\section*{Introduction}
\noindent False or misleading information is a fundamental problem for people trying to form an accurate understanding of the world \cite{Lazer1094}. 
There has been widespread concern in recent years about the spread of misinformation and ``fake news'' -- for example during Brexit and the 2016 US Presidential Election \cite{Lazer1094}, the COVID-19 pandemic \cite{doi:10.1177/0956797620939054,Weste1912444117}, and the 2020 US Presidential election \cite{PennyMisinfoRev} -- among policy makers, scientists, and the general public, leading to an explosion of academic research on this topic.

Most prior work has focused on news \textit{consumers}, asking empirical questions such as: Who reads, shares, and believes fake news \cite{Guesseaau4586,GuessNHB,Grinberg374}? Why do people fall for fake news \cite{PENNYCOOK201939,BagoJEP,MosNatComm}? What interventions can combat fake news \cite{PennyNature,Guess15536,doi:10.1177/0956797617714579,10.1371/journal.pone.0175799}? Although substantial progress has been made on these questions regarding fake news consumers, relatively little work has considered the other side of the misinformation problem: \textit{producers} of fake news \cite{Gentzkow,10.1257/jep.31.2.211}. The dynamics of the media ecosystem
are driven by a feedback loop between consumers and producers  \cite{Bak-Colemane2025764118,Stewart19,BakColComm}. Inferences about the revealed preferences of consumers 
that fail to account for these dynamics are likely to be misleading. 
To fully understand the spread of fake news, we must therefore examine the strategic considerations that arise on both sides of the news market \cite{Bak-Colemane2025764118}. 

To this end, we develop a game-theoretic model for the dynamics of news dissemination and engagement. Specifically, we study an asymmetric ``fake news game'' played between news outlets, who can choose to publish true or false information, and news consumers, who can choose whether or not to engage with stories from a given outlet.
We focus on the dissemination strategies of outlets that try to generate engagement with either true or false stories, and the news engagement strategies of
readers who seek to consume true news but operate under cognitive constraints.



We show that by the strategic dissemination of true and false stories in a characteristic pattern, publishers seeking to spread fake news 
can successfully induce a substantial level of engagement with misinformation among truth-seeking readers who try to maximize their consumption of true stories. In particular, our model predicts that publishers who seek to spread fake news will adopt strategies that: i) include a mix of true  and false  stories over time and ii) increase the rate of disseminating misinformation as the level of engagement they receive from a reader increases. Building on the game-theoretic concept of extortion strategies in repeated games \cite{Akin2,Hilbe:2013aa,Press:2012fk,Stewart:2012ys,Stewart:2013fk,Milinski19,CushmanExtort}, we call this type of behavior a ``coercive'' dissemination strategy.

Our model predicts that, as a result of such coercive dissemination, outlets seeking to spread fake news will induce their readers to have greater engagement with false stories than with true stories, creating an apparent preference for fake news among their readers; whereas outlets seeking to spread accurate information will produce the opposite pattern. Importantly we predict that this trend will hold even when all readers in fact value accuracy, and false headlines are no more ``attractive'' (e.g. due to novelty, emotional appeal, etc) than accurate headlines. We verify these predictions in the empirical patterns of engagement observed among known fake news and mainstream media outlets, using $20,000$ accuracy ratings for $1,000$ news stories published across 40 outlets, along with Facebook engagement ratings for each story from Crowdtangle \cite{CrowTan}.



Finally we show that revealed preference can be an unreliable approach to determining readers' intentions. News sites that assume readers prefer to engage with false stories will tend to induce such an apparent preference, even when none exists. This effect is exacerbated when readers are inattentive to the accuracy of the news they consume, or when when news sources can target stories at specific groups of readers.
And so reducing the \emph{production} of misinformation by news outlets who simply seek to maximize engagement may require fixing incorrect assumptions about the preferences of their readers, as well as the habits of news consumption among readers themselves.
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\noindent \textbf{The fake news game:} 
We develop a theory of the production and consumption of misinformation using a game-theoretic approach. We consider an asymmetric, asynchronous, infinitely repeated coordination game, which we dub the ``fake news game'', in which a news transmitter (i.e.~a news outlet or a platform promoting news stories) chooses whether to transmit pieces of true or false information (i.e. news stories) while a receiver (i.e.~news consumer) chooses whether or not to engage with each piece of transmitted news (Figure 1).

\begin{figure}[th!] \centering \includegraphics[scale=0.175]{Figure1-Cartoon}
\caption{\small \textbf{The fake news game.} a) We developed the ``fake news game'' in which transmitters choose whether to share true or false news stories and receivers decide whether to engage with each story or not. We assume that receivers gain an increase in utility $B$ from engaging with accurate information but suffer a decrease in utility $C$ for engaging with misinformation. Thus the receivers are incentivized to seek true news stories and avoid fake news stories (we consider the alternate case in which receivers are incentivized to engage with fake stories in the SI Section 2).   
We assume that a transmitter chooses each subsequent story to be true or false based on the veracity of their previous story and the level of engagement it received. The number of independent receivers whose engagement a transmitter considers determines how precisely the transmitter can target its stories.  c) Receivers are assumed to update their strategy using a myopic, noisy optimization process, which may in general take account of the stories shared by multiple transmitters. }
\end{figure}

Receivers derive a utility benefit $B$ from engaging with true news stories, and they suffer a utility loss $C$ from engaging with false stories (Figure 1); they derive no utility payoff if they do not engage. This payoff structure reflects the preferences of news readers who prefer accurate information and seek to avoid falsehoods; in the SI (Section 2.2 and Section 3.6), we provide results for receivers who prefer falsehood or avoid accurate information. 

Similarly, transmitters derive benefit $b_t$ or $b_f$ from engagement by a receiver with a true or false story, and no payoff if the receiver does not engage. We first consider two types of news transmitters: a misinformation transmitter that seeks to promote fake news stories, so that $b_t=0$ and $b_f>0$, and a mainstream transmitter that seeks to promote accurate stories, so that $b_t>0$ and $b_f=0$. Later, we also consider a ``clickbait'' transmitter who simply aims to maximize engagement regardless of accuracy ($b_t=b_f>0$).
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\noindent \textbf{Transmitter strategies:} 
We model the dynamics of news dissemination and engagement as an optimization process, first specifying the strategy spaces available to transmitters and receivers.
A transmitter can choose to share either a true or false story with a probability that depends on the level of engagement their previous story generated. We assume that the transmitter response function is linear, such that a transmitter strategy $\mathbf{r}$ has the form 
\begin{eqnarray}
\nonumber r_{kt}&=&\alpha+\gamma\frac{k}{N}\\
r_{kf}&=&\beta+\theta\frac{k}{N}
\end{eqnarray}
where $r_{kj}$ defines the probability of sharing a true story given that the previous story was true or false ($j\in\{t,f\}$) and that $k$ out of $N$ targeted receivers chose to consume the story. 
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\noindent \textbf{Receiver strategies:} 
We assume that receivers can pay different levels, and different types, of attention when deciding whether to engage with a news story. They can attend to the likely veracity of the headline, and to their recent experience of engagement with the source that transmitted the story. In particular, a receiver engages with a story with probability $q^l_{ij}$ where
\begin{equation}
    q^l_{ij}=a_0\delta_{lt}+(1-a_0)[(1-a_1)p_0+a_1p_{ij}].
\end{equation}
The indices $i$ and $j$ describe the outcome of the previous news story shared by the transmitter: index $j\in\{t,f\}$ indicates whether the transmitter's \emph{previous} story was true or false. The index $l\in\{t,f\}$ indicates whether the transmitter's \emph{current} story is true or false, and index $i\in\{c,n\}$ indicates whether the receiver either consumed ($c$) or did not consume ($n$) the previous story. With probability $a_0$ the receiver
is able to directly assess the veracity of the current story, where $\delta_{lt}$ is the Kronecker delta, equal to 1 if the newly shared story is true, and 0 otherwise. If the receiver is unable to assess
the veracity of the current story (e.g.~due to lack of background knowledge, or failure to pay attention), then with probability $a_1$ they make their decision based on their previous experience of the news source, according to a behavioral strategy $\mathbf{p}=\{p_{ct},p_{cf},p_{nt},p_{nf}\}$, where $p_{ct}$ is the probability of engaging given that they chose to consume in the previous round and the story was true, and so on. Finally, if the reader attends to neither veracity nor past experience, they engage with the story with fixed probability $p_0$.
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\noindent \textbf{Dynamics of reader engagement:}
We begin by analyzing the dynamics of the fake news game for a pair of fixed transmitter and receiver strategies (see SI Section 1). We show that a transmitter using a strategy of the type described by Eq.~2 enforces a specific relationship between the proportion of transmitted stories that are false, $v_f$, and the overall rate at which receivers consume true and false stories, $v_{tc}$ and $v_{fc}$:

\begin{equation}
    v_f=\frac{1-\alpha}{1-\alpha+\beta}-\frac{\theta}{1-\alpha+\beta} v_{fc}-\frac{\gamma}{1-\alpha+\beta} v_{tc}
\end{equation}
Remarkably, this relationship between the false news transmission rate and the receiver's consumption of true and false news holds regardless of the receiver's strategy for engagement -- and therefore regardless of the receiver's preference for consuming true versus false news.
That is, the news site can unilaterally shape the dynamics of consumer engagement by constraining the way in which engagement and fake news transmission covary. For example, a transmitter can seek to generate greater engagement with false stories by ensuring that higher levels of receiver consumption always result in a higher rate of the transmitter sharing false stories. This kind of coercive strategy can be understood as a form of extortion \cite{CushmanExtort,Press:2012fk}. Critically, this means that false stories receiving more engagement than true stories (e.g.~Vosoughi \emph{et al.} 2018 \cite{Vosoughi1146}) does \textit{not} necessarily mean that false stories are actually more attractive to readers. Instead, this pattern could be entirely explained by transmitters using coercive strategies,   
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Having defined a strategy space for transmitters and receivers, and having described how a transmitter can unilaterally constrain the dynamics of the fake news game (Eq.~2), we now examine the dynamics of a receiver's engagement strategy  in response to a coercive transmitter. We study a single receiver who employs a noisy optimization process in an attempt to increase engagement with true stories (that is, to increase payoff given truth-seeking preferences, see Methods). We focus on a myopic optimization process in which the receiver compares the payoff they gain under their current strategy to that under a randomly selected alternate strategy, and tends to adopt the alternate strategy if it will increase their payoff. The case of a single transmitter and receiver corresponds to perfect microtargeting by the transmitter (i.e.~the transmitter is able to promote different types of news to specific receivers in direct response to their engagement habits). However our results also hold in the more realistic case where a transmitter is able to target their stories only at the level of a group of receivers (see SI Section 3.3).
In the SI (Section 1.4) we also consider the case where receivers are perfectly rational \cite{8563,10.2307/2296617}, and the case where receiver behavior is the product of social learning \cite{Traulsen:2006zr} (SI Section 3.2), where we find similar results as for the myopic optimization process. 

