#  Similarity of Information and Collective Action111This research was
supported in part by National Science Foundation award 2417694. Aditya
Kuvalekar acknowledges British Academy Small Grant SRG2223\231506. We are
grateful to Nageeb Ali, Marco Battaglini, Steve Callander, Mehmet Ekmekci,
Albin Erlanson, Daniel Garrett, Elliot Lipnowski, John Maxwell, Meg Meyer,
Emre Ozdenoren, Jacopo Perego, Doron Ravid, Phil Reny, Vasiliki Skreta, Yu Fu
Wong, Jidong Zhou, and conference audiences and seminar participants at
Bocconi University, CERGE-EI, University of Chicago, Collegio Alberto, Essex
University, HKUST, Kelley School of Business, Northwestern University, Oxford
University, Penn State University, Princeton University, Stony Brook Game
Theory meeting, SITE Political Economy, Toulouse School of Economics,
University College London, University of Warwick, and Yale University for
their helpful comments and suggestions. Basak: Indiana University, Kelley
School of Business, email<EMAIL_ADDRESS>Deb: New York University, email:
<EMAIL_ADDRESS>Kuvalekar: University of Essex, email:
<EMAIL_ADDRESS>

Deepal Basak Indiana University Joyee Deb New York University Aditya Kuvalekar
University of Essex

######  Abstract

We study a canonical collective action game with incomplete information.
Individuals attempt to coordinate to achieve a shared goal, while also facing
a temptation to free-ride. Consuming more similar information about the
fundamentals can help them coordinate, but it can also exacerbate free-riding.
Our main result shows that more similar information facilitates (impedes)
achieving a common goal when achieving the goal is sufficiently challenging
(easy). We apply this insight to show why insufficiently powerful
authoritarian governments may face larger protests when attempting to restrict
press freedom, and why informational diversity in committees is beneficial
when each vote carries more weight.

Keywords: collective action, information similarity, regime change

JEL Codes: D82, D83

##  1 Introduction

This paper addresses the following question: When does increased similarity of
information among participants help or harm participation in collective
action?

A collective action problem is a situation in which individuals want to
achieve a common goal but face a temptation to free-ride (see Tullock (1971),
Olson (1965)), because reaching the goal requires a sufficient number of
people to take a costly action, while the benefit accrues to all.222Examples
of collective action problems are ubiquitous: protests, regime changes,
boycotts, or voting in committees. See Palfrey and Rosenthal (1985), Taylor
and Ward (1982), Goeree and Holt (2005), Myatt and Wallace (2008), Diermeier
and Van Mieghem (2008), Shadmehr (2021), Dziuda et al. (2021) for some
examples. We consider collective action problems under incomplete information,
in which individuals face uncertainty about the benefits of reaching the goal,
and can privately learn about it. In such situations, an individual may not
take the costly action even after learning that reaching the goal is socially
beneficial. This is because her decision depends not only on what her private
information tells her about the state (fundamental uncertainty), but also on
what it tells her about the information that others have and thus what they
will do (strategic uncertainty). For instance, an individual’s private
information may make her believe, at the same time, that reaching the goal is
beneficial, but that others do not intend to take the costly action, rendering
her own costly action ineffective. We investigate what happens to
participation in collective action when information becomes more similar, in
the sense that agents know that others have private information with content
similar to their own.

Understanding the effects of changing similarity of information in strategic
environments is particularly important against the backdrop of the
extraordinary changes in how information is disseminated and consumed in
recent years. Algorithms steer individuals to news or video content based on
personal characteristics, resulting in like-minded people accessing the same
information from the same source. A plausible conjecture is that if people
with the same objectives now access the same information, it may be easier for
them to predict each other’s actions and attain better outcomes.

Our central observation is that greater similarity of information among
agents, _even those with identical preferences_ , can serve as a double-edged
sword in collective action games. On the one hand, if people believe that
others are more likely to have the same information as them, they may be able
to coordinate better to reach the common goal. On the other hand, the
temptation to free-ride may be exacerbated: if an agent knows that others have
the same information and predicts that they will take action, then she does
not need to take a costly action herself. Both these opposing effects have
been observed separately in empirical work on protests, an archetypal
collective action game. Enikolopov et al. (2020) find, using Russian data,
that in cities where individuals accessed news from the same media platform,
it led to more protests, and cities in which people were not all on the same
platform faced fewer protests. However, in an experiment on Hong Kong
protests, Cantoni et al. (2019) show essentially the opposite: knowledge of
others’ participation led to a stronger temptation to free-ride for potential
participants. These opposite effects motivate our research question: when is
information similarity helpful and when is it harmful in collective action
games? Our main results characterize when increased information similarity
helps or harms participation in a canonical regime-change game of incomplete
information. To illustrate the main forces, let us start with a simple
example.

Example: Abe and Bob are working on a project that is of high
($\bm{\theta}=1$) or mediocre ($\bm{\theta}=0$) quality with equal
probability. They do not know the project quality, but each receives a private
binary signal about it. For simplicity, we assume that a player never mistakes
a mediocre project for a high-quality one but may mistake a high-quality
project for a mediocre one. Formally, each player $i$ receives a binary signal
$\mathbf{X}_{i}$, such that $\mathbf{X}_{i}=0$ if $\bm{\theta}=0$, and if
$\bm{\theta}=1$, the signals are drawn from some exchangeable joint
distribution $\mathscr{P}^{1}$. Importantly, the signals might not be
conditionally independent.

After observing the private signals, the two players decide (independently and
simultaneously) whether to work or shirk. Working entails a cost of $c$. The
project is completed with certainty if both players work and completed with
probability $q$ if only one works. If a high-quality project is completed, the
players enjoy a benefit of $1$ each. If the project is mediocre or unfinished,
they both get $0$. Table 1 specifies the payoffs. For simplicity, suppose also
that $c$ is high enough that players will never work after receiving signal
$\mathbf{X}_{i}=0$.

|  | Bob
---|---|---
|  | work | shirk
Abe | work | $\theta-c,\theta-c$ | $q\theta-c,q\theta$
shirk | $q\theta,q\theta-c$ | $0,0$
Table 1: Payoff matrix

In this example, notice that while both players enjoy the benefit from the
completion of a high-quality project, each player may have an incentive to
free-ride even if he believes the project is high quality. We ask first
whether there is a symmetric, pure-strategy equilibrium $\sigma^{1}$ in which
a player works whenever $\mathbf{X}_{i}=1$—that is, when he knows the project
is of high quality. Then we ask, what happens if the realized private signals
that Abe and Bob observe become more similar, in the sense that each player
knows that his private signal is now a better predictor of the other player’s
signal (and his consequent action choice).

For a player to be willing to work after observing $\mathbf{X}_{i}=1$, we need

$\displaystyle(1-q)\mathscr{P}^{1}_{1}(\mathbf{X}_{-i}=1)+q\mathscr{P}^{1}_{1}(\mathbf{X}_{-i}=0)\geq
c.$

where $\mathscr{P}^{1}_{1}$ is the conditional distribution of
$\mathbf{X}_{-i}$ given that $\mathbf{X}_{i}=1$.333Due to exchangeability of
$\mathscr{P}^{1}$, this conditional distribution need not be indexed by $i$.
Intuitively, conditional on observing $\mathbf{X}_{i}=1$, player $i$ works
only if he believes he is pivotal (his effort makes a difference) with high-
enough probability. Recall that if a player works, then his marginal
contribution to the completion of the project is $(1-q)$ if the other player
also works and $q$ if the other player does not work. The marginal benefit
from effort depends on two primitives: (i) the extent to which individual
effort can make a difference, measured by $q$, and (ii) a player’s conditional
belief about his opponent’s signal (and implied equilibrium action). It is
easy to see that there are $c$ and $\mathscr{P}^{1}$ (when
$\mathscr{P}^{1}_{1}(\mathbf{X}_{-i}=1)<\frac{1}{2}$) such that the following
result holds:

> There exists $q^{*}\in(0,1)$ such that for all $q\in[q^{*},1]$, there is an
> equilibrium in which each player $i$ works whenever $\mathbf{X}_{i}=1$.444We
> provide a complete analysis of this example along with the parametric
> configurations supporting these results in Appendix B.6.

What happens when Abe’s and Bob’s signals become _more similar?_ , i.e., it
becomes more likely that Abe and Bob see the same signals? Formally, suppose
that when the project is high quality ($\bm{\theta}=1$), the probability that
both receive the same (different) signal increases (decreases) by $\alpha$.
They continue to receive a signal $0$ when $\bm{\theta}=0$. Moreover, suppose
that, for any given agent, the probability that he receives a signal $1$
conditional on $\bm{\theta}=1$ remains unchanged. Figure 1 illustrates such a
change in the joint distribution in state $\bm{\theta}=1$.

$0$$1$$0$$1$$+\alpha$$-\alpha$$-\alpha$$+\alpha$$X_{1}$$X_{2}$ Figure 1:
Increasing similarity when $\bm{\theta}=1$

How does this increased similarity in the realized private signals affect
players’ incentives to work in equilibrium? The answer to this question
depends on the value of $q$. A player works on a signal $1$ if

$\displaystyle(1-q)\underbrace{\mathscr{P}^{1}_{1}(\mathbf{X}_{-i}=1)}_{\uparrow\text{
with more
similarity}}+q\underbrace{\mathscr{P}^{1}_{1}(\mathbf{X}_{-i}=0)}_{\downarrow\text{
with more similarity}}\geq c.$

When $q$ is high, with high likelihood one player’s effort suffices for
project completion. In this case, after observing $\mathbf{X}_{i}=1$, a player
$i$ is less likely to be pivotal compared to before and has a stronger
incentive to free-ride. When $q$ is low, after observing $\mathbf{X}_{i}=1$, a
player $i$ assigns a higher probability to being pivotal and has a stronger
incentive to work. Indeed, there exists an $\alpha$ which makes
$\mathscr{P}^{1}_{1}(\mathbf{X}_{-i}=1)>\frac{1}{2}$ such that the following
holds:

> There exists $q^{**}\in(0,1)$ such that, with more similar information,
> there is an equilibrium in which each player $i$ works after observing
> $\mathbf{X}_{i}=1$ if and only if $q\in[0,q^{**}]$.

That is, after observing $\mathbf{X}_{i}=1$, under less similar signals,
agents are willing to work if and only if $q$ is sufficiently high. With more
similar information, the exact opposite is true: agents are willing to work if
and only if $q$ is sufficiently low. Intuitively, a low value of $q$ captures
environments in which not achieving coordination is the primary obstacle to
collective action. In these settings, increasing information similarity proves
beneficial. A high value of $q$ captures environments in which free-riding is
the primary obstacle to collective action. In these settings, increasing
information similarity may be harmful.

Of course this example has several simplifying features. With binary signals,
it was straightforward to define what it means to have more similar
information. Moreover, the parameter $q$ was a mechanical way of quantifying
the relative importance of coordination (versus preventing free-riding).
Finally, strategies are quite simple with binary signals and given no agent
participates after a signal $0$. But it turns out that the essential insights
from this example generalize.

In the baseline model, we consider a canonical regime-change game. There are
two states of the world: one in which regime change is beneficial for society,
and one in which it is not. There are two geographically dispersed groups of
uncertain size. The groups have identical preferences but have access to
different information about the state. Each group receives a signal about the
state, drawn from a finite set. Each individual sees only the signal of their
own group and decides whether to participate in an action aimed to bring about
regime change. Regime change is successful only if a (potentially random)
threshold number of agents participate. The benefit of a successful regime
change is public, but the cost of participation is borne only by the
participants. In this canonical setting, we ask whether increased similarity
of groups’ signals increases or decreases participation.

First, we need a notion that allows us to compare similarity between
information structures. We use an order of information similarity, analogous
to the one in the example above. We say information is _more similar_ or has a
_higher concentration along diagonal_ (CAD) when, conditional on observing
signal realization $x$, an agent believes it is more likely that others also
observed the signal realization $x$ and less likely that others observed a
signal realization different from $x$.555For two-dimensional signals, our CAD
order is equivalent to one proposed by Meyer (1990). Like her, we require that
the two signals have the same marginal distribution over states.

We then show that, analogous to low and high values of $q$ in the example, it
is possible to cleanly partition the general setting into _encouragement
environments_ , in which greater aggregate participation encourages
participation by making it more likely that an individual can make a
difference, and _discouragement environments_ , in which greater aggregate
participation discourages participation by making it less likely that an agent
can make a difference.

Our main results establish that more similar information in the CAD order
increases participation in encouragement environments but can reduce
participation in discouragement environments. Formally, for any strategy
profile, we define the participation (nonparticipation) set as the set of
signals for which agents participate (do not participate). Given any
information structure, our game can admit multiple equilibria, including,
possibly equilibria with strategies that are non-monotonic in signals. We
prove that more similar information enlarges the participation set in the
equilibrium with maximal participation in encouragement environments (Theorem
1) but can shrink it in discouragement environments (Theorem 2).

We also characterize these two environments. Under an intuitive single-
crossing condition, it turns out that resilient regimes, in which regime
change requires a large number of participants, constitute encouragement
environments and weak regimes, in which regime change requires a small number
of participants, constitute discouragement environments. So the effect of
changing information similarity is qualitatively different based on the
resilience of regimes. Recall the mixed empirical evidence regarding how
modern information technology has affected protests. Our results imply that
increased similarity of information may have facilitated greater participation
in collective action against regimes previously thought to be impregnable but,
at the same time, hindered movements with ex-ante easier goals.

We apply our framework to two applications. First, we study mass protests (as
in Shadmehr, 2021) and investigate the effect of increased information
similarity on the likelihood of a successful protest and consequent welfare.
Notice that even when increased similarity of information increases
participation, it may not improve welfare or the likelihood of success. This
is because increased similarity of information increases the probability that
everyone participates and the probability that no one participates, making the
overall effect on the probability of success unclear a priori. Welfare effects
are also ambiguous because even if increased participation makes regime change
more likely, the marginal social benefit may be lower than the marginal cost.
Specializing to an environment with a deterministic participation threshold
and Poisson uncertainty about group sizes we show that increasing similarity
can never lower the maximal probability of a successful protest or welfare in
encouragement environments but can do so in discouragement environments. These
findings have important implications for press freedom and authoritarian
governments. When authoritarian governments curb press freedoms, they
effectively reduce the similarity of information across individuals. Our
results suggest that it is precisely the resilient authoritarian governments
that benefit more from doing this. Less powerful regimes may face larger
protests and a higher likelihood of being overthrown when attempting to
restrict press freedom.

Second, we apply our framework to a setting of costly voting in a committee
(as in Palfrey and Rosenthal, 1985). We consider a monetary policy committee
with $n>2$ members who must vote on whether to raise interest rates. A rate
increase is implemented only if a threshold number of members vote for it.
Public opinion is against any rate increases, and committee member votes are
observed ex post, making it costly for any member to vote in favor of a rate
increase. Committee members base their decision on their private information
about whether a rate increase is warranted based on the state of the economy.
Their information can be more or less similar depending on their
backgrounds—academic backgrounds or areas of expertise that makes them focus
on different aspects of the available evidence. How do changes in member
diversity affect the voting outcome? In this setting, there is no uncertainty
about the number of participants or the vote threshold. And importantly, we
need to move beyond bivariate signals. We develop a natural analog of our CAD
order to compare multivariate binary signals, and use it to establish results
analogous to Theorems 1 and 2. We show that conditional on the rate increase
being warranted, a more diverse committee (with less similar information)
strengthens the incentive to vote correctly and increases the maximal
equilibrium number of votes in favor of a rate increase if each individual
vote carries enough weight—that is, fewer votes are required for a rate hike.
Conversely, if an individual vote carries very little weight, a more diverse
committee weakens the incentive to vote correctly.666Unlike Theorems 1 and 2,
however, with more than two groups, the characterization is only partial—that
is, not every environment can be classified as encouragement or
discouragement. We provide more details in Section 6. We also examine how
changes in information similarity influence the optimal voting threshold rule.

We conclude by examining some other extensions of our baseline model.

