Misinformation Regulation in the Presence of Competition between Social Media Platforms (Extended Version)

The game players are multiple users and a sender, who choose platform $\mathbf{P}\in\{\mathbf{A},\mathbf{B}\}$. The sender’s (respectively user $i$’s) choice of platform $\mathbf{A}$ or $\mathbf{B}$ is denoted as $\mathbf{P}_{S}$ ($\mathbf{P}_{i}$).

The sender’s goal is to persuade as many users as possible to adopt a specific belief about the state of the world, while accounting for users’ platform choices and updates of beliefs in response to its signaling policy. The sender thus acts as a propagandist, whose decision-making process is best described using the framework of Bayesian persuasion [17], as is done in [18].

The world state is represented by a random variable $w\in\{0,1\}$ with $\mu=Prob(w=1)<1/2$. User $i$ estimates the world state as $a_{i}\in\{0,1\}$ and gets payoff $c$ if $a_{i}=w=0$, payoff $1-c$ if $a_{i}=w=1$, or no payoff if $a_{i}\neq w$. The assumption is made that $\mu<c<1/2$, implying that the user’s optimal estimation is $a_{i}=0$ in absence of any news information. The sender is the only one who can observe the state of the world, and their objective is to have the user estimate $a_{i}=1$ regardless of the world state. To persuade users, the sender sends signal $s\in\{0,1\}$. However, the choice of signal is not entirely arbitrary, and the sender selects a level of deceitfulness, denoted as $\beta\in[0,1]$. The signal $s$ is generated probabilistically, such that $Prob(s=1|w=0)=\beta,Prob(s=1|w=1)=1$. This means that the sender reports the true news when the state of the world is $w=1$, but occasionally lies when the world state is unfavorable. User $i$ in platform $\mathbf{P}$ receives the signal with probability of $p_{i\mathbf{P}}$, and then makes their estimation $a_{i}$ based on the sender’s strategy $\beta$ and the realized signal $s$.

Now let us examine the optimal strategy for user $i$ with respect to their estimation $a_{i}$. If the user does not receive any signal, they should choose the default optimal estimation of $a_{i}=0$. If they receive signal $s=0$, the biased sender is reporting unfavorable news and the world state is without a doubt $w=0$. However, if they receive signal $s=1$, which could potentially be misleading, a more careful analysis is needed. In this case, the expected payoff for the user when choosing $a_{i}=1$ is given by

while for $a_{i}=0$, it is given by

Therefore, the user should trust signal $s=1$ and select $a_{i}=1$ if the sender’s deceitfulness satisfies $\displaystyle{\beta\leq\frac{\mu(1-c)}{(1-\mu)c}=:\beta^{\prime}.}$ Otherwise, they should distrust signal $s=1$ and select $a_{i}=0$. Overall, the user’s optimal strategy when receiving a signal is to trust any signal ($a_{i}=s$) if $\beta\leq\beta^{\prime}$, and to ignore any signal ($a_{i}=0$) otherwise. Therefore, their expected payoff (for estimating the world state correctly) in platform $\mathbf{P}$ is given by

We call $\Psi_{i\mathbf{P}}$ news consumption payoff. The sender’s expected utility $U$ is the total number of users who choose $a_{i}=1$, which can be expressed as

if $\beta\leq\beta^{\prime}$. Otherwise, the sender fails to persuade the users, and therefore $U=0$.

Before introducing social-network aspects of our model, let us consider a scenario with only one user and one platform, ${\mathbf{A}=\mathbf{P}_{i}=\mathbf{P}_{S}}$, and see the implications of the setting so far. If the sender chooses $\beta\leq\beta^{\prime}$, their utility will be $U=(\mu+(1-\mu)\beta)p_{i\mathbf{A}}.$ Since the utility function increases monotonically for $\beta\in[0,\beta^{\prime}]$, the sender’s optimal strategy is $\beta^{*}=\beta^{\prime}$. They prefer a deceitful strategy as long as the user trusts them.