We simulated strategy optimization for inattentive receivers ($a_0=a_1=0$) against $10^8$ randomly selected transmitter strategies $\mathbf{r}=\{\alpha,\beta,\gamma,\theta\}$ (Eq. 3). 
We measured the average rate at which the transmitter shares false stories with the receiver, $v_f$, and the average rate at which the receiver engages with false stories (probability of engaging per false story) and with true stories (probability of engaging per true story), $v_{fc}/v_f$ and $v_{tc}/v_t$, at equilibrium (see Methods). 

We define the most successful misinformation transmission strategies as those that i) shared false news more often than not, $v_f>0.5$ and ii) produced a false news engagement rate within the top 10\% of all transmitter strategies considered (see Methods and SI Figure S2). We consider alternate definitions, in which the minimum amount of fake news shared varies, in the SI (Figure S2). Likewise, we define successful mainstream transmission strategies as those that produce $v_t>0.5$ along with true news engagement within the top 10\% of all transmitter strategies considered. 
Note that mainstream transmitters, when faced with inattentive and myopic readers whose incentives are unknown, may also share a mix of true and false news stories as they attempt to induce engagement, just as misinformation sites may share accurate stories to draw readers in.

The most successful mainstream and misinformation dissemination strategies induce characteristic, and opposite, patterns of engagement among readers in our model (Figure~2). In particular, successful misinformation sites induce higher reader engagement with each false news story, as well as greater overall consumption of false news than true news (Table 1). This phenomenon arises even though we assume that receivers strictly prefer true news over false news, so there is no inherent appeal of false news stories under our model.

To understand this phenomenon we inspected the strategies of successful misinformation sites under our model. We find that all successful misinformation sites indeed use coercive strategies that enforce a positive correlation between engagement and false news transmission. A significant proportion ($72\%$, see Methods) use a type of extortion strategy \cite{Press:2012fk}
that enforces $v_f\geq v_{fc}+v_{tc}$, i.e.~successful misinformation site strategies tend to increase their false news output rapidly in response to increased engagement by a given user. However, if engagement drops, they tend to increase their output of true stories (to draw the user back in). As a result, they ``mash up'' true and fake stories as engagement fluctuates over time.


\begin{figure}[th!] \centering \includegraphics[scale=0.25]{MainCompFig1.pdf}
\caption{\small \textbf{Successful news transmission strategies --} To identify strategies that are successful at promoting engagement with true news stories (mainstream sites) or with fake news stories (misinformation sites), we randomly drew $10^8$ transmitter strategies (Eq.~2) and allowed a single receiver, incentivized to engage with true news, to optimize their engagement strategy (see Methods) over the course of $10^4$ interactions. The receiver was assumed to be inattentive ($a_0=a_1=0$) and optimization was assumed to be noisy ($\sigma=1$). a) We identified the transmitter strategies that  successfully promote engagement with true news stories (mainstream sites, blue), i.e those that produce a true news engagement rate within the 90th percentile of the $10^8$ transmitter strategies considered, as well as $v_t>0.5$. Similarly, we identified the transmitter strategies that successfully promote engagement with fake news (misinformation sites, red), i.e those that produce a fake news engagement rate within the 90th percentile, as well as $v_f>0.5$. We  plot the difference between engagement with true and fake stories, $v_{tc}/v_t-v_{fc}/v_f$, for all mainstream and misinformation sites. Mainstream site strategies induce engagement with accurate stories while misinformation site strategies induce the opposite effect. b) We also report the difference in overall consumption of true stories, $v_{tc}$, and consumption of fake stories $v_{fc}$. In all cases receiver mutations were local (see SI section 1.5) and we set receiver payoffs $B=2$ and $C=1$. Results for other parameter choices are shown in the SI Section 3.}
\end{figure}

The behavior of successful mainstream sites, that seek to generate engagement with true stories, shows the opposite pattern from of successful misinformation sites. All successful mainstream strategies enforce a \textit{negative} correlation between engagement and false news transmission.

\begin{figure}[th!] \centering \includegraphics[scale=0.2]{MainTable1.pdf}
\caption*{\small Table 1: \textbf{News consumption patterns.}  Summary of the engagement rates per story and overall consumption rates for mainstream and misinformation site transmission strategies shown in Figure 2.}
\end{figure}


A significant proportion ($56\%$, see Methods) use a strategy
that enforces $1-v_f\geq v_{fc}+v_{tc}$, i.e.~successful mainstream site strategies tend to decrease their output of false stories rapidly in response to increased engagement, but may share more false stories when engagement is low (in a misguided attempt to draw readers back in).
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\noindent \textbf{Empirical patterns of fake news engagement:}
Our analysis of the fake news game makes four key predictions about the dynamics of engagement with misinformation: i) misinformation sites that seek to spread fake news should nonetheless also share accurate information to induce engagement and ii)
successful misinformation sites will produce higher engagement with false than with true stories, even when consumers prefer truth to falsehood. In contrast iii) mainstream sites that seek to spread accurate news stories may nonetheless also share some misinformation in an attempt to induce engagement and iv)
successful mainstream sites will produce higher engagement with true than with false stories.

We tested these theoretical predictions against empirical data on engagement with news from publishers that prior work has determined to be either fake news or mainstream outlets \cite{Pennycook2521}. We sampled a total of $1,000$ articles from 40 outlets (20 fake news, 20 mainstream).
We selected the 25 most recently available news stories for each site from the Crowdtangle database (see Methods), providing a snapshot of the output from each site (and avoiding selecting on the dependent variable by only analyzing high-engagement articles \cite{Allen21}). For each story, we retrieved  Facebook engagement ratings from Crowdtangle \cite{CrowTan}, and assessed perceived accuracy by recruiting $1,000$ American participants from Amazon Mechanical Turk and having them each rate the accuracy of $20$ headlines (yielding a total of $20,000$ accuracy ratings)  (Figure 3). 

\begin{figure}[th!] \centering \includegraphics[scale=0.42]{NewMainFig3.pdf}
\caption{\small \textbf{Empirical patterns of news consumption and accuracy, for mainstream and misinformation sites --} We selected 20 mainstream and 20 fake news sites identified in previous studies of misinformation \cite{Pennycook2521}. Using Crowdtangle we selected the most recent 25 news stories for which engagement data on Facebook was available (see Methods). We then recruited American subjects from Amazon Mechanical Turk to assess headline accuracy ($20$ ratings per headline). a) Regression coefficient between accuracy and $\log_{10}$-engagement for mainstream news sites (blue) and misinformation sites (red)). There is a significant positive correlation between accuracy and engagement across mainstream sites (Fisher's combined test, $p=0.004$; Random effects meta-analysis \cite{DERSIMONIAN1986177,https://doi.org/10.1002/sim.1186}, $p=0.005$, standardized average correlation coefficient -- blue line -- $\mu=0.13$; See Methods and SI Section 3). However there is a significant negative correlation between accuracy and engagement across misinformation sites (Fisher's combined test, $p=0.005$; Random effects meta-analysis, $p=0.013$; standardized average correlation coefficient -- red line -- $\mu=-0.11$; See Methods and SI Section 3). b) Regression of individual site trust ratings \cite{DiasMisinfoRev} against standardized accuracy-engagement regression coefficient (see Methods and SI Section 3). More trusted sites tend to produce higher engagement with accurate information, while less trusted sites tend to produce higher engagement with less accurate information ($p=0.0008$, $r^2=0.25$). c) Distribution of accuracy ratings for mainstream sites (blue, mean=$5.05$, SD=$0.56$) and mainstream sites (red, mean=$4.27$, SD=$0.62$). As expected stories from misinformation sites are rated as significantly less accurate than stories from mainstream sites ($p<0.001$, two-sample t-test). }
\end{figure}

Our data show that mainstream news sites, which we assume seek to promote accurate information, tend to share headlines with higher accuracy ratings than fake news sites ($p<0.001$, Figure 3c). Importantly, however,  both mainstream and fake news sites show wide variation in headline accuracy. These observations are consistent with our theoretical predictions for transmitter strategies that mash up stories with high and low accuracy, for both mainstream and misinformation sites, as they vary their content to draw readers in. Critically, our model predicts opposite correlations between accuracy and engagement for the two types of transmitters (Figure 2). And indeed we observe 
opposite correlations for mainstream versus fake news sites: we find a significant negative correlation between engagement and perceived accuracy across fake news sites ($p=0.005$), whereas we find a significant positive correlation across mainstream sites ($p=0.004$). Taken together, these empirical results confirm key predictions of our model. 