###  1.1 Related Literature

A large literature going back to at least Hirshleifer (1971) studies how
exogenous change in the information environment changes agents’ incentives in
strategic environments. More recently, Morris and Shin (2002), Angeletos and
Pavan (2007), Bergemann and Morris (2013), Jensen (2018), and Mekonnen and
Vizcaíno (2022) have studied this question in a class of games with monotone
best responses (pure complementarity or substitutability). Monotonic best
responses imply that extremal equilibria are in monotone strategies (see, for
example, Van Zandt and Vives (2007)). This literature provides insights about
how changes in the exogenous information structure affect the monotone
equilibria and welfare. Since a canonical collective action game involves
coordination and free-riding motives, the best response is nonmonotonic,
making the established tools unsuitable. Moreover, much of the literature
cited above focuses on the effect of new public information. Importantly, we
study how the equilibrium set changes with _changes in similarity of
information_.

We consider an arbitrary signal structure and propose CAD as a natural order
of information similarity. The existing literature contains other measures of
the interdependence of joint distributions (for example, Müller and Stoyan
(2002), Meyer and Strulovici (2012)), but none are appropriate for comparing
the _conditional_ belief distributions that arise in strategic settings with
incomplete information. For the bivariate case (the focus of this paper), the
CAD order is the same as that in Meyer (1990). Like her, we consider
multivariate random variables with fixed marginal distributions while changing
the joint distribution. Clemen and Winkler (1985) and Cheng and Borgers (2024)
study how such changes impact the value of information and show that
informational diversity may be valuable. de Oliveira et al. (2023) consider an
environment with known marginal distributions but unknown joint distribution
to obtain the robustly optimal policy in a class of decision problems. Awaya
and Krishna (2022) study the effect of the interdependence of signals on
common learning as do Cripps et al. (2008). They show that essentially any
interdependence obstructs common learning.

We also contribute to the sizable literature on protests and voting. The
majority of the theoretical work on protest focuses on the coordination
aspect. However, some recent papers—for example, Shadmehr (2021), Dziuda et
al. (2021), and Park and Smyrniotis (2022)—incorporate free-riding and
construct cutoff equilibria. Dziuda et al. (2021) show that a lower required
participation threshold for success might not increase the likelihood of a
successful protest. Mutluer (2024) shows that a higher cost of participation
may lead to larger protest. These papers do not consider changing information
environments, which is our focus. Some recent empirical papers have studied
the effect of modern communication technologies on the size of protests. For
instance, Manacorda and Tesei (2020) show empirical evidence that mobile
phones facilitated protests in Africa, and Enikolopov et al. (2020) show that
the diffusion of an online social network increased protest turnout in Russia.
However, Cantoni et al. (2019) demonstrate, in a recent experiment about mass
protesters in Hong Kong, that the knowledge of others’ participation led to a
stronger temptation to free-ride for potential participants. These ambiguous
empirical results underscore the importance of our research question. We
provide a clear characterization of when information similarity helps and when
it hurts. Our results have implications for the effect of press freedom on
protests. Edmond (2013) considers a game of protest in which the regime can
manipulate information. However, in this game, the agents only have the
coordination motive and no free-riding motive. In our voting application, we
study how informational diversity affects voting incentives. Taylor and
Yildirim (2010) and Roesler (2022) study similar questions. In these papers,
if no other agent votes, then an agent can always get her desired outcome by
voting. In contrast, in our setup, if others do not vote, then an agent alone
can never get her desired outcome. Chemmanur and Fedaseyeu (2018) present a
model of voting on a corporate board that closely resembles our application.
In a recent paper, Kattwinkel and Winter (2023) characterize the optimal
voting rule under conditionally independent signals. We restrict attention to
simple threshold voting rules but show how information similarity affects the
optimal voting rule.

##  2 A Regime-Change Game

There are two states of the world: $\bm{\theta}\in\Theta=\\{0,1\\}$. In
$\bm{\theta}=1$, it is socially beneficial to change the regime. In
$\bm{\theta}=0$, it is not socially beneficial to change the regime. An
alternative interpretation of $\bm{\theta}$ is that it captures whether a
regime change is feasible ($\bm{\theta}=1$) or not ($\bm{\theta}=0$). Society
consists of $G$ groups. For most of the paper, we consider $G=2$. The analysis
for more than two groups is relegated to the online appendix. We introduce
population uncertainty à la Myerson (1998): the number of agents in any group
$g$, denoted by $\mathbf{N}_{g}$, is a $\mathbb{Z}_{+}$-valued random variable
with probability mass function $\eta(\cdot)$ and mean $N$. Agents do not
observe the size of their own group or other groups. We let $\eta^{A}(\cdot)$
denote the conditional probability mass function of $\mathbf{N}_{g}-1$
according to an agent in group $g$.777 In general, $\eta^{A}(\cdot)$ can be
different from $\eta(\cdot)$ because an agent may be more or less optimistic
about the size of her group, conditional on belonging to the group herself,
compared to what someone outside the group believes about the group size. As
an example, if $\eta(\cdot)$ were drawn from a Poisson distribution, then
$\eta^{A}(\cdot)$ and $\eta(\cdot)$ would coincide.

Each agent decides whether to take a costly action. Participating (choosing
$a=1$) costs $c>0$, while not participating ($a=0$) is costless. Regime change
occurs only if enough agents participate. We assume that the threshold
participation required to change the regime is an $\mathbb{N}$-valued random
variable, $\bm{\bar{n}}+1$, where $\bm{\bar{n}}$ follows probability mass
function $\phi(\cdot)$, and agents do not observe the realization of
$\bm{\bar{n}}+1$. We call $\bm{\bar{n}}$ the _resilience_ of the regime. We
allow population uncertainty and a random (unobserved) resilience because this
is more realistic in many regime-change settings. However, in Section 5 we
present an application with observable and deterministic group sizes and
participation threshold. We summarize the payoffs in the matrix below, in
which $\bar{n}+1$ denotes the realized resilience and $A$ denotes the number
of agents who participate.

| $A\geq\bar{n}+1$ | $A\leq\bar{n}$
---|---|---
$a=1$ | $\theta-c$ | $-c$
$a=0$ | $\theta$ | $0$

Note that regime change when it is not beneficial (in $\bm{\theta}=0$) entails
no additional costs beyond the costs of participation. This assumption can be
relaxed (see Section 6). Finally,
$\bm{\theta},\mathbf{N}_{1},\mathbf{N}_{2},\bm{\bar{n}}$ are independent.

###  2.1 Information Structure

Before deciding whether to participate, agents receive information about the
state of the world. There is a fixed, finite set of signals $\mathcal{X}$.
Each group receives a signal $\mathbf{X}_{g}$ drawn from $\mathcal{X}$, and
every agent in group $g$ observes only the signal received by their own group.
Let $\mathbf{X}:=(\mathbf{X}_{g})_{g\in\mathcal{G}}$. We denote the joint
distribution of $(\bm{\theta},\mathbf{X})$ by
$\mathscr{P}(\cdot)\in\Delta(\Theta\times\mathcal{X}^{G})$, and the
distribution of $\mathbf{X}$ conditional on $\bm{\theta}=\theta$ by
$\mathscr{P}^{\theta}\in\Delta(\mathcal{X}^{G})$. As we will describe in
Section 2.3, similarity of information in our context will simply be a measure
of interdependence of $\mathbf{X}$.

We assume that $\mathbf{X}_{1},\mathbf{X}_{2}$ are independent of
$\mathbf{N}_{1},\mathbf{N}_{2}$ and $\bm{\bar{n}}$. That is, a group’s signal
conveys information about the state to the agents but not about their or the
other group’s realized size or the threshold. Let
$\mu:=\mathscr{P}(\\{\bm{\theta}=1\\})$. We denote by
$\mathbb{P}^{\theta}_{g}\in\Delta(\mathcal{X})$ the distribution of
$\mathbf{X}_{g}$ given that $\bm{\theta}=\theta$. We assume that the
distribution of $\mathscr{P}^{\theta}(\cdot)$ is exchangeable so that we have
$\mathbb{P}^{\theta}_{g}=\mathbb{P}^{\theta}_{g^{\prime}}=:\mathbb{P}^{\theta}$
for all $g,g^{\prime}\in\mathcal{G}$ and all $\theta\in\Theta$.

Let $\mu(x):=\mathscr{P}(\\{\bm{\theta}=1\\}|\\{\mathbf{X}_{g}=x\\})$ denote
the posterior probability that any agent in group $g$ assigns to the state’s
being $1$ given a realized signal $x$. We assume that
$\mu:\mathcal{X}\to[0,1]$ is injective. Given group $g$, we let
$\mathbf{X}_{-g}$ be the random variable denoting the signal of the other
group. Let $\mathscr{P}^{\theta}_{x}\in\Delta(\mathcal{X})$ denote the
conditional distribution of $\mathbf{X}_{-g}$ given state $\bm{\theta}=\theta$
and $\mathbf{X}_{g}=x$. Since $\mathbb{P}^{\theta}$ is exchangeable,
$\mathscr{P}^{\theta}_{x}(\cdot)$ is the same for every group.

###  2.2 Strategies and Aggregate Participation

Strategies: A (pure) strategy of agent $i$ from group $g$ is a mapping,

$\sigma_{g}:\mathcal{X}\to\\{0,1\\}.$

That is, we assume symmetric strategies within a group. We restrict attention
to pure strategies throughout the paper, until Section 6, in which we discuss
the extension to mixed strategies. Given a strategy profile
$\sigma=(\sigma_{1},\sigma_{2})$, we define the _participation set of
$\sigma_{g}$ for group $g$_, denoted by $P(\sigma_{g})$, to be the set of
signals such that $\sigma_{g}(x)=1$. Analogously, we define the
_nonparticipation set of $\sigma$_, denoted by
$NP(\sigma_{g}):=\mathcal{X}\backslash P(\sigma_{g})$. When the dependence on
$\sigma_{g}$ is obvious, we denote $P(\sigma_{g})$ and $NP(\sigma_{g})$ by
$P_{g}$ and $NP_{g}$, respectively.
Aggregate participation: Since group size is random, we let $\mathbf{A}$
denote the random variable corresponding to the total number of participating
agents, given a strategy profile.

$\mathbf{A}:=\sum_{g=1}^{G}\mathbf{N}_{g}\sigma_{g}(\mathbf{X}_{g})$

We call $\mathbf{A}$ the aggregate participation. Notice that $\mathbf{A}$ is
$(\mathbf{N}_{1},\mathbf{N}_{2},\mathbf{X})$-measurable and depends on
$\sigma$. When this is obvious, we suppress $\sigma$. All agents in group $g$
receive the same signal and have the same belief. Let
$\mathbf{A}_{-g}(x_{g},x_{-g};\sigma)$ be the aggregate participation
according to an agent in group $g$, excluding herself. Then we have

$\displaystyle\mathbf{A}_{-g}(x_{g},x_{-g};\sigma)$ $\displaystyle:=$
$\displaystyle\mathbf{N}_{-g}\sigma_{-g}(x_{-g})+(\mathbf{N}_{g}-1)\sigma_{g}(x_{g})$
$\displaystyle=$ $\displaystyle\mathbf{N}_{-g}\mathbbm{1}_{x_{-g}\in
P_{-g}}+(\mathbf{N}_{g}-1)\mathbbm{1}_{x_{g}\in P_{g}}.$

Expected aggregate participation in $\bm{\theta}=1$:

For any $\sigma$, define

$\displaystyle\mathscr{V}(\sigma):=\mathbb{E}[\mathbf{A}(\mathbf{X};\sigma)|\bm{\theta}=1]$
(1)

to be the expected aggregate participation in state $1$, which is the state in
which it is beneficial to change the regime. For a fixed $\sigma$,
$\mathscr{V}(\sigma)$ depends only on the marginal distribution of the signals
and not on the joint distribution. So information similarity affects
$\mathscr{V}$ only by affecting the equilibrium $\sigma$. We state this in the
following lemma. The proof is in the appendix. With some abuse of notation,
let $\mathbb{P}^{1}(S)$ denote $\mathscr{P}(\mathbf{X}_{g}\in
S|\bm{\theta}=1)$.

######  Lemma 1:

For any $\sigma$, with associated participation sets
$(P_{g})_{g\in\mathcal{G}}$,

$\mathscr{V}(\sigma)=N\sum_{g=1}^{G}\mathbb{P}^{1}(P_{g}).$

Therefore, when $G=2$, we can say
$\mathscr{V}(\sigma)=2[\mathbb{P}^{1}(P_{1}\cup
P_{2})+\mathbb{P}^{1}(P_{1}\cap P_{2})]$.

Solution concept: We consider Bayes Nash equilibria in pure strategies. We do
not impose any additional structure on the equilibrium, such as monotone or
symmetric strategies. Multiple equilibria may exist, including one in which no
one participates regardless of the signal.

Notice that since we have a collective action problem, best responses are not
monotonic in aggregate participation, unlike in other regime-change games with
only strategic complementarities (Morris and Shin, 2002, for example).
Therefore, it is not clear whether equilibria can be ordered in any natural
way. This means we cannot use existing tools—such as those used in
supermodular games—directly. Let $\mathcal{E}(\mathscr{P})$ be the set of
strategy profiles that constitute an equilibrium under information structure
$\mathscr{P}$.

Given multiple equilibria, we focus on how increased similarity affects the
maximal possible participation in any equilibrium. Accordingly, we define the
following.

######  Definition 1 (Maximal Participation Equilibrium and Maximal
Equilibrium Aggregate Participation):

We say that an equilibrium $\sigma^{*}$ is a maximal participation equilibrium
if $\mathscr{V}(\sigma^{*})\geq\mathscr{V}(\sigma)$ for all
$\sigma\in\mathcal{E}(\mathscr{P})$.888Since the set of signals is finite and
we look at pure strategies, the existence of a maximal aggregate participation
equilibrium is guaranteed.  Let $\mathscr{V}^{*}(\mathscr{P})$ denote the
expected aggregate participation (in state 1) in the maximal participation
equilibrium given information structure $\mathscr{P}$, and call it the maximal
equilibrium aggregate participation.

Both the expected aggregate participation given a strategy
$\mathscr{V}(\sigma)$ and the maximal equilibrium aggregate participation for
an information structure $\mathscr{V}^{*}(\mathscr{P})$ are defined
_conditional on $\bm{\theta}=1$_; that is, when change is beneficial. For
brevity, henceforth, we do not mention this explicitly.
$\mathscr{V}(\cdot),\mathscr{V}^{*}(\cdot)$ also depend on other parameters,
such as $\bm{\bar{n}}$. We typically suppress this dependence and only make
the dependence on the information structure explicit.999Most of our results
about the expected aggregate participation remain unchanged if we used the ex-
ante expected aggregate participation rather than expected aggregate
participation when $\bm{\theta}=1$.

Arguably, instead of studying maximal aggregate participation, we could have
focused on the maximal equilibrium probability of successful collective action
or maximal welfare. One reason to focus on participation is its empirical
relevance. Recent work in political science on collective action measures
aggregate participation in mass protests.101010For instance, several empirical
studies (Enikolopov et al., 2020, e.g.), as well as popular press articles
(e.g., https://www.nytimes.com/2023/11/22/opinion/does-protest-work-
bevins.html) and even datasets that document protests over time (Clark and
Regan, 2016, e.g.) record turnout. In Section 4, we study also the effect of
information similarity on the probability of successful regime change and on
welfare (in the canonical case where $\mathbf{N}_{1},\mathbf{N}_{2}$ are
Poisson distributed), and show that results analogous to our main results
continue to hold.

###  2.3 A Measure of Information Similarity

Given our research question, we need a notion of informational similarity.
Agents make participation decisions based on their beliefs about the state of
the world and the expected aggregate participation. So they must reason about
the _conditional_ probability of others’ information given their own. We use
the following similarity order for two-dimensional random variables using such
conditional beliefs.

######  Definition 2 (Concentration Along Diagonal (CAD)):

Let $\mathcal{Y}\subset$ be a finite set. Let $Y$ and $\widehat{Y}$ be two
$\mathcal{Y}^{2}$-valued exchangeable random variables whose distributions are
given by $\mathscr{D}$ and $\widehat{\mathscr{D}}$, respectively. We say $Y$
is more similar than $\widehat{Y}$ in the CAD order, denoted by
$Y\succcurlyeq_{CAD}\widehat{Y}$ or
$\mathscr{D}\succcurlyeq_{CAD}\widehat{\mathscr{D}}$, if the following two
conditions hold.

