Optimal Echo Chambers

Learning from others is fundamental to making decisions. Yet when deciding whom to learn from, people often choose those with similar beliefs. Consumers of news from both traditional and social media prefer sources aligned with their political views, and even when money is on the line, investors tend to follow social media accounts whose views on the market match their own.²²2[79] provide experimental evidence that news consumers prefer sources that align with their political leanings. [54] find that Facebook users tend to have friends who share their political views, and thus end up seeing news they agree with more often. [69] find similar selective exposure on Twitter. [64] find that users of a social media site for stock investors tend to follow others whose sentiment (bullish/bearish) matches their own. [93] reviews the communications literature on selective exposure. The resulting echo chambers—cliques of people communicating only with those who agree with them—are often attributed to cognitive bias, at odds with accuracy. They have been blamed for polarization in beliefs and the spread of misinformation during a global pandemic, and framed as a threat to democracy as well as the stability of financial markets.³³3See (respectively) [67] (discussed further in footnote 25), [62], [95], and [96].

We provide an alternative explanation for echo chambers that requires no behavioral biases and matches the empirical evidence well, yet has very different policy implications than cognitive bias. If some information sources are less accurate than others but it is not clear which, it is reasonable to place more trust in those one agrees with. [71] show this is true for a binary state, but otherwise this insight remains largely absent from the literature, despite its relevance to a wide variety of learning environments.⁴⁴4[81] review the economics literature on echo chambers, but focus on segregation that does not improve the accuracy of beliefs. [75] survey psychology work on selective exposure, weighing confirmation of pre-existing attitudes against accuracy as competing motives. We extend this result to a richer context (a continuous state space), which allows us to develop counter-intuitive new insights about learning from others. We define echo chambers broadly, as environments in which people tend to encounter views similar to their own, and our findings apply to learning from news organizations as well as dialogue with peers.

Proposals to eliminate echo chambers (and messaging campaigns more generally) often aim to expose people to views that challenge their beliefs—for example, appealing to people to diversify their sources themselves, or even giving them less control over what appears in their social media feeds.⁵⁵5While some suggestion algorithms show users news stories similar to those they already like, others such as the BuzzFeed News feature “Outside Your Bubble” [91] are designed to show users a range of perspectives. See [87] for an appeal to self-diversify. It may seem natural that if somebody is misinformed and only paying attention to similar views, exposing them to more accurate information will bring their beliefs closer to the truth. But this doesn’t always work in practice—for example, exposing partisans to opposing political views can result in wider disagreement, rather than bringing views together.⁶⁶6[53] find that social media users incentivized to follow accounts they are politically opposed to polarize more than the control group. See Table 1 in Appendix A2 for further detail. Our theory can help resolve this seeming paradox. We show that if someone is unsure whom to trust, broadening the views they are exposed to can backfire, slowing their learning and undermining their trust in others.

In our model, an agent starts with a prior belief about the state of the world, which is real-valued. There exists a collection of signal sources, and some are more accurate and informative than others, though the agent is not sure which. The agent chooses a range to sample from (her echo chamber), and draws a signal realization from those within that range of her prior expectation. Finally, having observed the signal realization (but not the quality of its source), she takes an action and is rewarded based on how closely it matches the true state.

A key feature of our model is the agent’s ability to focus attention on a specific range of views, which applies to a wide range of real-world examples. Choosing to watch only a news channel with a known partisan slant is one example, or filtering the social media one consumes to avoid certain viewpoints. Selecting which article to read based on headlines could also qualify, or striking up a conversation with a new neighbor based on the political signs in their yard. Choosing to hear only news one agrees with will sometimes require an editorial intermediary to filter out signals that challenge one’s beliefs, though our results hold even if the agent’s control of the echo chamber is inexact (Appendix A1).⁷⁷7Another interpretation of such a “fuzzy” echo chamber is that the agent divides their attention across a spectrum of news sources, allocating more attention to some than others. And even in contexts without an intermediary filter, we show that people have incentives to coordinate and learn from those with similar views (Corollary 2).

Our main results are as follows. First, if there exist information sources of low enough accuracy, signal realizations close to one’s prior expectation are more likely to be accurate (Proposition 1). Accordingly, it can be optimal to pay more attention to viewpoints similar to one’s own (Proposition 2), as this screens out signals that are less accurate in expectation. This is true even if there are very few low-quality sources. The optimal echo chamber balances the credibility of views close to one’s own with the usefulness of those further away. Second, we find that signal realizations close to the prior expectation can in some cases move beliefs further than more contrary views, since the former are more credible (Proposition 4). The message most persuasive to someone may not be the truth, but instead something closer to their own beliefs. Third, removing the agent’s ability to focus on signals she agrees with not only makes her posterior beliefs less accurate in expectation, but can also cause her to be less influenced by the signal she receives (Proposition 3). Despite only providing a limited range of viewpoints, echo chambers may do more to correct mistaken beliefs than exposure to viewpoints outside the echo chamber, since only trusted views are heeded. And by improving the accuracy of posterior beliefs, echo chambers can reduce the dispersion of beliefs in a population (Corollary 1). To be clear, echo chambers prevent people from learning from those whose knowledge may be most useful to them. However, this may be impossible to remedy without first addressing the underlying friction: uncertainty about accuracy.