We compare these empirical results to the predictions of previously proposed theories for explaining engagement with misinformation \cite{10.1257/jep.31.2.211,Vosoughi1146,PennyNature} (SI Section 2). We show that, in contrast to our model, none of these theories predict the empirical pattern observed in Figure 3, in which more trustworthy sources generate greater engagement with more accurate stories, while less trustworthy sources generate the opposite pattern.
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\noindent \textbf{Coercion and revealed preference:}
Having empirically validated the key predictions of our model, we now explore the dynamics that lead transmitters to produce misinformation.
So far we have focused on the behavior of receivers optimizing in response to a fixed transmitter strategy. In reality both transmitters and receivers update their behavior in response to one another. In many cases transmitters may have no agenda other than generating engagement with the stories they share -- that is, they have no intrinsic desire to spread misinformation or true information, but instead only to increase consumption among their audience of receivers. However those consumption patterns in turn depend on the strategy of the transmitter (Figure 2).

We explored the behaviors of transmitters and receivers when both players co-optimize their strategies (Figure 4). We considered transmitters who seek only to maximize engagement among receivers ($b_f=b_t$), but make different assumptions about the type of news those receivers prefer. Transmitters who assume receivers prefer fake news (e.g.~because it is more novel) employ strategies that seek to reinforce increased engagement by increasing the amount of false stories they share in response (see SI Section 1). Similarly transmitters who assume receivers prefer true news employ strategies that seek to reinforce increased engagement by increasing the amount of true news they share in response. Transmitters who make no assumption about receivers preferences (and hence try out all possible strategies without bias) and transmitters who assume receivers prefer true news, 
both produce greater engagement with true than with false stories (Figure 4a). 
In contrast, transmitters who assume receivers prefer fake news, produce greater engagement with false than with true stories.
 


\begin{figure}[th!] \centering \includegraphics[scale=0.25]{NNNFigure5.pdf}
\caption{\small \textbf{Self-reinforcing fake news} We allowed both receivers and transmitters to optimize under the process described in Figure 2. Here we assume that transmitters seek only engagement with news, regardless of veracity ($b_f=b_t=1$). But transmitters draw strategies that reflect their assumptions about the receivers' preferences (see Methods). a) Time series of receiver engagement as receivers and transmitters co-optimize. We initialized with a transmitter that always shares true news and a receiver that never engages. We calculated the percentage difference in probability of engagement with a true versus a fake story, averaged across $10^4$ replicate simulations. When transmitters make no assumption (gray), or assume receivers prefer true news (blue), true news receives greater engagement, which reflects the underlying receiver preferences. When transmitters incorrectly assume receivers prefer fake news (red), however, false stories receive greater engagement -- that is, the assumption that receivers prefer fake news is self-reinforcing. b) We calculated the average engagement difference 
for different levels of receiver attention to payoffs, $\sigma$, and for different probabilities that receivers have prior knowledge about headline accuracy, $a_0$. It is only when attention to accuracy and attention to payoffs are low that fake news is self-reinforcing (red region).
In all cases receiver mutations were local (see SI Section 1.5) and receiver payoffs are set to $B=2$ and $C=1$. Transmitter mutations were global. Receiver attention to payoffs was set at $\sigma=1$, attention to accuracy at $a_0=0$ in panel a) with memory $a_1=0$. Transmitter attention to payoffs was set at $\sigma=100$.}
\end{figure}

As these results show, transmitters tend to elicit behavior among receivers that reinforce their assumptions about receiver preferences. We explored more broadly (Figure 4b) the conditions under which an assumed preference for fake news is self-reinforcing in this way (i.e.~leads receivers to engage more with false stories than with true). We find that fake news is self-reinforcing when attention to accuracy is low (i.e.~when receivers either cannot assess or do not pay attention to the accuracy of a headline) and when attention to payoffs is low. This suggests that increasing reader attention to the news they are consuming can reduce not only their own consumption of misinformation but also the production of misinformation by transmitters who make faulty assumptions about receiver preferences.

In the SI (Section 2.5) we also explore the impact of microtargetting and receiver attention to accuracy. We find that when microtargeting is low, the difference in engagement between false versus true stories declines. Thus the ability to target news stories at specific receivers, either by news sites directly or by social media algorithms, can exacerbate the apparent attractiveness of false news stories (Figure S5). We also explored the impact of three forms of receiver attention: i) attention to and/or prior knowledge of story accuracy, $a_0$, ii) memory of past interactions when deciding to engage with a news source, $a_1$, and  iii) long-term attention to payoff, parameterized as the amount of noise in the receiver optimization process, $\sigma$ (see Methods and SI Section 2). 
We find that increasing prior knowledge of or attention to headline accuracy reduces both the rate of engagement with and the overall consumption of false stories under a myopic optimization process (Figure S9). Similarly, increasing attention to payoffs reduces engagement and consumption of false stories (Figure S9).



\section*{Discussion}




 

Understanding why people engage with fake news has become increasingly urgent, as misinformation threatens to undermine democratic institutions and fuel populist movements and polarization, as well as undermining public health efforts and furthering the impact of the COVID-19 pandemic.
Here, we shed new light on misinformation by considering not just the consumers of fake news, but also the producers. We develop a theoretical framework to study the
 strategic interaction between readers and news sources.
Our analysis makes two key predictions for the dynamics of engagement with fake news. First, even a misinformation site committed to pushing as much fake news as possible into the information ecosystem should
-- when faced with readers who typically value accuracy -- publish a substantial amount of true news.
Second, misinformation site strategies that are successful in producing engagement among 
readers will typically produce patterns of engagement in which the false stories
receive more engagement than true stories receive, even when false stories have no greater inherent appeal over true stories. In contrast, mainstream sites that seek to promote engagement with accurate stories tend to produce the opposite engagement pattern. Both of these predictions are confirmed by our experiments

The dissemination strategies that induce these correlations are ``coercive'', since they effectively exert control over the behavior of receivers and push them towards the transmitter's preferred equilibrium, by increasing the amount of accurate information they share when engagement trends in the wrong direction. Strategies like these have been studied widely in other contexts, such as the repeated Prisoner's Dilemma \cite{Press:2012fk,Stewart:2012ys,Hilbe:2013aa,Stewart:2013fk,Milinski19} where an ``extortioner" can coerce their co-player into conceding a higher payoff to an ``extortioner'' than they receive themselves. In psychological settings, these types of behavioral strategies have been characterized as antisocial reciprocity, where they are associated with pro-self social behavior \cite{CushmanExtort}.


Our analysis also helps to resolve an apparent contradiction in prior empirical work. Prominent work has found that false stories get more engagement than true stories, and argues that this occurs because false stories are more novel (and thus more attractive to read) \cite{Vosoughi1146}. Yet other studies find that true stories get as much or more engagement than false stories \cite{Grinberg374,PENNYCOOK201939}. Our results suggest that these different patterns can be explained by the different \textit{sources} of news examined in the different studies. Among claims that have been fact-checked by Snopes (largely coming from misinformation sources) - as in  \cite{Vosoughi1146} - our model predicts that false news gets more engagement. Conversely,  this is not true when claims come from a more balanced set of sources, as in \cite{Grinberg374,PENNYCOOK201939}.
Of course, our findings do not exclude the possibility that novelty may also drive engagement with fake news \cite{Vosoughi1146}, but our work shows that high levels of fake news engagement can arise even in the absence of intrinsic novelty appeal or motivated reasoning, and especially when transmitters can microtarget their content. 

The idea that misinformation sites can successfully coerce readers into engaging with false stories 
has important practical implications. First, our findings suggest a new approach for identifying outlets that are seeking to spread misinformation: examining the relationship between articles' engagement and perceived accuracy. Outlets from which particularly implausible articles generate particularly high levels of engagement may be employing coercion strategies to spread misinformation. Observing such a relationship could be used as a signal that an outlet is intent on spreading misinformation, or making faulty assumptions about the preferences of readers.
Similarly, if such a relationship is observed in the patterns of engagement with inaccurate stories on a particular social media platform, this may suggest the platform uses coercive strategies to promote false news stories to users.
Second, the tendency to share fake news can become self-reinforcing on the part of transmitters: sites that seek to spread misinformation using coercion strategies can create the incorrect impression that readers prefer falsehoods. This pattern, if taken at face value, may then lead to increased false news production by  sites that do not have any particular agenda beyond simply maximizing engagement -- even when such sites would actually maximize engagement by publishing true articles. Third, we find that this apparent preference for false news can be reversed by encouraging readers to pay attention to key aspects of their news consumption habits. This observation reinforces the importance of (in)attention in combating misinformation.

In sum, we have demonstrated the importance of the feedback loop between news publishers and readers. We have shown that focusing only on reader behavior can lead to incorrect conclusions about readers' preferences, while considering publisher strategies can resolve apparent contradictions in the literature, and can highlight new directions for combating misinformation. News consumption does not occur in a vacuum, and to understand the dynamics of misinformation it is essential to explore supply as well as demand.    





\section*{Methods}

\noindent \textbf{Game dynamics:}
We consider a transmitter strategy $\mathbf{r}$ as described by Eq. 1 such that the probability of a transmitter sharing true news is $r_{ki}$, where  the last piece of news shared was consumed by $k$ receivers and was of type $i$. The possible types of news are true ($t$) or fake ($f$), so that $i\in\{t,f\}$, and in a population of $N$ receivers the amount of consumption is $k\in[0,N]$. The transmitter strategy space we consider is thus memory-1. When using a strategy of this type, a transmitter unilaterally enforces a  linear relationship between receiver consumption and the amount of fake news shared by the transmitter (see Methods and SI Section 1). This relationship is given by Eq. 3 and holds regardless of the receiver strategy $\mathbf{q}$, i.e. it holds for other types of receiver strategy than given by Eq. 2, including cases where the receiver strategy employs memory of their full history of interactions with the transmitter (see SI Section 2).