  1. 1.

$Y_{i}$ and $\widehat{Y}_{i}$ are identically distributed for all
$i\in\\{1,2\\}$.

  2. 2.

For $y\in\mathcal{Y}$ and $T\subseteq\mathcal{Y}$,

    1. (a)

$\mathscr{D}(Y_{2}\in T|Y_{1}=y)\geq\widehat{\mathscr{D}}(\widehat{Y_{2}}\in
T|\widehat{Y_{1}}=y)$ if $y\in T$.

    2. (b)

$\mathscr{D}(Y_{2}\in T|Y_{1}=y)\leq\widehat{\mathscr{D}}(\widehat{Y}_{2}\in
T|\widehat{Y}_{1}=y)$ if $y\notin T$.

Notice that, by exchangeability of the distributions, we can interchange
$Y_{1}$ and $Y_{2}$ in the definition. We use the CAD order to compare
$\mathscr{P}^{\theta}_{x}(\cdot)$, the beliefs of players conditional on a
state, and a realized signal, keeping the marginal beliefs conditional on
$\bm{\theta}=\theta$ by $\mathbb{P}^{\theta}$ unchanged.111111We analyze the
effect of increasing similarity of $\mathscr{P}^{1}_{x}(\cdot)$. This is
because, given the definitions of states and payoffs in our game, changing
$\mathscr{P}^{0}_{x}(\cdot)$ is not payoff relevant if marginals
$\mathbb{P}^{\theta}$ are unaltered. See Section 6 for a more detailed
discussion.

The CAD order captures the idea that when information becomes more similar,
any agent believes that it is now more likely that others received the same
signal as they did. Notice that agents face two types of uncertainty in our
environment: fundamental uncertainty about $\bm{\theta}$, and strategic
uncertainty about the other group’s information. Part 1 in the definition
means that an increase in CAD keeps the fundamental uncertainty unchanged, and
potentially varies only the strategic uncertainty.

The large literature that studies the value of _public_ information in
coordination games (for example, Morris and Shin, 2002) alters both
fundamental and strategic uncertainty at once since public information changes
both, the joint and the marginal distribution at once. Our approach of
exploring the implications of varying joint distributions while keeping the
marginal distributions fixed does arise in some existing papers. Clemen and
Winkler (1985) study the value of information in a class of decision problems
where the noises are jointly normally distributed. Like us, they fix the
marginal distribution and vary the correlation structure. Cheng and Borgers
(2024) consider a general information structure with fixed marginals while
varying correlations conditional on the state.121212de Oliveira et al. (2023)
also study a robust decision problem with known marginals and unknown
correlation that is similar in spirit. By keeping the fundamental uncertainty
unchanged, our formulation isolates the effect of strategic uncertainty. In
Section 6 we demonstrate how our results are robust to allowing the marginal
distributions to also change.

To gain more intuition about the notion of CAD, it is useful to consider a
simple numerical example of two information structures that are CAD-ordered.
Suppose $(\widehat{Y}_{1},\widehat{Y}_{2})$ is an information structure with
$\widehat{Y}_{i}$’s being conditionally independent. Now consider a new
information structure $Y$, with the same marginal distribution as
$\widehat{Y}$. But now the $Y_{i}$’s are conditionally independent with
probability $1-\varepsilon$ for some $\varepsilon>0$ and perfectly correlated
with probability $\varepsilon$. $Y$ is more similar than $\widehat{Y}$ in the
CAD order. To see how such a CAD increase in information similarity can arise
in practice, consider how people have converged on where they get information
from. For instance, with YouTube being the dominant player and algorithms
steering content to users, it becomes more likely (higher $\varepsilon$) that
people now view exactly the same content. This corresponds to a CAD increase.

An equivalent way of stating Definition 2 is that
$Y\succcurlyeq_{CAD}\widehat{Y}$ if the probability of events in which
$\widehat{Y}_{1},\widehat{Y}_{2}$ are _exactly equal_ must (weakly) increase
under $Y$, and the probability of events in which
$\widehat{Y}_{1},\widehat{Y}_{2}$ are unequal must (weakly) decrease under
$Y$. This alternate formulation is equivalent to an order by Meyer (1990). The
requirement that the two agents see exactly the same signal with a higher
probability may be too strong in some contexts, making the CAD order quite
incomplete. Indeed, two information structures $Y$ and $\widehat{Y}$ are not
comparable in the CAD order if the probability of events in which
$\widehat{Y}_{1},\widehat{Y}_{2}$ are _very close in value_ increased under
$Y$ and the probability of events in which $\widehat{Y}_{1},\widehat{Y}_{2}$
are _significantly different_ decreased. We discuss the implications of using
alternative, more complete orders after presenting our main results.

Finally, notice that we construct posteriors explicitly using the signals,
rather than model signals as posteriors themselves, as is now standard,
following Kamenica and Gentzkow (2011). Our assumption that $\mu(\cdot)$ is
injective implies that signals and posteriors are interchangeable. We could
have chosen signals as posteriors and started with a _feasible_ joint
distribution over posterior beliefs instead. With more than one agent,
characterizing the feasible joint distributions over posteriors is not
trivial. Recently, Arieli et al. (2021) characterize the set of feasible two-
dimensional joint distributions. By working with signals directly and
performing the CAD operations, we have feasible joint distributions by
construction.

##  3 Information Similarity and Participation

In this section, we present our main results that characterize how increased
similarity of information affects participation in equilibrium.

###  3.1 Preliminaries

In an equilibrium of our regime-change game, an agent is willing to bear the
cost of participation if and only if she believes that it is sufficiently
likely that a change is beneficial and that her participation will make a
difference.

Consider an agent in group $g\in\\{1,2\\}$. She believes that the size of the
other group is $\mathbf{N}_{-g}\sim\eta(.)$. However, that she belongs to
group $g$ may change her belief about the size of her own group. She believes
that $\mathbf{N}_{g}-1\sim\eta^{A}(\cdot)$. Let $\eta^{A}_{2}(\cdot)$ denote
her belief about $\mathbf{N}_{g}-1+\mathbf{N}_{-g}$.131313Since
$\mathbf{N}_{1}$ and $\mathbf{N}_{2}$ are i.i.d., this distribution does not
need to be indexed by group $g$. The agent’s participation incentive depends
on whether she expects her own participation to make a difference to the
protest’s outcome. Below, we define expressions $\Lambda_{\text{b}}$,
$\Lambda_{\text{o}}$, and $\Lambda_{-\text{o}}$, which denote, respectively,
the probabilities of an agent being pivotal when both groups participate, when
only her own group participates, and when only the other group participates.

$\Lambda_{\text{b}}:=\sum_{k=0}^{\infty}\phi(k)\eta^{A}_{2}(k),\
\Lambda_{\text{o}}:=\sum_{k=0}^{\infty}\phi(k)\eta^{A}(k),\
\Lambda_{-\text{o}}:=\sum_{k=0}^{\infty}\phi(k)\eta(k)$ (2)

In collective action games, agents take a costly action when they believe they
can make a difference (are pivotal). While models with pivotality are
ubiquitous in the voting literature, one common critique is that voters in
large electorates are unlikely to ever be pivotal. But in our setting with
uncertainty about the population size and the required threshold,
$\Lambda_{\text{b}}$, $\Lambda_{\text{o}}$, and $\Lambda_{-\text{o}}$ are
smooth functions of pivotal probabilities and reflect agents’ beliefs about
whether their participation can make a difference. So, $\Lambda_{\text{b}}$,
$\Lambda_{\text{o}}$, and $\Lambda_{-\text{o}}$ serve as a way of modeling
this incentive to avoid costly participation in any collective action game
with strategic uncertainty.

We assume that from any agent’s perspective, the probability of being pivotal
when only the other group participates is weakly smaller than the probability
of being pivotal when only her own group participates or when both groups
participate.141414This assumption is satisfied if, for example,
$\mathbf{N}_{1},\mathbf{N}_{2}$ are Poisson random variables, as in, for
example, Myerson (1998). In contrast, if $\mathbf{N}_{1},\mathbf{N}_{2}$ are
deterministic, say equal to $N$, then Assumption 1 is violated if
$Prob(\bm{\bar{n}}=N+1)>\max\\{Prob(\bm{\bar{n}}=N),Prob(\bm{\bar{n}}=2N\\}$.
Assumption 1 rules out cases in which an agent from a group that does not
participate has the strongest incentive to participate, knowing that no one in
his group will participate. We maintain the assumption below throughout the
paper.

######  Assumption 1:

$\max\\{\Lambda_{\text{o}},\Lambda_{\text{b}}\\}\geq\Lambda_{-\text{o}}$

We first write down the conditions for a strategy profile to be an
equilibrium.

######  Proposition 1:

A strategy profile $\sigma$ is an equilibrium if and only if for all
$g\in\\{1,2\\}$ and for all $x\in\mathcal{X}$,

$\displaystyle\mathscr{P}^{1}_{x}(P_{-g})\Lambda_{\text{b}}+(1-\mathscr{P}^{1}_{x}(P_{-g}))\Lambda_{\text{o}}\geq\frac{c}{\mu(x)}$
$\displaystyle\text{ if }x\in P_{g}$ (IC:P)
$\displaystyle\mathscr{P}^{1}_{x}(P_{-g})\Lambda_{-\text{o}}\leq\frac{c}{\mu(x)}$
$\displaystyle\text{ if }x\in NP_{g}.$ (IC:NP)

The intuition is straightforward. Consider an agent in group $g$ with signal
$x\in P_{g}$. If the other group also receives a signal in its participation
set, which occurs with probability $\mathscr{P}^{1}_{x}(P_{-g})$, then this
agent can make a difference with probability $\Lambda_{\text{b}}$. If the
other group does not receive a signal in $P_{-g}$, then the agent can make a
difference with probability $\Lambda_{\text{o}}$. (IC:P) simply says that the
agent has an incentive to incur the cost of participating if she believes she
can make a difference with a sufficiently high probability. The logic behind
(IC:NP) is similar.

To capture how the incentives change with similarity of information, it is
convenient to partition the model primitives into two environments.151515In
the knife-edge case with $\Lambda_{b}=\Lambda_{o}$, changing information
similarity has no impact on participation.

######  Definition 3:

[Encouragement/Discouragement] move to next line

  * •

We say we are in an encouragement environment if

$\Lambda_{\text{b}}>\Lambda_{\text{o}}.$ (E)

In this case, an agent is more likely to make a difference when both groups
participate than if only her own group participates; that is, higher aggregate
participation encourages participation.

  * •

We say we are in a discouragement environment if

$\Lambda_{\text{o}}>\Lambda_{\text{b}}.$ (D)

In this case, an agent is more likely to make a difference when only her group
participates than if both groups participate; that is, higher participation by
others discourages individual participation.

At first glance, encouragement and discouragement environments may seem to be
environments of strategic complementarity and substitutability, respectively.
This is not quite true. Discouragement environments do not feature strategic
substitutability, because a nonparticipating agent in group $g$ (with
$x_{g}\in NP_{g}$) has a _stronger_ incentive to participate if the other
group is more likely to participate.

Theorems 1 and 2 establish that if we compare information similarity in the
sense of CAD, then the above simple condition about primitives (whether
$\Lambda_{\text{b}}>\Lambda_{\text{o}}$ or
$\Lambda_{\text{o}}>\Lambda_{\text{b}}$) yields a complete characterization of
when increased information similarity facilitates or hinders participation.

###  3.2 Encouragement Environment

######  Theorem 1:

In encouragement environments, the maximal equilibrium aggregate participation
increases when information becomes more similar. That is,

$\mathscr{P}^{\theta}\succcurlyeq_{CAD}\widehat{\mathscr{P}}^{\theta}\
\text{for all}\
\theta\implies\mathscr{V}^{*}(\mathscr{P})\geq\mathscr{V}^{*}(\widehat{\mathscr{P}})\
\text{in}\ \eqref{enco}.$

This is true regardless of how $\mathscr{P}^{0}$ changes, as long as the
marginals $\mathbb{P}^{\theta}$ are unaltered.

The proof is in the appendix and proceeds in two steps. First, we show that in
encouragement environments, the maximal participation equilibrium must be in
symmetric strategies. Then we show that any maximal participation equilibrium
remains an equilibrium when information similarity increases. Suppose that in
the maximal equilibrium, in each group, an agent participates if and only if
$x\in P$. If information similarity increases, an agent with $x\in P$ now
assigns a higher probability that the other group also sees a signal $x\in P$
that induces them to participate. That is, $\mathscr{P}^{1}_{x}(P)$ increases
for $x\in P$. Since $\Lambda_{\text{b}}>\Lambda_{\text{o}}$, we can see from
(IC:P) that such an agent has an even stronger incentive to participate.
Analogously, if information similarity increases, a nonparticipating agent
with $x\in NP$ now assigns a lower probability that the other group sees $x\in
P$. That is, $\mathscr{P}^{1}_{x}(P)$ decreases for $x\in NP$. We can see from
(IC:NP) that a nonparticipating agent has an even weaker incentive to
participate.

###  3.3 Discouragement Environment

Next, we analyze discouragement environments, in which
$\Lambda_{\text{o}}>\Lambda_{\text{b}}$. The maximal equilibrium might no
longer be symmetric. Further, a symmetric equilibrium under
$\widehat{\mathscr{P}}$ might no longer be an equilibrium under $\mathscr{P}$
when $\mathscr{P}\succcurlyeq_{CAD}\widehat{\mathscr{P}}$. This alone does not
imply a smaller maximal equilibrium participation under $\mathscr{P}$, because
new equilibria may arise under $\mathscr{P}$ that were not sustainable under
$\widehat{\mathscr{P}}$. Given any information structure $\mathscr{P}$ and a
maximal equilibrium $\sigma^{*}$, we define a condition that describes why
$\sigma^{*}$ is maximal under $\mathscr{P}$.

######  Definition 4 (Condition M):

Let $\sigma^{*}$ be a maximal equilibrium for $\mathscr{P}$ with participation
sets $(P_{1}^{*},P_{2}^{*})$. We say that $\mathscr{P}$ satisfies condition M
if, for any strategy profile $\widehat{\sigma}$ with participation sets
$(\widehat{P}_{1},\widehat{P}_{2})$ such that
$\mathscr{V}(\widehat{\sigma})>\mathscr{V}(\sigma^{*})$,

at least one of the following holds.

  1. (M1)

$\exists x\in\widehat{P}_{1}\cap\widehat{P}_{2}$ such that

$\displaystyle\min_{i\in\\{1,2\\}}\left\\{\mathscr{P}^{1}_{x}(\widehat{P}_{i})\Lambda_{\text{b}}+(1-\mathscr{P}^{1}_{x}(\widehat{P}_{i}))\Lambda_{\text{o}}\right\\}<\frac{c}{\mu(x)}.$

Or

  2. (M2)

$\exists
x\in(\widehat{P}_{1}\cup\widehat{P}_{2})\backslash(\widehat{P}_{1}\cap\widehat{P}_{2})$
such that

$\displaystyle\mathbbm{1}_{x\in\widehat{P}_{1}}\mathscr{P}^{1}_{x}\left(\widehat{P}_{1}\right)+\mathbbm{1}_{x\in\widehat{P}_{2}}\mathscr{P}^{1}_{x}\left(\widehat{P}_{2}\right)>\frac{c}{\Lambda_{-\text{o}}\mu(x)}.$

Condition M says there are two reasons why any strategy profile
$\widehat{\sigma}$ with a larger expected participation than the maximal
equilibrium $\sigma^{*}$ fails to be an equilibrium. Either (IC:P) is violated
for some signal that prescribes both groups to participate under
$\widehat{\sigma}$, or (IC:NP) is violated at a signal at which exactly one
group is prescribed to participate under $\widehat{\sigma}$. Below we
establish that in discouragement environments, if the information structure
satisfies condition M, increasing information similarity can lead to lower
maximal equilibrium aggregate participation.

