Information often diffuses via communication between friends or social acquaintances. However, people transmit not only correct information, but also rumors, that is, false or imprecise information.
The virality of these rumors shapes public debates, sometimes involving significant personal and social costs. Take, as an example, the link between HIV and AIDS. Policies inspired by conspiracy theories denying this link are being held responsible for around 350, 000 deaths in South Africa between 2000 and 2005 (Chigwedere et al., 2008; Nattrass, 2008) . Furthermore, belief in these conspiracy theories is associated with less consistent use of condoms in the US (Bogart and Thorburn, 2005) .
While the diffusion of rumors is not a new phenomenon, its investigation has recently gained in prominence due to its perceived increase, which is associated with an increased reliance on virtual platforms for news consumption and communication. For example, studies have shown that in the 2016 US Presidential elections, online social platforms were more likely to facilitate the spread of "fake news" than mainstream media (Guess, Nagler and Tucker, 2019; Silverman, 2016; Allcott and Gentzkow, 2017) . Similarly, in a report by Cramer (2018) , around 40% of parents stated that they are regularly exposed to negative messages about vaccinations on social media. This has prompted demands on policy makers, news providers and online social platforms to counter the diffusion of rumors (Frenkel and Isaac, 2018) . Most recently, misinformation relating to the COVID-19 pandemic has caused responses from various online communication providers, all aiming to suppress its diffusion (Marr, 2020) .
However, there is little understanding about why rumors diffuse on social networks, which makes it difficult to reduce their propagation. In order to provide informed policy recommendations, it is important to consider individual choices and the associated equilibrium effects on the type of content that circulates. Indeed, rumors appear to diffuse even when proven, scientific, information is available that would debunk them (Silverman, 2016) ; and although exposure to correct information decreases the probability of believing misinformation (Porter, Wood and Kirby, 2018) , it is far from eradicating such beliefs. In fact, individuals exposed to correct information may nevertheless remain vocal proponents of rumors (Zollo et al., 2015) .
To fill this gap, the aim of the present paper is to understand how rumors diffuse by proposing a model in which two distinct messages about the state of the world (the truth and the rumor) circulate simultaneously when individuals can debunk the rumor.
In particular, we assume the society is partitioned in groups according to ideological predisposition with respect to rumors, which defines their bias. It has been documented on many topics that people who tend to believe in fake news can clearly be identified in the society and in online social communities (e.g., Bessi et al. 2016; Samantray and Pin 2019) . 1 Also, we assume all individuals are better off if they know the truth. Hence, they have an interest in debunking the rumor by exerting a costly verification effort.
One's verification effort is determined by the probability she attaches to her bias being correct, which in turn depends on the message she hears. Specifically, upon successful verification, individuals become aware of the true state of the world and accept it. If verification is unsuccessful, individuals believe the message in line with their bias, which is the message they think is more likely, when their bias is sufficiently strong. Thus, the message heard and the opinion held afterwards do not necessarily coincide.
Individuals communicate their opinion to their neighbors, but they cannot credibly reveal their bias, nor whether they verified. Socialization is characterized by a certain degree of homophily, which captures the likelihood that one interacts with people of the same initial bias.
We find that the rumor always survives alongside the truth. Hence, contrarily to what an epidemiological model would predict (Prakash et al., 2012) , false information may become endemic even if it can be debunked and all individuals seek the truth. Rumors survive unless agents are able and willing to perfectly verify all information that contradicts their bias, which can happen only if verification costs are zero, which is unlikely even in our digital era. Rumors do not need the help of malign or irrational agents to thrive.
The truth to rumor ratio depends on the level of verification and the degree of homophily, and not on other network characteristics nor the communication technology. Hence, new technologies for the consumption and the diffusion of information affect the relative diffusion of the two opinions only if they change these parameters. This stresses that an increase in rumor diffusion cannot be imputed to online social platforms per se.
However, new communication technologies might have indirectly facilitated the spread of rumors by congesting communication opportunities. As communicating becomes cheaper, the opportunity cost of verification increases. This naturally decreases verification, thus benefiting the rumor.
Additionally, new communication technologies might have increased homophily by offering less constrained meeting opportunities than offline socialization. 2 In our model, an increase in homophily implies that individuals become less confident in the messages they hear, which incentivizes verification. However, it also implies that individuals receive relatively more messages in line with their bias, which get verified less. We derive general conditions under which either of these effects dominates and discuss examples of verification functions for either case.
Our model delivers interesting insights on how to fight the diffusion of rumors. Specifically, it highlights that any policy which amounts to a onetime injection of truthful information, such as a time-limited information 2 Existing evidence suggests that online patterns of news consumption are no more segregated than offline ones (Gentzkow and Shapiro, 2011; Halberstam and Knight, 2016) . The levels of homophily in online communication are however extremely high (see, e.g., Zollo et al., 2017) , and online platforms appear to lead to more segregated communication patterns (Halberstam and Knight, 2016) , which is the channel on which we focus.
campaign, is ineffective in reducing the ratio of rumor to truth in the long run. Instead, policies should focus on lowering verification costs as this incentivizes individuals to verify information themselves.
Our results are robust to the introduction of stubborn individuals who never verify information and never change their opinion. Indeed, the presence of these individuals gives higher incentives to verify so as to completely offset any negative impact. Ignoring endogenous debunking would lead to dramatically different conclusions.