Our work contributes to a long theoretical literature on the tendency to choose confirmatory sources and the polarization of beliefs. A decision maker receiving a binary signal only wants to hear that their prior expectation is wrong if the signal is sufficiently strong to overwhelm their prior. As such, a signal that tends to confirm the receiver’s prior (but is very likely correct when it disagrees) can be preferable to an unbiased signal [59]. [94] shows that using such biased sources over time, while rational, can lead to polarization in beliefs. [86] extend this work by incorporating rational inattention. We show that even if the full spectrum of signal realizations is available, it can be optimal to focus attention on a subset.⁸⁸8See also [61], who study a stopping problem with two information sources, one biased towards each state, but no variation in quality. [78] extend this model to show that news consumers prefer to pay scarce attention to other consumers with similar beliefs and preferences, creating echo chambers when consumers learn from one another as well as biased primary sources.

Another strand of the literature features heterogeneity in the preferences or prior beliefs of those one is learning from. Heterogeneity in prior beliefs can lead to information segregation between groups [90]. Similarly, it may be preferable to learn from the opinions of those whose preferences are correlated with one’s own, if someone’s opinion reflects both what they know and what they want [97]. In models of persuasion, echo chambers can offer those with similar beliefs [84] or preferences [80] an opportunity to credibly share information they otherwise might have misrepresented in an attempt to influence others.⁹⁹9See also [73]. By contrast, the driving mechanism behind the confirmation bias in our paper is heterogeneity in signal accuracy.

[55] show that rational players can update their beliefs about a certain issue in opposite directions in response to the same signal if that signal is “equivocal”—if its likelihood depends on some other ancillary issue, about which the players may have different beliefs. Disagreement can also persist when people use different models [83], or face even slight uncertainty about signal distributions [50].

The antecedent model closest to ours is that of [71], who show that a Bayesian will infer that news sources are more accurate if their reports have agreed with her beliefs in the past. If producers of news value consumer perceptions of their accuracy, they will bias their reporting towards consumers’ beliefs to build trust. However, [71] do not model consumers’ choice of news source—the focus of our paper. And since their consumers are choosing between just two actions, news reports that confirm a consumer’s prior are useless, as they cannot change her action. By contrast, we are able to show (using a richer state and signal space) that confirmatory reports can do the most to change an agent’s views, and can thus be the most valuable.¹⁰¹⁰10Another distinction is that their bias takes the form of garbling: randomly mis-reporting when the state of the world does not match the consumer’s prior. As such, the biased reporting in [71] makes the consumer worse off—a sharp contrast to our finding that selective exposure to a different set of facts can improve expected welfare.

Our continuous state space is a natural way to think about similarity between different states,¹¹¹¹11For example, if the state of the world represents the optimal minimum wage policy, it is intuitive to think that $16 is more similar to $15 than $6. and we show that the key intuition from [71] carries over: signal realizations close to the prior expectation are of higher expected accuracy (Proposition 1).¹²¹²12This is also true in [72], whose model features normally distributed signals, discussed further below. For the remainder of our results, the richer state space is essential, allowing us to study echo chambers on a spectrum. Our paper is the first to study the essential trade-off between credibility and usefulness that affects learning from others whenever one is not certain about the accuracy of others’ information. The setup does not rely on biased signals (though we allow for them), as our agent can focus on confirmatory reports by sampling unbiased signals with realizations close to her prior expectation. Our results do not depend on discontinuities created by binary signals, or the boundedness of their support. And we do not require reputation built over time—even in our one-shot setup, credibility instantly accrues to viewpoints similar to the agent’s own.

We study a rational agent who simply wants to know the state of the world. By contrast, another strand of related literature departs from fully rational Bayesian updating of beliefs.¹³¹³13[88] study an agent who occasionally misinterprets unbiased signals when they disagree with the prior belief. [72] show that a slight bias in what is thought to be unbiased direct information can lead to polarization in trust of secondary sources. In their model an agent observes signals from all sources; we complement their findings by studying how, even without biased signals, one’s choice of sources interacts with mistrust to affect the accuracy and responsiveness of beliefs. [82] show that if boundedly rational agents update marginal beliefs but not a joint distribution, beliefs can diverge asymptotically in response to common information. [74] review work on avoiding free information; in our model, information has an opportunity cost. [92] models affective polarization, in which players’ distrust of others’ selflessness grows; our model has no misalignment of incentives. [66] study “persuasion bias”—a failure to account for repetition of information. [57] show polarization can occur when people sharing information misperceive others’ access to primary sources. However, experimental findings support the idea that selective exposure to similar views is motivated by seeking credible information sources under time or attention constraints. [68] find that limiting the number of information sources participants can review increases their tendency to select information that supports their prior views. [85] find that news articles that support participants’ prior attitudes are viewed as more credible. [98] find evidence that partisans don’t just dislike listening to those who are politically opposed, they sincerely believe them to be less informed. Even when selective exposure is associated with inaccurate beliefs (as [64] find of stock investors), this may be explained by those in echo chambers simply having lower budgets for information or attention. Relative to work that does seem to find irrational levels of confirmation bias (e.g. [76]¹⁴¹⁴14[76] finds that subjects learning about political facts do update beliefs, but more slowly than Bayes’ rule would suggest; in their experiment signal quality is known, unlike our setup.), fully rational models such as ours may be thought of as a relevant baseline on top of which psychological biases may play a role. Further, our model may explain why such biases arose—as a heuristic approximating a rational distrust of those with opposing views.

Finally, our paper can also be related to the literature on network formation. Social networks often exhibit homophily (the tendency of connected individuals to share traits) in many dimensions, including political affiliation: friends tend to agree on politics. [63] demonstrate this empirically, and show that networks of friendship or cooperation may form amongst politically aligned people as a social reward to motivate costly voting. We provide a complementary explanation for homophily in beliefs, beyond heterogeneity in preferences: people befriend those whose opinions they trust. And amongst similarly knowledgeable peers with different prior expectations, our model generates a pairwise-stable network of communication (Corollary 2).