Condition M is not a condition on the primitives of the model. However, a
straightforward sufficient condition that guarantees condition M is as
follows. Let $\mathcal{X}$ be ordered according to the posterior beliefs,
$\mu(x)$. Define $\widehat{x}:=\inf\\{x:\mu(x)\Lambda_{\text{o}}\geq c\\}$. If
$\widehat{\sigma}:=\mathbbm{1}_{\mathbf{X}\geq\widehat{x}}$ is an equilibrium
for some $\mathscr{P}$, then $\mathscr{P}$ satisfies condition M. The reason
is that regardless of the information structure, (IC:P) can never be satisfied
for $x<\widehat{x}$. Therefore, no equilibrium can have a larger expected
participation than $\widehat{\sigma}$. And therefore condition M is satisfied.
One easily verifies with a binary-signal example that this sufficient
condition is not vacuous. Equipped with this condition, we now present our
second main result, which establishes how increased information similarity can
reduce maximal aggregate participation.

######  Theorem 2:

In a discouragement environment that satisfies condition M, the maximal
equilibrium participation decreases when information becomes more similar.
That is,

$\mathscr{P}^{\theta}\succcurlyeq_{CAD}\widehat{\mathscr{P}}^{\theta},\
\text{for all}\
\theta\implies\mathscr{V}^{*}(\mathscr{P})\leq\mathscr{V}^{*}(\widehat{\mathscr{P}})\
\text{under}\ \eqref{disco}\ \text{if}\ \widehat{\mathscr{P}}\ \text{satisfies
condition}\ M.$

Moreover, the inequality can be strict. The result is true regardless of how
$\mathscr{P}^{0}$ changes, as long as the marginals $\mathbb{P}^{\theta}$ are
unaltered.

The proof is in the appendix. The argument involves two steps. First, we argue
that the maximal equilibrium might no longer be an equilibrium when
information becomes more similar. Let $\sigma^{*}$ with participation sets
$(P_{1}^{*},P_{2}^{*})$ be the maximal equilibrium under
$\widehat{\mathscr{P}}$. Consider a participating agent with a signal $x\in
P_{1}^{*}\cap P_{2}^{*}$. If information becomes more similar, this agent
assigns a higher probability to the event that the other group also receives a
signal in their respective participation set. However, unlike in encouragement
environments, this reduces her incentive to participate since
$\Lambda_{\text{o}}>\Lambda_{\text{b}}$. As a result, this agent’s (IC:P) may
be violated. Indeed, a nonparticipant’s (IC:NP) may also fail. Consider a
signal $x\in P_{1}^{*}\backslash P_{2}^{*}$. An agent in group $2$ who
receives such a signal is prescribed to not participate. However, with
increased similarity of information, this agent assigns a higher probability
that group $1$ will participate. This, in turn, makes her more likely to
participate, which may violate (IC:NP). So the maximal equilibrium
$\sigma^{*}$ under $\widehat{\mathscr{P}}$ may no longer be an equilibrium
under $\mathscr{P}$.

In the second step, we establish that no new equilibrium with larger expected
participation arises under $\mathscr{P}$. Suppose, for a contradiction, there
is an equilibrium $\sigma^{\prime}$ with
$\mathscr{V}(\sigma^{\prime})>\mathscr{V}(\sigma^{*})$. By the maximality of
$\sigma^{*}$, we know that $\sigma^{\prime}$ is not an equilibrium under
$\widehat{\mathscr{P}}$. By condition M, two cases arise. In case (i),
$\sigma^{\prime}$ is not an equilibrium under $\widehat{\mathscr{P}}$ because
an agent’s incentive to participate (IC:P) is violated at some signal $x\in
P_{1}^{\prime}\cap P_{2}^{\prime}$, where both groups are prescribed to
participate. With more similar information, (IC:P) would continue to be
violated. To see why, note that (IC:P) is a convex combination of
$\Lambda_{\text{b}}$ and $\Lambda_{\text{o}}$, and with more similar
information, she assigns a higher weight to $\Lambda_{\text{b}}$. Now if
(IC:P) was violated under $\widehat{\mathscr{P}}$, then it will also be
violated under $\mathscr{P}$ because in discouragement environments,
$\Lambda_{\text{b}}\leq\Lambda_{\text{o}}$. In case (ii), $\sigma^{\prime}$ is
not an equilibrium under $\widehat{\mathscr{P}}$ because (IC:NP) is violated
for some agent—say, from group $2$—with a signal in $(P_{1}^{\prime}\backslash
P_{2}^{\prime})$. Such an agent wishes to participate under
$\widehat{\mathscr{P}}$ because she assigns a high probability to the event
that group $1$ participates. Under $\mathscr{P}$, when information is more
similar, she has an even stronger incentive to participate (since she believes
that the other group is more likely to participate and her own group is not
going to participate). So, in both cases, if $\sigma^{\prime}$ is not an
equilibrium under $\widehat{\mathscr{P}}$, then it cannot be an equilibrium
under $\mathscr{P}$ either.

Discussion of Theorems 1 and 2: A few observations are worth highlighting.
First, the existing literature often restricts attention to monotone, cutoff
strategies for the sake of tractability. In a game of pure complementarity
(see Morris and Shin, 2002), the best and worst equilibria are in cutoff
strategies. However, with both complementarities and substitutabilities, this
need no longer be true. We do not impose such a restriction on strategies; we
allow nonmonotonic and asymmetric strategies. Also, we only assume a fixed
finite signal space without imposing any additional restrictions on the signal
structure. Consequently, there is little hope of characterizing all
equilibria. However, using an indirect approach, we can characterize the
effects of changing similarity of information on the set of equilibria and the
maximal ones. Also note that we restrict attention to pure strategies. In
encouragement environments, this is without loss: maximal equilibrium is in
symmetric pure strategies. In discouragement environments, this need not be
true. However, the qualitative insight that increased information similarity
can lead to lower participation in discouragement environments remains valid
even if the maximal equilibrium is in mixed strategies. See Section 6 for a
detailed discussion.

Second, we started by partitioning the underlying primitives into the
encouragement and discouragement environments to capture when coordination and
free-riding, respectively, are the primary hurdles to collective action. This
partition was independent of any information structure. Theorems 1 and 2 imply
that measuring information similarity using the notion of CAD yields the
intuitive economic insight that increasing information similarity helps
collective action exactly when coordination is the main challenge and can
hinder it when free-riding is the main challenge.

This raises the natural question whether our characterization can be obtained
using an order of similarity that is less demanding (more complete) than CAD.
Given information structures $Y$ and $\widehat{Y}$ with the same marginals, we
say $Y\succeq_{CCAD}\widehat{Y}$ if

$Prob(Y_{2}\in A|Y_{1}=x)\geq Prob(\widehat{Y}_{2}\in A|\widehat{Y}_{1}=x)$

for all $x$ and all _upper- and lower-contour sets $A$ containing $x$_.161616A
set $A$ is an upper-contour set if
$A=\\{z^{\prime}\in\mathcal{X}:z^{\prime}\geq z\\}$ for some
$z\in\mathcal{X}$. A similar definition applies for a lower-contour set.
Intuitively, increased similarity no longer means that there is a higher
chance of getting _exactly the same_ signal. Rather,
$Y\succeq_{CCAD}\widehat{Y}$ if $Y_{1}$, $Y_{2}$ are _close to each other in
value_ with a higher probability relative to
$\widehat{Y}_{1},\widehat{Y}_{2}$. We can obtain a result similar to Theorem 1
using this weaker order (under a mild regularity condition on the information
structure). The maximal equilibrium then has a cutoff structure in
encouragement environments: agents participate if they receive a signal above
a threshold. An increase in similarity even in this weaker order still implies
an increase in participation in encouragement environments. However, in
discouragement environments, maximal equilibria might not be in cutoff
strategies, and, in general, an analogous result to Theorem 2 does not hold.
The extant literature often allows only cutoff strategies—for example,
Shadmehr (2021). If we consider only cutoff strategies, then Theorem 2 holds
with the weaker order. In this sense, CAD, even though demanding, enables a
characterization of the basic trade-off in the paper without imposing
restrictions on equilibrium strategies.

Similarly, one might aim to characterize changing information similarity using
more familiar orders like the supermodular order or the PQD order (see Meyer
and Strulovici, 2012, for details). In two dimensions, the CAD order is
stronger than the supermodular order, which is equivalent to the PQD order. As
previously mentioned, encouragement and discouragement environments divide the
game into scenarios where strategic complementarities or strategic
substitutabilities prevail. It is straightforward to create examples where
increasing similarity in the supermodular order (and hence the PQD order)
results in lower participation in encouragement environments. Thus, the CAD
order provides a characterization that aligns with our initial intuition—that
increasing similarity aids coordination and exacerbates free-riding—something
that orders like the supermodular order do not seem to provide.

###  3.4 Encouragement, Discouragement, and Resilience

Roughly speaking, encouragement and discouragement environments are the
general analogs of the parameter $q$ in our example in the introduction. They
quantify the relative importance of coordination (versus preventing free-
riding). Recall that the definitions of these environments (Definition 3),
while based on primitives, were in terms of the population uncertainty and the
threshold uncertainty, making them somewhat abstract. Below we provide an
alternate characterization that shows that intuitively, a higher threshold for
regime change makes an encouragement and a lower threshold makes a
discouragement environment.

We say a regime with participation threshold $\bm{\bar{n}}^{\prime}$ is more
resilient than one with threshold $\bm{\bar{n}}$, denoted by
$\bm{\bar{n}}^{\prime}\succcurlyeq_{st}\bm{\bar{n}}$, if
$\bm{\bar{n}}^{\prime}$ first-order stochastically dominates $\bm{\bar{n}}$.

######  Proposition 2:

Suppose that a single-crossing assumption holds—that is, for all
$\bm{\bar{n}}^{\prime}\succcurlyeq_{st}\bm{\bar{n}}$, if
$(\Lambda_{\text{b}}-\Lambda_{\text{o}})(\bm{\bar{n}})\geq 0$ then
$(\Lambda_{\text{b}}-\Lambda_{\text{o}})(\bm{\bar{n}}^{\prime})\geq 0$. Define
$\bm{\bar{n}}$ such that $(\Lambda_{2}-\Lambda_{1})(\bm{\bar{n}})=0$.

  1. 1.

For any $\bm{\bar{n}}^{\prime}$ with
$\bm{\bar{n}}^{\prime}\succcurlyeq_{st}\bm{\bar{n}}$, we are in an
encouragement environment.

  2. 2.

For any $\bm{\bar{n}}^{\prime}$ with
$\bm{\bar{n}}\succcurlyeq_{st}\bm{\bar{n}}^{\prime}$, we are in a
discouragement environment.

The single-crossing assumption means that if an agent is more likely to make a
difference when both groups join than when only her group joins, this
continues to be true when regime change becomes more difficult—that is, if
$\Lambda_{\text{b}}>\Lambda_{\text{o}}$ for a given regime, then
$\Lambda_{\text{b}}>\Lambda_{\text{o}}$ also for more resilient regimes. The
assumption is satisfied, for example, if the group sizes
$\mathbf{N}_{1},\mathbf{N}_{2}$ are drawn from a Poisson distribution and the
resilience $\bm{\bar{n}}$ is deterministic. In this case,
$\Lambda_{\text{b}}>\Lambda_{\text{o}}$ if and only if
$\bar{n}>n^{*}=\frac{N}{\ln 2}$. Figure 2 illustrates encouragement and
discouragement environments for the Poisson example.

$N$$\bar{n}$$2N$$x$$\psi(\bar{n},x)$ (a) Encouragement (high $\bar{n}$)

$N$$\bar{n}$$2N$$x$$\psi(\bar{n},x)$ (b) Discouragement (low $\bar{n}$)

Figure 2: Encouragement and discouragement environments. Here,
$\psi(\bar{n},x)=\frac{e^{-x}x^{\bar{n}}}{\bar{n}!}$ is the Poisson pmf at
$\bar{n}$ with mean $x$. The peak of these pmfs occurs at $\bar{n}.$ In both
the environments, $N<\bar{n}<2N$. However, in encouragement environments,
$\bar{n}$ is much larger than $N$. Therefore,
$\Lambda_{\text{o}}=\psi(\bar{n},N)<\psi(\bar{n},2N)=\Lambda_{\text{b}}$ in
encouragement environments (left), while the reverse holds in discouragement
environments (right).

In words, the proposition states that sufficiently resilient regimes are
encouragement environments, and weak regimes are discouragement environments.
This result implies that the effect of changing information similarity is
qualitatively different based on the resilience of regimes: Increased
similarity of information facilitates greater participation against regimes
that are hard to change but can hurt participation against weaker regimes. For
instance, consider mass protests. The popular press often claims that mass
protests have becomes larger in modern times and attributes this to the change
in modern communication technology.171717See, for instance, The Economist
(2020). Our result suggests that this casual observation may be biased: While
popular media has highlighted how a more connected world enabled large
protests against regimes previously thought to be impregnable, they may have
overlooked how this same increased information similarity may have hindered
collective action in movements with ex ante easier goals.

##  4 Application 1: Success of Mass Protests

We can apply our framework directly to study mass protests. Theorems 1 and 2
characterize the effect of changes in information similarity on aggregate
participation in a mass protest but are silent about the effect on the
likelihood of a successful protest. A priori, increased participation might
not imply an increased probability of a successful regime change. To see this
quickly, suppose that $\overline{\sigma}$ is the equilibrium in which the
probability of successful protest when $\bm{\theta}=1$ is maximized. For
simplicity, restrict attention to symmetric strategies across groups, and let
$\overline{P}$ and $\overline{NP}$ denote respectively the participation and
nonparticipation sets. If information becomes more similar, the event
$\\{\mathbf{X}_{1}\in\overline{P},\mathbf{X}_{2}\in\overline{P}\\}$ occurs
more frequently when $\bm{\theta}=1$. This increases the probability of a
successful protest. However, the probability of the event
$\\{\mathbf{X}_{1}\in\overline{NP},\mathbf{X}_{2}\in\overline{NP}\\}$ also
increases, which means it is also more likely that neither group participates.
In other words, increased similarity of information increases the probability
of the event that both groups participate and the event that neither group
participates. The overall effect on the probability of a successful protest is
ambiguous.

To study the effect of increased information similarity on the probability of
success of a protest, we specialize to an environment in which the population
sizes $\mathbf{N}_{1},\mathbf{N}_{2}$ are Poisson distributed with mean $N$
and the threshold participation required for regime change is a constant
$\overline{n}$. We also consider symmetric strategies across the
groups.181818In encouragement environments, it is easy to argue that the
equilibrium with maximal probability of success is in symmetric strategies.
Therefore, we can obtain a similar result as Proposition 3 without restricting
to symmetric strategies. Results analogous to Theorems 1 and 2 about the
expected maximal participation still hold.191919We further show (in
Proposition 5 in Appendix A.5.1) that if a strategy profile constitutes a
maximal equilibrium under two information structures ranked according to CAD,
then _conditional on there being any participation_ , the maximal equilibrium
participation and the probability of a regime change are strictly higher under
the more similar information structure. However, the _unconditional_ size of
mass protests or probability of successful protests might not increase with
more similar information. Furthermore, Proposition 3 below shows that
increasing similarity can never lower the probability of a successful protest
in encouragement environments but can do so in discouragement environments.

For a given information structure $\mathscr{P}$ and equilibrium strategy
profile $\sigma$ we define

$\displaystyle\Pi(\sigma;\mathscr{P}):=$
$\displaystyle\mathscr{P}(\mathbf{A}\geq\bar{n}+1|\bm{\theta}=1)$
$\displaystyle\overline{\Pi}(\mathscr{P}):=$
$\displaystyle\max_{\sigma\in\mathcal{E}(\mathscr{P})}\Pi(\sigma).$

######  Definition 5 (Condition $M^{\prime}$ ):

Fix an information structure $\mathscr{P}$. Let $\overline{\sigma}$ be an
equilibrium maximum probability of success
($\bar{\Pi}(\mathscr{P})=\Pi(\overline{\sigma};\mathscr{P})$). We say that
$\mathscr{P}$ satisfies condition $M^{\prime}$ if, for any strategy profile
$\widehat{\sigma}$ with
$\Pi(\widehat{\sigma};\mathscr{P})>\bar{\Pi}(\mathscr{P})$, $\exists
x\in\widehat{P}:=P(\widehat{\sigma})$, such that

$\displaystyle\mathscr{P}^{1}_{x}(\widehat{P})\Lambda_{\text{b}}+(1-\mathscr{P}^{1}_{x}(\widehat{P}))\Lambda_{\text{o}}<\frac{c}{\mu(x)}.$

######  Proposition 3:

The maximum probability of successful protest increases in encouragement
environments, while it decreases in discouragement environments provided it
satisfies Condition $M^{\prime}$.