Our paper contributes to the recent literature that uses game theory to understand the diffusion of information in networks. 3 Bloch, Demange and Kranton (2018) find that, when there are partisans who try to diffuse false information, other individuals filter the diffusion of rumors by blocking messages coming from parts of the network with many partisans. While their paper focuses on strategic information transmission, we study debunking of rumors through costly verification. This introduces a different aspect of interaction: aggregate verification rates affect the incentives of individuals to verify. In this context, individuals cannot affect the steady state prevalence of the opinions, so there is no strategic motive to transmit them.
Our model uses a SIS framework, a class of models introduced to study the diffusion of viruses. 4 Following the seminal work of Kremer (1996) , some papers have studied how strategic decisions on protection or immunization affect diffusion of a disease (Chen and Toxvaerd, 2014; Goyal and Vigier, 2015; Toxvaerd, 2019; Talamàs and Vohra, 2020; Bizzarri, Panebianco and Pin, 2020) . In particular, Galeotti and Rogers (2013) study the effect of homophily in a model with two groups. In these papers, protection is a local public good. 5 In our model, verification of discordant messages is instead increasing in aggregate verification. 6 The novelty of our paper is that we model multiple states, and in particular, how the diffusion of the rumor (the disease) interacts with that of the truth. In Prakash et al. (2012) , two viruses spread simultaneously on a network and grant immunity against each other. They find that only the more contagious virus survives, i.e., rumors would die out. Relatedly, Campbell, Leister and Zenou (2019) and Tabasso (2019) study the simultaneous diffusion of two types of information that are non-contradictory. While also in these papers homophily may induce polarization, a denser network favors the minority message. A new feature of our model is that individuals can disbelieve a message they hear. This implies that not only rumors survive, but that both opinions equally share the benefits of increased communication.
More broadly, our paper relates to a recent literature on opinion dynamics on random graphs. Akbarpour, Malladi and Saberi (2018) and Sadler (2020) study information cascades, while Campbell (2013) and Carroni, Pin and Righi (2020) study monopoly pricing when consumers learn about products via word-of-mouth. In those papers, agents either adopt or not without the option of external verification, and non-adoption does not create any externality. In our model, one's decision depends instead on the level of (non-)verification in the economy.
We believe we are the first to study the strategic decision of individuals to verify what they hear when there are several messages diffusing in a network.
The paper proceeds as follows. Section 2 introduces the model. Section 3 presents the main analysis. Section 4 discusses the policy implications. Section 5 presents the model with stubborn agents. Section 6 concludes. All proofs are in the Appendix.
There exists an infinite population of individuals of mass 1, indexed by i, who are represented as nodes on a network. Time is continuous, indexed by t. There exist two messages m ∈ {0, 1} that individuals diffuse via word of mouth when they communicate with the network of their social contacts. These messages pertain to the state of the world, Φ ∈ {0, 1}. Without loss of generality, we denote Φ = 0 as the true state of the world, ex-ante unknown to the individuals. We refer accordingly to m = 0 as the "truth," and m = 1 as the "rumor". When individual i tells individual j her opinion transmitting message m, this reveals to j that there is a debate over the true state of the world and the set of values that Φ may take.
Individuals. All individuals of type b ∈ {0, 1}, where each type constitutes half of the population, have k meetings at each period of time t. 7
Regarding information, individuals may be in one of two states: either they are entirely unaware of the debate about the state of the world, in which case they are in state S (Susceptible), or they may hold an opinion about which value it takes, in which case they are in state I (Informed ).
Someone in state S transitions into state I in a period by hearing message m during a meeting. At each meeting, communication occurs at rate ν.
Individuals in state I die at rate δ and are replaced with individuals of the same type in state S. All our results hold also in the limit as δ goes to zero. Thus, we are able to capture the evolution of rumors which diffuse over whole lifespans (such as the HIV-AIDS or the vaccination-autism links) as well as more short-lived rumors that diffuse in a constant population. We assume that neither ν nor δ are affected by one's type or opinion.
Individuals' types are related to their informational bias. When an indi-vidual in state S hears message m from one of her contacts, she decides a verification effort. An individual's type determines which opinion she holds when there is no verification. In this case, an individual of type b is of the opinion that Φ = b upon receiving message m, irrespective of the identity of m. This is a simplifying assumption that captures in reduced form the following process. We assume below that individuals use Bayes' rule to update their beliefs on how likely it is that Φ = b upon receiving m. Since it is a corner solution, there exists a high enough prior y that will lead individuals who do not verify to hold the opinion in line with their type independently on the message received. Such value of y might be higher than 1/2 for same values of β; however, we assume y = 1/2 for tractability. 8 The degree of homophily in meetings, β, represents instead the meeting bias. In particular, we assume that a fraction β ∈ [0, 1] of meetings are with individuals of the same type b, while 1 − β are across types. Types are not observable, but β is common knowledge. For example, in the context of online social networks, it is natural to assume that by frequenting certain chat rooms or online groups, users do not know the identity/type of every specific user, but they have a reasonable estimate about how likely it is that they communicate with "like-minded" people.
Types do not affect payoffs from holding a specific opinion. All individuals in state I expect a (present value) lifetime utility of 1 if their opinion coincides with the true state of the world and 0 otherwise. We set the utility of being in state S to 0. 9 The interpretation is that being informed of the true state of the world allows individuals to choose correct actions given the state.
Utility and Verification. Individuals who hear m choose how much effort 8 We do not model how types originate. Rather, we consider them as broader than the specific debate regarding Φ and thus unlikely to be affected by it. As an example, types might be high vs. low trust in modern medicine. Individuals of these two types differ in their opinions on a variety of issues, of which safety of vaccinations is one.