Section 2 describes the baseline model and Section 3 derives the main results, which require little in terms of regularity assumptions. Section 4 assumes that the signal noise and prior are normally distributed to permit sharper and more intuitive results, which are illustrated graphically. Section 5 concludes by discussing policy implications and ideas for future work. Appendix A1 extends the baseline model to allow for sampling according to a normal distribution over signals, rather than uniform sampling within a certain radius. Throughout, proofs and ancillary results are relegated to Appendix A2.

Consider an agent choosing an action $a\in\mathbb{R}$. The state of the world $\omega\in\mathbb{R}$ is unknown, and the agent has a prior belief $\mu_{0}(\omega)$ which has finite mean $\omega_{0}$ and finite variance $\sigma_{0}^{2}$. The state could be economic growth over the next year, the current rate of global warming, or the number of hospitalizations caused by a new virus, to give a few examples. The agent wants to match her action to the state of the world: her utility is $u(a|\omega)=-(a-\omega)^{2}$.

There exists a continuum of signal sources. These might be peers or news organizations—both applications are discussed further below. A fraction $h$ are high quality and produce signal realizations distributed $f_{H}(s|\omega)$, while the remaining fraction $1-h$ are low-type, with signal realizations distributed $f_{L}(s|\omega)$. High-type signals are unbiased (the mean of $f_{H}\left(s|\omega\right)$ is $\omega$, the true state of the world), though low-type signals need not be,¹⁵¹⁵15The assumption that high-type signals are unbiased is not important, since the agent could simply de-mean the observed signals if necessary. and both $f_{H}$ and $f_{L}$ are strictly positive everywhere. We denote the variances of the high- and low-type signal distributions $\sigma_{H}^{2}$ and $\sigma_{L}^{2}$, and assume that they are finite. Define $f\left(s|\omega\right)\equiv hf_{H}\left(s|\omega\right)+\left(1-h\right)f_{L}\left(s|\omega\right)$ to be the total probability density of signal $s$ given state $\omega$.

A signal realization $s$ is a report that the state is $\omega=s$, and $\left(s-\omega\right)^{2}$ thus measures how inaccurate the report is. The key difference between the two types of signals is that low-type signals are less accurate (higher variance): $\text{E}\left(\left(s_{L}-\omega\right)^{2}\right)>\text{E}\left(\left(s_{H}-\omega\right)^{2}\right)$. This may be because low-type sources make worse estimates, or because they make more mistakes in recollection or reporting, or because they are more biased or deliberately deceitful. We also assume that low-type signals are (Blackwell) less informative. This should seem natural given that they are less accurate, but rules out the possibility that the agent somehow knows the low type sources well enough to unravel their mistakes or misreporting so as to end up with better inferences than can be made from the more accurate high-type signals. While inaccuracy is not the only way one signal can be less informative than another,¹⁶¹⁶16For example, low-type signals might instead be low-variance but completely unrelated to the true state, in which case sampling close to the agent’s prior expectation would not be an effective way of screening out low type signals. heterogeneity in accuracy is a feature of many important real-world learning environments—see e.g. [65] or [51]. We assume that the signal noise is orthogonal to the agent’s prior noise,¹⁷¹⁷17If the agent realized she were reading a news report that cited the same primary source she had used to form her prior, for example, she would want to disregard it in favor of new information. but make no restrictions on correlation between signals.¹⁸¹⁸18As such, it may be that for any given set of signal realizations, the low type signals are close to each other—that is, wrong in the same way.

We assume that low-type signals are from a family of densities parameterized by variance that satisfies the following regularity condition: for any state $\omega$,

Equation 1 says that the low-type density converges to zero pointwise as its variance goes to infinity. In other words, increasing the variance parameter does not simply make the tails longer, but can make any given region of the support arbitrarily unlikely. This property should not seem unnaturally restrictive—Eq. 1 holds for the normal distribution and the exponential distribution, among others.¹⁹¹⁹19Furthermore, even if the low-type signal distribution does not satisfy Eq. 1, its tails always will (otherwise the distribution would be improper). That is, there will always be a fixed radius around the true state outside which $f_{L}\left(s|\omega\right)\rightarrow 0$ as $\sigma_{L}\rightarrow\infty$. If the tails are less informative than the remainder of the signal distribution (low and high type combined), then we can relabel the tails the low type signals and proceed.

Before choosing an action, the agent can sample one signal. Its type (accuracy) cannot be credibly conveyed to the agent, but the agent can choose to sample only signal realizations within a radius $r\in\bar{\mathbb{R}}_{+}$ of her prior mean $\omega_{0}$. Given a chosen range, let $T_{r}$ denote the truncation operation. Then the distribution of the signal realization sampled is $T_{r}f(s|\omega)$, that is:

where $F(r|\omega)=\int_{-\infty}^{r}f(s|\omega)ds$ is the cumulative distribution function of $f(\cdot)$.

The timing is as follows.

1. 1.

   The state of the world $\omega$ is drawn from the agent’s prior belief distribution $\mu_{0}$, which has mean $\omega_{0}$ and variance $\sigma^{2}_{0}$.

2. 2.

   The agent chooses a radius $r$ to sample within.

3. 3.

   The agent observes a signal realization $s$, drawn from the distribution $T_{r}f\left(s|\omega\right)$.

4. 4.

   The agent takes an action $a\in\mathbb{R}$, and receives a payoff $u(a|\omega)=-(a-\omega)^{2}$.