Suppose the platform sets a regulation $\rho_{\mathbf{A}}$ that prohibits the sender from using a $\beta$ value greater than $\rho_{\mathbf{A}}$. This, for example, could be done by setting a content moderation guideline to de-platform a news source that reports misinformation often. In the absence of an alternative platform, the sender must comply with the regulation and restrict their deceitfulness to the range $\beta\in[0,\rho_{\mathbf{A}}]$ instead of $\beta\in[0,1]$, because a de-platformed sender’s utility would be zero. A relatively strict regulation $\rho_{\mathbf{A}}<\beta^{\prime}$ works effectively in reducing the sender’s optimal strategy $\beta^{*}$ from $\beta^{\prime}$ to $\rho_{\mathbf{A}}$, thus improving the quality of news information on the platform. Conversely, a loose regulation with $\rho_{\mathbf{A}}\geq\beta^{\prime}$ is meaningless, as the sender’s optimal strategy remains at $\beta^{*}=\beta^{\prime}$.

Users and the sender are connected via an undirected graph, where edges represent friend relationships. Given choices $\mathbf{P}_{i}$’s and $\mathbf{P}_{S}$, this graph induces two subgraphs containing actors that are on the same platform. We assume that the signal originating from the sender travels along the edges of the subgraph corresponding to the sender’s platfom choice and that user $i$ receives that message with probability $p_{i\mathbf{P}_{i}}=p^{k}$ if and only if $\mathbf{P}_{i}=\mathbf{P}_{S}$, and $i$ is $k$ edges away from $S$ in that subgraph. We call parameter $p$ information diffusiveness. Works such as [19] have shown that news dissemination follows tree-like broadcast patterns. To account for this, we assume that the receiving probability depends on the shortest path length in the subgraph.

Note that the probability $p_{i\mathbf{P}_{i}}$ depends on other users’ choices of platforms, and a user in platform $\mathbf{P}_{i}\neq\mathbf{P}_{S}$ does not receive the signal. See Figure 1 for an illustrated example.

The number of user $i$’s friends in platform $\mathbf{P}$ (i.e., the size of their neighborhood in the corresponding subgraph) is denoted by $N_{i\mathbf{P}}$. The payoff derived from social interactions with these friends is $\Phi_{i\mathbf{P}_{i}}=N_{i\mathbf{P}_{i}}b_{\mathbf{P}_{i}}$, where $b_{\mathbf{P}}$ is a parameter, controlled by $\mathbf{P}$, which determines the reward obtained for each social interaction in platform $\mathbf{P}$. The value of $b_{\mathbf{P}}$ should be thought of as increasing with every service or feature (e.g., Story/Fleet feature, birthday notification…) introduced by the platform to enhance a user’s experience interacting with peers. We call $b_{\mathbf{P}}$ the quality of social interaction. In real life, some friends may be more important than others, but the uniform $b_{\mathbf{P}}$ for all neighbors is a typical assumption in the literature on networked agents’ adoption of a new option with positive externality (e.g., [20]). All in all, user $i$’s utility in platform $\mathbf{P}$ is given by

Given the sender’s decision on $\mathbf{P}_{S}$ and $\beta\in[0,\beta^{\prime}]$, user $i$ should choose $\mathbf{P}_{i}$ such that $V_{i\mathbf{P}_{i}}\geq V_{i\mathbf{P}}$ for all $\mathbf{P}\in\{\mathbf{A},\mathbf{B}\}$. We say the users’ choices of platforms are in equilibrium if this condition is satisfied for all users. A trivial equilibrium is $\mathbf{P}_{i}=\mathbf{P}_{S}$ for all $i$, but there can be other equilibria.

Since we are interested in situations where a pre-existing, originally “dominant”, platform ($\mathbf{A}$) imposes regulations in the presence of a competing one ($\mathbf{B}$), it makes sense to specifically look for equilibria that occur as the limit of an “adoption process” carried out by the users. The process is described as follows. First, the sender chooses $\mathbf{P}_{S}$. Then, from the initial state with all users in $\mathbf{A}$, users repeatedly update their choices. In each iteration, users choose $\mathbf{P}_{i}=\operatorname{argmax}_{\mathbf{P}}{V_{i\mathbf{P}}}$ simultaneously, based on other users’ previous choices. Iterations continue until an equilibrium is reached, if any. We will discuss the convergence of this process next, noting for now that it is essentially a fictitious play-iteration (see, e.g., [20]) for the non-cooperative game played by the users with utilities $V_{i\mathbf{P}}$, once the sender’s choice is enacted.

