipulate the election if and only if

$$\lambda\cdot\max\left\{\frac{1-q_{\ell}}{q_{\ell}},1-q_{\ell}^{2}\right\}+(1-\lambda)\cdot\max\left\{\frac{1-q_{h}}{q_{h}},1-q_{h}^{2}\right\}\geq\frac{1}{2}.$$

Observe that, in contrast with Proposition 2, there is no non-monotonicity here: the left-hand-side of the inequality decreases with $q_{\ell}$ and with $\lambda$.

This last necessary and sufficient condition on the designer’s ability to manipulate the election extends in a straightforward manner to the variant with a continuous distribution of signal accuracies from Section 5.1 above. In this case, the designer can manipulate the election if and only if

$$\int_{1/2}^{1}\max\left\{\frac{1-q}{q},1-q^{2}\right\}\cdot f(q)dq\geq\frac{1}{2}.$$

What if the designer has even more finely grained information about voters, such that she knows not only each voter’s signal accuracy but also the realization of this exogenous signal? In this case, the designer can provide a strongly targeted signal, in which voters with different signal accuracies and different realizations get different signals.

For a given signal accuracy $q$, a voter with realization $a$ has interim belief $q$ about the probability the state is $A$, whereas a voter with realization $b$ has interim belief $1-q$ about this probability. If the designer knows these realizations, she faces a standard Bayesian persuasion problem relative to each one of these kinds voters. In order to maximize the probability that such voters vote for $A$, the designer should supply the former voter with no additional signal, and the latter voter with additional signal $(q,0)$. The latter signal here is the optimal signal from the standard Bayesian persuasion setting (Kamenica and Gentzkow, 2011), supplied to a voter who has belief $1-q$ and is will to vote for $A$ once the belief is above $1/2$. Given these signals, the former voter will always vote for $A$, whereas the latter will vote for $A$ with probability $\frac{1-q}{q}$ in state $\theta_{B}$. In state $\theta_{B}$, the total fraction of voters with accuracy $q$ who vote for $A$ is thus

$$(1-q)\cdot 1+q\cdot\frac{1-q}{q}=2(1-q).$$

For a homogeneous population in which all agents have accuracy $q$, the designer can manipulate the elections iff $2(1-q)\geq\frac{1}{2}\Leftrightarrow q\leq\frac{3}{4}.$ When there are two signals accuracies, $q_{\ell}$ and $q_{h}$, the designer can then manipulate the election if and only if

$$2\lambda\cdot(1-q_{\ell})+2(1-\lambda)\cdot(1-q_{h})\geq\frac{1}{2}.$$

When there is a continuous distribution of signal accuracies, as in Section 5.1 above, the designer can manipulate the election if and only if

$$2\int_{1/2}^{1}(1-q)f(q)dq\geq\frac{1}{2}.$$

Our analysis and results lead to a straightforward comparison of the effects of social media (through private persuasion) and traditional media (through public persuasion). For the latter, suppose that instead of each voter obtaining a conditionally independent realization of the designer’s signal, all voters obtain the same realization. Is such a public signal better or worse for the designer?

Observe that with a public signal, the designer can never manipulate the election with probability 1. However, she can always at least slightly increase the probability that voters vote for $A$ (with a positively biased signal as in Kamenica and Gentzkow, 2011). Thus, the answer to whether this is better than private persuasion depends on whether or not the designer can manipulate the election in the latter case. If she can, then private persuasion by social media is better. If she cannot, then public persuasion by traditional media is better.