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

So Sasaki, Cédric Langbort Manuscript received January 1, 2023; revised
January 2, 2023; accepted January 3, 2023. This work was supported by an ARO
MURI award, under agreement W911NF-20-1-0252. S. Sasaki and C. Langbort are
with Coordinated Science Laboratory, University of Illinois at Urbana-
Champaign, Urbana, IL 61801 USA (e-mail<EMAIL_ADDRESS>langbort@illinois.edu).

###### Abstract

Social media platforms have diverse content moderation policies, with many
prominent actors hesitant to impose strict regulations. A key reason for this
reluctance could be the competitive advantage that comes with lax regulation.
A popular platform that starts enforcing content moderation rules may fear
that it could lose users to less-regulated alternative platforms. Moreover, if
users continue harmful activities on other platforms, regulation ends up being
futile. This article examines the competitive aspect of content moderation by
considering the motivations of all involved players (platformer, news source,
and social media users), identifying the regulation policies sustained in
equilibrium, and evaluating the information quality available on each
platform. Applied to simple yet relevant social networks such as stochastic
block models, our model reveals the conditions for a popular platform to
enforce strict regulation without losing users. Effectiveness of regulation
depends on the diffusive property of news posts, friend interaction qualities
in social media, the sizes and cohesiveness of communities, and how much
sympathizers appreciate surprising news from influencers.

###### Index Terms:

Content moderation, Network game theory, Platform competition, Social media

##  I Introduction

In recent years, social media platforms have significantly changed their
content moderation policies. For instance, Jhaver et al. [1] reported that
Pinterest recently started actively blocking some search results for
controversial or debunked queries, while Reddit has taken action against
certain toxic communities by banning or quarantining them. Meta/Facebook has
also identified and flagged groups that engage in hate speech on its platform.
Different platforms have implemented widely varying policies, resulting in
different standards for what is considered acceptable speech. One of the most
visible examples of this was the different treatment of President Trump’s May
2020 messages by Twitter and Facebook, as reported by [2] and [3].

The difference in content moderation policies between social media platforms
is likely influenced by a combination of factors, including each company’s
interpretation of §230 of the US Communications Decency Act and the founders’
views on freedom of speech. However, it is also probable that economic
incentives play a role in shaping these policies. In fact, a platform’s
regulation, or lack thereof, can be perceived as a competitive advantage, as
users may choose to switch to a competitor if they feel that the moderation
rules in place amount to censorship and limit their access to the news and
content they value. This effect can be amplified in social networking markets,
where the positive externalities of network economics are typically present.
As more users join a particular platform, it becomes even more socially
advantageous for others to join as well, resulting in a winner-takes-all
effect, as has been well documented in the literature, such as [4]. Such a
threat of seeing its users leave en masse may play a role in reducing a
platformer’s interest in policing misinformation on its network. At the same
time, other actors, such as companies advertising on the network, may value
regulation (e.g., as a form of so-called ’brand safety’ [5, 6]) and introduce
incentives that, in contrast, may push the platformer towards more oversight.

The aim of this article is to investigate the competitive nature of content
moderation on simplified platform models by considering the motivations of
different players such as the platform owner, news sources, and social media
users. We purposefully omit the kind of pro-regulation stakeholders mentioned
above in this first study. This is because, in the context of our model, they
only make it easier for the platform to implement strict regulation, and we
are interested in characterizing the most restrictive such policy that can be
sustained in equilibrium.

More precisely, in our model, we examine the effect of regulation when
multiple users and a news source (also referred to as an influencer or sender)
choose platforms to maximize their individual utility. A user’s utility is
determined by their social interaction payoff and news consumption payoff.
Platform owners have the option to deplatform a deceptive news source based on
their policies. As a result, the news source must decide whether to comply
with the strict regulation of a popular platform or relocate to an alternative
platform to spread more biased information to a smaller audience.

This study is part of a larger body of research on platform governance, as
discussed in reviews [7] and [8]. We specifically examine how platforms can
prevent the spread of misinformation, such as through the preclusion of
accounts. Twitter account suspension following the 2020 U.S. presidential
election is examined in [9]. Our model also takes into account how users and
news sources react to the regulations. The way users perceive and interpret
content moderation is studied closely in [10]. Meanwhile, Horta Ribeiro et al.
[11] investigate how moderation policies affect harmful activity in the wider
web ecosystem, and whether this activity becomes more radicalized on
alternative platforms. Additionally, Innes and Innes [12] examine the intended
and unintended consequences of deplatforming interventions, such as the
emergence of “minion accounts” of a banned account.

Other recent works concerned with regulation in information diffusion include
[13, 14, 15], all of which place a heavier emphasis on the process underlying
dynamic propagation and/or the formation of opinions. Little research has been
conducted, however, in regard to the intensity of effective regulation that
mainstream platforms can implement without losing users. Understanding this
fundamental aspect can advance our conversation on the social responsibilities
of tech companies.

This article improves our previous work [16] in that 1) we give tighter
sufficient conditions in the propositions for simple social network structures
and 2) more realistic network structures (finite networks and random networks
with community structures) are investigated. In addition, 3) our analysis is
extended to networks consisting of heterogeneous users (sympathizers and non-
sympathizers). These improvements allow our model to analyze e.g., how
regulation should be adjusted depending on community sizes, in-community
cohesiveness, inter-community structures, and sympathizers near an influencer.

The rest of the paper proceeds as follows. First, we introduce our model of
information diffusion and regulation on a single platform. Next, we describe
the competitive process of platform migration among the sender and users, and
characterize the strictest regulation that can be imposed in equilibrium. We
then relate its properties to various network structures. Lastly, the analysis
scope is extended to heterogeneous users.

##  II Model

###  II-A News information from sender

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

$Prob(w=1|s=1)(1-c)=\frac{\mu}{\mu+(1-\mu)\beta}(1-c),$ (1)

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

$Prob(w=0|s=1)c=\frac{(1-\mu)\beta}{\mu+(1-\mu)\beta}c.$ (2)

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

$\Psi_{i\mathbf{P}}=\begin{cases}\mu p_{i\mathbf{P}}(1-c)+(1-\mu)(1-\beta
p_{i\mathbf{P}})c&\text{if}\beta\leq\beta^{\prime}\\\
(1-\mu)c&\text{if}\beta>\beta^{\prime}\end{cases}$ (3)

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

$\begin{array}[]{ll}U&=\mathbb{E}\bigg{[}\sum_{i}a_{i}\bigg{]}=\sum_{i}(\mu+(1-\mu)\beta)p_{i\mathbf{P}_{i}},\end{array}$
(4)

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 $\beta^{*}=\beta^{\prime}$.