This is the sole equilibrium we will consider going forward, when saying that a property holds “at (the) equilibrium.”

The main insights of our model remain relevant in situations where more than two platforms are competing, with one of them being initially dominant. In fact, the two-platform model can be considered as the worst case for this dominant platform because the key factor of platform competition is the positive externality. If users can disperse to many alternative platforms, it is difficult for each one to become competitive against the dominant one. Therefore, the dominant platform might be able to enforce stricter regulation.

In its current form, our model assumes that users adopt one platform at a time. In reality, users do not necessarily have to leave one platform to join another and can, a priori, consume news and interact with peers on multiple sites contemporaneously. If such multihoming occurs, the sender’s departure to another platform may not necessarily cause users to leave the dominant one, and the latter can in turn enforce any regulation without any risk of losing its user base.

However, if user $i$ has a finite time and/or attention budget to allocate between the various sites they join, and if they do not explicitly value diversity of platforms, they will de facto spend all that budget on the platform bringing them highest utility, i.e., move to $\mathbf{P}$ such that $V_{i\mathbf{P}}\geq V_{i\mathbf{P}^{\prime}}$ for all $\mathbf{P}^{\prime}$, as we have assumed in Section III.

Using a similar argument as the one we described for multihoming, one can contend that users do not have to get their news from a single source. In turn, if a specific source leaves the dominant platform, a user can just either continue relying on other sources or decide to adopt a new one. For this reasoning to hold, however, it is necessary that all news sources be seen as substitutable by the users. Assuming otherwise (as we do) and, in particular, that the news consumption payoff enters a user’s utility according to (5) thus implicitly posits that (a) they have already committed to a specific sender of interest, to the exclusion of all others and, (b) that this commitment is independent from the messages sent by the source.
These two points are consistent with the behavior of at least some categories of social media users, whose main motivations for following an influencer are “entertainment” and “out of habit”, as uncovered by recent research in the field of social media studies [21]. Conditions (a) and (b) are also likely to hold more generally if the news source has unique features that make it especially appealing or salient to users, even before receiving any message from it (e.g., if the source is a particularly visible public figure or media outlet, or a charismatic influencers). Another way in which the users’ ex-ante devotion to the source (encapsulated by (a) and (b)) manifests itself in our model is in the assumption that they know the parameter $\beta$. This could be the result of users having interacted with the source for a long enough time, and hence “knowing what to expect” from it, and/or $\beta$ being one of the salient features of the source that make it attractive to the users. This is also consistent with the literature on Bayesian Persuasion, which has been used before to model interactions between platforms and news consumers [17, 18, 22]. Note, however, that users do not necessarily heed the source’s signal but, rather, use their knowledge of $\beta$ to interpret the signals they receive.

An infinite linear network has users $i=0,1,2,\ldots$ with edges between $i$ and $i+1$. The news source can send a signal to user 0. If the sender and user $0,\ldots,k$ are in the same platform, user $k$ receives a signal with probability $p^{k}$. We will use index $k$ instead of $i$ to imply the user is $k$ edges away from the sender, although $k$ coincides with $i$ in the linear network.

If the sender opts for platform $\mathbf{A}$, all users remain in $\mathbf{A}$ since $V_{k\mathbf{A}}>V_{k\mathbf{B}}$. The sender’s utility function is

If $\mathbf{P}_{S}=\mathbf{B}$, some users may switch platforms. Let user $K$ be the last user who switches to $\mathbf{B}$. We have $V_{k\mathbf{A}}\leq V_{k\mathbf{B}}$ for $k\leq K$, and $V_{k\mathbf{A}}>V_{k\mathbf{B}}$ for $k>K$, where

for $k=K,K+1$. The sender thus maximizes

with $K$ satisfying the condition

It is worth noting that if $b_{\mathbf{A}}\leq b_{\mathbf{B}}$, then $K=\infty$.

