# A Population’s Feasible Posterior Beliefs111Itai acknowledges support from
the Ministry of Science and Technology (grant 19400214). Yakov acknowledges
support from the Israel Science Foundation (grant 2021296).

Itai Arieli, Yakov Babichenko Faculty of Industrial Engineering and
Management, Technion: The Israel Institute of Technology,
<EMAIL_ADDRESS>of Industrial Engineering and Management,
Technion: The Israel Institute of Technology<EMAIL_ADDRESS>
###### Abstract

We consider a population of Bayesian agents who share a common prior over some
finite state space and each agent is exposed to some information about the
state. We ask which distributions over empirical distributions of posteriors
beliefs in the population are feasible. We provide a necessary and sufficient
condition for feasibility. We apply this result in several domains. First, we
study the problem of maximizing the polarization of beliefs in a population.
Second, we provide a characterization of the feasible agent-symmetric product
distributions of posteriors. Finally, we study an instance of a private
Bayesian persuasion problem and provide a clean formula for the sender’s
optimal value.

##  1 Introduction

Economics is deeply concerned with the question of whether agents’ behaviors
can be rationalized in the sense that they can be explained as an equilibrium
behavior in a Bayesian model with rational agents. One such related question
is under what conditions agents’ beliefs, i.e., posterior distribution, are
compatible with Bayes’s rule? This question is well understood for the single
agent where by the splitting lemma ((Aumann et al., 1995; Blackwell, 1951))
feasibility simply means that the expectation of the posterior equals the
prior. But the characterization of feasible posterior distribution for two or
more agents is trickier.

The first paper to address the feasibility of posterior distributions for more
than one agent is (Dawid et al., 1995) who characterize feasible distributions
for two agents. Recently a growing number of papers have examined the
feasibility of posterior distributions and their connection to information
design problems. (Arieli et al., 2021) connect the notion of feasibility to
the agreement theorem of (Aumann, 1976) and to the related no-trade theorem of
(Milgrom and Stokey, 1982). A follow-up paper by (Morris, 2020) generalizes
the result in (Arieli et al., 2021) to an arbitrary finite state space.
Despite these elegant connections, in general, determining whether a given
distribution of posterior beliefs is feasible is a tough question.

In this work we study an _anonymous_ variant of the feasibility question. The
object whose feasibility we are willing to determine is a distribution over
the _empirical distributions of posterior beliefs in the population_ rather
then a distribution over _profiles of posteriors_. The difference between
these two variants is that a profile of posteriors indicates which agent holds
which posterior, while the anonymous version of the problem only reveals how
many agents hold each posterior without revealing their identity.

Surprisingly, we show that determining feasibility for the anonymous variant
of the problem is relatively easy, especially in the case where the number of
different posteriors for an individual in the population is low. Our
characterization (Theorem 1) provides a necessary and sufficient condition for
a distribution over empirical samples of a population to be feasible. The
characterization relies on the observation that conditional on a state the
frequencies of posterior beliefs in the population can be identified. We show
that every combination of spreads of these expected frequencies (one for each
state) is feasible. Unlike the feasibility characterization by (Arieli et al.,
2021) and (Morris, 2020), our characterization does not become more complex as
the number of agents in the population increases. Furthermore, the complexity
of our characterization does not increase with the number of states either.
However, the parameter of the number of possible posteriors in the population
has an effect on the complexity of our characterization.

As was mentioned above, the feasibility problem is an intriguing instance of
rationalization in the case where an econometrician observes the distribution
over empirical distributions of posteriors in the population. In the context
of rationalization, it is natural to consider another variant where the
econometrician observes a single sample; i.e., she observes a single empirical
distribution of posteriors and wants to determine whether it is possible that
the agents are Bayesian.222(Shmaya and Yariv, 2016; De Oliveira and Lamba,
2019) have studied Bayesian rationalization of the behavior of a single agent
in a repeated environment. (Molavi, 2021) shows that every empirical
distribution of posteriors can be rationalized. This question is equivalent to
understanding which empirical distributions of posteriors might belong to the
support of a feasible distribution. Our results can be viewed as an extension
of (Molavi, 2021); however, we go a step further and determine the entire set
of feasible distributions rather than focus on points that might belong to the
support.

We apply our main theorem in three different domains.

#### Polarization

The question of how polarized the posteriors of two Bayesian agents might be
has been previously studied. Different measures of polarization have been
suggested, including the distance between the posteriors (Dubins and Pitman,
1980), the distance to the power $\alpha$ (Arieli et al., 2021), and the
indicator function of the posterior’s distance being above some threshold
(Burdzy and Pal, 2019; Burdzy and Pitman, 2020). The case of a quadratic
distance can be naturally generalized to multiple agents by considering the
variance of the belief’s in the population. The maximal polarization for two
agents is obtained by revealing the state to one agent and providing no
information to the other agent ((Arieli et al., 2021)). We extend this result
to every population of agents of even size: the maximal polarization is
obtained by revealing the state to half of the population and providing no
information to the other half. Furthermore, we extend this result to an
arbitrary state space (not necessarily binary).

#### Private private information

A fascinating instantiation of the feasibility problem is that of product
distributions. If the distribution of belief profiles is a product
distribution then the information that agent $i$ receives does not affect her
belief about the posteriors of her peers. (He et al., 2021) refer to this
property as _private private information_ of the agents. The feasibility of
product distributions has been considered in (He et al., 2021; Arieli et al.,
2021; Gutmann et al., 1991a) and we currently have a good understanding of
this problem for the two-agent case. However, the feasibility of product
distributions for more then two agents currently does not have a
characterization. We contribute to this line of literature by providing a
characterization of feasible _agent-symmetric_ product distributions for an
arbitrary number of agents. This characterization takes a simple form in the
case where the marginals of the product distributions are binary-supported.

#### Private Bayesian Persuasion

The Bayesian persuasion setting is quite well understood in the case where the
sender communicates with the receivers publicly. In particular the value of
the sender is characterized by the notion of concavification (Kamenica and
Gentzkow, 2011; Aumann et al., 1995). However, if the sender is allowed to
communicate privately with each receiver, the characterization of the value is
known only in special cases. A particular case that has been studied in
(Arieli and Babichenko, 2019; Babichenko and Barman, 2016; Dughmi and Xu,
2019) is the following. A marketer (sender) tries to persuade agents
(receivers) to adopt a product whose quality is either high or low. (Arieli
and Babichenko, 2019; Babichenko and Barman, 2016; Dughmi and Xu, 2019)
characterize optimal policies for the sender in special cases of sender’s
utility function such as supermodular or anonymous submodular utilities. We
provide an additional natural instance of the problem where the optimal
sender’s value (and the optimal policy) can be characterized. We consider the
case with a homogeneous population of agents (i.e., all agents have an
identical utility function) and the sender’s utility function is a monotonic
anonymous function (i.e., a function of a fraction of the adopters).
Interestingly, similarly to the standard Bayesian persuasion model ((Kamenica
and Gentzkow, 2011)), our characterization also relies on the notion of
concavification. However, the concavification in our case is applied in a
different domain.