###  II-B Social Interaction and Platform Adoption

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.

Figure 1: Illustration of our model. Blue (red) nodes are in platform A (B).
Values below nodes present $p_{i\mathbf{P}_{i}}$. Note that user 6 is assumed
to receive the source’s signal with probability $p^{2}$ (because it is two
edges away from it) even though there are two paths connecting these vertices.
See text for details.

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

$V_{i\mathbf{P}}=\Phi_{i\mathbf{P}}+\Psi_{i\mathbf{P}}.$ (5)

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.

######  Proposition 1.

If 1) the social network is acyclic and platform $\mathbf{A}$ provides higher
social interaction quality than platform B, i.e.,
$b_{\mathbf{A}}>b_{\mathbf{B}}$, or 2) the social network is a finite graph,
then the adoption process converges to an equilibrium in a finite number of
iterations.

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

###### Proof.

We will prove the result for Condition 1) only, as the proof for 2) is
similar. If $\mathbf{P}_{S}=\mathbf{A}$, the initial state is the equilibrium.
So we only need to consider $\mathbf{P}_{S}=\mathbf{B}.$

As shown in the Appendix, users can switch from platform $\mathbf{A}$ to
$\mathbf{B}$ but not from $\mathbf{B}$ to $\mathbf{A}$. It is also shown that
for an acyclic graph, user $i$ that switches to platform $\mathbf{B}$ at
iteration $t$ is $t$ edges away from the sender and has
$p_{i\mathbf{B}}^{(t)}=p^{t}$, where $x^{(t)}$ denotes the value of $x$ at
iteration $t$. (This implies that such user $i$’s neighbor who is $t-1$ edges
away from the sender has to switch to $\mathbf{B}$ at iteration $t-1$. It also
implies that if a user $k$ edges away does not switch to $\mathbf{B}$ at
iteration $k$, then the user and its downstream users do not switch to
$\mathbf{B}$ forever.)

Suppose user $i$ switches to $\mathbf{B}$ at an iteration of sufficiently
large $t$. Since $p_{i\mathbf{A}}^{(t)}=0,p_{i\mathbf{B}}^{(t)}=p^{t}$, we
have $\Psi_{i\mathbf{B}}^{(t)}<\Psi_{i\mathbf{A}}^{(t)}+\epsilon$ for small
$\epsilon>0$. Since user $i$’s neighbors farther from the sender are still in
platform $\mathbf{A}$ and the neighbor closer to the sender is in
$\mathbf{B}$, $N_{i\mathbf{B}}^{(t)}=1$. Since
$b_{\mathbf{B}}<b_{\mathbf{A}}$, we have
$\Phi_{i\mathbf{B}}^{(t)}<\Phi_{i\mathbf{A}}^{(t)}$ if
$N_{i\mathbf{A}}^{(t)}\geq 1$. Therefore,
$V_{i\mathbf{B}}^{(t)}<V_{i\mathbf{A}}^{(t)}$ if $N_{i\mathbf{A}}^{(t)}\geq
1$. Since user $i$ adopts $\mathbf{B}$ at iteration $t$ (i.e.,
$V_{i\mathbf{B}}^{(t)}\geq V_{i\mathbf{A}}^{(t)}$), this means
$N_{i\mathbf{A}}^{(t)}=0$. Therefore, user $i$ is a leaf node. Hence, no user
switches to $\mathbf{B}$ at iteration $\tau\geq t+1$.

∎

##  III Strictest Effective Regulation

We now focus on the scenario where platforms have the ability to regulate the
level of deceitfulness of the sender. Specifically, we assume that platform
$\mathbf{A}$ enforces a restriction $\rho_{\mathbf{A}}\in[0,1]$ on the
sender’s choice of $\beta$, limiting the range to $[0,\rho_{\mathbf{A}}]$. In
contrast, $\mathbf{B}$ has no such restriction, i.e., $\rho_{\mathbf{B}}=1$,
and the sender can choose any $\beta\in[0,1]$ if they choose $\mathbf{B}$. In
practice, such regulation can be implemented if the source has been present on
the platform long enough to enable it to fact-check its messages over a period
of time, thus obtaining an empirical estimate of $\beta$.

######  Definition 1.

Platform $\mathbf{A}$’s regulation $\rho_{\mathbf{A}}$ is said to be effective
if it decreases the sender’s deceitfulness in platform $\mathbf{A}$ at
equilibrium. The strictest effective regulation $\rho_{SE}$ is the minimum of
effective regulation $\rho_{\mathbf{A}}$.

In other words, Definition 1 states that $\rho_{\mathbf{A}}$ is effective if
in the presence of the constraint $\beta\in[0,\rho_{\mathbf{A}}]$, (1) the
sender stays in $\mathbf{A}$ and (2) the chosen value $\beta^{*}$ is less than
if $\rho_{\mathbf{A}}$ were equal to 1 (i.e., $\beta^{*}<\beta^{\prime}$). Our
primary interest lies in identifying the strictest effective regulation
$\rho_{SE}$. With this regulation, all users remain in platform $\mathbf{A}$.

Assuming that either condition 1) or 2) in Proposition 1 holds, we let
$U_{\mathbf{P}}$ be the sender’s expected utility in platform $\mathbf{P}$.
When users take the optimal strategies, the sender considers $U_{\mathbf{P}}$
as a function of $\beta$. We denote $U^{*}_{\mathbf{P}}$ as the maximum value
of $U_{\mathbf{P}}$ for $\beta\in[0,\rho_{\mathbf{P}}]$.

If the sender chooses platform $\mathbf{A}$, all users should choose
$\mathbf{A}$ at the equilibrium. If $\sum_{i}p_{i\mathbf{A}}<\infty$,
according to (4), $U_{\mathbf{A}}$ increases linearly with
$\beta\in[0,\min\\{\rho_{\mathbf{A}},\beta^{\prime}\\}]$. Thus, the sender’s
optimal level of deceitfulness is
$\beta^{*}=\min\\{\rho_{\mathbf{A}},\beta^{\prime}\\}$. On the other hand, if
$\sum_{i}p_{i\mathbf{A}}=\infty$, then $U^{*}_{\mathbf{A}}=\infty$ for any
$\beta\in[0,\min\\{\rho_{\mathbf{A}},\beta^{\prime}\\}]$.