Condition (11) can be rewritten equivalently in two ways,

where

Note that $F(K),G(\beta)$ are decreasing functions, and the equalities in the conditions hold at the equilibrium. We thus have

Using these $U_{\mathbf{A}}(\beta)$ and $U^{*}_{\mathbf{B}}$, the strictest effective regulation can be designed. In particular, (7) becomes

The application of Proposition 2 to this situation is illustrated in Figure 2. For all subsequent numerical experiments, we use the following parameter values unless otherwise specified: $\mu=0.2,c=0.3,p=0.9,b_{\mathbf{A}}-b_{\mathbf{B}}=0.01$. (The value of $\mu$ does not affect the claims of this article qualitatively, and we will study different values of the other parameters in the following sections.) When the sender chooses platform $\mathbf{B}$, the utility attains maximum $U^{*}_{\mathbf{B}}$ at $\beta=\beta^{*}_{\mathbf{B}}$. If the sender selects $\beta>\beta^{*}_{\mathbf{B}}$, the signal is biased towards the sender’s preference, but few users move to $\mathbf{B}$ due to the poor news quality. If $\beta<\beta^{*}_{\mathbf{B}}$, many users switch to $\mathbf{B}$ to follow the sender, but the signal is not biased in favor of the sender. On the other hand, when the sender chooses $\mathbf{A}$, all users remain in the platform. The sender’s utility, $U_{\mathbf{A}}$, increases for $\beta\leq\min\{\rho_{\mathbf{A}},\beta^{\prime}\}$, and therefore achieves the maximum value at $\beta=\min\{\rho_{\mathbf{A}},\beta^{\prime}\}$. Proposition 2 indicates, with these parameters, the strictest effective regulation is $\rho_{SE}$ such that $U_{\mathbf{A}}(\rho_{SE})=U^{*}_{\mathbf{B}}$. If platform $\mathbf{A}$ enforces a stricter regulation $\rho_{\mathbf{A}}<\rho_{SE}$, then the sender’s utility becomes lower than in $\mathbf{B}$. In other words, the sender would rather send more biased information to fewer people in the alternative platform than comply with the regulation to gain more potential receivers, and thus platform $\mathbf{A}$ would lose some users. On the other hand, regulation $\rho_{\mathbf{A}}\in[\rho_{SE},\beta^{\prime}]$ works effectively, making the sender less deceptive, and the platform does not lose any users.

We will now inspect for what $b_{\mathbf{A}},b_{\mathbf{B}},p$, platform $\mathbf{A}$ can enforce the strict regulation $\rho_{\mathbf{A}}=0$. Our previous work [16] proved a sufficient condition for $\rho_{SE}=0$:

We will improve this proposition with a tighter bound.

Let $b_{\mathbf{A}}\geq b_{\mathbf{B}}\geq 0$. Suppose that the sender chooses $\mathbf{B}$. At the equilibrium, the sender’s best strategy satisfies $\beta^{*}_{\mathbf{B}}=F(K)$. The strictest effective regulation $\rho_{SE}$ takes the value of zero if and only if $U_{\mathbf{A}}(\beta=0)\geq U_{\mathbf{B}}(\beta^{*}_{\mathbf{B}})$, i.e.,

or

Therefore, using $f_{K}(p)$ defined here, we have the following.

In Figure 3(a), the strictest effective regulation is calculated for different $p$ and $b_{\mathbf{A}}$, fixing $b_{\mathbf{B}}=0$. The white curve presents $b_{\mathbf{A}}=\max_{K}f_{K}(p)+b_{\mathbf{B}}$ as in Proposition 4. Above this curve, platformer $\mathbf{A}$ can enforce any regulation without losing users.

We now consider a finite linear network with users $0,1,\ldots,n-1$. Note that $K$ does not take the value of $n-2$ since user $n-1$ always chooses the same platform as user $n-2$. Following the argument in Appendix, we let

Then Proposition 4 becomes

Panel (d) in Figure 3 shows the strictest effective regulation for a finite linear network. As a rule, and as is evident from the figure, in the case of linear networks, platform $\mathbf{A}$ can impose any regulation on the sender as long as its quality of social interaction is sufficiently larger than that of platform $\mathbf{B}$.