We illustrate the key ideas of our main result via an example in Section 2.
Section 3 proceeds with the model and the main result. Section 4 studies the
implications of our results to maximal polarization of beliefs in a
population. In Section 5 we discuss the connections of our anonymous version
of feasibility with the previously studied (non-anonymous) version and present
an application to private private information. Finally, Section 6 demonstrates
an application in a Bayesian persuasion framework.

##  2 Warm-up Example

Let $\Omega=\\{0,1\\}$ be a binary state space with a common prior
$\mu=\frac{1}{2}$. We identify $\Delta(\Omega)$ with $[0,1]$, which indicates
the probability that is assigned to $\omega=1$. An information structure for
$n$ agents comprises $n$ measurable sets $\\{S_{i}\\}_{i=1}^{n}$ as well as a
probability distribution $G\in\Delta(\Omega\times S_{1}\times\cdots\times
S_{n})$. We note that almost surely any realization of $n$ signals
$(s_{1},\ldots,s_{n})$ defines $n$ posterior beliefs $G(\omega|s_{i})$ for
$i\in[n]$. In the example of the present section we assume that
$G(\omega|s_{i})\in\\{\frac{1}{4},\frac{3}{4}\\}$ almost surely for every
$i\in[n]$; namely, any information that an agent is exposed to results in a
posterior of either $\frac{1}{4}$ or $\frac{3}{4}$. The information structure
$G$ and the signal realization $s=(s_{1},\ldots,s_{n})$ induce a _distribution
of posteriors in the population_. Since we have assumed that there are only
two possible posteriors we can identify the distribution of posteriors in the
population with a fraction $H_{s}\in\\{0,\frac{1}{n},\frac{2}{n},...,1\\}$,
where $H_{s}=\frac{k}{n}$ indicates that $k$ agents in the population have the
high posterior $\frac{3}{4}$, and $n-k$ agents have the low posterior
$\frac{1}{4}$.

The information structure $G$ induces a distribution over distributions of
posteriors in the population
$\mathbf{P}_{G}\in\Delta(\\{0,\frac{1}{n},\frac{2}{n},...,1\\})$. We will call
such a $\mathbf{P}_{G}$ the _population’s empirical posteriors_ for short. The
population’s empirical posteriors anonymize the identity of the individuals
and considers only the empirical distributions of the belief distribution.

A population’s empirical posteriors $\mathbf{P}$ is called _feasible_ if
$\mathbf{P}=\mathbf{P}_{G}$ for some information structure of $n$ agents. A
characterization of the feasible population’s empirical posteriors is given in
the following proposition. We denote by $\delta_{c}$ the Dirac measure on an
element $c$.

######  Proposition 1.

A population’s empirical posteriors
$\mathbf{P}\in\Delta(\\{0,\frac{1}{n},\frac{2}{n},...,1\\})$ is feasible if
and only if $\mathbf{P}$ is a mean-preserving spread of the distribution
$\frac{1}{2}\delta_{\frac{1}{4}}+\frac{1}{2}\delta_{\frac{3}{4}}$.

To demonstrate the applicability of Proposition 1 consider, for instance, a
population of $n=9$ agents. The uniform distribution
$U(\\{0,\frac{1}{9},\frac{2}{9},...,1\\})$ is feasible because
$U(\\{0,\frac{1}{9},\frac{2}{9},...,1\\})$ is a mean-preserving spread of
$\frac{1}{2}\delta_{\frac{1}{4}}+\frac{1}{2}\delta_{\frac{3}{4}}=U(\\{\frac{1}{4},\frac{3}{4}\\})$;
that is, there exists an information structure such that with probability
$\frac{1}{10}$ the distribution of posteriors in the population will be $i$
agents with posterior $\frac{3}{4}$ and $9-i$ agents with posterior
$\frac{1}{4}$ for every $i=0,1,...,9$.

The uniform distribution $U(\\{\frac{1}{9},\frac{2}{9},...,\frac{8}{9}\\})$ is
infeasible because $U(\\{\frac{1}{9},\frac{2}{9},...,\frac{8}{9}\\})$ is not a
mean-preserving spread of
$\frac{1}{2}\delta_{\frac{1}{4}}+\frac{1}{2}\delta_{\frac{3}{4}}=U(\\{\frac{1}{4},\frac{3}{4}\\})$;
that is, there is no information structure such that with probability
$\frac{1}{8}$ the distribution of posteriors in the population will be $i$
agents with posterior $\frac{3}{4}$ and $9-i$ agents with posterior
$\frac{1}{4}$ for every $i=1,2,...,8$.

Proposition 1 is a special case of our main result, which characterizes the
feasibility of an arbitrary distribution. We find it useful to present the
proof of Proposition 1 because it contains the main proof ideas of the general
case (Theorem 1).

###### Proof of Proposition 1.

First, we show that every feasible distribution must be a mean-preserving
spread of $\frac{1}{2}\delta_{\frac{1}{4}}+\frac{1}{2}\delta_{\frac{3}{4}}$.
Conditional on state $\omega=1$, every agent $i\in[n]$ has a posterior of
$\frac{3}{4}$ with probability $\frac{3}{4}$. Therefore, conditional on state
$\omega=1$, the expected number of agents with the high posterior
$\frac{3}{4}$ is $\frac{3}{4}$. Similarly, conditional on state $\omega=0$,
the expected number of agents with the high posterior $\frac{3}{4}$ is
$\frac{1}{4}$. Denoting by $\mathbf{P}^{\omega}$ the population’s posteriors
conditional on $\omega$ we obtain a decomposition of $\mathbf{P}$ as
$\mathbf{P}=\frac{1}{2}\mathbf{P}^{0}+\frac{1}{2}\mathbf{P}^{1}$ where
$\mathbb{E}[\mathbf{P}^{0}]=\frac{1}{4}$ and
$\mathbb{E}[\mathbf{P}^{1}]=\frac{3}{4}$. We refer to $\mathbf{P}^{0}$ as the
spread of $\delta\frac{1}{4}$, we refer to $\mathbf{P}^{1}$ as the spread of
$\delta\frac{3}{4}$, and we get that $\mathbf{P}$ is a mean-preserving spread
of $\frac{1}{2}\delta_{\frac{1}{4}}+\frac{1}{2}\delta_{\frac{3}{4}}$.