To see this, encode the state of the repeated game in round $j$ as $v_{ki}^j$ where $k$ is the number  of receivers who consume in round $j$ and $i$ is the type of news shared by the transmitter. The probability that the game is in state $v_{ki}^{j+1}$ at round $j+1$ is then given by

\begin{equation}
v_{kt}^{j+1}=\sum^N_{m=0}\sum_{l\in\{t,f\}}v_{ml}^jr_{ml}\sum_{A\in F_k}\prod_{\alpha\in A}q_\alpha\prod_{\beta\in A^C}(1-q_\beta)
\end{equation}
\\
where $\text{Pr}(K=k)=\sum_{A\in F_k}\prod_{\alpha\in A}q^\alpha\prod_{\beta\in A^C}(1-q^\beta)$ is the Poisson Binomial distribution describing the probability that $k$ receivers engage, where $F_k$ encodes the possible subsets of $k$ integers that can be selected from $N$, i.e. the possible combinations of receivers who can consume a piece of news to produce overall consumption level $k$. The set $A$ is just a particular such subset. Note that we allow each transmitter strategy $q_\alpha$ and $q_\beta$ to be different and depend on the full history of play of the receivers and transmitter i.e. we place no particular constraint on the receiver strategy used. The master equation for $v_{kf}$ is defined in the equivalent way with $(1-r_{ml})$ in place of $r_{ml}$ in Eq. 4.



At equilibrium the solution to Eq. 4 must satisfy $v_{ki}^{j+1}=v_{ki}^j=v_{ki}$. We show in the SI (Section 1) that a transmitter strategy $\mathbf{r}$ of the form Eq. 2 results in an equilibrium solution satisfying Eq. 3. Since we assume an infinitely repeated game the dynamics are guaranteed to converge to this equilibrium. Note that we assume that receiver and transmitter actions are subject to a small execution error at rate $\varepsilon$, which ensures that the Markov process described by Eq. 4 has no absorbing states, and a unique stationary distribution.   
\\
\\
\textbf{Coercive and extortionate strategies:}
We define a coercive strategy as a strategy which assuredly increases the amount of news served to receivers of a given type as the receivers increase their engagement. Thus a coercive fake news strategy increases the rate of sharing fake news stories as receivers increase their engagement. For this to be true, the coefficients $\frac{\theta}{1-\alpha+\beta}$ and $\frac{\gamma}{1-\alpha+\beta}$ in Eq. 3 must be negative. 

In order for a strategy to be viable Eq.2 must produce viable probabilities, i.e. $r_{ki}\in[0,1]$. This in turn requires $0\leq \alpha\leq 1$, $0\leq \beta \leq 1$, $-\alpha\leq \gamma \leq 1-\alpha$ and $-\beta\leq \theta \leq 1-\beta$. A viable coercive fake news strategy thus requires $-\alpha\leq\gamma<0$ and $-\beta\leq\theta<0$. Equivalently, a coercive mainstream news (i.e. true news) strategy requires $0<\gamma\leq1-\alpha$ and $0<\theta\leq 1-\beta$.

We can further define an extortionate fake news strategy as one that ensures the amount of fake news shared always exceeds the amount of news consumed i.e. $v_f>v_{fc}+v_{tc}$. For a fake news site an extortion strategy has the form $\mathbf{r}_-^*=\big\{\theta=-\beta,1-\alpha-\beta<\gamma\leq0\big\}$ (see SI Section 1). Conversely, an extortion strategy for a mainstream site ensures that the amount of true news shared always exceeds the amount of news consumed i.e. $v_t>v_{fc}+v_{tc}$. For a mainstream site an extortion strategy has the form $\mathbf{r}^*_+=\big\{\gamma=1-\alpha, 0\leq\theta<1-\alpha-\beta\big\}$ . 
\\
\\
\textbf{Simulations:}
We performed simulations to determine the most successful transmitter strategies for fake and mainstream sites. We selected a transmitter strategy $\mathbf{r}=\{\alpha,\beta,\gamma,\theta\}$ (Eq. 2) with each parameter drawn uniformly from the interval required to produce a viable strategy. We initialized a receiver with a strategy that does not consume any news from the transmitter, i.e. $\mathbf{q}=\{0,0,0,0\}$ (Eq. 1) which was then allowed to update under a local optimization process (see SI Section 1). For each set of receiver and transmitter strategies, we  calculated the stationary distribution for transmission and consumption of news stories $\mathbf{v}=\{v_{tc},v_{tn},v_{fc},v_{fn}\}$ under the iterated fake news game. For a single receiver the stationary distribution can be found explicitly. For a larger population of receivers we simulated $10^4$ rounds of the game, which allowed us to estimate the stationary distribution numerically. In all simulations we assume that both transmitters and receivers experience execution errors \cite{Stewart17558} with probability $\varepsilon=10^{-3}$.

We used the stationary distribution $\mathbf{v}$ to calculate receiver payoffs at equilibrium. We then allowed receiver strategies to update under a local optimization process (see SI Section 1.5 for further details). After  a burn in period of $10^4$ update events we measured the average engagement and consumption rate of receivers over an additional $10^4$ update events.

Transmitters were identified as employing extortion strategies, $r_+^*$ or $r_-^*$, if their strategy lay within a $\Delta$-neighborhood of a ``true'' extortion strategy \cite{Stewart:2013fk,Hilbe:2013aa}, with  $\Delta=0.05$. Significant over-representation of extortion strategies among the most successful fake and mainstream sites (i.e those strategies that successfully produce engagement with fake and true stories respectively) was then determined by comparing the prevalence of the strategy in our data to the null distribution, i.e. the rate at which such strategies are randomly drawn. Significance level quoted is $p<0.01$.
\\
\\
\textbf{Experiments:} We asked participants to evaluate the accuracy of 1,000 articles across 40 mainstream and fake news sites, gathered from Crowdtangle. The study was approved by MIT COUHES (protocol 1806400195).
\\
\\
\textbf{Participants.} From 22 to 29 November 2020, we recruited 1,000 participants from Amazon Mechanical Turk who met the following three criteria: located in the US, more than 100 studies completed on the platform, and more than 95 per cent of them approved. The sample included 576 males and 424 females, with a mean age of 38.49 years (min. 19; max. 96). Median completion time was 4 minutes and 55 seconds.
\\ 
\\
\textbf{Materials.} We used Crowdtangle to gather the 25 most recently available news stories from 40 media outlets (i.e. 1,000 articles; 500 posted by 20 fake news sites and 500 posted by 20 mainstream sites), along with the headline, lede, date of publication, link and level of engagement. See Supplementary Information Section 3 for additional details. We then used this data to present the participants outlined above with 20 article headlines and ledes, drawn randomly from within one of the two media outlet subsets, and asked them to assess the accuracy of the information they were faced with. Specifically, they were asked "Do you think this story is true?", to which they responded on a seven-point scale from ``Definitely NO'' to ``Definitely YES''. The study concluded with seven demographic questions (age, gender, education, political conservativeness on social and economic issues, political position, and political preference) and a section to leave comments. 

\clearpage

\section*{Supplementary Information}

In this supplement we present a formal analysis of the fake news game, along with additional details of the modeling framework. We then compare our model to three prior models of bias and misinformation in news production, followed by robustness checks of our findings to alternate model formulations and assumptions. Finally we present additional details of experiments and meta-analysis.

\section*{Formal analysis of the fake news game}

The stage game payoff matrix for the fake news game is described in Table S1 for a single receiver and a single transmitter. We consider an infinitely iterated version of this game, which in general is played by a single transmitter and $N$ receivers, as described in the Methods section of the main text. This iterated game describes the long term dynamics of a receiver being served news stories by a transmitter.

\begin{figure}[th!] \centering \includegraphics[scale=0.2]{TableS1.pdf}
\caption*{\small Table S1: \textbf{ Stage game payoff matrix} }
\end{figure}

\subsection*{Stationary distribution}

The stationary distribution for the infinitely repeated game is encoded by the vector of probabilities $\mathbf{v}$. Each entry $v_{ki}\in [0,1]$ describes the stationary probability that the game is in state $(k,i)$, where $k \in \{0, 1, 2, \ldots N\}$ gives the number of receivers who choose to consume the piece of news shared by the transmitted, and $i\in \{t,f\}$ encodes the type of news shared. The master equation for the Markov process associated with the infinitely repeated game is given by Eq. 4 of the main text. At equilibrium, for a true news item $i=t$, this equation has the form
m
\begin{equation}\tag{5}
v_{kt}=\sum^N_{m=0}\sum_{l\in\{t,f\}}v_{ml}r_{ml}\sum_{A\in F_k}\prod_{\alpha\in A}q_\alpha\prod_{\beta\in A^C}(1-q_\beta).
\end{equation}
An analogous equation holds for $i=f$ by replacing $r_{ml}$ with $(1-r_{ml})$.

In order to solve for the stationary distribution of this process, marginalized over all possible consumption values $k$, we first note that the overall rate of sharing fake news by the transmitter is

\begin{equation}\tag{6}
v_{f}=\sum_{k=0}^N v_{kf}
\end{equation}
\\
and $v_t=1-v_f$ is the overall rate of production of true news by the transmitter. Applying this definition to Eq. 5 we recover

\begin{equation}\tag{7}
v_{t}=\sum^N_{m=0}\sum_{l\in\{t,f\}}v_{ml}r_{ml}
\end{equation}
\\
since in summing over all possible engagement levels $k$, we sum over the probabilities of all possible combinations of receiver consumption which necessarily total 1. If we now replace $r_{ml}$ in Eq. 7 with the form given in Eq. 2 we recover

\begin{equation}\tag{8}
v_{t}=\sum^N_{m=0}v_{mt}\left(\alpha+\gamma\frac{m}{N}\right)+v_{mf}\left(\beta+\theta\frac{m}{N}\right).
\end{equation}
\\
Now note that 
\begin{equation}\tag{9}
v_{tc}=\sum^N_{m=0}v_{mt}\frac{k}{N}
\end{equation}
\\
is the overall rate of receivers consuming true news. And so Eq. 8 can be written as

\begin{equation}\tag{10}
v_{t}=\alpha v_t+\gamma v_{tc}+\beta v_f+\theta v_{fc}
\end{equation}
\\
Replacing $v_f=1-v_t$ and rearranging gives us the relationship between the proportion of transmitted stories that are false and the overall rate at which receivers consume true and false stories, presented in Eq 3 of the main text. 