In this section, we consider a star-chain network, where hub user $k$ has $r-1$ leaf nodes and two hub users $k-1,k+1$ as their neighbors. The news source sends the signal to hub user 0. The cohesion of blocking clusters is $\frac{r}{r+1}$. If hub users $0,1,\ldots,k$ are in the same platform as the sender, hub user $k$ receives the signal with probability $p^{k}$ (and their leaf nodes receive with probability $p^{k+1}$).

The news source’s utility in platform $\mathbf{A}$ is

Suppose the sender chooses platform $\mathbf{B}$. Let $K$ be defined as an integer such that at the equilibrium, hub user $k\leq K$ chooses platform $\mathbf{B}$ and hub user $k>K$ chooses $\mathbf{A}$. Note that a peripheral user must always follow their hub user’s choice of platform. The sender in $\mathbf{B}$ maximizes their expected utility

subject to ¹¹1For hub user $k=K,K+1$, consider $$\begin{array}[]{ll}&V_{k\mathbf{A}}=rb_{\mathbf{A}}+(1-\mu)c,\\ &V_{k\mathbf{B}}=b_{\mathbf{B}}+\mu p^{k}(1-c)+(1-\mu)(1-\beta p^{k})c.\end{array}$$

Unlike linear network cases, even if the social interaction quality in platform $\mathbf{A}$ is lower than $\mathbf{B}$, users may prefer the former as it enables them to communicate with many friends. Indeed, Proposition 4 becomes

The proof follows the same argument as the linear network. The proposition indicates that platform $\mathbf{A}$ needs relatively low quality of social interactions to enforce strict regulations when hub users have many peripheral users. This is because users with many friends have high inertia to switch platforms. Since users have more friends in platform $\mathbf{A}$ and higher incentive to stay there, the dominant platform has more power to affect players’ behaviors. The higher the cohesion of blocking clusters is, the easier it becomes to enforce strict regulations.

Panel (b) in Figure 3 presents numerical results for $r=2,b_{\mathbf{B}}=0$. Compared to panel (a), the white curve is lower in the $(p,b_{\mathbf{A}})$-plane, and the dark area is larger in this panel. This means that platform $\mathbf{A}$ is more capable of enforcing the strict regulation $\rho_{\mathbf{A}}=0$ than in the linear network case.

For a finite star-chain with $n$ hub users, the counterpart of Proposition 5 also holds with some modification.²²2The farthest hub user can stay at platform $\mathbf{A}$ even when the other hub users are in $\mathbf{B}$, while in a finite linear network, the farthest user always chooses the same platform as the second farthest user. Letting

for $K\leq n-1$, Proposition 5 becomes

Panel (e) in the figure presents the result for $n=5$.

In an $r$-regular tree network, each user is connected to $r$ child nodes. The news source sends the signal to the root node (user 0). The blocking cluster cohesion is $\frac{r}{r+1}$ like for the star-chain network, but as the distance from the sender increases, the number of users grows exponentially. If users $0,1,\ldots,k$ along a path are in the same platform as the sender, then the probability that user $k$ receives the signal is $p_{k\mathbf{P}_{k}}=p^{k}$.

In platform $\mathbf{A}$, the sender’s expected utility is

The sender in platform $\mathbf{B}$ maximizes the utility

subject to condition (21). Proposition 6 holds when replacing function $f_{K}$ by $h_{K}$,

In addition, we can show the following.

When a signal is diffusive, even lower quality of social interaction is sufficient for strict regulation because of exponentially many potential receivers. Distant users are relatively important to the sender, and the dominant platform becomes powerful. As long as the social interaction quality satisfies $rb_{\mathbf{A}}>b_{\mathbf{B}}$, platform $\mathbf{A}$ can enforce any regulation if the signal’s reproduction rate $rp$ is greater than one. Lower degree trees require higher $p$ for strict regulation.

Panel (c) in Figure 3 shows the strictest effective regulation for $r=2,b_{\mathbf{B}}=0$. In the regular tree (c) and the star chain (b), platform $\mathbf{A}$ is more capable of enforcing strict regulation than in the linear network (a), due to cohesive blocking clusters. In the regular tree (c), platform $\mathbf{A}$ is even more capable than in the star chain (b) due to the number of the potential receivers. If $p>1/r$, a regular tree has $\rho_{SE}=0$ for any $b_{\mathbf{A}}\geq 0$.