Conversely, let
$\mathbf{P}=\frac{1}{2}\mathbf{Q}^{0}+\frac{1}{2}\mathbf{Q}^{1}$ where
$\mathbb{E}[\mathbf{Q}^{0}]=\frac{1}{4}$ and
$\mathbb{E}[\mathbf{Q}^{1}]=\frac{3}{4}$ be a mean-preserving spread of
$\frac{1}{2}\delta_{\frac{1}{4}}+\frac{1}{2}\delta_{\frac{3}{4}}$. Since
$\mathbf{P}\in\Delta(\\{0,\frac{1}{n},...,1\\})$ we also have
$\mathbf{Q}^{\omega}\in\Delta(\\{0,\frac{1}{n},...,1\\})$. Denote
$\alpha^{\omega}_{k}=\mathbf{Q}^{\omega}(\\{\frac{k}{n}\\})$. We define an
information structure $G$ as follows. In state $\omega$ we draw an integer $k$
with probability $\alpha^{\omega}_{k}$, then we choose a subset $A\subset[n]$
of size $|A|=k$ uniformly at random (among all possible subsets of size $k$),
and finally we send a high signal $h$ to all agents in the subset $A$ and a
low signal $l$ to the remaining agents. We need to show that the posterior of
an agent who has observed a signal $h$ is $\frac{3}{4}$ and the posterior of
an agent who has observed a signal $l$ is $\frac{1}{4}$. In state $\omega$ the
signal $h$ is obtained with probability
$\sum_{i=0}^{n}\alpha^{\omega}_{k}\frac{k}{n}=\mathbb{E}[\mathbf{Q}^{\omega}]$
because $\alpha^{\omega}_{k}$ is the probability that the subset is of size
$k$ and since the subset is random the probability that agent $i$ is included
is $\frac{k}{n}$. Therefore, the posterior of an agent who has observed signal
$h$ equals

$\frac{\mathbb{E}[\mathbf{Q}^{1}]}{\mathbb{E}[\mathbf{Q}^{0}]+\mathbb{E}[\mathbf{Q}^{1}]}=\frac{3}{4}.$

Similar considerations show that the posterior of an agent who has observed a
signal $l$ is $\frac{1}{4}$. ∎

##  3 Model and the Main Result

It turns out that the characterization of feasibility from Section 2 can be
naturally extended to a much more general case. Neither the restriction to
binary possible posteriors nor the restriction to a binary state space are
needed. We start by introducing the general model.

For a set $A$ we denote by $\Delta_{n}(A)\subset\Delta(A)$ the set of
empirical samples of size $n$, i.e.,
$\Delta_{n}(A)=\\{\sum_{k\in[n]}\frac{1}{n}\delta_{a^{k}}:a^{k}\in A\text{ for
}k\in[n]\\}$ (note that any $a\in A$ may appear in the sum more than once).

Let $\Omega=\\{1,\ldots,m\\}$ be a finite state space. An information
structure for $n$ agents comprises $n$ measurable sets $\\{S_{j}\\}_{j=1}^{n}$
as well as a probability distribution $G\in\Delta(\Omega\times
S_{1}\times\cdots\times S_{n})$. We denote by $\mathcal{G}_{n}$ the set of all
information structures of $n$ agents. The marginal distribution on $\Omega$
forms the prior; i.e., the prior $\mu\in\Delta(\Omega)$ satisfies
$\mu_{i}=G(\omega=i)$ for any state $\omega\in\Omega$. Any realization of $n$
signals $(s_{1},\ldots,s_{n})$ defines almost surely $n$ posterior beliefs
$G(\omega|s_{j})$ for $j\in[n]$. Hence $G$ and the signal realization
$s=(s_{1},\ldots,s_{n})$ induce a _distribution of posteriors in the
population_ $H_{s}\in\Delta_{n}(\Delta(\Omega))$ that is given by
$\sum_{j\in[n]}\frac{1}{n}\delta_{G(\omega|s_{j})}$.

The information structure $G$ induces a distribution over distributions of
posteriors in the population
$\mathbf{P}_{G}\in\Delta(\Delta_{n}(\Delta(\Omega)))$, which we call the
_population’s empirical posteriors_ for short.

In the case where $S_{j}$ is finite for every $j$, we let $G(s)$ be the
unconditional probability of a signal vector $s\in S_{1}\times\cdots\times
S_{n}$. In this case we can write

$\mathbf{P}_{G}=\sum_{s\in S_{1}\times\cdots\times S_{n}}G(s)\delta_{H_{s}},$
(1)

where $\delta_{H_{s}}\in\Delta_{n}(\Delta(\Omega))$ is a Dirac measure on
$H_{s}$.

######  Definition 1.

A distribution $\mathbf{P}\in\Delta(\Delta_{n}(\Delta(\Omega)))$ is called
_feasible_ for a prior $\mu\in\Delta(\Omega)$ if $\mathbf{P}=\mathbf{P}_{G}$
for some information structure $G\in\mathcal{G}_{n}$. We denote by
$\mathcal{P}_{n}(\mu)\subseteq\Delta(\Delta_{n}(\Delta(\Omega)))$ the set of
all feasible distributions for a prior $\mu\in\Delta(\Omega)$ and $n$ agents.

Note that if the signal sets $\\{S_{j}\\}_{j=1}^{n}$ are finite,
$\mathbf{P}_{G}$ assigns probability one to measures
$H_{s}\in\Delta_{n}(\Delta(\Omega))$ that are supported on some finite set
$\\{y_{1},\ldots,y_{m}\\}\subseteq\Delta(\Omega)$ (where
$m\leq\sum_{j}|S_{j}|$). In general, we say that $\mathbf{P}\in\mathcal{P}$ is
_finitely supported_ if there exists a finite set $Y\subseteq\Delta(\Omega)$
of posteriors such that $\mathbf{P}$ assigns probability one to the set of
measures $\Delta(\Delta_{n}(Y))$.

###  3.1 Main Result

For any $\mathbf{P}\in\Delta(\Delta_{n}(\Delta(\Omega)))$ we let
$\tilde{\mathbf{P}}\in\Delta(\Delta(\Omega))$ be the _expected_ measure that
$\mathbf{P}$ induces on $\Delta(\Omega)$. That is, for every Borel subset
$B\subseteq\Delta(\Omega)$,

$\tilde{\mathbf{P}}(B)=\int_{\Delta(\Omega)}H(B)\mathrm{d}\mathbf{P}(H).$

In particular, if $\mathbf{P}$ is finitely supported we can write
$\mathbf{P}=\sum_{j=1}^{k}a_{j}H_{j}$ (where
$H_{j}\in\Delta_{n}(\Delta(\Omega))$, $a_{j}\geq 0$ and $\sum_{j}a_{j}=1$);
then

$\tilde{\mathbf{P}}(B)=\sum_{j=1}^{k}a_{j}H_{j}(B).$

We define $m$ other distributions
$\tilde{\mathbf{P}}^{\omega}\in\Delta(\Delta(\Omega))$ for $\omega\in\Omega$
as follows:

$\mathrm{d}\tilde{\mathbf{P}}^{\omega}(x)=\frac{1}{\mu_{\omega}}x_{\omega}\mathrm{d}\tilde{\mathbf{P}}(x).$

If $\mathbf{P}$ is finitely supported we simply have
$\tilde{\mathbf{P}}^{\omega}(x)=\frac{x_{\omega}\tilde{\mathbf{P}}(x)}{\mu_{\omega}}$.