\subsection*{Definition of coercion and extortion strategies}

We have defined a coercive transmitter strategy as one that always serves more news of a particular type to receivers who consume more news. Thus a coercive fake news strategy serves more fake stories to receivers who consume more of the transmitter's stories. This is described as ``coercive'' because receivers who seek to increase their benefit from true stories by engaging more, in fact receive increasing amounts misinformation. In contrast, receivers who consume little of the transmitter's stories are targeted with more true stories and thus are incentivized to engage more. The conditions for a strategy to be coercive are $\gamma>0$ and $\theta>0$ (see Methods). 

We define an extortion strategy as a particular type of coercive strategy that additionally enforces $v_f>v_{c}$ (for fake news sites), where $v_c=v_{tc}+v_{fc}$ is the overall rate of consumption by receivers. Similarly, for mainstream sites extortion strategies must enforce $v_t>v_{c}$ . The term ``extortion'' was previously introduced in reference to analogous strategies studied in the Prisoner's Dilemma \cite{Press:2012fk,Stewart:2013fk,Stewart:2012ys,Hilbe:2013aa}. Extortion strategies in the fake news game constitute a strong form of coercion. In particular the extortion strategies studied here ensure that the rate of increase in fake (or true) news exceeds the rate of increase of overall consumption by receivers. 

\subsection*{Necessary and sufficient conditions for extortion}

The condition for a fake news site to be in extortioner is that $v_f>v_{c}$ for all receiver strategies. In order to find necessary and sufficient conditions for a strategy to enforce such a relationship, first define the following parameters \cite{Stewart:2013fk,Akin2}

\begin{equation*}
\kappa=\frac{1 -  \alpha - (\gamma - \theta)/2}{1 - \alpha + \beta + \theta}
\end{equation*}

\begin{equation*}
\lambda=-\frac{(\gamma - \theta)/2}{1 - \alpha + \beta + \theta}
\end{equation*}

\begin{equation}\tag{11}
\chi=-\frac{\gamma + \theta}{2(1 -  \alpha + \beta - (\gamma - \theta)/2)}
\end{equation}
where $|\chi|\leq1$ is required to produce a viable strategy (see Methods). Note that the denominator in Eq 11, namely $(1 -  \alpha + \beta - (\gamma - \theta)/2)$, is required to be positive to produce a viable strategy (see Methods) and thus any fake news promoting coercive strategy necessarily has $\chi>0$.

Substituting Eq. 11 into Eq. 3 produces the following expression

\begin{equation}\tag{12}
v_f-\kappa+\lambda(v_{tn}+v_{fc})=\chi\left[v_c-\kappa+\lambda(v_{tn}+v_{fc})\right]
\end{equation}
\\
where $v_{tn}$ is the rate at which true news is shared but not consumed. Eq. 12 is convenient because it allows us to state the conditions for extortion as i) $\chi>0$ (which is necessary for the strategy to be coercive) and ii) $v_c\leq\kappa+\lambda(v_{tn}+v_{fc})$ for all possible receiver strategies (which is necessary to ensure $v_{f}>v_c$, given that $|\chi|\leq 1$). If these two conditions are met, the strategy is coercive and $v_f>v_c$ for all receiver strategies, and so the strategy is an extortioner.

The first condition is met provided $\gamma + \theta<0$ (since $\gamma<1-\alpha$ is required to produce a viable strategy). To asses the second condition note that ii) implies $v_f+v_c\leq 2\kappa-2\lambda(v_{tn}+v_{fc})$. Also note that $v_f+v_c=1+v_{fc}-v_{tn}$ and so condition ii) is equivalent to

\begin{equation}\tag{13}
1+v_{fc}-v_{tn}\leq 2\kappa-2\lambda(v_{tn}+v_{fc})
\end{equation}
\\
All that remains is to inspect the limiting cases, where this conditions is most stringent. A receiver who never consumes enforces $v_{fc}=0$ and Eq. 13 becomes  $1-v_{tn}\leq 2\kappa-2\lambda v_{tn}$ which is guaranteed to hold provided 

\begin{equation*}
\kappa \geq 1/2
\end{equation*}

and

\begin{equation}\tag{14a}
\kappa\geq \lambda
\end{equation}
\\
The other limiting case of a receiver who always consumes enforces $v_{tn}=0$ and generates conditions 

\begin{equation*}
\kappa \geq 1/2
\end{equation*}
and
\begin{equation}\tag{14b}
\kappa\geq 1+\lambda.
\end{equation}

Eq. 14b is the more stringent of these two conditions and thus it provides necessary and sufficient conditions for extortion. Substituting back from Eq. 11 these conditions become

\begin{equation*}
\theta=-\beta
\end{equation*}
and
\begin{equation}\tag{15}
\alpha + \beta+\gamma \leq 1.
\end{equation}

The equivalent condition for a mainstream transmitter is
\begin{equation*}
\gamma=1-\alpha
\end{equation*}
and
\begin{equation}\tag{16}
\alpha + \beta+\theta \leq 1. 
\end{equation}
\\
These are the conditions used to define extortion strategies discussed in the main text. 



\subsection*{Nash equilibria in the fake news game}

Our model assumes inattentive receivers who noisily optimization their engagement strategies as they seek to consume true stories. Such a process does not necessarily lead to optimal receivers behavior, or even to behaviors predicted by standard solution concepts such as Nash equilibria. However, it is useful to characterize Nash equilibria for the fake news game. 
 


For a mainstream site, which seeks to promote accurate information, there is no conflict between the incentives of transmitter and receiver, and so they both should adopt a Nash equilibrium in which the receiver always engages and the transmitter only shares true stories.
For a fake news site, which seeks to promote false news, there is a conflict between the incentives of transmitter and receiver. 

The Nash equilibria for the two-player fake news game can be easily characterized using the Folk Theorem for infinitely iterated games \cite{8563}. The minimax value for receivers in the fake news game is 0, which is achieved by never consuming. If the receiver does consume, the transmitter can always hold them to a lower payoff by transmitting fake news, assuming that the cost of consuming fake news is positive, $C>0$. The minimax value for the transmitter is also 0, since we assume the payoff to the transmitter of fake news being consumed is $b_f\geq0$ and also the payoff from true news being consumed is $b_t\geq 0$, whereas the payoff when no news is consumed is 0, and so the transmitter's minimax profile is uniquely determined by a receiver strategy that never consumes (Table S1). Thus any feasible payoff profile that provides a positive payoff to both players is individually rational.

\begin{figure}[th!] \centering \includegraphics[scale=0.2]{FigureNS11.pdf}
\caption*{\small Figure S1: \textbf{Feasible, individually rational payoffs for the fake news game --} The set of feasible payoffs (blue region) and the subset of individually rational payoffs for both players (striped region) for a fake news game with  $b_f>b_t$ meaning that the transmitter prefers to generate consumption with fake news. Individually rational payoffs for both players only arise when both players receive a positive payoff. An equilibrium in which the receiver always consumes and the transmitter always shares fake news, $(-C,b_f)$ is feasible but not individually rational.}
\end{figure}

Any individually rational, feasible payoff is a Nash equilibrium of the game (Figure S1), and, since the game is assumed infinite, can always be enforced by players employing a grim trigger strategy. That is, if the receiver deviates from the prescribed course of play, the transmitter switches to only sharing fake news. Any benefit gained by the receiver from deviating is lost in the long run, since the best they can do is to cease consuming and receive zero payoff in every subsequent round. Conversely, if the transmitter deviates from the equilibrium, the receiver can switch to never consuming news, with the same result that any benefit to the transmitter gained by deviating eventually lost. As discussed above, the set of individually rational, feasible payoffs (Figure S1) in the fake news game corresponds to cases in which both transmitter and receiver have positive payoffs. Although a fake news site receives  payoffs  $b_t=0$ and $b_f>0$, this nonetheless means that they must share a mixture of both true and fake news, since the payoff that results from only sharing fake news produces a negative payoff to the receiver.

To explore this point further, consider a fake news site who simply transmits true stories with fixed probability $r$ each round. The payoff to a receiver who consumes news from the receiver at equilibrium rate $v_c$ is simply $w_t=v_c(Br-C(1-r))$, which is positive provided  $r>\frac{C}{B+C}$. If this condition is met, the receiver's payoff increases with consumption, and so an equilibrium at which the receiver always consumes and the transmitter shares true stories with probability $r>\frac{C}{B+C}$ can be enforced as a Nash equilibrium of the fake news game (e.g. by employing a grim trigger, as discussed above). This illustrates the need, in a general sense, for a fake news site to ``mash up'' true and fake stories to produce engagement. Finally, note that if $b_t>0$, $r=1$ and $v_c=1$ produces positive payoffs for both players, and can also be enforced as a Nash equilibrium -- i.e. mainstream news sites do not need to mash up true and fake stories when faced with rational receivers who value true news. 

Thus our model predicts that misinformation sites should always share a mixture of true and fake stories if their readers value accuracy, regardless of whether those readers are rational (as per this Nash equilibrium calculation), or lazy and irrational (as per our simulations above).