The agent can sample the whole space of signal realizations by choosing $r=\infty$.²⁰²⁰20In this case, let $T_{\infty}$ simply be the identity operator. We call this strategy “un-censored” sampling, since the agent updates her beliefs based on the un-truncated realization of $f(s|\omega)$. However, while increasing the sampling radius $r$ is beneficial whenever the state of the world is far from the prior mean $\omega_{0}$, signals far from $\omega_{0}$ are (ex ante) more likely to come from low-quality sources (see Proposition 1). Therefore, restricting the sampling radius may maximize the informativeness of the signal. Crucially, the way to think about censoring in this model is not that the agent receives and discards signal realizations outside radius $r$. After all, a defining feature of an echo chamber is that one does not hear what is outside it. Rather, by censoring the agent guarantees that the signal realization drawn will not be outside the sampling radius.²¹²¹21If signals are biased, censoring to an asymmetric interval around $\omega_{0}$ might (if allowed) yield higher utility than the symmetric radius $(\omega_{0}-r,\omega_{0}+r)$ required by our setup. Nonetheless, for simplicity we only allow a symmetric interval.

There are two distinct applications of this model: learning from peers, and learning from news organizations. As a model of learning from peers, the sampling radius $r$ can be thought of as the range of views the agent is willing to entertain from those she consults before making a decision. On social media, this might be effected by setting filters to block certain viewpoints one disagrees with. Offline, it might entail joining a club of like-minded people, or even choosing a city to live in based on how its residents tend to vote. Having selected a group of peers, the agent finds out what one of them says the state is (e.g. by scrolling through social media, or asking a neighbor) before making her decision.

Our model can also be applied to choosing a news organization, as follows. There exist many sources (experts, scientific studies, etc.) that a news organization might cite, some more biased or inaccurate than others. The agent chooses a news organization known to only publish reports that fall within a certain range of the political spectrum and pays no attention to others, the budget constraint on signals leading to a form of rational inattention.²²²²22[78] study a similar rational inattention problem. Since their setup involves a binary state, in their model only signals that challenge the agent’s prior are useful, and there is no uncertainty about accuracy. Notice that the news organization’s editor plays a crucial role here, by only reporting signal realizations within a certain range such that others never reach the agent. This may be because the editor prioritizes certain pieces of evidence, to promote an agenda or (as in [71]) to build trust—it is not costless to digest and summarize all the evidence about a particular topic. Alternatively, the editors themselves may not be as familiar with other evidence: writers may only pitch them stories that match the reputation of the organization, and editors may only develop relationships with expert sources that tend to confirm their views.

In both applications, the agent’s behavior can be considered “selective exposure,” and the limited range of signals the agent is willing to consider an “information silo” or “echo chamber.” In our model these phenomena arise as a rational response to realistic frictions: limited sampling capacity and uncertainty about the quality of signals. In both applications, the agent is aware that views outside her echo chamber exist, but given limited time or attention she may not be interested in hearing them. Appendix A1 shows that the ability to perfectly exclude certain signal realizations is not essential—our results still hold if the agent has inexact control and is only able to put more sampling weight on some signals than others.

Given the problem described above, let $a^{\star}\left(s,r\right)$ be the agent’s optimal action after having chosen sampling radius $r$ and then received signal realization $s$. We denote the agent’s optimal choice of sampling radius $r^{\star}$. To study the choice of censoring policy $r$, we also define the actions that the agent would choose if there was no uncertainty about source quality: for $q\in\{H,L\}$, let $a_{q}^{\star}\left(s,r\right)$ be the optimal action given signal realization $s$ and sampling radius $r$ if all signals are type $q$.

We now proceed to the main results of the paper. Proposition 1 shows that if low-type signals are inaccurate enough, signal realizations close to the prior expectation are likely to come from high types.

[70]²³²³23This working paper preceding [71] provides detailed formal results referenced in the published version. show that with a binary state space “a source is judged to be of higher quality when its reports are more consistent with the agent’s prior beliefs,” to which our Proposition 1 provides an analogue for a continuous state space. Despite our one-shot setup (another feature distinguishing our model from repeated games such as [71] and [97]), this expectation of quality is immediately useful to the agent. Specifically, echo chambers can aid inference given the constraints. Proposition 2 shows that if low types are inaccurate enough, sampling only signal realizations close to one’s prior expectation can make posterior beliefs more accurate, and improve welfare.

The essence of the proof (Appendix A2) is that if low types are inaccurate enough, the agent can set a sampling radius $r$ such that almost all signal realizations from high types are within $r$ and almost all signal realizations from low types are outside $r$. Censoring thus approximates the expected utility of receiving only high-type signals, which is preferable to receiving both types (Lemma 4). Low-type signal inaccuracy need not be extreme for our results to hold, though. For example, censoring can still be optimal even when low-type signals are just as accurate as the agent’s own prior expectation.²⁴²⁴24Both Propositions 1 and 2 do, however, make use of the lack of a bound on inaccuracy, which would not be the case if the state/signal space were restricted to a finite interval.

Next, we provide two results that apply our model to populations of agents. First, since selective exposure to confirmatory views can move individuals’ beliefs further towards the truth, echo chambers will reduce the expected variance of posterior beliefs in a population. Echo chambers are often assumed to increase polarization of beliefs, but we show that if they are used as a tool to filter out inaccurate information then they can actually reduce it.²⁵²⁵25[67] provide experimental evidence that people’s beliefs diverge after reading opposing articles, and more so if they chose the articles themselves based on the headlines. This is consistent with our model, which predicts that disparate signals can move beliefs in opposite directions, and that trusted news matching one’s prior can be more persuasive. Since the two articles used in their experiment were not random draws but rather selected specifically to represent two opposing viewpoints, the overall effect of selective exposure to news cannot be inferred from their experiment. In fact, since the articles were specifically chosen to represent views not typically associated with their sources, it is possible that they actually fostered convergence of beliefs amongst readers outside the experimental setting, if readers chose what to read based on source rather than headlines.