Finally, let $\mathbf{P}^{\prime}\in\Delta((\Delta(\Delta(\Omega)))$ be the
the measure that is defined as follows:

$\mathbf{P}^{\prime}=\sum_{\omega\in[m]}\mu_{\omega}\delta_{\tilde{\mathbf{P}}^{\omega}},$

where $\delta_{\tilde{\mathbf{P}}^{\omega}}\in\Delta((\Delta(\Delta(\Omega)))$
are Dirac measures. Our main result states the following:

######  Theorem 1.

A distribution $\mathbf{P}\in\Delta(\Delta_{n}(\Delta(\Omega)))$ is feasible
for $n$ agents and a prior $\mu\in\Delta(\Omega)$ if and only if $\mathbf{P}$
is a mean-preserving spread of $\mathbf{P}^{\prime}$.

The proof follows a similar logic to the proof of Proposition 1; however, the
general case differs from the binary-posterior case in the following respect.
A key observation in the proof of Theorem 1, as well as the proof of
Proposition 1, is that the expected distribution of posteriors conditional on
a state $\omega$ can be identified. In the binary-posterior case, conditional
on any state $\omega$, _every individual_ in the population has on average the
high posterior ($\frac{3}{4}$ in Proposition 1) with a fixed probability that
is determined by the two possible posteriors. In the case of more than two
posteriors we cannot deduce such a strong statement about each individual in
the population.333Consider, for instance, a binary-state space with three
possible posteriors $0<a<b<\mu<c<0$. Agent 1 has only two possible posteriors
$a$ and $c$. Agent 2 has only two possible posteriors $b$ and $c$. Obviously,
agents 1 and 2 will have different distributions of posteriors conditional on
any state. Nevertheless, it turns out that the _average_ distribution of
posteriors over the population conditional on a state can be identified. To do
so, we simply do the exercise of deducing the average _conditional_
($\tilde{\mathbf{P}}^{\omega}$) distribution of posteriors from the average
_ex-ante_ distribution of posteriors ($\tilde{\mathbf{P}}$). The validity of
the above deduction follows from the linearity of expectation and the
linearity of the operator that maps an ex-ante distribution of posteriors to a
conditional distribution of posteriors. Once we show that conditional on a
state $\omega\in\Omega$ the average distribution of posteriors must be
$\tilde{\mathbf{P}}^{\omega}$, the necessity for the mean-preserving spread
(of $\mathbf{P}^{\prime}$) condition immediately follows. To prove sufficiency
we construct an information structure with the population’s empirical
posteriors $\mathbf{P}$ using similar arguments to those of Proposition 1.

###  3.2 The binary-state binary-posterior case

In the case where the state space is binary and the posteriors get only two
values $a,b$ where $0\leq a<\mu<b\leq 1$, the distribution $\mathbf{P}$ is an
element of $\Delta(\Delta_{n}(\\{a,b\\}))$ or simply an element of
$\Delta(\\{0,\frac{1}{n},...,1\\})$ that indicates the weight of $b$ in the
distribution. In the latter representation of the population’s empirical
posteriors the terms $\tilde{\mathbf{P}}^{\omega}$ for $\omega=0,1$ represent
the expected fraction of agents that have the high posterior conditional on a
state. This expectation equals the probability of an agent having the high
posterior in the _single-agent_ case. Simple calculations show that
$\tilde{\mathbf{P}}_{1}=\frac{(\mu-a)b}{(b-a)\mu}$ and
$\tilde{\mathbf{P}}_{0}=\frac{(\mu-a)(1-b)}{(b-a)(1-\mu)}$.

Thus, we deduce the following corollary:

######  Corollary 1.

A population’s empirical posteriors
$\mathbf{P}\in\Delta(\\{0,\frac{1}{n},\frac{2}{n},...,1\\})$ is feasible for
the prior $\mu$ and the posteriors $a<\mu<b$ if and only if $\mathbf{P}$ is a
mean-preserving spread of

$\mu\delta_{\frac{(\mu-a)b}{(b-a)\mu}}+(1-\mu)\delta_{\frac{(\mu-a)(1-b)}{(b-a)(1-\mu)}}.$

The fact that $\mu_{P}$ and $\tilde{\mathbf{P}}^{\omega}$ can be deduced
directly from the values of the posteriors $a$ and $b$ is a special feature of
the binary-posterior case.444Those properties hold for the case of binary
posteriors and an arbitrary state space. In the general case, the terms
$\mu_{P}$ and $\tilde{\mathbf{P}}^{\omega}$ are not determined by the support
of the posteriors but rather are determined by the distribution $\mathbf{P}$
as described above in the definitions of these terms.

We also notice that for single-dimensional distributions it is easy to
determine whether the distribution is a mean-preserving spread of a given
binary-support distribution as follows. Let
$\mathbf{P}\in\Delta(\\{0,\frac{1}{n},\frac{2}{n},...,1\\})$ be a distribution
that assigns a probability of $p_{i}$ to $\frac{i}{n}$ and let
$\alpha\delta_{a}+(1-\alpha)\delta_{b}$ be a binary-support distribution with
$a<b$. We consider the _$\alpha$ -quantile distribution of $\mathbf{P}$_ that
is defined as follows. We let $k$ be the index such that
$\sum_{i<k}p_{i}\leq\alpha<\sum_{i\leq k}p_{i}$. We let
$q_{k}=\alpha-\sum_{i<k}p_{i}$. The $\alpha$-quantile distribution of
$\mathbf{P}$ is denoted by $\mathbf{P}|_{\alpha}$ and is defined to be the
distribution that assigns a probability of $\frac{p_{i}}{\alpha}$ to
$\frac{i}{n}$ for every $i<k$ and assigns a probability of
$\frac{q_{k}}{\alpha}$ to $\frac{k}{n}$. Simply speaking,
$\mathbf{P}|_{\alpha}$ is the distribution of $\mathbf{P}$ conditional on the
event that the realization belongs to the lower $\alpha$-quantile of
$\mathbf{P}$. The following lemma characterizes the mean-preserving spread
property.