9 Transitioning into and out of this state is not affected by individual choices and therefore irrelevant for marginal considerations.
to exert in verification. Exerting effort α ∈ [0, ∞) allows an individual to determine that Φ = 0 with probability x(α) where x(·) : R + → [0, 1]. With the complement probability, the result of the verification process is inconclusive, so that an individual will hold an opinion in line with her bias. The marginal costs of exerting verification effort are given by c. 10 The only assumption that we make on x(·) is that it is concave. This implies that the function can be discontinuous only in α = 0, that the limits
x(α) exist and that x(·) is strictly increasing (and hence invertible). Moreover, the derivative dx(α)/dα does not exist only in a countable and non-dense set of points. When it exists, it is decreasing and lim α→∞ dx(α)/dα = 0. The limit lim α→0 + dx(α)/dα exists and we call itd. For every point x ∈ [x,x] we can define the subderivative (Rockafellar, 1970; Greenberg and Pierskalla, 1973) 
if x =x and dx(α) dα exists (and is null).
This correspondence is upper hemicontinuous, non-empty, closed, and convex. Moreover it is strictly decreasing and invertible, in the sense that any
; g is continuous and weakly decreasing. A functional form that we will use as an illustrative example is x(α) = 1− e −α . This function results from the following verification process. Consider an individual who searches for information, which consists of n realizations leading to an answer with probability p n per realization. Hence, the search gives at least one answer with probability (1 − p n /n) n . If np n = α as the number of realizations n goes to infinity and the success of each realization goes to zero, this probability converges to 1 − e −α . 11
Another functional form we will discuss is the piece-wise linear case of
with s > 0 andx ∈ (0, 1). This function results from the following verification process. Consider an individual who searches for information on external sources, such as Wikipedia. For each unit of effort or time that she invests in the search, she gets a new piece of information at a rate s. The process continues until all the available information, denoted byx, is collected. If verification fails or the agent does not verify, while the individual holds the opinion that Φ = b, she knows that there is a chance that this is incorrect. She thus forms an expectation of being correct conditional on the message she heard, P r t (b = Φ|m).
The timeline is as follows. An individual i who hears message m at time t chooses effort level α it and becomes aware of the true state of the world with probability x(α it ). Having formed her opinion of the true state of the world, she takes an action a it to match it. The payoff from this action materializes at the end of her lifetime, which she expects to be in 1/δ periods, with a present value of 1 if she has correctly matched the true state of the world and 0 otherwise. While she is alive, the individual communicates to her social contacts a message in line with her opinion at rate ν.
Our assumption that payoffs materialize at the end of an individual's life captures the reality of many rumors for which it is not realistically possible for individuals to infer the true state of the world (and hence update their opinion) by taking an action. Examples of such cases include the link between AIDS and HIV, or being of the opinion that vaccinations are safe. In these instances, information is generally received much earlier than any decisionsuch as having protected sex or vaccinating your child-is taken. Payoffs may materialize only a long time after the action has been taken, or, as for vaccinations, individuals cannot calculate the probability of adverse effects relying on their own experiences. 12 The utility of individual i at time t, having heard message m and choosing effort α it is then
(1)
Verification occurs at rate x(α it ). If it fails or there is no verification, one's bias is correct with probability P r t (b = Φ|m). The last term represents the cost of verification. Individual i thus chooses her verification effort such that
where g is the inverse of the subderivative of x. It follows from (2) that equilibrium verification effort is independent of the identity of i. However, we allow for the optimal verification effort to depend on the message received. We denote by α b t the equilibrium effort individuals take in verifying a message in line with their type, i.e., m = b and by α −b t the equilibrium effort to verify m = b. To simplify the formal exposition, we call ≡ x(α b * ) and h ≡ x(α −b * ) the verification rates of same-type and different-type messages. 13 Transmission Process. Individuals in state I communicate a message in line with their opinion when they meet others. A meeting between two individuals is described by a link. The associated network is realized every period. Formally, we model the mean-field approximation of the system, whereby each agent takes as given overall verification levels and information 12 In the case of infinitely lived agents, we may assume instead that payoffs materialize at time T , after all individuals have become informed and taken their respective actions. 13 We show later that in equilibrium it will always be the case that ≤ h.
prevalence as she is infinitesimal among a continuum of individuals. Denote by ρ b m,t ( t , h t ) the proportion of type b individuals that hold opinion in line with m at time t. 14 Note that ρ 0 1,t ( t , h t ) = 0, as individuals of type b = 0 may only hold opinion 0, irrespective of verification. The overall prevalence of opinion in line with m in the population at time t is then equal to ρ m,t ( t , h t ), and calculated as follows
With some abuse of notation, we sometimes suppress the dependence on ( t , h t ) and denote the prevalence of opinions ρ m,t , for m = {0, 1}. We assume that the per contact transmission rate, ν, is sufficiently small that each individual in state S who meets another in state I becomes informed through meeting k neighbors in an independent fashion, so information transmits at rate kν. The laws of motion governing the transmission of the system are:
∂ρ 1
These expressions describe the evolution of opinions in line with m = 0 and m = 1 within the two groups b ∈ {0, 1}. Take for example expression (5), which describes how m = 0 evolves in group b = 0. The first term describes the mass of individuals who start holding opinion Φ = 0 at time t:
The proportion of susceptible individuals of type 0, (1 − ρ 0 0,t ), which receive either m, which occurs at rate νk βρ 0 0,t + (1 − β)(ρ 1 0,t + ρ 1 1,t ) . The second (negative) term, indicates that a proportion δ of the individuals of type 0 that are informed revert to state S. Note that for this group, verification rates are irrelevant. The interpretation of equations (6) and (7) is equivalent.