This observation allows us to characterize an effective regulation. Firstly,
if $\rho_{\mathbf{A}}\geq\beta^{\prime}$, then the regulation has no impact on
the sender’s behavior as the sender would have chosen a lower level of
deceitfulness. In other words, the regulation is too lenient to be effective.
Secondly, if regulation $\rho_{\mathbf{A}}<\beta^{\prime}$ satisfies
$U_{\mathbf{A}}(\rho_{\mathbf{A}})\geq U^{*}_{\mathbf{B}}$, then the sender
should remain in platform $\mathbf{A}$ since it would not gain more utility in
platform $\mathbf{B}$. Note that when $\rho_{\mathbf{A}}$ takes the lowest
value that satisfies the inequality, we have

$(\mu+(1-\mu)\rho_{\mathbf{A}})\sum_{i}p_{i\mathbf{A}}=U^{*}_{\mathbf{B}},$
(6)

where $p_{i\mathbf{A}}$ corresponds to $\mathbf{P}_{S}=\mathbf{A}$. Lastly, if
regulation $\rho_{\mathbf{A}}<\beta^{\prime}$ satisfies
$U_{\mathbf{A}}(\rho_{\mathbf{A}})<U^{*}_{\mathbf{B}}$, then the sender should
switch to $\mathbf{B}$ for higher utility. The regulation is too strict, and
the sender would prefer to move to the alternative platform with some of its
followers.

Rearranging these cases from the viewpoint of designing the strictest
effective regulation $\rho_{SE}$ in platform $\mathbf{A}$, we have the
following proposition.

######  Proposition 2.

If $U_{\mathbf{A}}(\beta^{\prime})\leq U^{*}_{\mathbf{B}}$, effective
regulation does not exist. If $U_{\mathbf{A}}(0)>U^{*}_{\mathbf{B}}$, platform
$\mathbf{A}$ can enforce any strict regulation effectively, i.e.,
$\rho_{SE}=0$. If $U_{\mathbf{A}}(0)\leq
U^{*}_{\mathbf{B}}<U_{\mathbf{A}}(\beta^{\prime})$, regulation should be
moderate and

$\rho_{SE}=\frac{1}{1-\mu}\bigg{(}\frac{U^{*}_{\mathbf{B}}}{\sum_{i}p_{i\mathbf{A}}}-\mu\bigg{)},$
(7)

where $p_{i\mathbf{A}}$ is for $\mathbf{P}_{S}=\mathbf{A}$.

##  IV Discussion of model assumptions

Having now presented the main features of our model, and before
particularizing its high-level predictions to specific networks, we are in a
good position to discuss its central assumptions and simplifications in more
details.

###  IV-A Multiple platforms

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.

###  IV-B Multihoming

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.

###  IV-C Sender strategy and multiple senders

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.

##  V Applications to Prototypical Networks

In this section, we apply the main result of Section sec3 to prototypical
network structures. To maintain analytical tractability and be able to
interpret results qualitatively, we focus on simple network structures with
some relevance to actual social media platforms. In particular, we study
linear networks, star-chain networks, regular trees, and stochastic block
models for the following reasons.

We analyze a linear network as it presents an intuitive analytical
representation of our model. In many social media settings, news dissemination
takes place with a tree-like hub-spoke structure, often embedded within a
larger social interaction network [23, 19]. A star-chain network is built on
this structure and possesses cohesive blocking clusters, a crucial element for
platform adoption as defined below.

######  Definition 2.

A cluster, or a set of users, has cohesion $x$ if each member has at least
fraction $x$ of their neighbors within the set. The cohesion of a cluster is
the maximum of such $x$.

Without news consumption payoffs, platform adoption process would be governed
by cluster cohesion. As shown in [20], a “blocking cluster,” that has the
highest cohesion in a graph would determine whether the adoption to the whole
population occurs or not. A regular tree network exhibits, in addition to
cohesive blocking clusters, the exponentially spreading communication feature,
making it a good benchmark to gauge the relative importance of both
characteristics (especially when compared to the star chain).

Large literature (e.g., [24], [25]) has demonstrated that homophily is encoded
in common social networks. Stochastic block model is a random network suited
to express such networks with community structures. Analysis of this
parameterized random network is useful because players may not know the exact
structure of a large network, and even if they do, computing exact solutions
is prohibitively expensive. Instead, an approximate solution based on a few
primary factors may be more practical. Coordination games on stochastic block
model have been explored in prior works (e.g., [26], [27]).

###  V-A Linear Network

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

$U_{\mathbf{A}}(\beta)=\sum_{k=0}^{\infty}(\mu+(1-\mu)\beta)p^{k}=\frac{\mu+(1-\mu)\beta}{1-p}.$
(8)

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

$\begin{array}[]{ll}&V_{k\mathbf{A}}=b_{\mathbf{A}}+(1-\mu)c,\\\
&V_{k\mathbf{B}}=b_{\mathbf{B}}+\mu p^{k}(1-c)+(1-\mu)(1-\beta p^{k})c\\\
\end{array}$ (9)

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

$\begin{array}[]{ll}U_{\mathbf{B}}(\beta)&=\sum_{k=0}^{K(\beta)}(\mu+(1-\mu)\beta)p^{k}\end{array}$
(10)

with $K$ satisfying the condition

$V_{K+1,B}<b_{\mathbf{A}}+(1-\mu)c\leq V_{K,B}.$ (11)

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,

$F(K+1)<\beta\leq F(K),$ (12) $G(\beta)-1<K\leq G(\beta),$ (13)

where

$\begin{array}[]{ll}F(K)&=\frac{1}{(1-\mu)c}\bigg{(}\mu(1-c)-\frac{b_{\mathbf{A}}-b_{\mathbf{B}}}{p^{K}}\bigg{)},\\\
G(\beta)&=\frac{1}{\log
p}\log\bigg{(}\frac{b_{\mathbf{A}}-b_{\mathbf{B}}}{\mu(1-c)-(1-\mu)\beta
c}\bigg{)}.\end{array}$ (14)

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

$\begin{array}[]{ll}U_{\mathbf{B}}^{*}&=\max_{K\in\mathbb{N}}(\mu+(1-\mu)F(K))\frac{1-p^{K+1}}{1-p}\\\
&=\max_{\beta:G(\beta)\in\mathbb{N}}(\mu+(1-\mu)\beta)\frac{1-p^{G(\beta)+1}}{1-p}.\end{array}$
(15)

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

$\rho_{SE}=\frac{(1-p)U^{*}_{\mathbf{B}}-\mu}{1-\mu}.$ (16)

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.