######  Lemma 1.

A distribution $\mathbf{P}\in\Delta(\\{0,\frac{1}{n},\frac{2}{n},...,1\\})$ is
a mean-preserving spread of $\mathbf{B}=\alpha\delta_{a}+(1-\alpha)\delta_{b}$
if and only if $\mathbb{E}[\mathbf{P}]=\mathbb{E}[\mathbf{B}]$ and
$\mathbb{E}[\mathbf{P}|_{\alpha}]\leq a$.

###### Proof.

We denote by $F_{\mathbf{P}},F_{\mathbf{B}}$ the CDFs for $\mathbf{P}$ and
$\mathbf{B}$. We have that $\mathbf{P}$ is a mean preserving spread of
$\mathbf{B}$ if and only if $\mathbb{E}[\mathbf{P}]=\mathbb{E}[\mathbf{B}]$
and for every $t\in[0,1]$ we have
$\int_{0}^{t}F_{\mathbf{P}}(x)dx\geq\int_{0}^{t}F_{\mathbf{B}}(x)dx$.
Moreover, it is well known that it suffices to check this inequality only for
points $t$ such that $F_{\mathbf{P}}(t)$ crosses $F_{\mathbf{B}}(t)$. In our
case $F_{\mathbf{B}}$ is piecewise constant with values $0,\alpha,1$ in the
corresponding segments $[0,a),[a,b),[b,1]$. Therefore, in our case it is
sufficient to check the inequality at the single point $t=\frac{k}{n}$.

Note that the CDF of $\mathbf{P}|_{\alpha}$ is
$F_{\mathbf{P}|_{\alpha}}(x)=\alpha^{-1}F_{\mathbf{P}}(x)$. Hence we have

$\int_{0}^{\frac{k}{n}}F_{\mathbf{P}}(x)dx=\alpha\int_{0}^{\frac{k}{n}}F_{\mathbf{P}|_{\alpha}}(x)dx=\alpha\left(\frac{k}{n}-\int_{0}^{\frac{k}{n}}(1-F_{\mathbf{P}|_{\alpha}}(x))dx\right)=\alpha\left(\frac{k}{n}-\mathbb{E}[\mathbf{P}|_{\alpha}]\right).$

We also have $\int_{0}^{\frac{k}{n}}F_{\mathbf{B}}(x)dx=\alpha(\frac{k}{n}-a)$
and therefore
$\int_{0}^{\frac{k}{n}}F_{\mathbf{P}}(x)dx\geq\int_{0}^{\frac{k}{n}}F_{\mathbf{B}}(x)dx$
if and only if $\mathbb{E}[\mathbf{P}|_{\alpha}]\leq a$. ∎

###  3.3 Proof of Theorem 1

Let $\mathbf{P}\in\Delta(\Delta_{n}(\Delta(\Omega))$ be a mean-preserving
spread of $\mathbf{P}^{\prime}$. Namely, there exist $m$ distributions
$\mathbf{Q}^{\omega}\in\Delta(\Delta(\Omega))$ with expectation
$\tilde{\mathbf{P}}^{\omega}$ such that

$\mathbf{P}=\sum_{\omega\in\Omega}\mu_{\omega}\mathbf{Q}^{\omega}.$

We now define a signaling structure $G$ that implements $\mathbf{P}$ as
follows. Let $S_{j}=\Delta(\Omega)$ for every agent $j$ and let
$G(\omega)=\mu_{i}$ for every state $i\in\Omega$. We define $G(s|\omega)$ by
coupling this measure with $\mathbf{Q}^{\omega}$ as follows. We draw
$H\in\Delta_{n}(\Delta(\Omega))$ according to $\mathbf{Q}^{\omega}$. Every
such $H$ has the form $H=\sum_{k\in[n]}\frac{1}{n}\delta_{x^{k}}$ where
$x^{k}\in\Delta(\Omega)$. For every realized $H$ we draw a permutation
$\pi:[n]\to[n]$ uniformly at random across all permutations and send signal
$x^{\pi(j)}$ to every agent $j$. Thus, overall the probability that agent $j$
observes the signal $x^{k}$ is $\frac{1}{n}$ for every $j,k\in[n]$.

We claim that $\mathbf{P}_{G}=\mathbf{P}$. To see this, by the construction,
it is enough to show that $G(\omega|s_{j}=x)=x_{\omega}$ for every agent
$j\in[n]$ and state $\omega\in\Omega$. We note first that since $H$ is drawn
according to $\mathbf{Q}^{\omega}$ and since the expectation of
$\mathbf{Q}^{\omega}$ is $\tilde{\mathbf{P}}^{\omega}$ we have that
$G(s_{j}\in B|\omega)=\tilde{\mathbf{P}}^{\omega}(B)$ for every Borel
measurable set $B\subset\Delta(\Omega)$. Therefore, by Bayes’s rule, for every
$\omega\in\Omega,$ we have that

$\displaystyle
G(\omega|s_{j}=x)=\frac{\mu_{\omega}\frac{\mathrm{d}\tilde{\mathbf{P}}^{\omega}}{\mathrm{d}\tilde{\mathbf{P}}}(x)}{\sum_{\omega^{\prime}\in\Omega}\mu_{\omega^{\prime}}\frac{\mathrm{d}\tilde{\mathbf{P}}^{\omega^{\prime}}}{\mathrm{d}\tilde{\mathbf{P}}}(x)}$
(2)

with probability 1.

We recall that
$\mathrm{d}\tilde{\mathbf{P}}^{\omega}(x)=\frac{x_{\omega}}{\mu_{\omega}}\mathrm{d}\tilde{\mathbf{P}}(x).$
Therefore,
$\frac{\mu_{\omega^{\prime}}\mathrm{d}\tilde{\mathbf{P}}_{\omega^{\prime}}(x)}{\mathrm{d}\tilde{\mathbf{P}}(x)}=x_{\omega^{\prime}}$
for every state $\omega^{\prime}\in\Omega$. Hence, Equation (2) implies that
$G(\omega|s_{j}=x)=\frac{x_{\omega}}{\sum_{\omega^{\prime}\in\Omega}x_{\omega^{\prime}}}=x_{\omega}$
as desired.

We now turn to the converse direction; namely, we show that every feasible
$\mathbf{P}\in\mathcal{P}_{n}$ is a mean-preserving spread of
$\mathbf{P}^{\prime}$. For clarity of exposition we prove this statement only
for the case where $\mathbf{P}$ is finitely supported. The extension of the
proof to arbitrary distributions $\mathbf{P}$ is relegated to the Appendix A
and essentially follows from the fact that any distribution $\mathbf{P}$ can
be approximated (in the weak* topology) by finitely supported ones.