Equilibrium. We are interested in the equilibrium of the model in the steady state. In an equilibrium, the profile (α b , α −b ) satisfies equation (2) for both m = b and m = b. The model is in steady state if equations (5), (6) and (7) are equal to zero. An equilibrium is stable if the system defined by (5), (6) and (7) is asymptotically stable in the equilibrium coordinates.
We now derive the steady state prevalence of truth and rumor in the population, taking verification as given. We then solve for equilibrium verification.
We are interested in the steady state of the system, that is, the situation in which the prevalence of the truth and of the rumor are constant over time. We remove the time subscript t to indicate the steady state value of variables.
The effects of ν and δ can be summarized by a unique parameter, which we define as λ = ν/δ and label it as the effective diffusion rate. We denote by ι b the proportion of type b ∈ {0, 1} individuals in state I (irrespective of their opinions) with ι = (ι 0 + ι 1 )/2 as overall information prevalence. With this notation in place, we can state the following.
Remark 1 There always exists a steady state in which the prevalence of both the truth and the rumor are zero.
The existence of a zero steady state is a trivial aspect of the SIS model: if no individual ever transmits an information, nobody can ever become informed. The more interesting questions are under which conditions one (or more) positive steady states exist, and which steady states are stable.
Proposition 1 For given verification rates, if λk > 1, there exists a unique steady state of the economy, which is positive and stable, and with ι 0 = ι 1 = ι = 1 − 1/(λk). If λk ≤ 1, only the zero steady state is stable.
Thus, in both groups the proportion of informed individuals is the same, equal to information prevalence in the overall population. This proportion is as in a standard SIS model. Indeed, as each m informs individuals about the set of possible states that Φ can take, we can set m ≡ Φ ∈ {0, 1}, and solve the model as a single-state SIS model with two groups.
While Proposition 1 is important in establishing the steady state proportion of informed individuals, the crucial question is the extent to which the rumor diffuses. In this regard, it is also interesting to consider how this relates to the diffusion of the truth. Proposition 2 addresses these questions.
Proposition 2 For any positive and exogenous levels of verification l and h, there exists a unique positive steady state prevalence for each opinion, ρ 0 and ρ 1 . This is the unique stable steady state if and only if λk > 1. Furthermore: i) ρ 0 and ρ 1 are increasing in the effective diffusion rate λ and in the number of meetings k; ii) the truth to rumor ratio ρ 0 /ρ 1 is greater than 1, increasing in both verification rates, l and h, and independent of k and λ; iii) the truth to rumor ratio, ρ 0 /ρ 1 , is decreasing in homophily if individuals verify more a message not in line with their bias, and increasing otherwise.
The proof of Proposition 2 rests on deriving the steady state prevalence of truth and rumor:
These equations show that the steady states of opinions inherit uniqueness and stability from ι. Since for λk ≤ 1, neither opinion is endemic, we focus in the remainder of the paper on the case λk > 1. Equation (9) highlights that, unless h = 1-which is too costly given the assumptions on x(·)-both opinions survive. Thus, rumors survive in the long run even if verification is possible and all individuals are better off if their opinion is correct.
Equations (8) and (9) show that with zero verification effort, ρ 0 = ρ 1 = ι/2, and the truth prevalence increases in any form of verification while rumor prevalence decreases. Hence, for any positive amount of verification the truth exhibits a larger prevalence than the rumor.
The insights that Proposition 2 offers into potential relationships between online social networks and the diffusion of rumors are rich. One factor through which online social networks allegedly stimulate the diffusion of rumors is the ease with which messages can be communicated, and the number of people receiving them. Thus, one generally expects them to have increased k, ν, or both. Proposition 2 shows that our model's predictions are in line with this view. However, it stresses that the truth and the rumor equally benefit from an increase of the ease of communication due to an increase in the number of meetings k or in the effective diffusion rate λ.
This result has several implications. First, while empirical studies on the impact of online communication often focus on the diffusion of rumors alone (e.g., Bessi et al., 2016) , a comparison with the diffusion of truthful messages would be of a greater interest. Second, if rumors have indeed become more prevalent in relative terms, this cannot be explained by online social networks increasing communication rates per se. We discuss in Section 4 how ease of communication might have indirect effects on relative rumor prevalence.
The role of homophily is similarly intriguing, as high degrees of homophily are commonly associated with an increased diffusion of rumors as people are likely to hear only messages in line with their bias ("echo chambers"). In fact, in our model the impact of homophily on the diffusion of truth and rumor depends entirely on how likely individuals are to verify m = b as opposed to m = b. Given verification rates, homophily indeed benefits the rumor and harms the truth if individuals are more likely to verify messages against their bias than those aligned with their bias. While such behavior appears intuitive, it motivates us to study endogenous verification in Section 3.2.
Overall, our results indicate that verification rates of both messages are the single-most important variables in determining the truth to rumor ratio in the network, both directly and indirectly through determining the impact of homophily. In fact, no other variable can affect this ratio in the long run.