Figure 2: The sender’s expected utility in a linear network

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 $\rho_{SE}=0$:

######  Proposition 3.

[16] For an infinite linear network, if
$b_{\mathbf{A}}-b_{\mathbf{B}}>\mu(1-c)$ or
$b_{\mathbf{A}}-b_{\mathbf{B}}>\mu(1-c)^{2}/p$, then the strictest effective
regulation is $\rho_{SE}=0$.

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.,

$\sum_{k=0}^{\infty}\mu
p^{k}\geq\sum_{k=0}^{K}(\mu+(1-\mu)\beta^{*}_{\mathbf{B}})p^{k},$ (17)

or

$b_{\mathbf{A}}-b_{\mathbf{B}}\geq\mu p^{K}-\frac{\mu
cp^{K}}{1-p^{K+1}}=:f_{K}(p).$ (18)

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

######  Proposition 4.

For an infinite linear network, the strictest effective regulation is
$\rho_{SE}=0$ if $b_{\mathbf{A}}-b_{\mathbf{B}}\geq\max_{K}f_{K}(p)$.

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.

Figure 3:  The strictest effective regulation $\rho_{SE}$ in (a) linear
network, (b) star-chain network, and (c) regular-tree network. Respectively,
(d), (e), (f) are the finite cases. In panel (a), the white curve is computed
using the formula of Proposition 4. From this proposition, the strictest
effective regulation is guaranteed to be $\rho_{SE}=0$ above this curve in the
$(p,b_{\mathbf{A}})$-plane. The orange curve plays the same role, but based on
the results of Proposition 3. The color map is computed by (15) and
Proposition 2. The white/orange curves and color maps in other panels are
computed similarly.

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

$\begin{array}[]{ll}f_{n,K}(p):=\mu p^{K}-(1-p^{n})\frac{\mu
cp^{K}}{1-p^{K+1}}&\text{for }K<n-2\\\ f_{n,n-2}(p):=\mu(1-c)p^{n-2}&\text{for
}K=n-1.\end{array}$

Then Proposition 4 becomes

######  Proposition 5.

For a finite linear network, the strictest effective regulation is
$\rho_{SE}=0$ if
$b_{\mathbf{A}}-b_{\mathbf{B}}\geq\max_{K=0,1,\ldots,n-2}f_{n,K}(p)$.

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 $\mathbf{B}$.

###  V-B Star-chain Network

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

$U_{\mathbf{A}}(\beta)=\sum_{k=0}^{\infty}(\mu+(1-\mu)\beta)(p^{k}+(r-1)p^{k+1}).$
(19)

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

$\begin{array}[]{ll}U_{\mathbf{B}}(\beta)=\sum_{k=0}^{K(\beta)}(\mu+(1-\mu)\beta)(p^{k}+(r-1)p^{k+1})\end{array}$
(20)

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

$V_{K+1,B}<rb_{\mathbf{A}}+(1-\mu)c\leq V_{K,B}.$ (21)

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

######  Proposition 6.

For an infinite star chain, the strictest effective regulation is
$\rho_{SE}=0$ if $rb_{\mathbf{A}}-b_{\mathbf{B}}\geq\max_{K}f_{K}(p)$.

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

$g_{n,K}(p):=\mu p^{K}-(1-p^{n})\frac{\mu cp^{K}}{1-p^{K+1}}$ (22)

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

######  Proposition 7.

For a finite star chain, the strictest effective regulation is $\rho_{SE}=0$
if $rb_{\mathbf{A}}-b_{\mathbf{B}}\geq\max_{K=0,1,\ldots,n-1}g_{n,K}(p)$.

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

###  V-C Regular Tree Network

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

$\begin{array}[]{ll}U_{\mathbf{A}}(\beta)&=\sum_{k=0}^{\infty}(\mu+(1-\mu)\beta)p^{k}r^{k}.\end{array}$
(23)

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

$\begin{array}[]{ll}U_{\mathbf{B}}(\beta)&=\sum_{k=0}^{K(\beta)}(\mu+(1-\mu)\beta)p^{k}r^{k}\end{array}$
(24)

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

$h_{K}(p):=\mu p^{K}-\frac{\mu cp^{K}}{1-(pr)^{K+1}}.$ (25)

In addition, we can show the following.

######  Proposition 8.

For an infinite regular tree, if $rb_{\mathbf{A}}>b_{\mathbf{B}}$ and $pr>1$,
then $\rho_{SE}=0$.

###### Proof.

When $pr>1$, the expected number of users who receive the sender’s signal is
infinite. On the other hand, with $rb_{\mathbf{A}}>b_{\mathbf{B}}$, at most
finite users move to platform $\mathbf{B}$. Therefore, the sender’s utility is
infinity in platform $\mathbf{A}$ and finite in $\mathbf{B}$. By Proposition
2, since $U_{\mathbf{A}}(0)>U^{*}_{\mathbf{B}}$, the sender should always
choose platform $\mathbf{A}$. Any strict regulation thus works effectively,
i.e., $\rho_{SE}=0$. ∎

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

$\begin{array}[]{ll}h_{n,K}(p):=\mu p^{K}-(1-(pr)^{n})\frac{\mu
cp^{K}}{1-(pr)^{K+1}}&\text{for }K<n-2,\\\
h_{n,n-2}(p):=\mu(1-c)p^{n-2}&\text{for }K=n-1.\end{array}$

######  Proposition 9.

For a finite regular tree, the strictest effective regulation is $\rho_{SE}=0$
if $rb_{\mathbf{A}}-b_{\mathbf{B}}\geq\max_{K=0,1,\ldots,n-2}h_{n,K}(p)$.

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.