We assume that $\mathbf{P}=\mathbf{P}_{G}$ is induced by an information
structure $G\in\Delta(\Omega\times S_{1}\times\cdots\times S_{n})$ with finite
signal sets $S_{i}$ for every $i\in[n]$ (the finite support assumption). In
addition to the distribution $\mathbf{P}$, we next define a few other
auxiliary measures. First, for every state $\omega\in\Omega$ let
$\mathbf{P}^{\omega}_{G}\in\Delta(\Delta_{n}(\Delta(\Omega)))$ be the
conditional distribution of $H_{s}$ conditional on state $\omega$. That is,
similar to Equation (1),

$\mathbf{P}^{\omega}_{G}=\sum_{s\in S_{1}\times\cdots\times
S_{n}}G(s|\omega)\delta_{H_{s}}.$

It follows that

$\mathbf{P}=\sum_{\omega\in\Omega}\mu_{\omega}\mathbf{P}^{\omega}_{G}.$ (3)

For every $\omega\in\Omega$, let $\tilde{\mathbf{P}}^{\omega}_{G}$ be the
expectation of $\mathbf{P}^{\omega}_{G}$. By definition, the measure
$\mathbf{P}^{\omega}_{G}$ is a mean-preserving spread of
$\delta_{\tilde{\mathbf{P}}^{\omega}_{G}}$. Therefore, in particular, by
Equation (3) $\mathbf{P}$ is a mean-preserving spread of the measure
$\sum_{\omega\in\Omega}\mu_{\omega}\delta_{\tilde{\mathbf{P}}^{\omega}_{G}}.$
In order to complete the proof we need to show that
$\tilde{\mathbf{P}}_{\omega}$ as defined above equals
$\tilde{\mathbf{P}}^{\omega}_{G}$ for every state $\omega\in\Omega$.

For each signal $s_{j}\in S_{j}$ let $G(\ |s_{j})\in\Delta(\Omega)$ be the
conditional probability on $\Omega$ given the signal $s_{j}$. Let
$H_{s_{j}}=\delta_{G(\ |s_{j})}.$ Using this notation, for any vector of
realized signals $s=(s_{1},\ldots,s_{n})$, one can write the measure $H_{s}$
as follows:

$H_{s}=\frac{1}{n}\sum_{j=1}^{n}H_{s_{j}}.$

Therefore, we have

$\mathbf{P}=\sum_{s\in S_{1}\times\cdots\times
S_{n}}G(s)\frac{1}{n}\sum_{j=1}^{n}H_{s_{j}}.$

Hence, by definition of $\tilde{\mathbf{P}}$, for every $x\in\Delta(\Omega)$
we have

$\tilde{\mathbf{P}}(\\{x\\})=\sum_{s\in S_{1}\times\cdots\times
S_{n}}G(s)\frac{1}{n}\sum_{j=1}^{n}H_{s_{j}}(x)=\frac{1}{n}\sum_{j=1}^{n}\sum_{s_{j}\in
S_{j},G(\ |s_{j})=x}G(s_{j}).$

Similarly, for any state $\omega\in\Omega$,

$\tilde{\mathbf{P}}^{\omega}_{G}(\\{x\\})=\sum_{s\in S_{1}\times\cdots\times
S_{n}}G(s|\omega)\frac{1}{n}\sum_{j=1}^{n}H_{s_{j}}(x)=\frac{1}{n}\sum_{j=1}^{n}\sum_{s_{j}\in
S_{j},G(\ |s_{j})=x}G(s_{j}|\omega).$

It follows from Bayes’s rule that if $G(\omega|s_{j})=x_{\omega}$, then
$G(s_{j}|\omega)=x_{\omega}\frac{G(s_{j})}{\mu_{\omega}}.$ Hence
$\tilde{\mathbf{P}}^{\omega}_{G}(\\{x\\})=\frac{x_{\omega}}{\mu_{\omega}}\tilde{\mathbf{P}}(\\{x\\})$
for every $x\in\Delta(\Omega)$. Thus
$d\tilde{\mathbf{P}}_{G,\omega}(\\{x\\})=\frac{x_{\omega}}{\mu_{\omega}}d\tilde{\mathbf{P}}(\\{x\\}).$
Therefore, for every $\omega\in\Omega$ it holds that
$\tilde{\mathbf{P}}^{\omega}_{G}=\tilde{\mathbf{P}}^{\omega}$, as desired.

##  4 Polarization

The question of how polarized the posteriors of Bayesian agents might be has
been studied in (Dubins and Pitman, 1980), (Burdzy and Pitman, 2020), and
(Arieli et al., 2021). These papers focuse on the case of two receivers. In
(Dubins and Pitman, 1980) polarization is measured by the absolute distance
$|x_{1}-x_{2}|$ and more generally in (Arieli et al., 2021) polarization is
measured by the function $|x_{1}-x_{2}|^{\alpha}$ for some $\alpha>0$, where
$x_{i}\in[0,1]$ denotes the posteriors of agent $i=1,2$. The case of
$\alpha=2$ is a particularly convenient case to study from a technical
perspective. It is also a natural case to study for the following descriptive
reason: it has a natural extension to an arbitrary number of agents as we show
below.

Given a distribution of posteriors in the population $H\in\Delta_{n}([0,1])$,
we can measure polarization via $\operatorname{Var}(H)$. Note that in the
special case of $n=2$ with posteriors $x_{1},x_{2}$ the variance is
$\frac{(x_{1}-x_{2})^{2}}{4}$, which (up to a factor of 4) is exactly the case
of $\alpha=2$.

The results of (Dubins and Pitman, 1980) for $\alpha=1$ and of (Arieli et al.,
2020) for $\alpha\leq 2$ show that for the case of two agents the information
structure that maximizes polarization is the one that reveals the state to one
agent and provides no information to the other agent.555It remains an open
problem to specify the information structure that maximizes polarization for
$\alpha>2$, even for the case of two agents. This information structure
achieves an expected variance of $\frac{\mu(1-\mu)}{4}$ for information
structures with prior $\mu\in[0,1]$. We show that the same result is true for
every even number of players: the information structure that maximizes
polarization is the one that reveals the state to half of the agents and
provides no information to the other half of the agents.

The polarization of $G$ is the expected variance of the distribution of
posteriors in the population. That is, for any $G\in\mathcal{G}$ let
$V_{G}=\int_{\Delta([0,1])}\operatorname{Var}(H)\mathrm{d}G(H)$.

######  Proposition 2.

Let $\mu\in[0,1]$ be the prior. For every even $n$, the information structure
that maximizes the polarization of $n$ agents is the one that reveals the
state to half of the agents and reveals no information to the other half. The
expected polarization equals $\frac{\mu(1-\mu)}{4}$.