Individuals choose verification according to equation (2). Individuals are aware of the transmission process; hence, when they first hear a message, they can calculate the probability that b = Φ conditional on having heard it. We assume that they perform this calculation using Bayes' rule, and that they have unbiased priors of being of type b = Φ of .5. 15
Updating is not entirely Bayesian as individuals do not update their beliefs in case they receive additional messages. This greatly simplifies our analysis. However, this assumption is not overly restrictive. The only group affected by it are type 1 individuals who did not successfully verify the first message they heard and hence hold opinion m = 1. Thus, our results on rumor prevalence are upper bounds with respect to a fully Bayesian model.
Using (8) and (9), these probabilities in steady state are equal to
It follows immediately that P r(b = Φ|m = b) is increasing in verification of either message, while P r(b = Φ|m = b) is decreasing. Indeed, more verification implies that more truth circulates, increasing the credibility of the message received. Hence, more verification implies that an agent's bias is more likely to be correct if she hears a message in line with this bias, and less so if she hears the opposite one. Furthermore, P r(b = Φ|m = b) and P r(b = Φ|m = b) are equal if and only if no verification occurs, when they are both 1/2. In this case, both opinions are equally prevalent and receiving either message does not allow individuals to update their beliefs about how likely their bias is correct. (10) and (11) in equation (2) allows us to derive the following result regarding equilibrium verification rates.
Proposition 4 In any equilibrium, α b ≤ α −b , with equality if and only if both are zero. Equilibrium verification is independent of the number of meetings, k, and the effective diffusion rate, λ. Any equilibrium takes one of the following forms: i) If c ≥d/2, there is no verification and the truth to rumor ratio is 1. ii) If c ∈ [c,d/2), individuals verify only messages against their bias. iii) If c < c, in any equilibrium both messages are verified. Furthermore, c is increasing in homophily, β.
Finally, the equilibrium is stable if and only if, either it coincides with a zero steady state and λk ≤ 1, or it is positive and λk > 1.
The intuition behind these results is the following. When verification is very costly, no message is verified; as a result, there is an equal mass of individuals holding each of the two opinions. As verification costs decrease, individuals first verify the message against their bias. Hence, agents of type 1 hearing message 0, disbelief the rumor, whose prevalence decreases, while type 0 individuals always believe the truth, independently of verification. When verification costs decrease further, individuals verify both messages, but they put a higher effort in verifying the one against their bias.
The threshold of verification cost below which both messages are verified is increasing in homophily. Indeed, when individuals are more likely to meet people with the same bias, they attach a lower informational content in hearing a message in line with their bias, thereby triggering more verification.
Finally, Proposition 4 provides conditions for the stability of the endogenous equilibrium that come directly from Proposition 1, as these conditions are independent from those determining verification.
However, when the verification efforts are endogenous the condition for stability does not guarantee uniqueness. We present below two simple and intuitive examples in which uniqueness is also guaranteed. We leave the mathematical derivations for these examples to the Appendix.
Let x(α) = 1 − e −α . In this case ∆(α) is a function for all α > 0, and we have the simple linear condition g(x) = 1 − x. There exists a unique equilibrium. If c ≥ 1/2, no message is verified. Otherwise, (i) Individuals always verify messages against their bias. (iii) When positive, verification efforts of both messages are strictly increasing in homophily.
Consider the piece-wise linear case 
The truth to rumor ratio is the fraction of the prevalence of both opinions among informed individuals. i.e., ρ 0 /ρ 1 . As discussed above, Proposition 4 already offers insights into the importance of homophily for the truth to rumor ratio, stressing that more homophily might trigger people to verify. We now derive the effect on homophily also within each equilibrium class. The truth to rumor ratio derives from equations (8) and (9) as follows
Expression (12) highlights the two opposing effects of homophily on the truth to rumor ratio. First, there is a direct effect: higher homophily implies a larger exposure to messages in line with one's bias, which are verified less. Hence, keeping verification constant, this tends to decrease the truth to rumor ratio. The indirect effect instead acts through changes in verification rates.
An increase in those exerts a positive force on the truth to rumor ratio. The following allows to better understand these two effects.
Proposition 5 In equilibrium, the truth to rumor ratio ρ 0 /ρ 1 is equal to
This result shows that the function L, representing the subderivative of the verification of the message in line with one's bias, is a sufficient statistic to study the truth to rumor ratio. While L depends on the verification of both messages, this result is useful in deriving additional intuitions. Indeed, the effect of homophily on the truth to rumor ratio is
When only the message against one's bias is verified, l is constant, and so is L. This happens when c ∈ (c,d/2), i.e., when c > 2L(0). Hence, an increase in homophily decreases the truth to rumor ratio as long as it does not increase c so much as to trigger verification of both messages. If instead both verification rates are positive, the first term of (13) is always positive. On the contrary, the sign of the second term depends on ∂L/∂β, that is, on how the derivative of l with respect to the verification effort changes with homophily. While this term cannot be signed in general, it highlights precisely that homophily has a negative impact on the truth to rumor ratio whenever verification does not significantly increase as a result of an increase in homophily.
In fact, while l is increasing and concave in homophily both with an exponential and piece-wise linear verification function, the effect of homophily differs in the two examples.
If x(α) = 1 − e −α as in Example 1, the truth to rumor ratio is weakly increasing in homophily. It is strictly increasing if > 0. If x(·) be piece-wise linear as in Example 2, the truth to rumor ratio is weakly decreasing in homophily. It is decreasing if > 0.