###  V-D Stochastic Block Model

Figure 4:  The strictest effective regulation $\rho_{SE}$ for (a1, a2)
community chains and (b1, b2) complete graphs of communities. The community
chain has a middle community of (a1) low or (a2) high intra-community
tightness. Parameter values are as described in the main text. The color map
is the simulation results of 50 random graphs. The white curves calculated by
Proposition 10 fit the results of the simulation, which does not use
Assumptions 1–3 for the proposition. The dashed curves represent the base case
with medium intra-community tightness (see Figure 6 panel (b)). With high $p$,
platform $\mathbf{A}$ can enforce strict regulation more easily (i.e., the
white curve is lower) in (a2) than (a1). For the complete graphs of
communities, the sender is in a (b1) small or (b2) large community. The white
curves calculated by Proposition 10 again fit the color maps (the simulation
results for 50 random graphs), which are computed without Assumptions 1–3 for
the proposition. With low $p$, platform $\mathbf{A}$ can enforce strict
regulation more easily in (b2) than (b1).

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.

######  Assumption 1.

If a user moves to platform $\mathbf{B}$, then the other users in the same
community do so too.

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.

######  Assumption 2.

Communities move to platform $\mathbf{B}$ in the order of community
$1,2,\ldots,M$ (if they relocate).

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.

######  Assumption 3.

Friendship across communities is rare (i.e., $\theta_{JJ^{\prime}}\approx
0,J\neq J^{\prime}$), and the first-relocating user in a community has a
single friend in the precedingly-relocated communities.

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.

Under Assumptions 1–3, the strictest effective regulation is ${\rho_{SE}=0}$
if

$n_{J}\theta_{JJ}b_{\mathbf{A}}-b_{\mathbf{B}}\geq\mu(1-c)p_{\phi(J)\mathbf{B}}-\frac{\sum_{L=J+1}^{M}R_{L}}{\sum_{L=1}^{J}R_{L}}\mu
cp_{\phi(J)\mathbf{B}}.$

for all $J=1,\ldots,M$, where $R_{J}$ denotes the expected number of signal
receivers in community $J$ when the community is in the same platform as the
sender.

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.

####  V-D1 Community-chain-type Scaling

Our first example is $p_{\phi(J)\mathbf{B}}$ and $R_{J}$ decaying with $J$
according to

$\displaystyle p_{\phi(J)\mathbf{B}}$ $\displaystyle=\begin{cases}p&J=1\\\
p^{2J-2}&2\leq J\leq M,\end{cases}$ (26) $\displaystyle R_{J}$
$\displaystyle=n_{J}p^{2J-1}.$

As explained heuristically in the Appendix, such a scaling can be achieved in
a so-called “chain of communities” network. Numerical simulations also
presented in Appendix seem to indicate that Assumptions 1–3 hold for these
networks when the intra-community tightness (i.e., probability $\theta_{JJ}$)
is high, which makes them good representatives of the class of networks for
which (26) holds and Proposition 10 is applicable. Note that this is all that
is needed of the network to perform our analysis.

For example, consider a base model with $n_{1}=n_{2}=n_{3}=30$ and
$\theta_{JJ^{\prime}}$ given by

$\theta=\begin{pmatrix}\frac{3}{4}&\frac{4}{900}&0\\\
\frac{4}{900}&\frac{3}{4}&\frac{4}{900}\\\ 0&\frac{4}{900}&\frac{3}{4}\\\
\end{pmatrix}.$ (27)

We utilize Proposition 10 to analyze how the regulation depends on the in-
between community’s “tightness.” For variants of the basic parameter sets,
instead of $\theta_{22}=3/4$ for the second community, we consider
$\theta_{22}=1/2$ (case (a1)) and $\theta_{22}=1$ (case (a2)).

Figure 4 shows the strictest regulation $\rho_{SE}$ for these networks. The
simulation does not use Assumptions 1-3 a priori, which Proposition 10 is
based on. The white curves indicate that our mathematical analysis of
Proposition 10 fits the simulation results (the color map) of the
corresponding physical model. In particular, with high $p$, platform
$\mathbf{A}$ can enforce strict regulation more easily for (a2), featuring a
tightly connected intermediate community, than for (a1), which contains a
loosely connected one. This is in agreement with our qualitative
understanding. When $p$ is high, the sender’s influence on distant users
becomes relatively important. Since a tightly connected intermediate community
blocks their platform migration, it helps platform $\mathbf{A}$ hold more
power to impose regulation.

####  V-D2 Community-star-type Scaling

Another example is given by $p_{\phi(J)\mathbf{B}}$ and $R_{J}$ that, in
contrast, do not decay with $J$:

$\displaystyle p_{\phi(J)\mathbf{B}}$ $\displaystyle=\begin{cases}p&J=1\\\
p^{2}&2\leq J\leq M,\end{cases}$ (28) $\displaystyle R_{J}$
$\displaystyle=\begin{cases}n_{1}p&J=1\\\ n_{J}p^{3}&2\leq J\leq
M.\end{cases}$

As explained again in the Appendix, such scalings can be obtained for a star
and a complete graph of communities (while satisfying Assumptions 1–3), when
the sender is in community $1$ and $\theta_{1J}>0$ for all $J=2,\ldots,M$. We
assume, without loss of generality, $n_{2}\theta_{22}\leq
n_{3}\theta_{33}\leq\ldots\leq n_{M}\theta_{MM}$. This implies that the sender
can attract the first users in community $2,\ldots,J-1$ more easily than
community $J$ since they have less friends within community. Assumption 2 is
thus satisfied.

Simulations are conducted for complete graphs of communities with the sender
at different communities. The model has four communities. In one simulation
setting, the sender is in a small community, and
$n_{1}=20,n_{2}=30,n_{3}=40,n_{4}=50$. In another, the sender is in a large
community, and $n_{1}=50,n_{2}=20,n_{3}=30,n_{4}=40$. The other parameters are

$\theta=\begin{pmatrix}\frac{3}{4}&\frac{4}{n_{1}n_{2}}&\frac{4}{n_{1}n_{3}}&\frac{4}{n_{1}n_{4}}\\\
\frac{4}{n_{2}n_{1}}&\frac{3}{4}&\frac{4}{n_{2}n_{3}}&\frac{4}{n_{2}n_{4}}\\\
\frac{4}{n_{3}n_{1}}&\frac{4}{n_{3}n_{2}}&\frac{3}{4}&\frac{4}{n_{3}n_{4}}\\\
\frac{4}{n_{4}n_{1}}&\frac{4}{n_{4}n_{2}}&\frac{4}{n_{4}n_{3}}&\frac{3}{4}\\\
\end{pmatrix}$ (29)

so that the network structure itself remains identical in the two settings.