For every odd $n$, the maximal polarization is bounded between

$\left(1-\frac{1}{n^{2}}\right)\frac{\mu(1-\mu)}{4}\leq\max_{G\in\mathcal{G}_{n}}V_{G}\leq\frac{\mu(1-\mu)}{4}.$

For an odd number of agents, Proposition 2 provides asymptotically tight
bounds when $n\to\infty$. It remains an open problem to provide the precise
characterization of polarization for an odd number of agents. Interestingly,
the idea of sending full information or no information to each one of the
agents is no longer optimal for an odd number of agents as demonstrated in the
following example.

######  Example 1.

For $n=3$ and prior $\mu=\frac{1}{2}$, consider the information structure
where the state is revealed to agent $3$, and agents $i=1,2$ receive the
signals $s^{i}_{1},s^{i}_{2}$ according to the following distribution
conditional on a state:

$\omega=0$ |  | $\omega=1$
---|---|---
| $s_{1}^{2}$ | $s_{2}^{2}$ |  |  | $s_{1}^{2}$ | $s_{2}^{1}$
$s_{1}^{1}$ | 1/3 | 1/3 |  | $s_{1}^{1}$ | 0 | 1/3
$s_{2}^{1}$ | 1/3 | 0 |  | $s_{2}^{1}$ | 1/3 | 1/3

Agent $i$’s posterior after observing $s^{i}_{1}$ is $1/3$; her posterior
after observing $s^{i}_{2}$ is $2/3$. The resulting distribution over
posterior profiles is uniform over the profiles:

$\\{(\frac{1}{3},\frac{1}{3},0),(\frac{1}{3},\frac{2}{3},0),(\frac{2}{3},\frac{1}{3},0),(\frac{1}{3},\frac{2}{3},1),(\frac{2}{3},\frac{1}{3},1),(\frac{2}{3},\frac{2}{3},1)\\}.$

The expected variance is
$\frac{2}{6}\frac{2}{81}+\frac{4}{6}\frac{2}{27}=\frac{14}{243}$, which is
above $\frac{1}{18}$, where $\frac{1}{18}$ is the maximal expected variance
that can be obtained by sending to every player either full information or no
information about the state.

###### Proof of Proposition 2.

First we show that $V_{G}\leq\frac{\mu(1-\mu)}{4}$ for every number of agents
$n$ (odd or even). This statement can be equivalently stated with respect to a
feasible population’s empirical posteriors. We denote
$V_{\mathbf{P}}=\int_{\Delta([0,1])}\operatorname{Var}(H)\mathrm{d}\mathbf{P}(H)$
and we trivially have
$\max_{G\in\mathcal{G}_{n}}V_{G}=\max_{\mathbf{P}\in\mathcal{P}_{n}(\mu)}V_{\mathbf{P}}$.

Let $\mathbf{P}\in\mathcal{P}_{n}(\mu).$ By Theorem 1 we can decompose
$\mathbf{P}=\mu_{\mathbf{P}}\mathbf{P}_{1}+(1-\mu_{\mathbf{P}})\mathbf{P}_{0}$
where $\mathbb{E}[\mathbf{P}^{\omega}]=\tilde{\mathbf{P}}^{\omega}$.
Therefore,
$V_{\mathbf{P}}=\mu_{\mathbf{P}}V_{\mathbf{P}_{1}}+(1-\mu_{\mathbf{P}})V_{\mathbf{P}_{0}}.$
Let
$\mathbf{P}^{\prime}=\mu_{\mathbf{P}}\delta_{\tilde{\mathbf{P}}_{1}}+(1-\mu_{\mathbf{P}})\delta_{\tilde{\mathbf{P}}_{0}}$.
Note that since $\mathbf{P}\in\mathcal{P}(\mu)$ it also holds that
$\mathbf{P}^{\prime}\in\mathcal{P}(\mu).$ We claim that $V_{\mathbf{P}}\leq
V_{\mathbf{P}^{\prime}}$. To see this, note that for any $\omega=0,1$,

$\displaystyle
V_{\mathbf{P}^{\omega}}=\int_{\Delta([0,1])}\operatorname{Var}(H)\mathrm{d}\mathbf{P}^{\omega}(H)=\int_{\Delta([0,1])}\mathbb{E}_{H}[x^{2}]-(\mathbb{E}_{H}[x])^{2}\mathrm{d}\mathbf{P}^{\omega}(H)=$
(4)
$\displaystyle\int_{\Delta([0,1])}\mathbb{E}_{H}[x^{2}]\mathrm{d}\mathbf{P}^{\omega}(H)-\int_{\Delta([0,1])}(\mathbb{E}_{H}[x])^{2}\mathrm{d}\mathbf{P}^{\omega}(H)=$
$\displaystyle\mathbb{E}_{\tilde{\mathbf{P}}^{\omega}}[x^{2}]-\int_{\Delta([0,1])}(\mathbb{E}_{H}[x])^{2}\mathrm{d}\mathbf{P}(H)\leq$
(5)
$\displaystyle\mathbb{E}_{\tilde{\mathbf{P}}^{\omega}}[x^{2}]-(\mathbb{E}_{\tilde{\mathbf{P}}^{\omega}}[x]))^{2}=\operatorname{Var}(\delta_{\tilde{\mathbf{P}}^{\omega}})$
(6)

Note that equation (4) follows from
$\operatorname{Var}(H)=\mathbb{E}_{H}[x^{2}]-(\mathbb{E}_{H}[x])^{2}$.
Equation (5) follows from the fact that the operation
$\mathbb{E}_{H}(x^{2}):\Delta(\Delta([0,1]))\to\mathbb{R}$ is a linear
operator and so for every $\mathbf{Q}\in\Delta(\Delta([0,1]))$ we have that
$\int_{\Delta([0,1])}\mathbb{E}_{H}[x^{2}]\mathrm{d}\mathbf{Q}(H)=\mathbb{E}_{\tilde{\mathbf{Q}}}[x^{2}]$.
The inequality in (6) follows from Jensen’s inequality. Since
$V_{\mathbf{P}^{\prime}}=\mu_{\mathbf{P}}V_{\delta_{\tilde{\mathbf{P}}_{1}}}+(1-\mu_{\mathbf{P}})V_{\delta_{\mathbf{P}_{0}}}$
it follows that $V_{\mathbf{P}^{\prime}}\geq V_{\mathbf{P}}.$

Note that $\tilde{\mathbf{P}}_{1},\tilde{\mathbf{P}}_{0}$ satisfy

$\mathrm{d}\tilde{\mathbf{P}}_{1}=\frac{x}{\mu}\mathrm{d}\tilde{\mathbf{P}}\text{
and }\mathrm{d}\tilde{\mathbf{P}}_{0}=\frac{1-x}{1-\mu}d\tilde{\mathbf{P}},$

for some $\tilde{\mathbf{P}}\in\Delta([0,1])$ with expectation $\mu$, and
hence the variances can be written as