Note that Corollary 1 does not require agents to start with the same prior expectation, nor does it require each agent to choose the same sampling radius. However, it does not apply within subgroups whose beliefs are not correct on average. For example, the more moderate members of a like-minded subgroup might change their beliefs most in response to incoming news, thus increasing disagreement within the subgroup while reducing it in the aggregate population.

Second, our results can be applied to network formation amongst peers. Suppose each signal source is another agent facing a similar problem. Each agent has beliefs about the state of the world, though some (high types) have more accurate expectations than others. Each chooses a radius that defines a set of peers whose expectations are within a certain distance from hers, and then learns the expectation of one of these peers before taking her action, with payoffs as before. Here the agents and their choices form a graph, in which each agent is a node, and a directed edge from $i$ to $j$ indicates that $j$ is included in the set of trusted peers that $i$ will sample from. In this context, censoring results in homophily in beliefs: links (edges) are more common between like-minded peers, a phenomenon observed in many real-world settings (see e.g. [63]). Furthermore, for similarly knowledgeable agents, the subgraph of optimal echo chambers is pairwise-stable: all trust is reciprocated (Corollary 2).

To be clear, the stability of this pairwise trust might not persist into the future, were we to extend our model to include multiple periods of learning and link formation. However, this is realistic: people’s beliefs change as they learn about the world, as does the set of peers they trust. Also, note that Corollary 2 implies that in settings of peer learning, no intermediary is necessary to filter out untrustworthy signals. Simply having a coordinating mechanism to meet up with like-minded people, be it explicit (joining a social media group that supports a political candidate) or incidental (moving to a city that tends to vote a certain way) can suffice.

These main results make no demands on the distributions of the prior or signals beyond finite variance and pointwise convergence (Equation 1). Simply having low-type sources that are inaccurate enough—even if they are rare—is enough to make signals close to one’s prior expectation better quality in expectation. As such, focusing attention on views similar to one’s own can yield the most accurate posterior beliefs given the constraints, reduce disagreement in a population, and generate networks of mutual trust.

To further understand the mechanics of the model, this section restricts the prior distribution $\mu_{0}$ and signal distributions $f_{L}$ and $f_{H}$ to be normally distributed, with $\sigma_{H}^{2}<\sigma_{L}^{2}$. This permits cleaner results, and sharper statements about how the optimal action changes with the parameters as well as with the agent’s choice of censoring policy.

To illustrate the results in this section, consider the following example. Assume the variances of the prior and signals are $\sigma_{0}^{2}=1$, $\sigma_{H}^{2}=\frac{1}{2}$, and $\sigma_{L}^{2}=3$. The prior mean is zero ($\omega_{0}=0$) and half of the sources are high-type ($h=\frac{1}{2}$). Figure 1 shows the prior expected distribution of signal realizations (that is, marginalizing over all possible states of the world).

With normal signals these ex-ante distributions are also normal, with mean $\omega_{0}$ and variance $\tilde{\sigma}_{q}^{2}\equiv\sigma_{0}^{2}+\sigma_{q}^{2}$ for $q\in\{L,H\}$, so that $\tilde{\sigma}_{H}^{2}<\tilde{\sigma}_{L}^{2}$. This implies that high-quality signals are expected to be concentrated more around the prior expectation so long as $\sigma_{H}^{2}<\sigma_{L}^{2}$ (by contrast, Proposition 1 potentially required extreme low-type variance to reach a similar conclusion). Thus, censoring to a finite radius results in a greater chance of drawing a high-type signal ($r^{\star}\approx 2.35$ is shown here, which Figure 2 will show is optimal).

Figure 2 compares the expected utility of censoring to radius $r$ (solid line) to the expected utility of not censoring (dotted line), in all cases assuming the agent plays the optimal action given the signal received as per Equation 8. At radius $r=0$, the agent is only sampling signal realizations exactly equal to the prior expectation of 0, and not learning anything. Expected utility is thus simply equal to the negative prior variance, since utility is quadratic loss. Radius $r^{\star}\approx 2.35$ maximizes the expected utility of censoring, at a level higher than the expected utility of not censoring. As $r$ goes to infinity, the expected utility with censoring approaches the expected utility with no censoring, as the radius of attention grows to encompass the full signal realization space.

Signal realizations close to the prior expectation are more credible but less useful, since they do not lead the agent to change her beliefs much. Signal realizations far from the prior expectation are useful if true, but less credible. Figure 2 illustrates this key tradeoff: balancing credibility with usefulness. Note that this tradeoff is hard to illustrate in the binary state/signal space used in much of the literature. With a binary state, learning that someone agrees with you strengthens your trust in the accuracy of their beliefs, but is not immediately useful as it will not change your action.

Figure 3 provides an alternative way to illustrate this tradeoff, by comparing what happens with the total variance of the signal and its correlation with the state of the world as the radius increases. Increasing $r$ increases the variance of the signal received, including valuable information about the state. Beyond the optimal radius $r^{\star}$, however, the correlation of the signal with the state diminishes as a low-type source becomes more likely.

To illustrate the conditions under which a finite sampling radius is optimal, we proceed to explore the agent’s best response in this normal framework. With normal signals, standard results in Bayesian inference yield that the optimal action given source quality $a_{q}^{*}(s,\infty)$ is a linear convex combination of the prior expectation and the received signal. The weights are given by the relative accuracy of the prior and of the signal:

This linearity between the signal and the action is broken when the agent chooses a finite sampling radius $r^{*}<\infty$, but the un-censored benchmark will nonetheless be useful to reaching the following three insights. First, conditional on the source type, the optimal action given a finite radius is more responsive to the signal than if it had come from an unbounded sampling radius. In fact, the signal is heeded more whenever the sampling radius is smaller, as Proposition 3 will show.