  1. 1.

If $\overline{n}>n^{*}$ and
$\mathscr{P}^{1}\succcurlyeq_{CAD}\widehat{\mathscr{P}}^{1}$, then
$\overline{\Pi}(\mathscr{P})\geq\overline{\Pi}(\widehat{\mathscr{P}})$.

  2. 2.

If $\overline{n}<n^{*}$ and $\widehat{\mathscr{P}}$ satisfies Condition
$M^{\prime}$, then
$\overline{\Pi}(\mathscr{P})\leq\overline{\Pi}(\widehat{\mathscr{P}})$.

An analogous result holds for welfare (see Appendix A.5). These findings have
important implications for press freedom and authoritarian governments. Edmond
(2013) analyzes a setting in which citizens protest and an authoritarian
government manipulates the information that citizens see. While the author
focuses on the coordination aspect, our setup also considers the free-riding
incentive. In today’s world, even when an authoritarian government can stop
the mainstream media from reporting about unfavorable policies, it is unlikely
to prevent people from learning about them. Therefore, the main effect of
curbing press freedom is that individuals are less likely to believe that
others have seen the same information, or, in other words, information
similarity across agents is reduced. Our theory suggests that it is the
powerful (that is, resilient) authoritarian governments that benefit more from
curbing press freedom. Less powerful regimes may face larger protests and a
higher likelihood of being overthrown when attempting to restrict press
freedom.

##  5 Application 2: Costly Voting in Committees

Next, we apply our framework to a quite different context: voting in
committees (à la Palfrey and Rosenthal, 1985). Consider members of a monetary
policy committee who must vote on whether to raise interest rates. Voting in
favor of an interest rate increase is costly for committee members because
votes are public and the public wants the status quo to be maintained.
Committee members base their decision on their private information about how a
rate increase affects the economy. An interest rate increase is implemented if
and only if a threshold number of members vote in favor of it. In this
setting, members’ private information can exhibit different levels of
similarity of information depending on the diversity of their backgrounds—for
example, different academic expertise may lead members to focus on different
aspects of the available empirical evidence. What effect do changes in
diversity of the committee have on voting outcomes? We can apply our tools to
characterize when increased information similarity (less diversity among
committee members) strengthens or weakens the incentive to vote in favor of a
rate increase based on evidence, and how it affects the choice of the optimal
voting threshold.

Suppose the committee comprises $G>2$ members who must vote for or against a
rate increase. The economy is in one of two possible states: a rate increase
is either unnecessary ($\bm{\theta}=0$) or warranted ($\bm{\theta}=1$). Each
member $i$ privately receives a noisy binary signal
$\mathbf{X}_{i}\in\mathcal{X}=\\{0,1\\}$ about the state $\bm{\theta}$.
Conditional on state $\bm{\theta}$, the signals are drawn from a joint
distribution $\mathscr{P}^{\theta}$. These signals can be interpreted as each
committee member’s understanding of the available data given their background
and area of expertise. Information among committee members is more similar if
their backgrounds are more similar. The rate increase is implemented if and
only if more than $\bar{n}+1$ members cast votes in favor of the rate
increase. Intuitively, we say each individual member vote carries little (a
lot of) weight if $\bar{n}$ is high (low). If a rate increase is implemented,
the members get a payoff of $\theta$. Voting in favor of a rate increase costs
$c$, interpreted as the cost of facing public hostility ex post. For
simplicity, we assume $\frac{c}{\mu(0)}>1$ to guarantee that in any
equilibrium, a member who believes that $\mathbf{X}_{i}=0$ never votes in
favor of a hike. We also restrict attention to symmetric strategies.

Notice that now there is no uncertainty about the number of participants or
the participation threshold. Further, importantly, with $G>2$ agents, we need
a notion of information similarity to compare random variables with $G>2$
dimensions. We extend our notion of CAD as follows. Let $I=\sum_{j\in
G}\mathbbm{1}_{\mathbf{X}_{j}=1}$ denote the number of committee members who
receive signal $\mathbf{X}_{j}=1$, and let $I_{-i}=\sum_{j\in
G\setminus\\{i\\}}\mathbbm{1}_{\mathbf{X}_{j}=1}$ denote the number of members
other than $i$ who receive signal $\mathbf{X}_{j}=1$.

######  Definition 6:

We say $\mathscr{P}^{1}\succcurlyeq_{CAD}\widehat{\mathscr{P}}^{1}$ if there
exists $k^{*}\in\\{0,1,\ldots,G-2\\}$ such that

$\gamma^{1}_{1}(k)\leq\widehat{\gamma}^{1}_{1}(k)\ \text{for all}\ k\leq
k^{*}$ $\text{and}\ \gamma^{1}_{1}(k)\geq\widehat{\gamma}^{1}_{1}(k)\
\text{for all}\ k>k^{*}.$

We call $k^{*}$ the index of sign change between $\mathscr{P}^{1}$ and
$\widehat{\mathscr{P}}^{1}$.

To understand the above definition, consider a member who receives a signal
$1$. If information similarity increases, then conditional on the rate
increase being warranted ($\bm{\theta}=1$), each member assigns a higher
probability to more than $k^{*}$ others having also observed signal $1$ and a
lower probability to fewer than $k^{*}$ others having observed the opposite
signal $0$.202020Notice that we do not restrict the conditional beliefs after
$\mathbf{X}_{i}=0$. This is because, given $c>\mu(0)$, an agent never casts
votes for a rate hike after observing $\mathbf{X}_{i}=0$. For more general
environments, we would need restrictions on the conditional beliefs after any
signal realization. In a companion paper, we present extensions of the CAD
order for more than two dimensions and study their implications for
equilibrium behavior in a class of binary-action games.

Consider a strategy profile, denoted by $\sigma^{1}$, in which members vote in
favor of a rate increase whenever they receive $\mathbf{X}_{i}=1$. For
$\sigma^{1}$ to be an equilibrium, we need

$\gamma^{1}_{1}(\bar{n})\geq\frac{c}{\mu(1)},$ (IC-voting)

where $\gamma^{1}_{1}$ is the probability distribution function of $I_{-i}$
conditional on $\bm{\theta}=1$ and $\mathbf{X}_{i}=1$, and $\mu(1)$ is the
posterior belief upon receiving signal $\mathbf{X}_{i}=1$. Proposition 4
characterizes how increased similarity of information affects the maximal
equilibrium (the equilibrium with the largest number of votes in favor of a
rate hike conditional on it being warranted). The only candidate equilibrium
with any votes for a rate hike is $\sigma^{1}$. Therefore, studying maximal
equilibria reduces to starting out with a $\sigma^{1}$ that is an equilibrium
and checking whether it remains an equilibrium when information similarity
increases.

######  Proposition 4:

Fix $\bar{n}$, the threshold number of votes required to implement a rate
increase. Suppose
$\mathscr{P}^{1}\succcurlyeq_{CAD}\widehat{\mathscr{P}}^{1}$, and let $k^{*}$
be the associated index of sign change between $\mathscr{P}^{1}$ and
$\widehat{\mathscr{P}}^{1}$. Suppose
$\sigma^{1}\in\mathcal{E}(\widehat{\mathscr{P}})$.

  1. 1.

If $k^{*}<\bar{n}$, then $\sigma^{1}\in\mathcal{E}(\mathscr{P})$.

  2. 2.

If $k^{*}\geq\bar{n}$, then it is possible that
$\sigma^{1}\notin\mathcal{E}(\mathscr{P})$.

Let us analyze the incentives of a committee member $i$ who believes that a
rate increase is warranted based on her signal $\mathbf{X}_{i}=1$. We consider
two extreme cases. First, suppose that individual member votes have as little
weight as possible—that is, a unanimous vote in favor is required to increase
rates (that is, $\bar{n}+1=G)$. Then member $i$’s vote is relevant only when
all the others also vote for a hike. This requires that all others also have
received the same signal. With increased information similarity,
$\gamma^{1}_{1}(G-1)$ increases, which makes it more likely that her vote is
relevant. This increases her incentive to vote in favor of a rate increase
conditional on the signal $\mathbf{X}_{i}=1$, regardless of $k^{*}$. Next,
suppose that individual member votes have the highest possible weight—that is,
just one vote is required to implement the rate increase ($\bar{n}+1=1$). Then
member $i$’s vote is relevant only if all the others vote against the interest
rate hike. Increased information similarity means $\gamma^{1}_{1}(0)$
decreases. The incentive to vote for a hike in this case is diminished,
regardless of $k^{*}$. In general, how increased information similarity
affects a member’s incentive to vote will depend on $k^{*}$. Intuitively,
conditional on the rate increase being warranted, a more diverse committee
increases (reduces) the maximal equilibrium number of votes in favor of a rate
increase if each individual vote has sufficiently high (low) weight.

This leads to an interesting normative question about the design of optimal
voting rules in committees: how do changes in the diversity of a committee
affect the choice of the optimal voting threshold rule?212121In a recent
paper, Kattwinkel and Winter (2023) study the optimal decision mechanism for
juries, allowing for general mechanisms but keeping the information structure
fixed with independent signals across jurors. Our framework considers a
special class of decision mechanisms, namely those in which a minimum
threshold number of votes is required for a decision, and asks how the optimal
rule changes with a changing information structure. The interested reader can
refer to Appendix A.7 for details.

##  6 Discussion

Finally, we discuss some extensions. Formal results are in the online
appendix.

##### Mixed strategies:

So far we have restricted our attention to pure strategies. This is without
loss in encouragement environments because equilibria with maximal
participation are necessarily pure and symmetric. However, in discouragement
environments, maximal equilibria may be in mixed strategies. The main
technical challenge of working with mixed strategies is that the maximal
equilibrium can involve mixing on some signals with different probabilities,
and it is no longer possible to partition $\mathcal{X}$ into participation and
nonparticipation sets $P$ and $NP$. Much of our analysis examined how
$\mathscr{P}^{1}_{x}(P)$ changes for any $x$. But when agents mix with
different probabilities after different signals, we need to understand how
$\mathscr{P}^{1}_{x}(x^{\prime})$ changes for each
$x,x^{\prime}\in\mathcal{X}$. Developing tools to completely characterize how
the set of mixed-strategy equilibria varies with information similarity is
left for future work. However, below, we show that even when we allow for
mixing, the main qualitative insight that increasing information similarity
can lead to lower maximal equilibrium participation in discouragement
environments remains valid.

For the sake of exposition, suppose $\mathbf{N}_{1},\mathbf{N}_{2}$ are
Poisson distributed and $\bm{\bar{n}}$ is a constant, and suppose the maximal
equilibrium is symmetric and exhibits mixing on exactly one signal. That is,
$\mathcal{X}$ can be partitioned into three sets: $P,NP$, and $\\{x^{*}\\}$.
$P$ and $NP$ have their usual meanings, while on $x^{*}$ agents mix—that is,
participate with probability $\beta<1$. Then, the IC for participation at
$x^{*}$ is

$\displaystyle\widehat{\mathscr{P}}^{1}_{x^{*}}(x^{*})\psi(\bar{n},2\beta
N)+\widehat{\mathscr{P}}^{1}_{x^{*}}(P)\psi(\bar{n},(1+\beta)N)+\widehat{\mathscr{P}}^{1}_{x^{*}}(NP)\psi(\bar{n},\beta
N)=\frac{c}{\mu(x^{*})}.$ (IC-P-Mix)

Recall that $\psi(k,x)=\frac{e^{-x}x^{k}}{k!}$ is the Poisson probability at
$k$ with mean $x$. Suppose that information becomes more similar. How this
affects the maximal equilibrium depends on both the type of change in the
information structure and how the three pivotal probabilities in (IC-P-
Mix)—$\psi(\bar{n},2\beta N),\psi(\bar{n},(1+\beta)N)$, and
$\psi(\bar{n},\beta N)$—are ranked.

$\beta N$$2\beta N$$(1+\beta)N$$\bar{n}$$x$$\psi(\bar{n},x)$ Figure 3: Pivotal
probabilities

To see this, suppose that $\psi(\bar{n},\beta N)>\psi(\bar{n},2\beta N)$ (as
in Figure 3) and $\mathscr{P}^{1}\succcurlyeq_{CAD}\widehat{\mathscr{P}}^{1}$.
In particular, suppose that
$\mathscr{P}^{1}_{x^{*}}(P)=\widehat{\mathscr{P}}^{1}_{x^{*}}(P)$, while
$\mathscr{P}^{1}_{x^{*}}(NP)<\widehat{\mathscr{P}}^{1}_{x^{*}}(NP)$. Then,
$\mathscr{P}^{1}_{x^{*}}$ assigns more weight on $\psi(\bar{n},2\beta N)$ and
less on $\psi(\bar{n},\beta N)$. Therefore, the LHS of (IC-P-Mix) will fall
under $\mathscr{P}$, violating the constraint. More importantly, as Figure 3
shows, (IC-P-Mix) cannot hold for any higher $\beta$ under
$\mathscr{P}$.222222It is not necessary that $\beta N>\bar{n}$ as in Figure 3
for (IC-P-Mix) to not be satisfied for any higher $\beta$. If
$2\psi_{2}(\bar{n},2\beta N)+\psi_{1}(\bar{n},\beta N)<0$, then (IC-P-Mix)
cannot be satisfied for a higher $\beta$ under $\widehat{\mathscr{P}}$. Here
$\psi_{1}(k,x)=\frac{\partial\psi(k,x)}{\partial k}$ and
$\psi_{2}(k,x)=\frac{\partial\psi(k,x)}{\partial x}$. Therefore, an
equilibrium in which players participate after observing signals in $P$ and
mix on observing $x^{*}$ must exhibit lower participation under $\mathscr{P}$.
Essentially, a player with a signal $x^{*}$ anticipates higher participation
under $\mathscr{P}$. In discouragement environments, this weakens his
incentive to participate.

In contrast, suppose that
$\mathscr{P}^{1}_{x^{*}}(NP)=\widehat{\mathscr{P}}^{1}_{x^{*}}(NP)$ and
$\mathscr{P}^{1}_{x^{*}}(P)<\widehat{\mathscr{P}}^{1}_{x^{*}}(P)$. In this
case, we assign more weight on $\psi(\bar{n},2\beta N)$ and less on
$\psi(\bar{n},(1+\beta)N)$. Therefore, the LHS of (IC-P-Mix) _increases_ under
$\mathscr{P}$. More importantly, if both $\psi(\bar{n},2\beta N)$ and
$\psi(\bar{n},(1+\beta)N)$ are decreasing in $\beta$ (as in Figure 3), then
(IC-P-Mix) will hold with an equality for a higher $\beta$. Therefore, maximal
participation will _increase_ provided other equilibrium conditions continue
to hold, as will be the case if the change in information similarity is small.
Unlike in the previous case, here an agent with a signal $x^{*}$ anticipates
lower participation by others under $\mathscr{P}$. In discouragement
environments, this increases his incentive to participate.

The key insight is that the impact of information similarity on incentives is
fundamentally linked to players’ beliefs about expected participation based on
their signals. In discouragement environments, higher participation reduces a
player’s likelihood of being pivotal, weakening her incentive to participate.
When players use mixed strategies, they adjust to the increased expected
participation—and the resulting reduced incentive to participate—by lowering
their own mixing probability. Thus, whether increased similarity aids or
hinders participation in mixed-strategy equilibria depends on how it affects
expected participation by others. This insight also applies when agents mix
over multiple signals.

##### State-dependent changes in similarity:

There is a public discourse about how undesirable autocratic regimes restrict
the flow of information among citizens, making coordination harder. We can ask
how participation in a mass protest changes if the similarity of information
is state dependent: information similarity is not allowed to increase in one
state, say, when the regime is autocratic, but increases in the other. Our
baseline model cannot directly answer this question. We interpret
$\bm{\theta}=1$ ($\bm{\theta}=0$) as the feasibility (infeasibility) of
collective action, and so changes in information similarity in state $0$ do
not affect agents’ incentives. But our methods can be readily applied to study
settings with state-dependent changes by simply relabeling the states—say,
$\Theta=\\{-1,1\\}$—so that a regime change is welfare improving in state
$\bm{\theta}=1$ and welfare reducing in $\bm{\theta}=-1$. Proposition 6 in the
online appendix shows that we can derive qualitatively different results in
this case: If information similarity increases only in good regimes (when
regime change is welfare reducing) and is unchanged in autocratic regimes
(when regime change is welfare improving), then expected participation falls
in encouragement environments and increases in discouragement environments.