\subsection*{Modelling local optimization}

For the most part we do not assume that receivers in the fake news game are perfectly rational, but rather update their behavior according to a local optimization process \cite{Traulsen:2006zr}. Under this process a receiver temporarily adopts an alternate strategy $j$ to the one they are currently employing, $i$, and compares the payoff they receive with the new strategy to that received under the old strategy. They then adopt the new strategy, and discard the old, with probability $\pi_{i\to j}$ determined by a Fermi function.

\begin{equation}\tag{17}
\pi_{i\to j}=\frac{1}{1+\exp[\sigma(w_i-w_j)]}
\end{equation}
\\
where $w_i$ is the payoff received under strategy $i$, and $\sigma$ determines the level of attention the player pays to their payoffs. If $\sigma<1$ little attention is paid to payoffs and the optimization process is very noisy, with receivers frequently adopting suboptimal consumption strategies. If $\sigma\gg1$ a high level of attention is paid to payoffs, and only strategies that improve payoffs are adopted.

When choosing the alternate strategy we assume that $j$ is small perturbation $\Delta_{\mu}\in[0,0.05]$ to the current strategy, such that each element of the receiver strategy  $\mathbf{p}=\{p_0,p_{ct},p_{cf},p_{nt},p_{nf}\}$ is increased or decrease by an independently drawn $\Delta_{\mu}$, with the constraint that each element must remain a viable probability (i.e. $p_{ij}\in[0,1]$).

\subsection*{Modelling optimization through social learning}

In addition to local optimization, receivers whose strategies are updated through a process of cultural evolution via payoff-biased imitation \cite{Traulsen:2006zr}. Under this model, a pair of receivers, $i$ and $j$, are selected at random from the population of receivers. Receiver $i$ then adopts the strategy of receiver $j$ with probability

\begin{equation}\tag{18}
\phi_{i\to j}=\frac{1}{1+\exp[\sigma(w_i-w_j)]}
\end{equation}
\\
In addition, mutations are introduced to the receiver population at rate $\mu$, under which a player spontaneously adopts a news strategy drawn uniformly from the space of possible receiver strategies. Note that Eq. 17 and Eq. 18 are identical, but here $j$ is drawn from the strategies present in the population, while in Eq. 17, $j$ is chosen as a perturbation to the current strategy. As such, the process of copying other players' strategies tends to result in many different players using the same strategy, compared to the case where each player independently optimizes their own strategy. 

\clearpage




\section*{Comparison to previous models}

In this section we compare our model to three previously proposed theories of misinformation engagement and/or production \cite{10.1257/jep.31.2.211,Vosoughi1146,PennyNature}. We ask whether the theories presented in these papers can explain 
our empirical results (main text Figure 3). That is, we ask whether our proposed theory is uniquely predictive of the observed data. We find that, indeed, the other theories cannot reproduce the observed patterns

\subsection*{Vosoughi et. al. 2018}

Vosoughi et. al. \cite{Vosoughi1146} present empirical results showing that, among news stories that were questionable enough to get fact-checked -- by snopes.com, politifact.com, factcheck.org, truthor-fiction.com, hoax-slayer.com, or urbanlegends.about.com -- false news spread further on social networks and persisted longer. They also find that false news was more novel, in an information-theoretic sense, than true news. The authors are careful to note that ``Although we cannot claim that novelty causes retweets or that novelty is the only reason why false news is retweeted more often, we do find that false news is more novel and that novel information is more likely to be retweeted.''

This suggests that readers derive benefit from engaging with misinformation due to its novelty. However such an explanation, absent feedback between receiver and transmitter sharing strategy, would predict greater engagement with false stories among both fake and mainstream sites
, which is not consistent with our empirical findings (main text Figure 3).

\subsection*{Pennycook et. al 2021}

Pennycook et. al \cite{PennyNature} investigate the role of inattention in shaping consumer engagement with misinformation. The authors present a model in which receiver utility is derived from a consumer's  preferences and the level of attention they pay to a topic. Fitting the model parameters to experimental data they find
`
the best-fit preference parameters indicate that participants value accuracy as much as or more than partisanship. Thus, they would be unlikely to share false but politically concordant content if they were attending to accuracy and partisanship''
suggesting a model in which consumers prefer true news to false news,
and occasionally share  misinformation accidentally, due to inattention. This account would therefore predict
greater engagement with true stories than false stories, regardless of whether to producer is a fake news or mainstream sit
, which is inconsistent with our empirical findings (main text Figure 3).







\subsection*{Allcott and Gentzkow 2017}
Allcott and Gentzkow \cite{10.1257/jep.31.2.211} develop a model of supply and demand of fake news. Under this model  ``Fake news arises in equilibrium because it is cheaper to provide than precise signals, because consumers cannot costlessly infer accuracy, and because consumers may enjoy partisan news.''

This model considers consumer decisions at the level of the news outlet (e.g. ``shall I subscribe to this newspaper or not?'') rather than at the level of the individual story, and thus does not make direct predictions about our key empirical outcome, the relative rate of engagement for true versus false news within a given outlet. However, the model's implications can be extrapolated to our setting. 
On the demand side, this means that consumers/receivers either (i) engage with fake news accidentally due to inattention or (ii) derive benefit from engaging with fake news due to e.g. partisan preference. Inattention alone, absent feedback between transmitter and receiver sharing strategy, leads in the limit to equal levels of engagement with true and false news. Similar to the case of a consumer preference for novelty discussed above, a partisan preference for false news, absent feedback between receiver and transmitter sharing strategy, would predict greater engagement with false stories among both fake and mainstream sites, which is not consistent with our empirical findings (main text Figure 3). 


On the supply side, this theory posits that transmitters broadcast false news because it is cheaper to produce than true news (i.e.~the net payoff for transmitters from receivers engaging with false news is greater than it is for true news).
By this logic, it is payoff maximizing for transmitters to produce false news so long as the engagement advantage of true news over false news is sufficiently small (or non-existent) -- i.e.~due to receivers being inattentive and/or preference for falsehoods. This explanation for the production of fake news, while entirely plausible, makes no prediction for the different patterns of engagement among consumers with mainstream versus fake news sources, which is our key empirical result (main text Figure 3). 







\clearpage


\section*{Model robustness}

In this section we perform robustness checks on our model, and show that our results hold as we alter our model specification. We show that i) our results are robust to different definitions of ``successful'' transmitters ii) our results are robust to different models of receiver learning, iii) our results are robust to different receiver group sizes but require a moderate to high degree of transmitter microtargetting, iv) our results require low levels of receiver attention, v) our results are robust to competition between transmitters and vi) our results are robust to changes to consumer demand for misinformation.

\subsection*{Identifying successful transmitter strategies}

In order to find successful mainstream and fake news transmission strategies we generated $10^8$ receiver strategies as described in Methods. We then defined successful fake news transmitters as those which achieved i) in the 90th percentile for fake news engagement, and ii) a higher rate of transmitting fake than true news, $v_f>0.5$. Similarly, we defined successful mainstream transmitters as those in the 90th percentile for true news engagement who transmit a higher rate of true news than fake, $v_t>0.5$. These definitions were used to select the strategies show in Figure 2 of the main text. The full, unfiltered distribution of engagement and consumption for the generated strategies is shown in Figure S2a-b. The distributions have a weak bias towards true news consumption ($p<0.001$) reflecting the noisy optimization process of the receivers, who prefer to consume true and avoid fake news.

\begin{figure}[th!] \centering \includegraphics[scale=0.2]{FigureS12.pdf}
\caption*{\small Figure S2: \textbf{Full distribution of engagement and consumption --} Shown are the distributions of the difference in engagement (a) and consumption (b) with true vs fake stories for $10^8$ transmitter strategies. We define successful fake news strategies as those with $v_f>0.5$ and fake news engagement, $v_{fc}/v_f\geq0.704$ which is the 90th percentile among all observed strategies.  Similarly we define successful mainstream strategies as those with $v_t>0.5$ and fake news engagement, $v_{tc}/v_t\geq0.758$ which is the 90th percentile among all observed strategies. c) We calculated the average difference in engagement as a function of the threshold rate of sharing the transmitter's proffered news type used when to determine which transmitter strategies are successful. We see that both mainstream (blue) and fake news sites (red) produce an engagement difference that is insensitive to this choice of threshold. d) However the overall consumption rate is highly sensitive to the threshold, so that when the threshold is low the preferred news type is consumed less frequently than the non-preferred type. This arises because many successful sites produce high levels of engagement with their preferred news type while sharing it rarely.}
\end{figure}

We also explored the effect of changing the definition of ``successful'' transmitters of true and fake news. In the main text we look at sites in the top 10\% of engagement, who share their preferred news type greater than 50\% of the time. In Figure S2c-d we vary this threshold and look at the average difference between engagement and consumption. We see that engagement difference is insensitive to the threshold choice, whereas consumption difference reverses direction when the threshold is low, indicating that there are many news sites who are good at producing engagement with true or fake news while also sharing such stories rarely.

\clearpage


\subsection*{The effect of receiver learning model choice}

We studied the effect of different receiver learning model choice. In particular we explored social learning via a model of cultural evolution, as described in Section 1.6 of this Supplement, on the level of engagement and consumption in a population of receivers. Figure S3 shows the impact of group size on engagement and consumption of fake and true news with a single fake news transmitter as in Figure 4 of the main text. Assuming that the rate of strategy mutations is $\mu=1/N$ where $N$ is the size of the group of receivers, we see that increasing group size has no effect on consumption and only a modest effect on engagement, tending to increase true news engagement and reduce fake news engagement, without reversing the pattern of higher engagement with fake stories than with true.

\begin{figure}[th!] \centering \includegraphics[scale=0.5]{FigureNS5.pdf}
\caption*{\small Figure S3: \textbf{Cultural evolution of receiver strategy --} a) A group of receivers evolving in response to a single transmitter, under the same process as described is Section 1.6 of the Supplement. We vary the size of the receiver group (x-axis) and calculate the equilibrium level of engagement with fake (orange) and true (purple) stories. b) We also calculate the equilibrium level of consumption. In all cases receiver mutations were local (see Methods) and we set receiver payoffs $B=2$ and $C=1$. Here misinformation sites use a fixed coercive strategy $\mathbf{r}=\{1.0,0.1,-0.9,-0.1\}$ see Methods}
\end{figure}


\clearpage

\subsection*{The effect of receiver group size and transmitter microtargetting}


We studied the effect of relaxing the assumption, employed in the main text, that a transmitter can directly target each receiver, in response to his or her engagement with the previous news article. We show that the effects observed -- in particular the apparent attractiveness of fake news -- hold under a wider range of conditions in which a transmitter targets groups of individuals. 