In Figure 4 (b1, b2), the white curves indicate that our mathematical analysis
of Proposition 10 is again consistent with the simulation results, which do
not presume the Assumptions 1–3. With low $p$, platform $\mathbf{A}$ can
enforce strict regulation more easily if the sender is in a large community
(panel (b2)) than if it is in a small one (panel (b1)). This goes along with
our qualitative intuition. When $p$ is low, the sender’s influence on nearby
agents becomes relatively important. Since it is difficult for the sender to
persuade a large community into platform $\mathbf{B}$, platform $\mathbf{A}$
can take advantage of this to impose regulation.

##  VI Influencer and Neighboring Sympathizers

The previous section demonstrated how the strictest effective regulation is
affected by network structure. In particular, we saw how the presence of
cohesive clusters can preclude new platform adoption (and hence help the
platformer impose stricter regulation), and how the location of pivotal users
(who determine the power balance between the sender and the platformer)
changes depending on information diffusiveness.

In this section, we consider yet another effect, namely heterogeneity in the
users’ preference for their received signal. More precisely, from now on, we
assume users differ in how much they appreciate the news information received
from the sender and in how biased they are toward unorthodox views.

This is captured by the payoff $c_{i}$ of user $i$ (replacing the common value
$c$ used up to now). A user with low $c_{i}$ gets high utility when they
correctly estimate the surprising world state, i.e., $a_{i}=w=1$. User $i$
with high $c_{i}$ gets high utility when they correctly estimate the
unsurprising world state, i.e., $a_{i}=w=0$. Therefore users with low $c_{i}$,
having high $\beta^{\prime}$, tend to trust a deceitful sender and bet on the
unlikely world state. Note that with the standing assumption of
$0<\mu<c_{i}<1/2$, their default estimation is still $a_{i}=0$. We call users
with low $c_{i}$ sympathizers, and high $c_{i}$ non-sympathizers.

Figure 5: The strictest effective regulation $\rho_{SE}$ for infinite linear
networks. Panel (b) is the basic case with $c_{i}=0.3$ for all $i$. For panel
(a), users close to the sender are sympathizers ($c_{i}=0.21$ for $i=0,1,2$)
and distant users are non-sympathizers ($c_{i}=0.4$ for $i\geq 3$). Panel (c)
is the opposite ($c_{i}=0.4$ for $i=0,1,2$ and $c_{i}=0.21$ for $i\geq 3$).
The white curves are the parameter threshold for $\rho_{SE}=0$ as in
Proposition 4, and the dashed curves represent that in the base case (b) for
reference. In case (a), it is difficult for platform $\mathbf{A}$ to enforce
strict regulation (i.e., the color map is bright and the white curve is high)
when $p$ is low. The color map is computed similarly to Figure 3, not using
any random procedure.

Figure 6: The strictest effective regulation $\rho_{SE}$ for a community
chain. Panel (b) is the basic case with $c_{J}=0.3$ for all $J=1,2,3$. For
panel (a), users in the sender’s community are sympathizers ($c_{1}=0.21$) and
distant users are the same as the basic case ($c_{2}=c_{3}=0.3$). Panel (c)
has non-sympathizers nearby ($c_{1}=0.4$ and $c_{2}=c_{3}=0.3$). The color map
shows the simulation result for 50 samples of the random graph. The white
curves are the parameter threshold for $\rho_{SE}=0$ as in Proposition 10, and
the dashed curves represent that in the base case (b) for reference. In case
(a), it is difficult for platform $\mathbf{A}$ to enforce strict regulation
(i.e., the color map is bright and the white curve is high) when $p$ is low.

###  VI-A Sympathizers in Linear Network

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

$\tilde{f}_{K}(p):=\mu p^{K}-\frac{\mu c_{K}p^{K}}{1-p^{K+1}}.$ (30)

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.

###  VI-B Community of Sympathizers

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.

######  Proposition 11.

Under Assumptions 1–3, the strictest effective regulation is $\rho_{SE}=0$ if

$n_{J}\theta_{JJ}b_{\mathbf{A}}-b_{\mathbf{B}}\geq\mu(1-c_{J})p_{\phi(J)\mathbf{B}}-\frac{\sum_{L=J+1}^{M}R_{L}}{\sum_{L=1}^{J}R_{L}}\mu
c_{J}p_{\phi(J)\mathbf{B}}$

for all $J=1,\ldots,M$.

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.

##  VII Discussion and Conclusion

This article provided a novel perspective on the design of regulation policies
that a dominant platform can enforce without reducing its user base, thus
shedding light on the theoretical efficacy of such misinformation-mitigation
measures, and the responsibility of mainstream platforms.

More precisely, we identified the strictest level of regulation a platform can
enforce on an information source without losing users to a non-regulated
competitor. We focused on simple yet relevant network structures, such as
stochastic block models, and used the model to make predictions in a variety
of scenarios. In particular, we showed that a tightly connected community
prevents new platform adoption thereby helping the currently dominant platform
impose strict regulation, especially if the information is diffusive. We also
showed that a news source in a large community tends to comply with strict
regulation in the popular platform if the information is not diffusive, while
a news source in a small community is more likely to relocate to an
alternative platform. Finally, a news source surrounded by sympathizers who
appreciate the information it emits exerts a greater indirect influence over
the platform’s content moderation policy than one surrounded by non-
sympathizers. However, when information is diffusive, this effect diminishes,
allowing the platform to enforce stricter regulations.

## Appendices

###  VII-A Lemmata for Proposition 1

######  Lemma 1.

Suppose $\mathbf{P}_{S}=\mathbf{B}$. In the adoption process, users can switch
from platform $\mathbf{A}$ to $\mathbf{B}$ but not from $\mathbf{B}$ to
$\mathbf{A}$.