##### More than two groups:

With two groups, comparison in the CAD order allowed for a complete
characterization of when more similar information increased or decreased
participation. Our application of voting illustrates that our results extend
qualitatively to some settings with more than two agents. However, in general,
with more than two groups, no order of similarity can yield a complete
characterization. This is because the probability of being pivotal in a
regime-change game is often quasi-concave—first increasing in the number of
groups with similar information, and then decreasing—and random variables are
ranked according to the quasi-concave order if and only if they have the same
distribution.232323To be ranked according to the quasi-concave order means
that the expectations of any quasi-concave function are ranked. To see why
this ranking implies the same distribution, notice that
$f(x):=\mathbbm{1}_{x\geq z}$ and $f(x):=\mathbbm{1}_{x\leq z}$ are both
quasi-concave. This means that in general, the participation incentives will
not be ranked as we vary information similarity. In the online appendix, we
propose a notion of similarity of $n$-dimensional random variables and show
that results similar to Theorems 1 and 2 are valid, even though the
characterization is partial. We use an $n$-dimensional order due to Meyer
(1990), which is stronger than the one in Section 5.

##### Informativeness of turnout:

In many settings, individuals use protests and petitions to convey dispersed
private information to policymakers, and in turn, policymakers use the
observed participation in protests or petition to _infer_ the state of the
world and then decide whether to change policy. Our model abstracts from this,
as regime change occurs whenever turnout exceeds an exogenous threshold. We
consider a version of our model in which a strategic policymaker observes the
realized turnout, updates her belief about the state of the world, and then
decides whether to change the regime. Recent works by Battaglini (2017) and
Ekmekci and Lauermann (2022) use a similar setup and focus on the
informational role of turnout. We can apply our framework to ask: does
increased similarity of information affect the informational content of
participation and improve information aggregation? We first show that
informativeness of equilibria can decrease with more similar
information.242424This is consistent with current public discourse. For
instance, in a piece about technology and protests in the _Atlantic_ , Zeynep
Tufekci writes, “Protests are signals: ‘We are unhappy, and we won’t put up
with things the way they are.’ But for that to work, the ‘We won’t put up with
it’ part has to be credible. Nowadays, large protests sometimes lack such
credibility, especially because digital technologies have made them so much
easier to organize.”(See
https://www.theatlantic.com/technology/archive/2020/06/why-protests-
work/613420/) The intuition is that when information becomes more similar,
holding the strategies fixed, the policymaker wants to lower the threshold,
and this has two opposing effects. On the one hand, a lower threshold
encourages more participation because individuals are more likely to make a
difference. On the other hand, a lower threshold exacerbates free-riding. We
also show that when the threshold belief at which the policymaker changes the
regime is not extreme (in an intermediate range), increasing information
similarity can enable information aggregation that would have been impossible
under conditionally independent signals.

##### Information design:

A natural question is how a designer might choose the optimal level of
similarity of information, given a certain objective. In the online appendix,
we derive the information structure that maximizes expected participation when
regime change is beneficial (in $\bm{\theta}=1$). We show that in
encouragement environments, the optimal information structure is full
correlation: both groups receive identical signals. In discouragement
environments, interior levels of similarity—neither conditionally independent
signals nor full correlation—can be optimal if the conditionally independent
signals do not satisfy condition M. Our analysis restricts attention to
information structures that are (weakly) more similar than conditionally
independent signals. More generally, in discouragement environments, some
negative interdependence may be desirable.

##### Changing marginal distributions:

Throughout we restrict attention to changes in the joint distribution while
keeping the marginal distributions unchanged. This assumption ensures that our
results are not driven by the changes in the fundamental uncertainty about
$\bm{\theta}$. However, in practice, increases in information similarity can
simultaneously endow agents with more information about the fundamentals. It
is straightforward to provide examples when more similar information can lead
to lower participation in the discouragement environment (as in Theorem 2)
even when the marginal distributions change. In Appendix B.5, we define an
order of similarity that does not require marginals to be unchanged. Using
that, Proposition 11 delivers a similar result to Theorem 1 if we further
require that the groups’ marginal distributions over posteriors are ranked in
the “more spread-out order” when $\bm{\theta}=1$.252525Without this condition,
in encouragement environments, information that is both more similar and more
Blackwell-informative informative about $\bm{\theta}$ can lead to a lower
expected participation. Examples are available with the authors.

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##  Appendix A Appendix: Proofs

###  A.1 Proof of Lemma 1

###### Proof.

Let $\vec{x}=(x_{1},x_{2},\ldots x_{G})$ be a profile of signal realizations.
By definition,

$\displaystyle\mathscr{V}(\sigma)=$
$\displaystyle\sum_{\vec{x}\in\mathcal{X}^{G}}\mathscr{P}^{1}(\vec{x})\left[\sum_{g=1}^{G}N\mathbbm{1}_{x_{g}\in
P_{g}}\right]$ $\displaystyle=$
$\displaystyle\sum_{\vec{x}\in\mathcal{X}^{G}}\sum_{g=1}^{G}\mathscr{P}^{1}(\vec{x})N\mathbbm{1}_{x_{g}\in
P_{g}}$ $\displaystyle=$
$\displaystyle\sum_{g=1}^{G}\sum_{x_{g}\in\mathcal{X}}\mathbb{P}^{1}_{g}(x_{g})N\mathbbm{1}_{x_{g}\in
P_{g}}$ $\displaystyle=$ $\displaystyle N\sum_{g=1}^{G}\mathbb{P}^{1}(P_{g}).$

where the last equality is due to the exchangeability of the distribution. ∎

###  A.2 Proof of Proposition 1

###### Proof.

Consider the payoff difference for any agent between participating and not.
Let $u_{g}(a,x;\sigma)$ be the expected payoff of an agent from group $g$ by
playing action $a$ given that $\mathbf{X}_{g}=x$, and the players are
following $\sigma$. We define the net expected payoff from participation as

$\Delta_{g}(x;\sigma):=u_{g}(1,x;\sigma)-u_{g}(0,x;\sigma)$

Consider an agent in group $g$. Suppose that $\mathbf{X}_{g}=x$. If she
participates, she incurs a cost $c$ and gets a positive payoff only if the
regime change is beneficial ($\bm{\theta}=1$) and at least $\bm{\bar{n}}$
other agents participate. If she does not participate, then she gets a
positive payoff only if $\bm{\theta}=1$ and the turnout is at least
$\bm{\bar{n}}+1$ without her participation. So, assuming that players play
according to $\sigma$, we have

$\Delta_{g}(x;\sigma)$
$=\mathscr{P}\left(\\{\bm{\theta}=1\\}\bigcap\\{\mathbf{A}_{-g}\geq\bm{\bar{n}}\\}\bigg{|}\mathbf{X}_{g}=x\right)-c-\mathscr{P}\left(\\{\bm{\theta}=1\\}\bigcap\\{\mathbf{A}_{-g}\geq\bm{\bar{n}}+1\\}\bigg{|}\mathbf{X}_{g}=x\right)$
$=\mathscr{P}(\bm{\theta}=1|\mathbf{X}_{g}=x)\Big{[}\mathscr{P}(\mathbf{A}_{-g}\geq\bm{\bar{n}}|\mathbf{X}_{g}=x,\bm{\theta}=1)-\mathscr{P}(\mathbf{A}_{-g}\geq\bm{\bar{n}}+1|\mathbf{X}_{g}=x,\bm{\theta}=1)\Big{]}-c$
$=\mu(x)\mathscr{P}(\mathbf{A}_{-g}=\bm{\bar{n}}|\bm{\theta}=1,\mathbf{X}_{g}=x)-c$

To simplify the above expression further, consider two cases. (i) Suppose
$x\in P_{g}$. If the realized signal for the other group $x_{-g}\in P_{-g}$,
then $\mathbf{A}_{-g}=\mathbf{N}_{g}-1+\mathbf{N}_{-g}$, and if $x_{-g}\notin
P_{-g}$, then $\mathbf{A}_{-g}=\mathbf{N}_{g}-1$. (ii) Suppose $x\notin
P_{g}$. Then, $x_{-g}\in P_{-g}$ $\implies$ $\mathbf{A}_{-g}=\mathbf{N}_{-g}$,
and $x_{-g}\notin P_{-g}$ $\implies$ $\mathbf{A}_{-g}=0$. Therefore,

$\displaystyle\Delta_{g}(x;\sigma)=$
$\displaystyle\begin{cases}\mu(x)\Big{[}\mathscr{P}^{1}_{x}(P_{-g})Prob\left(\mathbf{N}_{g}-1+\mathbf{N}_{-g}=\bm{\bar{n}}\mid\bm{\theta}=1,\mathbf{X}_{g}=x\right)\\\
\quad+\mathscr{P}^{1}_{x}(NP_{-g})Prob(\mathbf{N}_{g}-1=\bm{\bar{n}}\mid\bm{\theta}=1,\mathbf{X}_{g}=x)\Big{]}&\text{
if }x\in P_{g}\\\
\mu(x)\mathscr{P}^{1}_{x}(P_{-g})Prob(\mathbf{N}_{-g}=\bm{\bar{n}}\mid\bm{\theta}=1,\mathbf{X}_{g}=x)&\text{
if }x\notin P_{g}\end{cases}$

Finally, $\sigma$ is an equilibrium if, for all $x\in\mathcal{X}$ and for all
$g\in\\{1,2\\}$,

  1. 1.

$\sigma_{g}(x)=1\implies\Delta_{g}(x;\sigma)\geq 0$.

  2. 2.

$\sigma_{g}(x)=0\implies\Delta_{g}(x;\sigma)\leq 0$.

The expression in the proposition follows from noting that according to an
agent in group $g$, $\mathbf{N}_{g}-1+\mathbf{N}_{-g}\sim\eta^{A}_{2}(\cdot)$,
$\mathbf{N}_{g}-1\sim\eta^{A}(\cdot)$, $\mathbf{N}_{-g}\sim\eta(\cdot)$, and
$\bm{\bar{n}}\sim\phi(\cdot)$.

∎

###  A.3 Proof of Theorem 1

###### Proof.

We prove this result using two steps. In Step 1, we establish that the maximal
equilibrium is symmetric. In step 2, we show that the symmetric equilibrium
under $\widehat{\mathscr{P}}^{1}$ remains an equilibrium under
$\mathscr{P}^{1}$.

###### Step 1:

With some abuse of notation, we say that
$(P_{1},P_{2})\in\mathcal{E}(\mathscr{P})$ to mean that
$\sigma:=(\mathbbm{1}_{\mathbf{X}_{1}\in P_{1}},\mathbbm{1}_{\mathbf{X}_{2}\in
P_{2}})\in\mathcal{E}(\mathscr{P})$.

######  Lemma 2:

There is a unique maximal equilibrium in encouragement environments, and it is
symmetric.

###### Proof.

Suppose that $\sigma$ is some asymmetric equilibrium with participation sets
$P_{1}$ and $P_{2}$ for groups $1$ and $2$ respectively. We show that
$\exists$ a symmetric equilibrium $\widehat{\sigma}$ with a participation set
$P\supseteq P_{1}\cup P_{2}$.

For any set $S$, define

$\displaystyle\mathcal{T}(S):=S\bigcup\left\\{x\in\mathcal{X}:\mathscr{P}^{1}_{x}(S){\Lambda_{-\text{o}}}\geq\frac{c}{\mu(x)}\right\\}.$
(3)

In words, $\mathcal{T}(S)$ adds those signals to $S$ (if there are any) at
which an agent wants to participate if he believes that his group will not
participate but the other group will participate if they receive a signal in
$S$.

######  Claim 1:

Let $\sigma=(\mathbbm{1}_{P_{1}},\mathbbm{1}_{P_{2}})$ be a strategy profile
such that (IC:P) is satisfied for all $x\in P_{1}\cup P_{2}$ given that
players follow $\sigma$. Then, for all $x\in\mathcal{T}(P_{1}\cup P_{2})$,

$\displaystyle\mathscr{P}^{1}_{x}(\mathcal{T}(P_{1}\cup
P_{2}))\Lambda_{\text{b}}+(1-\mathscr{P}^{1}_{x}(\mathcal{T}(P_{1}\cup
P_{2})))\Lambda_{\text{o}}\geq\frac{c}{\mu(x)}$

###### Proof.

Since $(P_{1},P_{2})\in\mathcal{E}(\mathscr{P})$, (IC:P) implies,

$\displaystyle\mathscr{P}^{1}_{x}(P_{2})\Lambda_{\text{b}}+(1-\mathscr{P}^{1}_{x}(P_{2}))\Lambda_{\text{o}}\geq\frac{c}{\mu(x)}$
$\displaystyle\text{ if }x\in P_{1}$
$\displaystyle\mathscr{P}^{1}_{x}(P_{1})\Lambda_{\text{b}}+(1-\mathscr{P}^{1}_{x}(P_{2}))\Lambda_{\text{o}}\geq\frac{c}{\mu(x)}$
$\displaystyle\text{ if }x\in P_{2}$

Since $\Lambda_{\text{b}}>\Lambda_{\text{o}}$ and $\mathscr{P}^{1}_{x}(\cdot)$
is monotonic (in the set inclusion order),

$\displaystyle\mathscr{P}^{1}_{x}(P_{1}\cup
P_{2})\Lambda_{\text{b}}+(1-\mathscr{P}^{1}_{x}(P_{1}\cup
P_{2}))\Lambda_{\text{o}}\geq\frac{c}{\mu(x)}\quad\forall x\in P_{1}\cup
P_{2}$

If $\mathcal{T}(P_{1}\cup P_{2})\neq P_{1}\cup P_{2}$, then, for all
$x\in\mathcal{T}(P_{1}\cup P_{2})\backslash(P_{1}\cup P_{2})$, we have,

$\displaystyle\mathscr{P}^{1}_{x}(P_{1}\cup P_{2}){\Lambda_{-\text{o}}}$
$\displaystyle\geq\frac{c}{\mu(x)}$
$\displaystyle\implies\mathscr{P}^{1}_{x}(P_{1}\cup
P_{2})\Lambda_{\text{b}}+(1-\mathscr{P}^{1}_{x}(P_{1}\cup
P_{2}))\Lambda_{\text{o}}$ $\displaystyle\geq\frac{c}{\mu(x)}$

where the inequality is due to $\Lambda_{\text{b}}>\Lambda_{\text{o}}>0$ and
$\max\\{\Lambda_{\text{b}},\Lambda_{\text{o}}\\}\geq\Lambda_{-\text{o}}$ (see
assumption 1). ∎

Define, $\mathcal{T}^{*}(P_{1}\cup
P_{2}):=\mathcal{T}^{|\mathcal{X}|}(P_{1},P_{2})$. First, notice that
$\mathcal{T}(\cdot)$ is an increasing (in the set-inclusion order) map.
Therefore, either $\mathcal{T}^{*}(S)=\mathcal{X}$ (due to the finiteness of
$\mathcal{X}$), or $S\subseteq\mathcal{T}^{*}(S)\subset\mathcal{X}$ for any
$S$. If $\mathcal{T}^{*}(P_{1}\cup P_{2})\neq\mathcal{X}$, then, by
definition, (IC:NP) is satisfied for all $x\notin\mathcal{T}^{*}(P_{1}\cup
P_{2})$ when players play
$(\mathbbm{1}_{\mathbf{X}_{1}\in\mathcal{T}^{*}(P_{1}\cup
P_{2})},\mathbbm{1}_{\mathbf{X}_{2}\in\mathcal{T}^{*}(P_{1}\cup P_{2})})$.
Moreover, (IC:P) is satisfied when both the groups play $a=1$ on
$\mathcal{T}^{*}(P_{1}\cup P_{2})$ by Claim 1. Therefore, given any
equilibrium $(P_{1},P_{2})$, $\mathcal{T}^{*}(P_{1}\cup P_{2})$ is a larger
symmetric equilibrium.