While the optimal action conditional on the source type is not linear, it is monotonically increasing, so $a_{q}^{*}(s,r)$ is a non-linear pivot of $a_{q}^{*}(s,\infty)$ around the prior expectation (note these policy functions necessarily intersect when $s=\omega_{0}$) for any $q$ and any finite $r$.

While Lemma 1 seems intuitive, it does rely on regularity properties afforded by the normal distribution—namely, symmetry and quasi-concavity. [60] provide clear examples illustrating why both are necessary.

Second, the ex-post belief that a signal is high quality (which determines the weights on the source-specific policy functions) is independent of the sampling radius, and monotonically decreasing in the distance between the prior expectation and the signal realization received.

Lemma 2 says that the more the signal received disagrees with the agent’s prior, the less likely it is perceived to be high-quality information. With the regularity properties of normal distributions, this is true everywhere in the signal space as long as $\sigma_{L}^{2}>\sigma_{H}^{2}$. (Proposition 1 is similar in spirit, but requires the low-type signal noise $\sigma_{L}^{2}$ to be high enough, and reaches a weaker conclusion.) Note that this principle can be applied to other contexts as well, such as scientific experiments. If you step on a scale and it tells you that you weigh a tenth of what you expected, most likely you need a new scale.

Lastly, the un-censored benchmark is useful because the optimal action is more responsive to the signal for any finite $r$. In fact, the smaller the sampling radius, the more the agent heeds the signal received.

Intuitively, a signal realization that is different from your prior expectation is more indicative of an extreme state of the world if it comes from your echo chamber than if it comes from an un-censored sample.²⁶²⁶26Note that, from Lemma 2, we should not interpret from Proposition 3 that echo chambers make the agent more confident that the received signal comes from a high-type source; rather, it is more indicative of an extreme state. In fact, the intuition of the proof is that restricting signal realizations to a censoring radius generates a posterior belief (given signal $s$) that is equivalent to the posterior that would be achieved if the agent had a higher-variance prior and was drawing an un-censored signal. The effect is to put more weight on the signal and less on the prior expectation, as illustrated in Equation 5 for each source type; this effect applies to the convex combination as well, since the weights do not depend on the sampling range. In this sense, echo chambers facilitate compromise.

Figure 4 illustrates these pieces put together for the aforementioned parameters. The dotted blue line in the middle is the optimal action (posterior expected state) with no censoring. Per Lemma 3, this is a mixture of the optimal action with low types only and the optimal action with high types only, given by the dotted red and green lines respectively. The weight is determined by the chance of a high type for a given signal, given by the grey shading (right axis). By Lemma 2, higher signals are less likely to come from high types, so the optimal action starts off closer to the upper high-types-only ray at the origin but approaches the lower low-types-only ray as the signal $s$ goes to infinity.

The solid blue line in the middle shows the expected state if the agent samples only within the optimal radius $r^{\star}\approx 2.35$. This is again a weighted mixture of the solid red and green lines below and above, which represent the low-types-only and high-types-only optimal actions given censoring to $r^{\star}$. Note that the agent is more responsive to signals from the echo chamber.

To sum up, in the normal framework, the optimal action is the weighted average between the response given only high types and the response given only low types. Each of these policies is a non-linear pivot of an un-censored policy which admits a simple linear closed-form. The weights on each of type-specific policy do not depend on the chosen radius,²⁷²⁷27This property is not exclusive to normal signals—it is shown for the general framework in Appendix A2. and the weight on the high-type-only response steadily decreases as the signal gets further from the prior expectation.²⁸²⁸28This monotonicity in the likelihood of a high type comes from the fact that the prior expected distributions of signals satisfy the monotone likelihood property—see Appendix A2 for a more general treatment Finally, the agent is more responsive to signals from smaller sampling radii.

An interesting property arising from signals with uncertain quality is that the optimal action (especially with no censoring) can exhibit non-monotonicity. Since signals further from the agent’s prior expectation are less trustworthy, they can also be less persuasive. So it may be the case that there are regions of the signal space in which the agent’s posterior belief is decreasing in the signal received—a more extreme signal results in a less extreme action.²⁹²⁹29[97] derives a similar non-monotonicity owing to correlation in preferences.

In the normal case, the non-monotonicity occurs even though both components that are averaged, $a_{H}^{*}(\cdot)$ and $a_{L}^{*}(\cdot)$, are monotonically increasing. The average is non-monotonic because the weights are not constant. For example in Figure 4, a signal realization of $s=2$ prompts a higher action than $s=4$, because the agent knows that the latter almost surely came from a low-type source. In other words, a moderate view can elicit a greater response than an extreme one, because it is more trustworthy. Proposition 4 gives conditions for this non-monotonicity to occur.

The uncertainty about the quality of a signal is key for the optimal response to be non-monotonic. So long as $h$ is in the interior, and the difference in quality between high-types and low types is big enough, the transition from an optimal response close to the high-type-only response to one close to the low-types-only response will imply some non-monotonicity. In the normal case this gap is completely dictated by the difference in variances, but non-monotonic responses can also be expected in more general settings.

Finally, with normal signals, great enough uncertainty about the signal’s quality is not only sufficient, but also a necessary condition for censoring to be optimal.