First we studied the effect of varying the size of the group of receivers engaging with a single transmitter, when all receivers use the same strategy to engage with the transmitter, with optimization taking place at the level of the group rather than at the level of the individual, as in Figure 5 of the main text. Figure S4 shows that, in such a scenario, group size has no impact on either engagement or consumption even as group size varies across two orders of magnitude.

\begin{figure}[th!] \centering \includegraphics[scale=0.5]{FigureNS6.pdf}
\caption*{\small Figure S4: \textbf{Homogenous receiver groups --} a) A group of receivers optimizing in response to a single transmitter, under the same process as described for main text Figure 2, with the extension that that optimization occurs at the level of the group rather than the individual. We vary the size of the receiver group (x-axis) and calculate the equilibrium level of engagement with fake (orange) and true (purple) stories. b) We also calculate the equilibrium level of consumption. In all cases receiver mutations were local (see Methods) and we set receiver payoffs $B=2$ and $C=1$. Here misinformation sites use a fixed coercive strategy $\mathbf{r}=\{1.0,0.1,-0.9,-0.1\}$ see Methods}
\end{figure}



Third we explored the effect of microtargetting on receiver engagement with misinformation in the face of a coercive transmitter. The engagement dynamics between a single transmitter and single receiver corresponds to the case of direct micro-targeting by a transmitter, in the sense that the transmitter makes their decision about what type of news to share (e.g.~by promoting stories on social media) with a receiver based solely on the engagement (or not) of that specific receiver. In reality, this degree of precision is unrealistic, but as we have shown above (Figure S3-S4) the same effects also hold when a transmitter's audience is a group of receivers who employ similar behavioral strategies. 

In order to explore how micro-targeting impacts the engagement and consumption habits of receivers interacting with a coercive fake news strategy, we define the strength of micro-targeting as $M=1/G$, where $G$ is the number of independent individuals or groups to whom the transmitter promotes the same news. Here a ``group'' refers to a newsfeed, a social media page, or distribution list to which the transmitter promotes content. A group may contain only a single individual, or be composed of many people who share the same interest. When a transmitter promotes the same content to many such groups ($M\ll1$), each group updates their engagement independently in response, and so the strength of micro-targeting is low: a change in the behavior of any one group has a small impact on the overall engagement with the transmitter (Main Text Eq.~2), so the transmitter is not very responsive to a given receiver. When the strength of micro-targeting is high (e.g. when there is only one transmitter and one receiver, $M=1$) the transmitter is highly responsive to the engagement pattern of a given receiver.

Figure S5 shows the effect of varying the degree of micro-targeting available to the transmitter. We find that when micro-targeting is low the difference in engagement between false versus true stories declines, until the apparent attractiveness of fake news disappears entirely. Thus the ability to target news stories at specific groups of receivers, either by news sites directly or by social media algorithms, can substantially contribute to the apparent attractiveness of false news stories.



\begin{figure}[th!] \centering \includegraphics[scale=0.6]{NNNFigure4.pdf}
\caption*{\small Figure S5: \textbf{Micro-targeting facilitates engagement with fake news}  a) A single misinformation site micro-targeting multiple independent receivers. The strength of micro-targeting $M$ (x-axis) quantifies the responsiveness of the transmitter to each of the target groups of receivers (see main text). Each target receiver group is assumed to independently optimize their engagement strategy as described previously (Figure 2). The misinformation site uses the same transmission strategy for all the groups included in their targeting. The less precise the targeting of the transmitter (lower values of $M$), the smaller the difference in engagement between true and false news stories. b) However, the overall consumption rate remains unchanged with $M$.  In all cases receiver strategic mutations are local (see Methods) and receiver payoffs are $B=2$ and $C=1$. Other parameter choices are explored in the SI. Here the misinformation site uses a fixed coercive strategy $\mathbf{r}=\{1.0,0.1,-0.9,-0.1\}$
see Methods}
\end{figure}



\clearpage


\subsection*{The effect of different types of receiver attention}

We studied the effects of varying different types of receiver attention to news content and to transmitter behavior. In the analysis presented in Figure 2 of the main text we assume that receivers are inattentive to both prior experience with a transmitter and prior information about the veracity of each headline, i.e. $a_0=a_1=0$ in Eq. 1. In Figure S6 we repeat the same analysis assuming receivers are attentive to past experience, i.e. $a_1=1$ in Eq. 1. We see that, as in Figure 2, the most successful fake news strategies tend to produce greater consumption and engagement with fake stories than with true. Conversely, the most successful mainstream strategies produce more engagement and consumption of true stories than false.

\begin{figure}[th!] \centering \includegraphics[scale=0.2]{FigureNS3.pdf}
\caption*{\small Figure S6: \textbf{Successful news transmission strategies with receiver memory --} To identify strategies that are successful at promoting engagement with true news stories (mainstream sites) or with fake news stories (misinformation sites), we randomly drew $10^8$ transmitter strategies (Eq.~2) and allowed a single receiver, incentivized to engage with true news, to optimize their engagement strategy (see Methods) over the course of $10^4$ interactions. The receiver was assumed to be inattentive to accuracy ($a_0=0$) but attentive to past stories ($a_1=1$) and optimization was assumed to be noisy ($\sigma=1$). a) We identified the transmitter strategies that  successfully promote engagement with true news stories (mainstream sites, blue), i.e those that produce a true news engagement rate within the 90th percentile from the $10^8$ transmitter strategies considered, as well as $v_f<0.5$. Similarly, we identified the transmitter strategies that successfully promote engagement with fake news (misinformation sites, red), i.e those that produce a fake news engagement rate within the 90th percentile, as well as $v_f>0.5$. We  plot the difference between engagement with true and fake stories, $v_{tc}/v_t-v_{fc}/v_f$, for all mainstream and misinformation sites. Mainstream site strategies induce engagement with accurate stories while misinformation site strategies induce the opposite effect. b) We also report the difference in overall consumption of true stories, $v_{tc}$, and consumption of fake stories $v_{fc}$. c) Summary of the engagement rates per story and overall consumption rates for mainstream and misinformation site transmission strategies. In all cases receiver mutations were local and we set receiver payoffs $B=2$ and $C=1$. }
\end{figure}



The distribution of engagement and consumption for all $10^8$ transmitter strategies when receivers employ memory of their previous interaction ($a_1=1$) is shown in Figure S7. We see that the distribution of engagement is narrower than in Figure S2, when receivers are inattentive to memory, while the distribution of consumption follows a similar distribution in both cases.

\begin{figure}[th!] \centering \includegraphics[scale=0.2]{FigureNS4.pdf}
\caption*{\small Figure S7: \textbf{Full distribution of engagement and consumption with receiver memory --} Shown are the distributions of the difference in engagement (a) and consumption (b) with true vs fake stories for $10^8$ transmitter strategies. We define successful fake news strategies as those with $v_f>0.5$ and fake news engagement, $v_{fc}/v_f$ in the 90th percentile among all observed strategies.  Similarly we define successful mainstream strategies as those with $v_t>0.5$ and fake news engagement, $v_{tc}/v_t$ in the 90th percentile among all observed strategies.}
\end{figure}


We also considered (Figure S8) the effects of receivers conditioning their behavior on memory of previous interactions beyond simply the immediately preceding encounter. We assumed that receivers treated a transmitter as having shared fake news with them when employing their strategy $\mathbf{p}$ (Eq. 1) if the transmitter has shared a fake story in \emph{any} of the last $m$ rounds. We show engagement and consumption as a function of such memory in Eq. S7, where we see that longer memory of this type has only a weak effect on engagement and consumption, and does not by itself reverse the trend of successful fake news sites generating higher engagement and consumption of fake vs true stories among inattentive consumers.



\begin{figure}[th!] \centering \includegraphics[scale=0.5]{FigureNS7.pdf}
\caption*{\small Figure S8: \textbf{Receiver memory length --} a) A single receiver optimizing in response to a single transmitter, under the same process as described for main text Figure 2. The receiver uses memory of preceding rounds to decide whether the transmitter shares true stories (and so is labelled $t$) or fake stories (and so is labelled $f$). We assume that if the receiver identifies any item of fake news in their memory of the transmitter's shared stories, they label the transmitter as fake ($f$). We vary the length of this memory (x-axis) and calculate the equilibrium level of engagement with fake (orange) and true (purple) stories. b) We also calculate the equilibrium level of consumption. In all cases receiver mutations were local (see Methods) and we set receiver payoffs $B=2$ and $C=1$. Here misinformation sites use a fixed coercive strategy $\mathbf{r}=\{1.0,0.1,-0.9,-0.1\}$ see Methods}
\end{figure}


We also studied the effect of varying attention to headline accuracy in Eq. 1, against successful fake news promoting transmitter strategies. We varied attention to accuracy, $a_0$, memory of past interactions, $a_1$ and attention to payoffs, $\sigma$ among receivers. We also considered the case where receivers both prefer true news (as in the main text) and the case where they prefer fake news, i.e. derive benefit $B$ from interacting with fake news and pay cost $C$ for interacting with true news (Figure S9). 