Finally, let $(P,P)$ and $(P^{\prime},P^{\prime})$ be two maximal symmetric
equilibria with $P\neq P^{\prime}$. First, Claim 1 establishes that (IC:P) is
satisfied for all signals in $\mathcal{T}(P\cup P^{\prime})$, and hence for
all the signals in $\mathcal{T}^{*}(P\cup P^{\prime})$. By construction,
(IC:NP) is satisfied for all the signals outside of $\mathcal{T}^{*}(P\cup
P^{\prime})$. Therefore, $\mathcal{T}^{*}(P\cup P^{\prime})$ is an
equilibrium, and $P,P^{\prime}\subseteq\mathcal{T}^{*}(P\cup P^{\prime})$,
contradicting the maximality of $P,P^{\prime}$. Therefore, $P=P^{\prime}$,
i.e., there is a unique maximal equilibrium. ∎

###### Step 2:

By definition, $\mathscr{P}^{1}\succcurlyeq_{CAD}\widehat{\mathscr{P}}^{1}$
implies the following:

  1. 1.

$\mathscr{P}^{1}_{x}(P(\sigma))\geq\widehat{\mathscr{P}}^{1}_{x}(P(\sigma))$
for all $x\in P(\sigma)$.

  2. 2.

$\mathscr{P}^{1}_{x}(P(\sigma))\leq\widehat{\mathscr{P}}^{1}_{x}(P(\sigma))$
for all $x\in NP(\sigma)$.

Consider (IC:P), the incentive constraint for an agent who participates. The
left hand side is a convex combination of $\Lambda_{\text{b}}$ and
$\Lambda_{\text{o}}$. In encouragement environments, a higher weight on
$\Lambda_{\text{b}}$ increases the LHS of (IC:P), making the constraint easier
to satisfy. Therefore,

$\displaystyle\mathscr{P}^{1}_{x}(P(\sigma))\Lambda_{\text{b}}+(1-\mathscr{P}^{1}_{x}(P(\sigma)))\Lambda_{\text{o}}$
$\displaystyle\geq\widehat{\mathscr{P}}^{1}_{x}(P(\sigma))\Lambda_{\text{b}}+(1-\widehat{\mathscr{P}}^{1}_{x}(P(\sigma)))\Lambda_{\text{o}}$
$\displaystyle\geq\frac{c}{\mu(x)}\mbox{ }\text{ since
}\sigma\in\mathcal{E}(\widehat{\mathscr{P}}).$

Therefore, (IC:P) is satisfied for all $x\in P(\sigma)$ under $\mathscr{P}$.
Similarly, for all $x\in NP(\sigma)$,
$\mathscr{P}^{1}_{x}(P(\sigma))\leq\widehat{\mathscr{P}}^{1}_{x}(P(\sigma))$.
Therefore, (IC:NP) is satisfied for all $x\in NP(\sigma)$ under signal
$\mathscr{P}$. Therefore, $\sigma\in\mathcal{E}(\mathscr{P})$.

Finally, since $\mathscr{V}(\cdot)$ depends only the marginal distributions
(by Lemma 1)—which are the same in $\mathscr{P}$ and
$\widehat{\mathscr{P}}$—it follows that
$\mathscr{V}^{*}(\mathscr{P})\geq\mathscr{V}^{*}(\widehat{\mathscr{P}})$. ∎

###  A.4 Proof of Theorem 2

###### Proof of Theorem 2.

We prove this result using two steps. In step 1, we show that the maximal
equilibrium under $\widehat{\mathscr{P}}^{1}$ may no longer be an equilibrium
under $\mathscr{P}^{1}$. In step 2, we show that no larger equilibrium can
emerge when information becomes more similar.

###### Step 1:

Let $\sigma^{*}$ be a maximal equilibrium with the associated participation
sets $(P_{1}^{*},P_{2}^{*})$.

  1. Case 1.

$P_{1}^{*}=P_{2}^{*}$

Let $P:=P_{1}^{*}=P_{2}^{*}$. By definition,
$\mathscr{P}^{1}\succcurlyeq_{CAD}\widehat{\mathscr{P}}^{1}$ implies the
following:

    1. (a)

$\mathscr{P}^{1}_{x}(P)\geq\widehat{\mathscr{P}}^{1}_{x}(P)$ for all $x\in P$.

    2. (b)

$\mathscr{P}^{1}_{x}(P)\leq\widehat{\mathscr{P}}^{1}_{x}(P)$ for all $x\in
NP$.

Consider (IC:P), the incentive constraint for an agent who participates. The
left hand side is a convex combination of $\Lambda_{\text{b}}$ and
$\Lambda_{\text{o}}$. In discouragement environments, a higher weight on
$\Lambda_{\text{b}}$ decreases the LHS of (IC:P). Therefore, for all $x\in P$,

$\displaystyle\mathscr{P}^{1}_{x}(P)\Lambda_{\text{b}}+(1-\mathscr{P}^{1}_{x}(P))\Lambda_{\text{o}}\leq\widehat{\mathscr{P}}^{1}_{x}(P)\Lambda_{\text{b}}+(1-\widehat{\mathscr{P}}^{1}_{x}(P))\Lambda_{\text{o}}$

Therefore, (IC:P) may fail for some $x\in P$ under $\mathscr{P}$, in which
case $\sigma$ may no longer be in $\mathcal{E}(\mathscr{P})$.

  2. Case 2.

$P_{1}^{*}\neq P_{2}^{*}$

Then,
$\mathscr{P}^{1}_{x}(P_{i}^{*})\geq\widehat{\mathscr{P}}^{1}_{x}(P_{i}^{*})$
for all $i\in\\{1,2\\}$ and $x\in P_{i}^{*}$. Notice that at least one of
$P_{1}^{*}\setminus P_{2}^{*}$ and $P_{2}^{*}\setminus P_{1}^{*}$ is not
$\emptyset$. Let $P_{1}^{*}\backslash P_{2}^{*}\neq\emptyset$ wlog. Consider
some $x\in P_{1}^{*}\backslash P_{2}^{*}$. Agents in group $2$ must find it
incentive compatible to not participate when they receive a signal in
$P_{1}^{*}$. Therefore, we must have,

$\displaystyle\mathscr{P}^{1}_{x}(P_{1}^{*})\Lambda_{-\text{o}}\leq\frac{c}{\mu(x)}$

for all $x\in P_{1}^{*}$. However, since
$\mathscr{P}^{1}_{x}(P_{1}^{*})\geq\widehat{\mathscr{P}}^{1}_{x}(P_{1}^{*})$
for all $x\in P_{1}^{*}$, (IC:NP) is harder to satisfy for any $x\in
P_{1}^{*}\backslash P_{2}^{*}$, and for any $x\in P_{2}^{*}\backslash
P_{1}^{*}$. Hence, $\sigma^{*}$ may no longer be in
$\mathcal{E}(\mathscr{P})$.

###### Step 2:

Next, consider any
$(\widehat{P}_{1},\widehat{P}_{2})\geq(P_{1}^{*},P_{2}^{*})$. For
$(\widehat{P}_{1},\widehat{P}_{2})$ to be an equilibrium, one necessary
condition is (IC:NP) for signals in $\widehat{P}_{1}$ and $\widehat{P}_{2}$
for groups $2$ and $1$ respectively. That is, we need that, at least one of
the following holds:

$\displaystyle\mathscr{P}^{1}_{x}(\widehat{P}_{1})\Lambda_{-\text{o}}\leq\frac{c}{\mu(x)}$
$\displaystyle\text{ if }x\in\widehat{P}_{1}$
$\displaystyle\mathscr{P}^{1}_{x}(\widehat{P}_{2})\Lambda_{-\text{o}}\leq\frac{c}{\mu(x)}$
$\displaystyle\text{ if }x\in\widehat{P}_{2}$

Suppose that $\widehat{P}_{1}\cap\widehat{P}_{2}=\emptyset$. Since
$\widehat{\mathscr{P}}$ satisfies condition M (Definition 4), part $(M2)$
implies that, for some $x\in\widehat{P}_{1}\cup\widehat{P}_{2}$,

$\displaystyle\widehat{\mathscr{P}}^{1}_{x}(\widehat{P}_{1})\Lambda_{-\text{o}}>\frac{c}{\mu(x)}$
$\displaystyle\text{ if }x\in\widehat{P}_{1}$
$\displaystyle\widehat{\mathscr{P}}^{1}_{x}(\widehat{P}_{2})\Lambda_{-\text{o}}>\frac{c}{\mu(x)}$
$\displaystyle\text{ if }x\in\widehat{P}_{2}$

By CAD,
$\mathscr{P}^{1}_{x}(\widehat{P}_{1})\geq\widehat{\mathscr{P}}^{1}_{x}(\widehat{P}_{1})$
if $x\in\widehat{P}_{1}$ (and analogously for $\widehat{P}_{2}$). Therefore,
$(\widehat{P}_{1},\widehat{P}_{2})$ cannot be an equilibrium in $\mathscr{P}$
if $\widehat{P}_{1}\cap\widehat{P}_{2}=\emptyset$.

Suppose that $\widehat{P}_{1}\cap\widehat{P}_{2}\neq\emptyset$. Since
$\widehat{\mathscr{P}}$ satisfies Condition M, if $(M1)$ holds for some
$x\in\widehat{P}_{1}\cap\widehat{P}_{2}$, then,

$\displaystyle\min_{i\in\\{1,2\\}}\left\\{\widehat{\mathscr{P}}^{1}_{x}(\widehat{P}_{i})\Lambda_{\text{b}}+(1-\widehat{\mathscr{P}}^{1}_{x}(\widehat{P}_{i}))\Lambda_{\text{o}}\right\\}<\frac{c}{\mu(x)}$

By CAD,
$\mathscr{P}^{1}_{x}(\widehat{P}_{i})\geq\widehat{\mathscr{P}}^{1}_{x}(\widehat{P}_{i})$
for $i\in\\{1,2\\}$. Since $\Lambda_{\text{o}}>\Lambda_{\text{b}}$, this
implies that,

$\displaystyle\min_{i\in\\{1,2\\}}\left\\{\mathscr{P}^{1}_{x}(\widehat{P}_{i})\Lambda_{\text{b}}+(1-\mathscr{P}^{1}_{x}(\widehat{P}_{i}))\Lambda_{\text{o}}\right\\}<\frac{c}{\mu(x)}$

Therefore, (IC:P) fails for such an $x$.

Finally, if (IC:P) is satisfied for all
$x\in\widehat{P}_{1}\cap\widehat{P}_{2}$, then, by Condition M $(ii)$, the
exact same argument as in the case when
$\widehat{P}_{1}\cap\widehat{P}_{2}=\emptyset$ implies that
$(\widehat{P}_{1},\widehat{P}_{2})$ cannot be an equilibrium under
$\mathscr{P}$. Therefore, no larger equilibrium can exist under
$\widehat{\mathscr{P}}$, i.e.,
$\mathscr{V}^{*}(\widehat{\mathscr{P}})\leq\mathscr{V}(\mathscr{P})$. ∎

###  A.5 Proof of Proposition 3

We first derive a useful property of the CAD order.

######  Lemma 3:

If $\mathscr{D}\succcurlyeq_{CAD}\widehat{\mathscr{D}}$, then, for every
$T\subseteq\mathcal{Y}$, $\exists$ $\alpha_{T}\geq 0$ such that
$\mathscr{D}(T,T)=\widehat{\mathscr{D}}(T,T)+\alpha_{T}$, and
$\mathscr{D}(T,\mathcal{Y}\backslash
T)=\widehat{\mathscr{D}}(T,\mathcal{Y}\backslash T)-\alpha_{T}$.

###### Proof of Lemma 3.

Suppose $\mathcal{Y}\subset\mathbb{R}$ is finite, and $Y$ and $\widehat{Y}$
are two $\mathcal{Y}^{2}$-valued random variables with joint distributions
$\mathscr{D}$ and $\widehat{\mathscr{D}}$ respectively, and identical
marginals. Consider any two distinct points in the support of $Y$, say
$y_{j},y_{k}$. Define an “elementary transformation along identical intervals”
(ETI) as an operation in which, for some $\alpha>0$, we increase the
probability mass on points $(y_{j},y_{j})$ and $(y_{k},y_{k})$ each by
$\alpha$, and reduce the probability mass on $(y_{j},y_{k})$ and
$(y_{k},y_{j})$ each by $\alpha$. An alternative characterization of our CAD
order in two dimensions is that
$\mathscr{D}\succcurlyeq_{CAD}\widehat{\mathscr{D}}$ if and only if
$\mathscr{D}$ can be derived from $\widehat{\mathscr{D}}$ by a finite sequence
of ETIs. This follows from Meyer (1990) (Proposition 1). We use this
characterization to establish Lemma 3.

Let $(y_{i,k},y_{j,k})_{k}$, $k=1,\ldots n$, be a finite set of points in
$\mathcal{Y}^{2}$ describing a sequence of ETIs, each with a mass
$\alpha_{k}$, to obtain $\mathscr{D}$ from $\widehat{\mathscr{D}}$. Let the
resulting distribution after the $k$-th ETI be $\widehat{\mathscr{D}}_{k}$.
So, $\widehat{\mathscr{D}}_{1}=\widehat{\mathscr{D}}$ and
$\widehat{\mathscr{D}}_{n}=\mathscr{D}$. If $(y_{i,k},y_{j,k})\in T\times T$
or $(\mathcal{Y}\backslash T)\times(\mathcal{Y}\backslash T)$, then
$\widehat{\mathscr{D}}_{k}(T,T)=\widehat{\mathscr{D}}_{k-1}(T,T)$. On the
other hand, if exactly one of $\\{y_{i,k},y_{j,k}\\}$ is in $T$ for some $k$,
then
$\widehat{\mathscr{D}}_{k}(T,T)=\widehat{\mathscr{D}}_{k-1}(T,T)+\alpha_{k}$.
Therefore,
$\mathscr{D}(T,T)=\widehat{\mathscr{D}}(T,T)+\sum_{k=1}^{n}\alpha_{k}$. Since
an ETI leaves the marginal distribution unchanged, therefore
$\mathscr{D}(T,\mathcal{Y}\backslash
T)=\widehat{\mathscr{D}}(T,\mathcal{Y}\backslash T)-\sum_{k=1}^{n}\alpha_{k}$.
The lemma follows.

∎

###### Proof of Proposition 3.

Towards establishing $(1)$, suppose $\Lambda_{\text{b}}>\Lambda_{\text{o}}$
and $\mathscr{P}^{1}\succcurlyeq_{CAD}\widehat{\mathscr{P}}^{1}$. First, by
the argument in Theorem 1,
$\mathcal{E}(\widehat{\mathscr{P}})\subseteq\mathcal{E}(\mathscr{P})$.
Consider any equilibrium $\sigma\in\mathcal{E}(\widehat{\mathscr{P}})$, and
let the associated participation and nonparticipation sets be P and NP
respectively. By Lemma 3,
$\mathscr{P}^{1}(P,P)=\widehat{\mathscr{P}}^{1}(P,P)+\alpha$ and
$\mathscr{P}^{1}(P,NP)=\widehat{\mathscr{P}}^{1}(P,NP)-\alpha$ for some
$\alpha\geq 0$.262626$\mathscr{P}^{1}(T,T)$ is a shorthand to denote
$\mathscr{P}^{1}(\\{\mathbf{X}_{1}\in T,\mathbf{X}_{2}\in T\\})$. Therefore,

$\displaystyle\Pi(\sigma;\mathscr{P})=$
$\displaystyle(1-\Psi(\bar{n},2N))\mathscr{P}^{1}(P,P)+2((1-\Psi(\bar{n},N))\mathscr{P}^{1}(P,NP)$
$\displaystyle=$
$\displaystyle(1-\Psi(\bar{n},2N))(\widehat{\mathscr{P}}^{1}(P,P)+\alpha)+2((1-\Psi(\bar{n},N))(\widehat{\mathscr{P}}^{1}(P,NP)-\alpha)$

Simplifying, we get

$\displaystyle\Pi(\sigma;\mathscr{P})-\Pi(\sigma;\widehat{\mathscr{P}})=$
$\displaystyle\alpha\left(2\Psi(\bar{n},N)-\Psi(\bar{n},2N)-1\right)$

It is easy to check that $\frac{\partial\Psi(k,x)}{\partial x}=-\psi(k,x)$.
Defining $\Delta(N):=2\Psi(\bar{n},N)-\Psi(\bar{n},2N)-1$, we have
$\Delta^{\prime}(N)=2(\psi(\bar{n},2N)-\psi(\bar{n},N)>0$ by assumption. Also,
$\lim_{N\to 0}\Delta(N)=0$. Therefore, $\Delta(N)>0$ for any $N$ such that
$\Lambda_{\text{b}}>\Lambda_{\text{o}}$. Hence,
$\Pi(\sigma;\mathscr{P})\geq\Pi(\sigma;\widehat{\mathscr{P}})$ with the
inequality being strict whenever $\alpha>0$. Therefore,
$\overline{\Pi}(\mathscr{P})\geq\overline{\Pi}(\widehat{\mathscr{P}})$.