The intuition of the proof is that if there is no uncertainty about the quality of information, the signal received is normal, so the posterior expected state is a linear-convex combination of the prior and the signal. The expected utility is, thus, strictly monotonic and achieves its maximum at $r=\infty$. The continuity of the expected utility with respect to $\sigma_{H}^{2}$, $\sigma_{L}^{2}$, and $h$ completes the proof.

Given these results, what is the best way to disabuse the misinformed? Proposition 3 showed that echo chambers make the agent more responsive to signals, so allowing someone a signal from an echo chamber may move their beliefs further towards the truth than simply exposing them to views they don’t trust. Figure 5 plots the agent’s expected action as a function of the true state, for various censoring radii. Since the agent’s prior expectation is zero, the further the true state is from zero the more wrong the agent’s prior beliefs. While no radius dominates over all possible states, signals from the optimal echo chamber $r^{\star}$ yield a greater expected response than un-censored signals for all but the most extreme states.³⁰³⁰30If the prior expectation is very wrong (in this case, more than  2.5 standard deviations from the true state), then an un-censored signal is better ex post. But since the agent believes such a state is extremely unlikely, this is deemed a risk worth taking. So when source quality is uncertain, echo chambers may provide the best chance of correcting mistaken beliefs, since only trusted news is heeded. Note, however, that this result relies on the assumption that even within the echo chamber, signals are more likely to be close to the true state than further away.

This paper studies conditions under which echo chambers are rational. When some people’s beliefs are more accurate than others, it is reasonable to think those who agree with you are more likely to be among the well-informed (Proposition 1). So if time or attention is dear, it may be better to focus your attention on those with beliefs similar to your own (Proposition 2). As such, exposing people to contrary views may actually limit their willingness to change their own, if they have little reason to trust what they are hearing.

Importantly, this does not mean that echo chambers indicate a healthy environment for learning from others. On the contrary, they signal the presence of these two key frictions: people don’t have enough time for all the information out there, and aren’t sure who to trust. The selective exposure induced by these constraints may lead people to miss the information that would be most useful to them.

So what interventions can help? If high-quality information sources can be identified (perhaps marking the social media accounts of reporters from trusted news organizations, for example), then this would obviate the need for filtering based on signal content. Decreasing the proportion of low-quality sources (e.g. those spreading misinformation) would improve inference, though it would not preclude echo chambers—even if almost all sources are high quality, it can be preferable to focus attention on less extreme signals if the low-type signal variance is high enough (Proposition 2). Finally, improving the quality of low-type sources would reduce the incentive to filter them out. And easing the agent’s time or attention constraint would also reduce the need for filtering. However, these may be harder to translate into concrete policy recommendations.

Our work demonstrates that demand for information can differ depending on its content, not because of preferences but solely to facilitate inference. An important question not addressed here is how the supply side reacts to this phenomenon in equilibrium. Since signals closer to one’s prior expectation can elicit stronger reactions than extreme signals (Proposition 3), those seeking to persuade others with distant beliefs may find compromise more effective than accurately stating their views. However, it is not clear that such shading of signals will survive in equilibrium. Further exploration of the supply side could also entail extending the binary state space in [71] to a richer continuum to explore how consumer beliefs shape the range of views news producers³¹³¹31These may include news aggregators capable of tailoring their product based on consumer characteristics—see [77]. see fit to print.

Another interesting direction might be distributional concerns in a dynamic setting. While censoring can increase the accuracy of beliefs on average, it does make those with very inaccurate priors less likely to encounter high-quality information. If the radically misinformed are unable to escape their echo chambers and suffer outsize consequences or take actions that negatively affect others, this may outweigh any benefits of accuracy that selective exposure has for others.

Our results yield a novel, testable implication: restricting someone’s ability to sample information sources based on their content can reduce the accuracy of their beliefs. Understanding the constraints that lead people to focus their attention on confirmatory views can also shed new light on existing empirical findings and provide important guidance for future work. When those in an echo chamber have less accurate beliefs than those with a more diverse diet of news (as found by e.g. [64]), one interpretation is that the echo chamber is a costly mistake. But is this difference in outcomes driven by differences in cognitive bias, or differences in constraints? Our results show that some people might choose echo chambers because they face greater constraints on time or attention. If so, exposing them to more diverse viewpoints may actually hamper their learning. Distinguishing cognitive bias from constraints on time or attention may require randomization of information sources—otherwise, different choices may simply reflect different constraints.³²³²32[53] provide an example of such randomization. Which mechanism dominates is ultimately an empirical question, as is the degree to which beliefs are improved by selective exposure. Future work might seek to measure these forces in various contexts.

This section demonstrates that the main findings are not unique to the strategy space of the main model, which requires the agent to sample signals within a radius (with equal probability) and not sample at all outside the radius. Here we consider an alternative strategy space for the agent which permits a more gradual weighting of signals distant from her prior expectation, and does not rely on sampling strategies with bounded support. Suppose that instead of choosing a censoring radius $r$, the agent chooses a sampling function $\gamma(s)$ which is a normal distribution with mean $\omega_{0}$ and variance $\sigma_{\gamma}^{2}$ (Lemma 9 in Appendix A2 shows that it is optimal to choose the sampling mean equal to the prior mean regardless of the choice of variance).