\begin{figure}[th!] \centering \includegraphics[scale=0.2]{FigureS8.pdf}
\caption*{\small Figure S9: \textbf{Attention and news consumption --} We randomly drew $10^5$ misinformation strategies (see Methods) and allowed receivers to optimize as described in Figure 2 of the main text. (a-b) When receivers have no memory $a_1=0$, and pay little attention to long-term payoffs, $\sigma=1$, but prefer accurate information, we increased attention to accuracy in the optimization process ($a_0$) (x-axis), and calculated a) the average rate of engagement with true (purple) and fake (orange) stories. As attention to accuracy increases, overall engagement with fake stories decreases to zero, while engagement with true stories increases. b) Similarly, overall consumption rate of fake stories decreases with increasing attention to accuracy, while consumption of true stories increases. 
(c-d)  When receivers have no memory $a_1=0$, and no attention to accuracy, $a_0=0$ , but prefer accurate information, we increased attention to long-term payoffs in the optimization process ($\sigma$) (x-axis), and calculated c) the average rate of engagement with true (purple) and fake (orange) stories. As attention to payoffs increases, overall engagement with true and fake stories decreases to zero. d) Similarly, overall consumption rate of true and fake stories decreases with increasing attention to payoffs.
(e-h) Shows the same plots as in (a-d) except with memory of past interactions, i.e. $a_1=1$. (e-f) show similar patterns as (a-b) while (g-h) shows increasing engagement and consumption for true news (purple) with increasing attention to payoffs ($\sigma$).
(i-l) Shows the same plots as in (a-d) except with receivers preferring fake news, i.e. gaining benefit $B$ from interacting with a fake story and paying a cost $C$ from interacting with a true story. (i-j) show opposite patterns to (a-b) while (k-l) show k) increasing engagement for both true and fake news engagement with attention to payoffs, but l) declining consumption of true news.
(m-p) Shows the same plots as in (e-h) except with receivers preferring fake news, i.e. gaining benefit $B$ from interacting with a fake story and paying a cost $C$ from interacting with a true story. (m-n) show opposite patterns to (e-f) while (o-p) show opposite patterns to (g-h). 
In all cases we set attention to payoffs $\sigma=1$ and receiver payoffs $B=2$ and $C=1$ unless otherwise stated.}
\end{figure}

We see that in many cases increasing attention not only reduces engagement and consumption among receivers who prefer true news (Figure S9 panels a-h) it can also reverse the apparent preference for fake news induced by coercive transmitter strategies. However increased attention tends to have the opposite effect on engagement and consumption when receivers prefer fake news (Figure S9 panels i-p).




\clearpage


\subsection*{The effect of competition between news sources}



We studied the effect of competition between news sources, with a receiver facing a single misinformation site within in a population of otherwise mainstream sites (Figure S10). We see that the receiver tends to engage and consume more fake news from the misinformation site as the population size of mainstream sites grows. This makes intuitive sense, because the relative contribution of the misinformation site to the receiver's payoff necessarily declines as the population grows, and so avoiding fake news becomes less important to receiver payoff.

\begin{figure}[th!] \centering \includegraphics[scale=0.5]{FigureS11.pdf}
\caption*{\small Figure S10 \textbf{Groups of transmitters --} a) A single receiver optimizing in the face of a population of transmitters, under the same process as described for Figure 2. The transmitter population consists of a single misinformation site and $N-1$ mainstream sites. Increasing the number of mainstream sites increases engagement with true (purple) and fake (orange) news stories from the fake news transmitter as well as b) overall fake news consumption. In all cases receiver mutations were local (see Methods) and we set receiver payoffs $B=2$ and $C=1$. Other parameter choices are explored in the SI. Here misinformation sites use a fixed coercive strategy $\mathbf{r}=\{1.0,0.1,-0.9,-0.1\}$ and mainstream sites use a fixed coercive strategy $\mathbf{r}=\{0.9,0.0,0.1,0.9\}$ see Methods}
\end{figure}

\clearpage


\subsection*{The effect of supply and demand}


We studied the effects consumer demand on transmitter supply of misinformation. To do this we varied receiver payoff on consumption and engagement, shown in Figure S11 and Figure S12. In Figure S11 we fix the cost $C$ of consuming fake news, and vary the benefit $B$ of consuming true news, for different levels of attention to payoff, $\sigma$. We see that increasing benefits of true news often has only weak effects on consumption and engagement, but do tend to increase overall levels of both once benefits become sufficiently large. 


\begin{figure}[th!] \centering \includegraphics[scale=0.3]{FigureNS8.pdf}
\caption*{\small Figure S11: \textbf{Varying benefit of true news consumption --}  We randomly drew $10^5$ coercive fake news strategies (see Methods) and allowed receivers to optimize as described in Figure 2 of the main text. a) When attention to payoff is low, ($\sigma=0.1$), cost of engaging with fake news is fixed, ($C=1$), and receivers have no memory and no attention to story accuracy, $a_0=a_1=0$, we increased the benefit of engaging with true news stories $B$ in the optimization process (x-axis), and calculated the average rate of engagement with true (purple) and fake (orange) stories. As attention to payoffs increases, overall engagement slowly increases and the difference in engagement with true and fake stories declines. b) Similarly, consumption when attention to payoffs is low ($\sigma=0.1$) remains close to constant. 
c) When attention to payoffs is moderate ($\sigma=1$) the same pattern holds for engagement while d) consumption slowly increases, and the difference between true and fake news consumption also increases. e) When attention to payoffs is strong ($\sigma=10$) there is little difference between true and fake news engagement, and overall engagement increases with $B$ while f) overall consumption increases, and the difference between true and fake news consumption also increases. Unless otherwise stated parameters are the same as in Figure 4 of the main text.}
\end{figure}

In Figure S12 we fix the benefit $B$ of consuming true news, and vary the cost $C$ of consuming fake news, for different levels of attention to payoff, $\sigma$. We see that increasing costs of fake news has a strong effect on consumption and engagement, reducing both close to zero once costs become sufficiently large. This result is consistent with our game-theoretic expectation that higher costs require a fake news sharing strategy to include ever higher levels of true news to make engagement worthwhile for the receiver.

\begin{figure}[th!] \centering \includegraphics[scale=0.3]{FigureNS9.pdf}
\caption*{\small Figure S12: \textbf{Varying cost of true news consumption --}  We randomly drew $10^5$ coercive fake news strategies (see Methods) and allowed receivers to optimize as described in Figure 2 of the main text. a) When attention to payoff is low, ($\sigma=0.1$), benefit of engaging with true news is fixed, ($B=1$), and receivers have no memory and no attention to story accuracy, $a_0=a_1=0$, we increased the cost of engaging with fake news stories $C$ in the optimization process (x-axis), and calculated the average rate of engagement with true (purple) and fake (orange) stories. As attention to payoffs increases, overall engagement slowly decreases and the difference in engagement with true and fake stories declines. b) Similarly, consumption when attention to payoffs is low ($\sigma=0.1$) declines as $C$ increases. 
c) When attention to payoffs is moderate ($\sigma=1$) the same pattern holds for engagement and d) consumption. e) When attention to payoffs is strong ($\sigma=10$) both engagement and f) consumption decline to zero once $C>1$. Unless otherwise stated parameters are the same as in Figure 4 of the main text.}
\end{figure}






\clearpage

\section*{Supplementary information on experiments}

Here we provided additional details and analysis of our experimental findings.

\subsection*{Headline selection and engagement data}

We studied 20 mainstream and 20 fake news sources, used in previous studies of misinformation online \cite{Pennycook2521}. We selected the most recent 20 unique news headlines with engagement data available for Facebook via CrowdTangle. We excluded stories whose headlines did not constitute news (e.g. a quiz). We also studied the relationship between source trust and patterns of engagement without a priori groupings of mainstream and misinormation sites (main text Figure 3b). To do this we used trust ratings from \cite{DiasMisinfoRev}.

We gathered engagement data via CrowdTangle on the most recent 20 news headlines for each news source as available on November 22nd, 2020.This information was gathered by generating a historical report for each news site and gathering the 20 most recent news headlines in the database. The engagement information recorded for each story is the number of Facebook interactions reported by CrowdTangle at the the time. This number accounts for likes, comments and shares of posts that contain the URL for the article across the CrowdTangle database.

The list of headlines for each source, the headline accuracy rating, the publication date of the story and the Facebook engagement information recovered via CrowdTangle is available via github. 

\subsection*{Meta-analysis}

We performed a meta-analysis on the results of linear regressions of engagement against accuracy, for the 20 fake news and 20 mainstream sites. In order to determine whether there was a negative correlation between engagement and accuracy for fake news sites, we performed Fisher's combined test, using the p-values resulting from a single-tailed t-test from a regression of engagement against accuracy. we used the same procedure to determine whether there was a positive correlation between engagement and accuracy for mainstream news sites. 

To determine whether there was significant heterogeneity among fake news or mainstream sites we calculated the between study variance $\tau^2_{DL}$ using the  DerSimonian and Laird \cite{DERSIMONIAN1986177} moment-based estimate, and calculated the proportion of total variance attributable to between study effects, $I^2$, as well as the degree of homogeneity $Q_{RE}$ under a random-effects model \cite{https://doi.org/10.1002/sim.1186}. These results are summarized in Table S2 where we test regressions of $\log_{10}$-engagement vs accuracy, using both unstandardadized and standardized data, as well as regression using the rank of engagement and accuracy for each site. In all cases we find significant negative correlation for fake news and significant positive correlation for mainstream sites with $p<0.01$. However we find no significant between site heterogeneity for either $I^2$ or $Q_{RE}$ with $p<0.05$.

\begin{figure}[th!] \centering \includegraphics[scale=0.25]{TableS2.pdf}
\caption*{\small Table S2: \textbf{ Meta analysis --} The mean effect size $\hat{\mu}_{RE}$ for correlations with weights from a random effect model, as well as measures between study heterogeneity $I^2$ and $Q_{RE}$. Significance is indicated with an asterisk (*).}
\end{figure}