###### Proof.

We show by induction. Suppose this statement is true at iteration $t$. Then,
for each user $i$, the number of neighbors in platform $\mathbf{A}$ (resp.
$\mathbf{B}$) decreases (increases) or stays the same. This means
$\Phi_{i\mathbf{A}}^{(t)}\geq\Phi_{i\mathbf{A}}^{(t+1)}$,
$\Phi_{i\mathbf{B}}^{(t)}\leq\Phi_{i\mathbf{B}}^{(t+1)}$. Also,
$p_{i\mathbf{B}}^{(t)}\leq p_{i\mathbf{B}}^{(t+1)}$ because the shortest path
in the subgraph for platform $\mathbf{B}$ does not become longer. This means
$\Psi_{i\mathbf{B}}^{(t)}\leq\Psi_{i\mathbf{B}}^{(t+1)}$. Note that
$p_{i\mathbf{A}}^{(t)}=p_{i\mathbf{A}}^{(t+1)}=0$ and therefore
$\Psi_{i\mathbf{A}}^{(t)}=\Psi_{i\mathbf{A}}^{(t+1)}$. Hence,
$V_{i\mathbf{A}}^{(t)}\geq V_{i\mathbf{A}}^{(t+1)}$ and
$V_{i\mathbf{B}}^{(t)}\leq V_{i\mathbf{B}}^{(t+1)}$. Therefore, if
$V_{i\mathbf{B}}^{(t)}\geq V_{i\mathbf{A}}^{(t)}$, then
$V_{i\mathbf{B}}^{(t+1)}\geq V_{i\mathbf{A}}^{(t+1)}$. This means if a user
chooses $\mathbf{B}$ at iteration $t$, then chooses $\mathbf{B}$ at iteration
$t+1$ too. By induction, the statement is true at any iteration. ∎

######  Lemma 2.

Suppose $\mathbf{P}_{S}=\mathbf{B}$ and the network is acyclic. User $i$ that
switches to platform $\mathbf{B}$ at iteration $t$ is $t$ edges away from the
sender and has $p_{i\mathbf{B}}^{(t)}=p^{t}$.

###### Proof.

We show by induction. Suppose the statement is true for $1,2,\ldots,t$.

Let $S_{t}$ be the set of users who switched to $\mathbf{B}$ at iteration $t$.
Since the graph is acyclic, each user has only one neighbor upstream (closer
to the sender) and the other neighbors downstream (farther from the sender).
Therefore, the upstream neighbors of $S_{t}$ is already in $\mathbf{B}$ at
iteration $t$ (otherwise, $p_{i\mathbf{B}}^{(t)}=0$ for $i\in S_{t}$, which
contradicts the assumption). Therefore, the platform adoption of $S_{t}$
affects $N_{j\mathbf{P}}$ only if user $j$ is a downstream neighbor of $S_{t}$
or is already in $\mathbf{B}$.

By assumption, all users more than $t$ edges away from the sender are in
$\mathbf{A}$ at iteration $t$. Therefore, the platform adoption of $S_{t}$
affects $p_{j\mathbf{B}}$ only if user $j$ is a downstream neighbor of
$S_{t}$.

Therefore, the platform adoption of $S_{t}$ affects $V_{j\mathbf{P}}$ only if
user $j$ is a downstream neighbor of $S_{t}$ or is already in $\mathbf{B}$.
Therefore, if a user switches to $\mathbf{B}$ at iteration $t+1$, then it is a
downstream neighbor of $S_{t}$. A downstream neighbor of $S_{t}$ is $t+1$
edges away from the sender and has $p_{i\mathbf{B}}^{(t+1)}=p^{t+1}$.

By induction, the statement holds for any $t$. ∎

###  VII-B Finite Linear Network

We have

$\displaystyle U_{\mathbf{A}}(\beta)=\sum_{k=0}^{n-1}(\mu+(1-\mu)\beta)p^{k},$
(31) $\displaystyle
U_{\mathbf{B}}(\beta)=\sum_{k=0}^{K}(\mu+(1-\mu)\beta)p^{k}$

with $K<n-2$ satisfying

$V_{K+1,B}<b_{\mathbf{A}}+(1-\mu)c\leq V_{K,B}$ (32)

or $K=n-1$ if $b_{\mathbf{A}}+(1-\mu)c\leq V_{n-2,B}$. For the best strategy
in platform B, the sender takes $\beta^{*}_{\mathbf{B}}=F(K)$ for $K<n-2$ and
$\beta^{*}_{\mathbf{B}}=F(n-2)$ for $K=n-1$.

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.,

$\displaystyle\sum_{k=0}^{n-1}\mu
p^{k}\geq\sum_{k=0}^{K}(\mu+(1-\mu)F(K))p^{k}$ $\displaystyle\text{for }K<n-2$
$\displaystyle\sum_{k=0}^{n-1}\mu
p^{k}\geq\sum_{k=0}^{n-1}(\mu+(1-\mu)F(n-2))p^{k}$ $\displaystyle\text{for
}K=n-1.$

This leads to Proposition 5.

###  VII-C Derivation of Proposition 10

Suppose the sender chooses platform $\mathbf{B}$. The critical situation for
community $J$’s platform adoption is that the first-adopting user $\phi(J)$
chooses a platform when a friend from community $1,\ldots,J-1$ is in platform
$\mathbf{B}$ and other friends in community $J$ are in $\mathbf{A}$. This user
compares

$\displaystyle V_{\phi(J)\mathbf{A}}$
$\displaystyle=n_{J}\theta_{JJ}b_{\mathbf{A}}+(1-\mu)c\quad\mbox{ and}$
$\displaystyle V_{\phi(J)\mathbf{B}}$ $\displaystyle=b_{\mathbf{B}}+\mu
p_{\phi(J)\mathbf{B}}(1-c)+(1-\mu)(1-\beta p_{\phi(J)\mathbf{B}})c.$

Hence, users in community $J$ chooses $\mathbf{B}$ if $\beta\leq\beta_{J}$
where

$\beta_{J}:=\frac{1}{(1-\mu)c}\bigg{(}\mu(1-c)-\frac{n_{J}\theta_{JJ}b_{\mathbf{A}}-b_{\mathbf{B}}}{p_{\phi(J)\mathbf{B}}}\bigg{)}.$
(33)

This $\beta_{J}$ is the sender’s best strategy assuming that communities
$1,\ldots,J$ are in platform $\mathbf{B}$ and $J+1,\ldots,M$ are in
$\mathbf{A}$ at equilibrium. Then, the strictest effective regulation
$\rho_{SE}$ satisfies $U_{\mathbf{A}}(\rho_{SE})\geq
U_{\mathbf{B}}(\beta_{J})$, i.e.,

$(\mu+(1-\mu)\rho_{SE})\sum_{L=1}^{M}R_{L}\geq(\mu+(1-\mu)\beta_{J})\sum_{L=1}^{J}R_{L}.$
(34)

Plugging in $\rho_{SE}=0$ and the value of $\beta_{J}$, we obtain the
following necessary and sufficient condition for $\rho_{SE}=0$ at the
equilibrium where communities $1,\ldots,J-1$ are in Platform $\mathbf{B}$ and
the rest are in $\mathbf{A}$:

$n_{J}\theta_{JJ}b_{\mathbf{A}}-b_{\mathbf{B}}\geq\mu(1-c)p_{\phi(J)\mathbf{B}}-\frac{\sum_{L=J+1}^{M}R_{L}}{\sum_{L=1}^{J}R_{L}}\mu
cp_{\phi(J)\mathbf{B}}.$

Considering all possible assignments of communities to platforms at
equilibrium, we achieve Proposition 10.