The finite case follows the same argument as Proposition 5. Let

Note that Proposition 8 does not hold since the signal does not spread to infinitely many users. Panel (f) shows the result for a finite tree truncated at the fifth generation.

A stochastic block model is a random graph with $M$ communities. Community $J\in\{1,\ldots,M\}$ has $n_{J}$ users, and a pair of users in communities $J,J^{\prime}$ are friends with probability $\theta_{JJ^{\prime}}\in[0,1]$.

To provide analytical expressions, we restrict our study to stochastic block models on which the platform reallocation process satisfies the few assumptions mentioned below.

While this assumption may appear restrictive, we show through numerical simulations in Appendix that it is satisfied for some stochastic block models when $\theta_{JJ}$ is close to one.

We restrict our attention to network structures such that we can estimate the order of migration. Such network class includes various community structures, for instance a chain, a star graph, and a complete graph of communities as we will see in the following subsections.

The argument can be naturally extended to more connected first-relocating users, provided that the number of such external friends, not necessarily one, is known to us. Define user $\phi(J)$ as the first (potentially) relocating user in community $J$.

The following proposition is derived in the Appendix.

Proposition 10 emphasizes the role played by $p_{\phi(J)\mathbf{B}}$ and $R_{J}$ in enabling the platformer to apply regulation $\rho_{SE}=0$. These aggregate numbers can further be related to the basic structure and characteristics of the network if, e.g., they satisfy particular scaling laws. In that case, Proposition 10 can be used to gauge the effect of such parameters as well.

For an infinite linear network, our previous arguments can be reused with minor modifications. with $c$ replaced by $c_{k}$. Therefore, a counterpart of Proposition 4 holds, replacing function $f_{K}$ in (18) with

In the simulation for an infinite linear network, three users close to the sender are sympathizers with $c_{i}=0.21,i=0,1,2$, and the rest are non-sympathizers with $c_{i}=0.4,i\geq 3$. For the purpose of comparison, simulation is conducted also for the opposite pattern, $c_{i}=0.4,i=0,1,2$, and $c_{i}=0.21,i\geq 3$. As can be seen in Figure 6 (a), it is more difficult (compared to the homogeneous case) for the platformer to impose strict regulation when the users closest to the sender are sympathizers and $p$ is low. This can be understood intuitively by noting that, when information is not diffusive, nearby users are relatively important for the sender to decide its strategy, and therefore the nearby sympathizers give it greater power.

When the information is diffusive, on the other hand, users far from the sender are relatively important, and therefore the sender needs to be more truthful to please the distant non-sympathizers (and, hence, the solid curves dips below the dashed on for large $p$ in Figure 6 (a)).

The pattern observed in panel (a) is reversed in (c), as sympathizers become helpful to the sender when $p$ is high.

For stochastic block models, we consider that users in the same community have the same payoff, but that different communities have different payoffs. With a slight abuse of notation, let $c_{J}$ denote the payoff parameter for community $J$. Then, the counterpart of Proposition 10 holds as follows.

Simulation is conducted for the previous model of community chain, $n_{1}=n_{2}=n_{3}=30$ and (27). The basic model has $c_{J}=0.3$ for all $J=1,2,3$. Keeping all other parameters fixed, a variant model has $c_{1}=0.21$, meaning that the sender is in a community of sympathizers. For comparison, another variant has $c_{1}=0.4$, corresponding to non-sympathizers in the sender’s community. Figure 6 indicates that in the case that the sender has sympathizers nearby, it is difficult for platform $\mathbf{A}$ to enforce strict regulation when $p$ is low. In the case that non-sympathizers are nearby, on the other hand, platform $\mathbf{A}$ can enforce strict regulation easily when $p$ is low. Like linear network cases, when the information is not diffusive, the nearby users are relatively important for the sender to decide its strategy, and therefore the nearby sympathizers (non-sympathizers) bring advantage (disadvantage) to the sender.