For simplicity, throughout this section we also assume (as in Section 4) that the prior belief $\mu_{0}$ is normally distributed with mean $\omega_{0}$ and variance $\sigma_{0}^{2}$, and that the high- and low-type signal distributions are normal distributions centered on the true state $\omega$ with variances $\sigma_{H}^{2}$ and $\sigma_{L}^{2}$, respectively. Similar to the original model (see Lemma 3 in Appendix A2), the optimal action is separable. $a^{*}(s,\sigma_{\gamma}^{2})$ is the convex combination of $a^{\star}_{H}(\cdot)$ and $a^{\star}_{L}(\cdot)$, but each of these components is now also a convex combination of the signal and the prior expectation (Lemma 8 in Appendix A2 provides more details). That is, $a^{\star}_{q}(s,\sigma_{\gamma}^{2})=\alpha_{\gamma q}\omega_{0}+(1-\alpha_{\gamma q})s$ for $q\in\{H,L\}$, where the weight on the prior is a function of $\sigma_{\gamma}^{2}$:

Lemma 2 also holds with normal sampling so long as the sampling rule $\gamma(s)$ does not depend on the state $\omega$ or the quality of the signal $q$. If we let $\bar{\alpha}_{\gamma}=P(H|s)\alpha_{\gamma H}+P(L|s)\alpha_{\gamma L}$ the optimization problem becomes

where $g(s|\omega,\sigma_{\gamma}^{2})=\frac{\gamma(s)f(s|\omega)}{\int\gamma(s^{\prime})f(s^{\prime}|\omega)ds^{\prime}}$ is the distribution of signal realizations given the choice of $\gamma(s)$³³³³33Notice that any linear transformation of $\gamma(s)$ generates the same signal distribution $g(\cdot)$., in particular of $\sigma_{\gamma}^{2}$. The above integral admits a closed form which is best represented by defining the quality-specific weights:

and their average $\bar{\lambda}_{\gamma}=h\lambda_{\gamma H}+(1-h)\lambda_{\gamma L}$. With this notation at hand, the problem simplifies to:

where $\sigma_{\gamma q}^{2}$ are the variances of the sampling policy conditional on type $q$:

When the quality of information is uncertain, $h\in(0,1)$, and the difference in source quality is large enough, we can also have a global maximizer with $\sigma_{\gamma}^{2}<\infty$ (Lemma 10, Appendix A2). This complements Proposition 2 by stating the existence of other types of sampling strategies concentrated around the agent’s prior that outperform the uncensored/uniform sampling strategy whenever quality uncertainty is large enough. Moreover, similar to Proposition 5, in this setting meaningful quality uncertainty is also a necessary condition (Corollary 3, Appendix A2).

When $h\in\{0,1\}$ or (equivalently) $\sigma_{H}^{2}=\sigma_{L}^{2}$ (that is, there is no quality uncertainty), all the average variables are equal to the type-specific variables, and the second term of Equation 6 vanishes. In Appendix A2, Lemma 9 shows that in this case the objective function is equal to $-\sigma_{0}^{2}$, at $\sigma_{\gamma}^{2}=0$ , and at $\sigma_{\gamma}^{2}=\infty$, the objective takes the value of $-\left(\frac{\sigma_{q}^{2}}{\sigma_{q}^{2}+\sigma_{0}^{2}}\right)\sigma_{0}^{2}>-\sigma_{0}^{2}$ with a (possible) local minimum at $\sigma_{\gamma}^{2}=\frac{\sigma_{q}^{2}-\sigma_{0}^{2}}{\sigma_{q}^{2}+\sigma_{0}^{2}}$. In such a case, the objective does not admit an interior solution, and is maximized at $\sigma_{\gamma}^{2}=\infty$, which implies uniform sampling (i.e. $g(\cdot)=f(\cdot)$).

It is not possible to rank normal vs. uniform sampling in terms of agent welfare, even for a normally distributed prior and signals—there exist cases where each is preferable. However, this section demonstrates that the main results of the paper are not an artifact of the particular sampling strategy we explored. The existence of inaccurate enough low-type sources makes weighting signals closer to one’s prior expectation preferable.

[53] find that disagreement between partisans on Twitter (currently called X) increased more for a group randomly treated with incentives to follow politically opposed accounts. Using their data ([52]), Table 1 shows mean ideology (on a seven-point scale, where higher is more conservative) by initial party affiliation and treatment across the course of the study. Disagreement within the control group increased during the study (from 3.16 to 3.21), indicating the presence of forces other than treatment affecting disagreement between the parties. To be clear, our model does not generally predict increasing disagreement over time. However, it can explain why control group disagreement increased just half as much as treatment group disagreement. These results are consistent with trusted (control group) Twitter exposure having a liberal influence on the beliefs of both Democrats and Republicans, the latter of whom would otherwise have shifted to the right under the influence of other (non-Twitter) sources of news and opinion. By reducing trust in views encountered on Twitter, the treatment resulted in a greater increase in disagreement compared to the control group.

The remainder of this appendix provides further detail and results for our model, discussed in the main text. The agent’s posterior belief on the state of the world after receiving signal $s$ is:

Since the agent maximizes the expected value of $u(a|\omega)=-(a-\omega)^{2}$, the optimal action given a sampling strategy equals the posterior expected state. Another way of thinking about it is that since utility is quadratic loss, maximizing utility is equivalent to minimizing the expected variance of the posterior. Given a signal realization and the sampling strategy, the optimal action is:

Thus, the problem of choosing $r$ reduces to the following integral

To study the choice of censoring policy, $r$, we define the actions that the agent would choose if there was no uncertainty about source quality. Let

for source quality $q\in\{H,L\}$, where $\mu_{1}(\omega|s,r,q)=\frac{T_{r}f_{q}(s|\omega)\mu_{0}(\omega)}{\int T_{r}f_{q}(s|\omega^{\prime})\mu_{0}(\omega^{\prime})d\omega^{\prime}}.$

It will be useful to note that the optimal action given a signal realization $s$ and censoring radius $r$ is separable in the following sense.

We now derive the unsurprising result that expected utility is higher when all signals are high-type.

We now proceed with proving the results found in the main text.