###  VII-D Numerical Validation of Assumption 1

To justify Assumption 1 for Proposition 10, we conduct simulations, taking
community chains and complete graphs of communities for example.

For variants of the base model of community chain ($n_{1}=n_{2}=n_{3}=30$ and
(27)), instead of $\theta_{JJ}=3/4$, we consider $\theta_{JJ}=1/4,1/8,1/16$.
Figure 7 shows the number of users who choose platform $\mathbf{B}$ when
platform $\mathbf{A}$ sets too strict regulation
$\rho_{\mathbf{A}}<\rho_{SE}$. For a large $\theta_{JJ}$, the number of users
in platform $\mathbf{B}$ is 0, 30, 60, or 90, indicating that all users in a
community choose the same platform. On the other hand, for a small
$\theta_{JJ}$, users in a community do not necessarily choose the same
platform. The simulation result is for a single sample for the random
networks.

Similar results are achieved in simulation for the complete graph of
communities.

Figure 7: The number of users who choose platform $\mathbf{B}$ when platform
$\mathbf{A}$ imposes too strict regulation $\rho_{\mathbf{A}}<\rho_{SE}$. The
network is a community chain. For a large $\theta_{JJ}$, the number of users
in platform $\mathbf{B}$ is 0, 30, 60, or 90, indicating that all users in a
community choose the same platform. On the other hand, for a small
$\theta_{JJ}$, users in a community do not necessarily choose the same
platform.

To discuss this property of community migration, we introduce new metrics. Let
$N(J,\mathbf{P})$ be the number of users in community $J$ who choose platform
$\mathbf{P}$. Then, define the number of users’ irregular choices as

$\sum_{J=1}^{M}\min\\{N(J,\mathbf{A}),N(J,\mathbf{B})\\}.$ (35)

This takes the value of zero when all users in each community choose the same
platform, and the maximum value $\sum_{J}n_{J}/2$ when half users in each
community choose a platform different from the other half.

Figure 8 presents the number of users’ irregular choices. The simulation is
conducted for 50 samples, fixing $p=0.7$ and $b_{\mathbf{A}}=0.002$. As in
Figures 7, the results consistently indicate that when users are tightly
connected within community (i.e., $\theta_{JJ}$ is high), all users in a
community choose the same platform. This simulation result supports Assumption
1 for Proposition 10.

Figure 8: The number of users’ irregular choices. The value of 0 means that
all users in a community choose the same platform.

###  VII-E Community Structures Applicable to Proposition 10

####  VII-E1 Community Chain

The chain of communities is introduced as a physical model that satisfies
(26).

Heuristically, a possible way to obtain such a scaling is by considering a
community chain with

$\theta_{JJ}\approx 1,$ (36)
$n_{J}\theta_{JJ^{\prime}}<1<n_{J}n_{J^{\prime}}\theta_{JJ^{\prime}}\ \text{
for }\ J^{\prime}=J-1,J+1,$ (37)
$n_{J}n_{J-1}\theta_{JJ-1}n_{J+1}\theta_{JJ+1}<1.$ (38)

Roughly speaking, (36) means that most user pairs in the same community are
friends; (37) means that although communities are connected, most users do not
have friends in other communities; (38) means that in most cases, no user in
community $J$ has friends both in community $J-1$ and community $J+1$. Under
these conditions, it should be intuitive that most users are two edges away
from the nearest neighbor in an adjacent community, thus leading to (26).

Assumption 1 is validated for (36) in simulation in Appendix D. The chain
structure of the network guarantees Assumption 2, and Assumption 3 is
satisfied due to (37).

####  VII-E2 Star/Complete Graph of Communities

In order to obtain (28), consider a stochastic block model with the sender in
community 1 and $\theta_{1J}>0$ for all $J=2,\ldots,M$. In addition to (36),
we assume

$n_{1}\theta_{1J}<1,\ n_{J}\theta_{1J}<1,\ n_{1}n_{J}\theta_{1J}>1,$ (39)
$\theta_{JJ^{\prime}}\approx 0\ \text{ for }\ J\neq 1,J^{\prime}\neq 1,J\neq
J^{\prime}.$ (40)

Equation (39) means that in most cases, a user in community $J$ (resp. 1) does
not have friends in community 1 (resp. $J$), but there are some inter-
community links between 1 and $J$. Equation (40) means that edges between
peripheral communities do not exist or are negligible. These assumptions
indicate that the first-adopting user $\phi(J)$ in community $J\geq 2$ is two-
edges away from the sender and most others are three-edges away; hence (28).

Assumption 1 is validated for (36) in simulation in Appendix D. Assumption 2
is satisfied because $n_{2}\theta_{22}<n_{3}\theta_{33}<n_{4}\theta_{44}$. The
combination of (39) and (40) satisfies Assumption 3.

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|  So Sasaki is working toward the Ph.D. degree with the University of
Illinois at Urbana–Champaign, Urbana, IL, USA. He is currently with the
Decision and Control Group, Coordinated Science Lab, University of Illinois at
Urbana–Champaign. His research interests include game, optimization, control,
graph theory, and machine learning, with application to social media.
---|---
|  Cédric Langbort received the Ph.D. degree in Theoretical and Applied
Mechanics from Cornell University, Ithaca, NY, USA, in 2005. He is currently a
Professor of Aerospace Engineering with the University of Illinois at
Urbana–Champaign, Champaign, IL, USA. He is also with the Decision and Control
Group, Coordinated Science Lab (CSL), where he works on the applications of
control, game, and optimization theory to a variety of fields.
---|---