Towards establishing $(2)$, suppose that
$\overline{\Pi}(\mathscr{P})>\overline{\Pi}(\widehat{\mathscr{P}})$,
$\mathscr{P}^{1}\succcurlyeq_{CAD}\widehat{\mathscr{P}}^{1}$, and
$\widehat{\mathscr{P}}^{1}$ satisfies Condition M (Probability). Let
$\sigma^{*}$ and $\widehat{\sigma}^{*}$ be the equilibria with maximum
probabilities of success under $\mathscr{P}$ and $\widehat{\mathscr{P}}$
respectively.

First, we argue that
$\Pi(\sigma^{*};\widehat{\mathscr{P}})>\Pi(\widehat{\sigma}^{*};\widehat{\mathscr{P}})$.
Suppose not. By the previous argument, we have that, _for any strategy profile
$\sigma$,_, $\Pi(\sigma;\mathscr{P})<\Pi(\sigma;\widehat{\mathscr{P}})$
whenever $\Lambda_{\text{o}}>\Lambda_{\text{b}}$. Therefore,

$\displaystyle\Pi(\widehat{\sigma}^{*};\widehat{\mathscr{P}})<\Pi(\sigma^{*};\mathscr{P})\leq\Pi(\sigma^{*};\widehat{\mathscr{P}})\leq\Pi(\widehat{\sigma}^{*};\widehat{\mathscr{P}});$

a contradiction.

But, if
$\Pi(\sigma^{*};\widehat{\mathscr{P}})>\Pi(\widehat{\sigma}^{*};\widehat{\mathscr{P}})$,
then, by Condition-M (Probability),
$\sigma^{*}\notin\mathcal{E}(\mathscr{P})$. Therefore, it cannot be an
equilibrium with maximum probability under $\mathscr{P}$. ∎

While Proposition 3 characterizes the effect of increasing interdependence on
the probability of success when $\bm{\theta}=1$, it raises a closely related
question of how increasing similarity affects overall welfare—the ex-ante
expected utility of a representative agent in any group $g$.

To this end, define welfare given an information structure and a strategy
profile, and maximal welfare achievable in any equilibrium $\mathscr{W}^{*}$
as follows.272727Exchangeability of the distribution of $\mathbf{X}$ implies
that welfare does not depend on group identity. The following is a corollary
of Proposition 3

$\displaystyle\mathscr{W}(\sigma,\mathscr{P}):=$
$\displaystyle\mu_{1}\Pi(\sigma;\mathscr{P})-c\mathbb{E}_{\mathscr{P}}\left[\sigma(\mathbf{X}_{g})\right]$
$\displaystyle\mathscr{W}^{*}(\mathscr{P}):=$
$\displaystyle\max_{\sigma\in\mathcal{E}(\mathscr{P})}\mathscr{W}(\sigma,\mathscr{P})$

The above expression makes it clear that welfare unambiguously increases
whenever $\Lambda_{\text{b}}>\Lambda_{\text{o}}$ since $\Pi(\sigma;\cdot)$
increases with similarity for any $\sigma$ due to Proposition 3. Analogous
reasoning (along with an analogous Condition M (Probability) replaced by
Condition M (Welfare)) establishes that welfare, too, decreases under
discouragement environments. We choose not to present it as a formal
proposition.

####  A.5.1 Turnout conditional on protests

In the main text, we focus on how information similarity affects the
equilibrium participation. However, even when we fix an equilibrium,
information similarity makes the participation more coordinated. Accordingly,
conditional on there being a protest, we may see that the protests are more
likely to bring about social changes. The following proposition formalizes
this intuition.

######  Proposition 5:

Suppose $\mathscr{P}^{1}\succcurlyeq_{CAD}\widehat{\mathscr{P}}^{1}$, and
$\sigma\in\mathcal{E}(\mathscr{P})\cap\mathcal{E}(\widehat{\mathscr{P}})$ and
$\sigma$ is symmetric. Then,

$\mathscr{P}^{1}\left[\mathbf{A}(\sigma)>\bar{n}\Big{|}\mathbf{A}(\sigma)>0\right]\geq\widehat{\mathscr{P}}^{1}\left[\mathbf{A}(\sigma)>\bar{n}\Big{|}\mathbf{A}(\sigma)>0\right].$

###### Proof.

Let $H_{2}(\cdot),H_{1}(\cdot)$ be the CDFs of $\mathbf{N}_{1}+\mathbf{N}_{2}$
and $\mathbf{N}_{1}$ (and $\mathbf{N}_{2}$ by symmetry) respectively. Suppose
that $\sigma\in\mathcal{E}(\mathscr{P})\cap\mathcal{E}(\widehat{\mathscr{P}})$
and $\mathscr{P}^{1}\succcurlyeq_{CAD}\widehat{\mathscr{P}}^{1}$. Under
$\widehat{\mathscr{P}}$, the probability of there being a protest at all when
$\bm{\theta}=1$ is
$\mathscr{P}^{1}(\\{\mathbf{A}>0\\})=1-\widehat{\mathscr{P}}^{1}(NP,NP)$. Here
$\mathscr{P}^{1}(NP,NP)$ means $\mathscr{P}^{1}(\\{\mathbf{X}_{1}\in
NP(\sigma),\mathbf{X}_{2}\in NP(\sigma)\\})$. Therefore,

$\mathscr{P}^{1}\left(\\{\mathbf{A}(\sigma)>\bar{n})\\}\Big{|}\\{\mathbf{A}(\sigma)>0\\}\right)=\frac{\left[(1-H_{2}(\bar{n}))\mathscr{P}^{1}(P,P)+2(1-H_{1}(\bar{n}))\mathscr{P}^{1}(P,NP)\right]}{1-\mathscr{P}^{1}(NP,NP)}.$

Then, by Lemma 3, there exists $\alpha>0$ such that

$\displaystyle\mathscr{P}^{1}\left(\\{\mathbf{A}(\sigma)>\bar{n})\\}\Big{|}\\{\mathbf{A}(\sigma)>0\\}\right)$
$\displaystyle=\frac{\left[(1-H_{2}(\bar{n}))(\widehat{\mathscr{P}}^{1}(P,P)+\alpha)+2(1-H_{1}(\bar{n}))(\widehat{\mathscr{P}}^{1}(P,NP)-\alpha)\right]}{1-\widehat{\mathscr{P}}^{1}(NP,NP)-\alpha}$
$\displaystyle=\underbrace{\frac{\widehat{\mathscr{P}}^{1}(P,P)+\alpha}{1-\widehat{\mathscr{P}}^{1}(NP,NP)-\alpha}}_{\textrm{Increasing
in
}\alpha}\left(1-H_{2}(\bar{n})\right)+\underbrace{\left(1-\frac{\widehat{\mathscr{P}}^{1}(P,P)+\alpha}{1-\widehat{\mathscr{P}}^{1}(NP,NP)-\alpha}\right)}_{\textrm{Decreasing
in }\alpha}\left(1-H_{1}(\bar{n})\right))$

The equality follows from noting that
$(\widehat{\mathscr{P}}^{1}(P,P)+\alpha)+2(\widehat{\mathscr{P}}^{1}(P,NP)-\alpha)=1-\widehat{\mathscr{P}}^{1}(NP,NP)-\alpha$.
Notice that the above expression is a convex combination of $1-H_{2}(\bar{n})$
and $1-H_{1}(\bar{n})$. Since $\mathbf{N}_{1},\mathbf{N}_{2}\geq 0$ a.s.,
$H_{2}(\cdot)\leq H_{1}(\cdot)$. Therefore, the LHS puts a larger weight on
the larger term for any $\alpha\geq 0$, which implies

$\displaystyle\mathscr{P}^{1}\left(\\{\mathbf{A}(\sigma)>\bar{n})\\}\Big{|}\\{\mathbf{A}(\sigma)>0\\}\right)$
$\displaystyle>\frac{\widehat{\mathscr{P}}^{1}(P,P)}{1-\widehat{\mathscr{P}}^{1}(NP,NP)}\left(1-H_{2}(\bar{n})\right)+\left(1-\frac{\widehat{\mathscr{P}}^{1}(P,P)}{1-\widehat{\mathscr{P}}^{1}(NP,NP)}\right)\left(1-H_{1}(\bar{n})\right)$
$\displaystyle=\widehat{\mathscr{P}}^{1}\left(\\{\mathbf{A}(\sigma)>\bar{n})\\}\Big{|}\\{\mathbf{A}(\sigma)>0\\}\right).$

∎

###  A.6 Proof of Proposition 4

###### Proof.

Suppose that $k^{*}<\bar{n}$. Then,
$\sigma^{1}\in\mathcal{E}(\widehat{\mathscr{P}})$ implies that
$\widehat{\gamma}^{1}_{1}(\bar{n})\geq\frac{c}{\mu(1)}$. By CAD,
$\gamma^{1}_{1}(\bar{n})\geq\widehat{\gamma}^{1}_{1}(\bar{n})$. Therefore,
(IC-voting) continues to be satisfied under $\mathscr{P}$, and hence
$\sigma^{1}\in\mathcal{E}(\mathscr{P})$.

On the other hand, if $k^{*}\geq\bar{n}$, then
$\gamma^{1}_{1}(\bar{n})\leq\widehat{\gamma}^{1}_{1}(\bar{n})$. Therefore, it
is possible that $\sigma^{1}\in\mathcal{E}(\widehat{\mathscr{P}})$ but
$\sigma^{1}\notin\mathcal{E}(\mathscr{P})$. Hence, part $(2)$ of the
proposition follows. ∎

###  A.7 Optimal voting rule from Section 5

Since there are multiple equilibria, we assume the maximal equilibrium is
played, and the optimal vote threshold is one that maximizes the probability
that a rate increase is implemented conditional on it being warranted; that
is, the optimal threshold $\bar{n}^{*}(\mathscr{P})$ is given by

$\bar{n}^{*}(\mathscr{P}):=\operatorname{argmax}_{\bar{n}}\mathscr{P}^{1}(I\geq\bar{n}+1|\bm{\theta}=1),$

subject to the (IC-voting) constraint.282828The qualitative argument is
unchanged if we assume a small negative payoff from raising rates when not
necessary to do so. Essentially, this formulation assumes that the cost of
inflation due to failure to raise rates when required far outweighs a
contemporaneous loss in output. In the absence of the incentive constraint,
the lowest possible $\bar{n}$ would be optimal. However, a low $\bar{n}$
reduces an individual member’s incentive to vote in favor of a rate increase
(conditional on private information and the increase being warranted). So the
optimal rule is the lowest vote threshold that satisfies the incentive
constraint. We can ask how this optimal threshold varies with committee
diversity.

###### Proof.

Consider $k^{*}<\bar{n}^{*}(\widehat{\mathscr{P}})$. Then, by definition,

$\gamma^{1}_{1}(\bar{n}^{*}(\widehat{\mathscr{P}}))\geq\widehat{\gamma}^{1}_{1}(\bar{n}^{*}(\widehat{\mathscr{P}}))\geq\frac{c}{\mu(1)}.$

This means under the same policy threshold
$\bar{n}^{*}(\widehat{\mathscr{P}})$, the incentive constraint is satisfied
even under more similar experiences ($\mathscr{P}$). Since the designer’s
objective $\mathscr{P}^{1}(I\geq\bar{n}+1)$ is decreasing in $\bar{n}$, we
have $\bar{n}^{*}(\mathscr{P})\leq\bar{n}^{*}(\widehat{\mathscr{P}})$.

Next, consider $k^{*}\geq\bar{n}^{*}(\widehat{\mathscr{P}})$. Recall that
$\bar{n}^{*}(\widehat{\mathscr{P}})$ is the lowest $\bar{n}$ that satisfies
the incentive constraint under $\widehat{\mathscr{P}}$. Therefore, for any
$\bar{n}<\bar{n}^{*}(\widehat{\mathscr{P}})$,

$\widehat{\gamma}^{1}_{1}(\bar{n})<\frac{c}{\mu(1)}.$

Since $k^{*}\geq\bar{n}^{*}(\widehat{\mathscr{P}})>\bar{n}$, by definition,

$\gamma^{1}_{1}(\bar{n})\leq\widehat{\gamma}^{1}_{1}(\bar{n})<\frac{c}{\mu(1)}.$

This means for any policy $\bar{n}<\bar{n}^{*}(\widehat{\mathscr{P}})$, under
more similar experiences ($\mathscr{P}$), the incentive constraint does not
hold. Moreover, since
$\gamma^{1}_{1}(\bar{n}^{*}(\widehat{\mathscr{P}}))\leq\widehat{\gamma}^{1}_{1}(\bar{n}^{*}(\widehat{\mathscr{P}}))$,
even under policy $\bar{n}^{*}(\widehat{\mathscr{P}})$, the incentive
constraint may no longer be satisfied under more similar experiences
($\mathscr{P}$). Therefore,
$\bar{n}^{*}(\mathscr{P})\geq\bar{n}^{*}(\widehat{\mathscr{P}})$. ∎

##  Appendix B Online Appendix (Not for publication)

###  B.1 On state-dependent changes in similarity

Suppose that $\Theta=\\{-1,1\\}$. We keep the model unchanged in all respects
otherwise. Now a regime change when $\bm{\theta}=-1$ is welfare-reducing. We
demonstrate how increasing similarity in state $\bm{\theta}=-1$ has the
opposite effects from what we showed in the main paper. To this end, we
specialize to symmetric strategies. Then, given any strategy $\sigma$, we have
the associated participation and nonparticipation sets given by $P(\sigma)$
and $NP(\sigma)$ respectively.

For $\sigma$ to be an equilibrium, the IC constraints for protesting and not-
protesting are:

$\displaystyle\mu(x)\left[\mathscr{P}^{1}_{x}(P)\Lambda_{\text{b}}+(1-\mathscr{P}^{1}_{x}(P))\Lambda_{\text{o}}\right]$
$\displaystyle-$
$\displaystyle(1-\mu(x))\left[\mathscr{P}^{-1}_{x}(P)\Lambda_{\text{b}}+(1-\mathscr{P}^{-1}_{x}(P))\Lambda_{\text{o}}\right]\geq
c$ $\displaystyle\text{ if }x\in P$ (IC:P-S)
$\displaystyle\mu(x)\mathscr{P}^{1}_{x}(P)\Lambda_{-\text{o}}-(1-\mu(x))\mathscr{P}^{-1}_{x}(P)\Lambda_{-\text{o}}\leq
c$ $\displaystyle\text{ if }x\notin P$ (IC:NP-S)

The only difference from our benchmark setup is the second term in the
incentive constraints. This captures the probability of being pivotal in state
$\bm{\theta}=-1$. It is straightforward to see that an increase in similarity
in state $1$ (i.e., CAD increases of $\mathscr{P}^{1}$) has the same impact as
in the main paper (for the natural modification of Condition M for this
environment). But now consider the effects of increases in similarity in
$\mathscr{P}^{-1}$. We can interpret

$\mathscr{P}^{-1}_{x}(P)\Lambda_{\text{b}}+(1-\mathscr{P}^{-1}_{x}(P))\Lambda_{\text{o}}$

as the cost of making a difference in state $\theta=-1$ for a participant. In
encouragement (discouragement) environments, an increase in similarity