The intuition behind these conflicting results is the following. In the piece-wise linear case, as the verification rate of messages against one's bias remains fixed, the verification of messages in line with one's bias will have to increase more than in the case where both verification rates may adjust. Therefore, in the former case the positive effect of more verification overcomes the negative effect of more messages coming from the own group, while this does not happen in the latter.
Furthermore, whenever the verification function delivers a unique stable equilibrium as in these two examples, time-limited injections of either message can only affect the truth to rumor ratio for a limited amount of time and not in the long run. Thus, even if we consider endogenous verification choices, rumors cannot simply be debunked by increasing the prevalence of the truthful message in the network for a limited time (e.g., through an information campaign). Therefore, in the following section we discuss how changes in underlying parameters affect the incentives of individuals to verify, such as the costs of verification.
Our paper has been motivated by ongoing demands on policy makers and online network platforms to fight the spread of rumors. Much of the political discussion rests on the role that online social networks play in the diffusion of rumors. In the following, we address the most commonly cited arguments as to how these platforms are related to the diffusion of rumors.
Online social platforms offer individuals the chance to self-select in more homogeneous groups along several dimensions than offline communication, as meeting opportunities are less constrained online. Evidence that communication patterns are more segregated online than offline is found, e.g., in Halberstam and Knight (2016) for the case of Twitter. Similarly, Zollo et al. (2017) find user interaction with Facebook posts to be highly segregated, with levels that are above the most segregated offline communication networks found by Gentzkow and Shapiro (2011) . As long as these dimensions are correlated with views of the world, the widespread use of online social platforms implies an increase in homophily. This in turn implies that individuals are more likely to hear messages in line with their views, resulting in "echo-chambers". It is usually assumed that this mechanism is one of the culprits for the increase in the diffusion of rumors in the past decade.
Our preceding analysis make the mechanism behind this logic explicit, stressing under which conditions it is valid. As meetings are more homophilous, on the one hand, verification of both messages increase, as individuals realize that they hear more biased messages; on the other hand, individuals hear fewer discordant messages, that are verified less. Rumors thrive when this second effect dominates.
As online social platforms facilitate communication, they are commonly conjectured to have led to denser networks or to have increased the transmission rate of information. Indeed, increases in communication-whether modeled as increases in k or in λ-increase the measure of informed individuals, but it does not affect verification rates (Proposition 4). 16 At the same time, we have shown that without such an impact the truth to rumor ratio is unaffected by changes in these parameters. Therefore, while ease of communication leads to an absolute increase in rumor prevalence, this in itself does not imply a relative increase.
It is possible, however, that the costs of verification, c, instead depend positively on the amount of communication that occurs in the network. Such congestion/information overload effects occur naturally if we consider verification costs as time costs, which seems a plausible scenario. In that case, through increasing communication, online social networks may increase verification costs and increase rumor prevalence also in relative terms.
To fix ideas, consider individuals to have a fixed time endowment H in each period, which they can spend on communication (νk), verification (α), and on independent leisure activities y. Assume that agents obtain utility u(y) from their leisure activities, where u(·) is a continuous, increasing and concave function. Its concavity captures the decreasing marginal utility of private leisure. Agent i's utility at time t can then be expressed as
and optimal verification effort is chosen again according to equation (2) if we define c = du/dα i . In this case, any increase in νk increases verification costs and lowers the equilibrium verification rates. This setup furthermore implies that marginal costs of verification are increasing and convex in α. 17 We conclude that if increased communication creates congestion effects in verification, online social networks may indeed have increased not only the absolute but also the relative rumor prevalence.
The ease with which rumors can be debunked by agents in the network has been an important aspect in discussions on policy interventions. Many established news outlets, such as The New York Times or Le Monde, publish guides on how to recognize false information, and have introduced newsrooms where fake news is identified and debunked (Roose, 2018) . On various online social platforms, for example Facebook, disputed information may be "flagged" as such (Maidenberg, 2018) by independent fact checkers.
In our model, any form of fact checking that reduces verification costs unambiguously increases the truth to rumor ratio. It follows that fact checking has to be designed in such a way that it truly reduces verification costs.
We believe this is not always the case in the methods that have been proposed. For example, Facebook experimented with assigning "flags" to posts simply stating that they had been disputed. This practice has been abandoned in 2017 as it proved less effective to stop the spread of rumors than expected, with some evidence that it might even have promoted the spread of such posts. Instead, an alternative was suggested whereby next to a disputed post a user would see various links to the articles disputing it (Silverman, 2016) . In the context of our model, the earlier policy would have amounted to nothing more than injecting truthful messages into the network, and as such would indeed have been expected to be unsuccessful in the long run. The updated policy instead, by providing links to the sources of the dispute, can credibly claim to lower verification costs.
Overall, our model corroborates that one of the most intuitive policies to deter rumors-incentivizing individuals to verify information-is also the one most likely to succeed as long as it is carefully designed.
It is possible that the diffusion of rumors in social media is primarily driven by individuals that diffuse them independently of whether they are confronted with factual evidence disproving them. We now study the implication of the existence of such stubborn individuals on the diffusion of messages.
We introduce stubborn individuals as people who never verify, always hold an opinion in line with their stance and transmit messages accordingly. 18 We assume that a fraction γ of individuals of each type are stubborn. This does not impact the evolution of information in group 0, as both stubborn and non-stubborn individuals are always of the opinion that Φ = 0. However, there are fewer individuals of type 1 who may verify any message they hear. We then derive the following Proposition.
Proposition 6 Assume a fraction γ of the population consists of stubborn individuals. The prevalence of the truth and the rumor among the remaining 1 − γ individuals are unaffected by γ and remain as in the baseline model.
The driving force behind this result is the endogeneity of verification. Indeed, the existence of stubborn individuals implies that non-stubborn individuals place a higher probability of being told the rumor. This therefore leads to a higher degree of verification among them, which offsets the negative impact that stubborn individuals have on the relative prevalence of truth and rumor. Formally, in the proof of Proposition 6 we show that the prevalence of both opinions can be re-written as in the baseline model with verification ratesx(α) = (1 − γ)x(α) instead of the original x(α)'s. This leads to the equilibriumx(α) being unaffected by stubborn individuals. 19 18 Alternatively, for all β < 1, we can think of stubborn individuals as having a prior of 1 of being biased towards the true state of the world, thus they never update. If β = 1, individuals can never meet someone with the opposing bias. Hence, hearing the opposite message is conclusive proof that one's bias is wrong, contradicting the certain prior.
19 Note that this does not imply that verification efforts α adjust linearly.
In this paper, we model how a correct message and a rumor diffuse in a population of individuals who seek the truth. Individuals verify the correctness of the messages they hear based on the probability that what they hear is correct. They are biased in the sense that, if they do not verify, they hold the opinion that adheres to their view of the world. We find that the rumor survives for any positive cost of verification. New communication technologies increase its absolute prevalence, however, its prevalence relative to the truth depends exclusively on verification and the degree of homophily in meetings. While verification costs have obvious effects, the impact of homophily is more nuanced. On the one hand, higher homophily leads to higher verification rates. On the other, an echo chambers effect emerges: individuals are more exposed to messages in line with their bias, which are verified less. The virality of rumors increases when the second effect dominates.
We employ our results to discuss policies to debunk rumors. Crucially, while injections of truthful messages are ineffective in debunking the rumor in the long run, successful policy interventions revolve around incentivizing individuals to verify. In the light of our model then, new communication technologies played a role in making rumors more viral only if they increased homophily in meetings or verification costs-due to, for example, information overload.
In our model, we assume that there only exist two opposing opinions and messages. This excludes the possibility that malevolent agents may aim to decrease the spread of the truth by creating new opposing messages over time. Similarly, biases may not be fixed but depend on past communication, as in Bolletta and Pin (2019). The analysis of these phenomena seems a promising avenue for future research.
Proof of Proposition 1. To prove Proposition 1, we first combine equations (6) and (7) to analyze the evolution of ι 1 :
In steady state, information prevalence in either group then is
from which it is obvious that there exists one steady state in which ι 0 = ι 1 = ι = 1 − 1/(λk). We now show that this steady state is unique. From equations (A-1) and (A-2), we can show that in any steady state with positive information prevalence it must be that
If ι 1 > ι 0 , then the RHS of this equation is positive while the LHS is negative, and vice versa for ι 1 < ι 0 . It can only hold if ι 0 = ι 1 = ι. Finally, either both groups have positive information prevalence, or neither. Deriving the Jacobian of the differential system reveals that both eigenvalues are negative at the positive steady state if and only if λk > 1. The steady state of zero information prevalence has instead two negative eigenvalues if and only if λk < 1. This concludes the proof of Proposition 1.
Proof of Proposition 2. To derive equations (8) and (9), we employ the fact that from equation (5), ρ 0 0 = ι 0 . By Proposition 1, ι 1 = ι 0 = 1 − 1/(λk). By plugging in these values into the steady states of equations (6) and (7), equations (8) and (9) obtain. Simple derivatives of (8) and (9) show that the prevalence of the truth is increasing, while the prevalence of the rumor is decreasing, in and h. Hence, the lowest value that ρ 0 can take, and the highest value of ρ 1 , is at = h = 0, when they are both equal to ι. The truth always has at least as high a prevalence as the rumor. This concludes the proof of Proposition 2.
Proof of Proposition 3. The conditions for any equilibrium can be written as conditions on two numbers , h ∈ [x,x] such that
Define G the function from [x,x] 2 that applies to the left-hand side of the above two equations a couple ( , h) and then applies function g to both the solutions. G is a continuous function from [x,x] 2 to itself. By Brouwer's fixed point theorem we have an equilibrium characterized by a couple ( , h) that is a fixed point of G. This concludes the proof of Proposition 3.
Proof of Proposition 4. The result that in equilibrium, α b ≤ α −b , with equality if and only if both are zero follows directly from the fact that P r(b = Φ|m = b) ≥ P r(b = Φ|m = b) and the preceding discussion regarding their equality at zero verification rates. Substituting the steady state values of P r(b = Φ|m = b) and P r(b = Φ|m = b) into equation (2) yields the following conditions describing equilibrium verification efforts:
It follows that neither k nor λ affect α b and α −b . Substituting α b = α −b = 0 in equations (A-3) and (A-4) shows that whenever c ≥d/2, there exists a unique equilibrium, and, in this, individuals exert zero verification efforts. The results on information prevalence and lack of impact of homophily follow. For α −b > 0, we need c <d/2. By substituting α b = 0, α −b > 0 into equation (A-3), we find that α b = 0 is an equilibrium at α −b > 0 if and only if
where c is increasing in β and decreasing in h. When α b = 0, the equilibrium condition for α −b becomes
i.e., α −b is decreasing in β. Furthermore, in this equilibrium, the truth to rumor ratio is decreasing in β as long as α b remains null. Hence, the overall effect of β on c is positive. This means that an increase in homophily may trigger verification of both messages as for all c < c, any equilibrium consists of both α b > 0 and α −b > 0. It also implies that at β = 0, there does not exist an equilibrium in which α b > 0 and α −b > 0.
Finally, as λ and k do not affect verification rates, stability follows from Proposition 1. This concludes the proof of Proposition 4.
Computations for Example 1. If x(α) = 1 − e −α , ∂x(α)/∂α = 1 − x(α).
If c ≥ 0.5, there is no verification and ∂x(0)/∂α = 1. We first discuss the stability of the possible steady states. When α b = α −b = 0, equations (A-3) and/or (A-4) hold with inequality. Local stability is then guaranteed as zero efforts are best replies even after small deviations. Denote the RHS of (A-3) and (A-4) by y and z, respectively. When (A-3) and (A-4) hold with equality,
The corresponding Jacobian matrix is
(1 − h) 2 .
All eigenvalues are negative and thus all equilibria are stable as
If we solve for and h and define E ≡ 3 − 2c + β(3c − 2), we find:
We write h as a function of as this is more convenient for further derivations. After some algebra, we can write ρ 0 /ρ 1 as ρ 0 ρ 1 = 3 + β(2 − 5c) − (1 + 4β(1 + β) + 4c − 6β(1 + 2β)c + (2 − 3β) 2 c 2 ) 2c (A-7) The derivative of its numerator with respect to c is −5β + −2 + 3β + 6β 2 − (2 − 3β) 2 c 1 + 4β(1 + β) + 4c − 6β(1 + 2β)c + (2 − 3β) 2 c 2
Since the derivative with respect to c of the second term of this expression is positive, as it is equal to 8β (6β 2 + β − 2) ((2 − 3β) 2 c 2 − 6β(2β + 1)c + 4β(β + 1) + 4c + 1) 3/2 , let us study the derivative of the numerator of ρ 0 /ρ 1 at c = 0. Then, this is equal to −1 − 2b 2 + 1/(1 + 2b), that is negative. Hence, the numerator of ρ 0 /ρ 1 is decreasing in c. Since the denominator is increasing in c, then ρ 0 /ρ 1 is decreasing in c.
Computations for Example 2. The piece-wise linear verification function is the following
It is easy to see that
• if c ≥ s/2, the equilibrium is such that = 0 and h = 0;
• if c < s/2 and s ≥ c(2−x(1−β))
, the equilibrium is such that =x and h =x. If we put these last two expressions into the formula of the truth to rumor ratio, from (12), then H cancels out and we obtain exactly
The results follow from the fact that if L ∈ ∆( ) then g(L) = . Finally, note that from (12) the truth to rumor ratio when α b = 0, i.e., l = x, is
Remember from Proposition 4 that α −b is decreasing in β while c is increasing in β when l = x. Hence the truth to rumor ratio is unambiguously decreasing in homophily if α b stays at 0. This concludes the proof of Proposition 5.
Computations for Example 3. From equation (A-8) we have the expression of the truth to rumor ratio, with respect to some L ∈ ∆( ), but when x(·) is differentiable in an interior point, then L is just the derivative of x(·) in the point where it attains the value of . In the exponential verification function x(α) = 1 − e −α , if we are in the corner solution where = 0 and h = 0, then the truth to rumor ratio is one and remains so if β changes marginally. If only = 0, then Proposition 5 applies. If instead the equilibrium is interior, then we have the value of from expression (A-5). If we plug it into equation (A-8) and we take the derivative with respect to β, we get ∂(ρ 0 /ρ 1 )/∂β = 2 − 5c + D, where D = −2 − β(2 − 3c) 2 + 3c + 6c 2 2 (1 + 4β(1 + β) + 4c − 6β(1 + 2β)c + (2 − 3β) 2 c 2 ) .
Note that, if c < .5, ∂D ∂β = 8c(−2 + c + 6c 2 ) 2[(1 + 4β(1 + β) + 4c − 6β(1 + 2β)c + (2 − 3β) 2 c 2 )] 3 2 < 0.
So, ∂(ρ 0 /ρ 1 )/∂β is strictly decreasing in β and, at β = .5, it is equal to −4c(1 − 2c)/(4 − c), that is negative for c < .5. This implies that ρ 0 /ρ 1 is strictly decreasing in β.
In the piece-wise linear, as in Example 2, the corner solutions = h = 0 and = h =x are again insensitive to marginal changes in β. In an interior solution where c ≥ s/2 and L = s, the expression of ρ 0 /ρ 1 from (A-8) is increasing in β since ∂(ρ 0 /ρ 1 )/β = −4 + 2s/c, which is strictly positive.
Proof of Proposition 6. As individuals of type 0 are always of the opinion that Φ = 0, γ is irrelevant for ρ 0,t . We separately consider stubborn and non-stubborn individuals of type 1. All stubborn individuals of this type hold opinion Φ = 1 and we denote the corresponding prevalence as ρ γ 1,t . We denote the prevalence of opinion b among non-stubborn individuals of type 1 as ρ 1−γ b,t and the proportion of informed non-stubborn (stubborn) individuals as ι 1−γ t (ι γ t ). As in the benchmark model, ι 0 t = ρ 0 0,t , but now ι 1 t = γι γ t + (1 − γ)[ι 1−γ 0,t + ι 1−γ 1,t ]. The total prevalence of both opinions is