$\displaystyle\operatorname{Var}(\tilde{\mathbf{P}}_{1})$
$\displaystyle=\int_{[0,1]}\frac{x^{3}}{\mu}\mathrm{d}\tilde{\mathbf{P}}(x)-\left(\int_{[0,1]}\frac{x^{2}}{\mu}\mathrm{d}\tilde{\mathbf{P}}(x)\right)^{2}$
$\displaystyle\operatorname{Var}(\tilde{\mathbf{P}}_{0})$
$\displaystyle=\int_{[0,1]}\frac{x^{2}(1-x)}{1-\mu}\mathrm{d}\tilde{\mathbf{P}}(x)-\left(\int_{[0,1]}\frac{x(1-x)}{1-\mu}\mathrm{d}\tilde{\mathbf{P}}(x)\right)^{2}$

Therefore,

$\displaystyle
V_{\mathbf{P}}\leq\mu\operatorname{Var}(\tilde{\mathbf{P}}_{1})+(1-\mu)\operatorname{Var}(\tilde{\mathbf{P}}_{0})=$
$\displaystyle\int_{[0,1]}x^{3}+x^{2}(1-x)\mathrm{d}\tilde{\mathbf{P}}(x)-\frac{1}{\mu}\left(\int_{[0,1]}x^{2}\mathrm{d}\tilde{\mathbf{P}}(x)\right)^{2}-\frac{1}{1-\mu}\left(\int_{[0,1]}x(1-x)\mathrm{d}\tilde{\mathbf{P}}(x)\right)^{2}=$
$\displaystyle\int_{[0,1]}x^{2}\mathrm{d}\tilde{\mathbf{P}}(x)-\frac{1}{\mu}\left(\int_{[0,1]}x^{2}\mathrm{d}\tilde{\mathbf{P}}(x)\right)^{2}-\frac{1}{1-\mu}\left(\mu-\int_{[0,1]}x^{2}\mathrm{d}\tilde{\mathbf{P}}(x)\right)^{2}=$
$\displaystyle(\frac{1+\mu}{1-\mu})\int_{[0,1]}x^{2}\mathrm{d}\tilde{\mathbf{P}}(x)-\frac{1}{\mu(1-\mu)}\left(\int_{[0,1]}x^{2}\mathrm{d}\tilde{\mathbf{P}}(x)\right)^{2}-\frac{\mu^{2}}{1-\mu}.$

Denote $A=\int_{[0,1]}x^{2}\mathrm{d}\tilde{\mathbf{P}}(x)$ and we obtain
$V_{\mathbf{P}}\leq(\frac{1+\mu}{1-\mu})A-\frac{1}{\mu(1-\mu)}A^{2}-\frac{\mu^{2}}{1-\mu}$.
This is a degree 2 polynomial whose global maximum is $\frac{\mu(1-\mu)}{4}$
that is obtained at $A=\frac{\mu(1+\mu)}{2}$.

To show that $\frac{\mu(1-\mu)}{4}$ is achievable for an even number of
players, we calculate the expected variance for the information structure that
reveals the state to $\frac{n}{2}$ of the players and reveals nothing to the
remaining $\frac{n}{2}$ players and get precisely $\frac{\mu(1-\mu)}{4}$.

To show that $(1-\frac{1}{n^{2}})\frac{\mu(1-\mu)}{4}$ is achievable for an
odd number of players, we calculate the expected variance for the information
structure that reveals the state to $\frac{n+1}{2}$ of the players and reveals
no information to the remaining $\frac{n-1}{2}$ players and get the desired
expression. ∎

###  4.1 Multiple States

For an arbitrary finite state space $\Omega$, one might measure polarization
of a distribution of posteriors in the population via
$\operatorname{Pol}(H)=\mathbb{E}_{x\sim H}[\|x-\mathbb{E}[H]\|_{2}^{2}]$,
where $x$ and $\mathbb{E}[H]$ are $|\Omega|$-dimensional vectors of beliefs in
the simplex. Note that
$\operatorname{Pol}(H)=\sum_{\omega\in\Omega}\operatorname{Var}(X_{\omega})$,
where $X_{\omega}$ denotes the random variable of the posterior probability
that is assigned to state $\omega\in\Omega$. Therefore, for an arbitrary state
space we obtain the following corollary.

######  Corollary 2.

For every prior $\mu\in\Delta(\Omega)$ and every even $n$ the information
structure that maximizes the expected polarization (i.e., $\mathbb{E}_{H\sim
P}\operatorname{Pol}(H)$) of $n$ agents is the one that reveals the state to
half of the agents and reveals no information to the other half.

The corollary simply follows from the fact that
$\operatorname{Pol}(H)=\sum_{\omega\in\Omega}\operatorname{Var}(X_{\omega})$
and the described information structure simultaneously maximizes
$\operatorname{Var}(X_{\omega})$ for all $\omega\in\Omega$.

##  5 Vectorial Feasibility

In this section we connect our results on the feasibility of distributions
over _empirical distributions of beliefs_ with the notion of the feasibility
of distributions over _vectors of beliefs_. As mentioned in the Introduction,
the latter notion of feasibility has been previously studied in (Dawid et al.,
1995; Arieli et al., 2021; He et al., 2021). To distinguish between the two
notions we refer to the latter notion as _vectorial feasibility_.

The model is similar to the one presented in Section 3. Given an information
structure $I\in\Delta(\Omega\times S_{1}\times...\times S_{n})$, every
realization of signals $(s_{i})_{i\in[n]}$ induces a vector of beliefs
$(x_{i}(s_{i}))_{i\in[n]}$. Hence, every information structure induces a
distribution over vectors of beliefs
$\mathbf{V}_{I}\in\Delta((\Delta(\Omega)^{n})$.

######  Definition 2.

A distribution $\mathbf{V}\in\Delta((\Delta(\Omega)^{n})$ is _vector feasible_
if there exists $I$ such that $\mathbf{V}=\mathbf{V}_{I}$.

Every distribution $\mathbf{V}\in\Delta((\Delta(\Omega)^{n})$ induces a
distribution $\mathbf{P}(\mathbf{V})\in\Delta(\Delta_{n}(\Delta(\Omega)))$
simply because every vector of beliefs induces an empirical distribution of
beliefs in the population. We observe that the feasibility of
$\mathbf{P}(\mathbf{V})$ is a necessary but insufficient condition for the
feasibility of $\mathbf{V}$. Indeed this condition is necessary, because if
there exists no $I$ that will induce the distribution of empirical
distributions $\mathbf{P}(\mathbf{V})$ then $\mathbf{V}$ cannot be implemented
by any $I$. It is also not hard to see that the feasibility of
$\mathbf{P}(\mathbf{V})$ is an insufficient condition for the feasibility
of666To construct a counterexample consider $|\Omega|=2$, $n=2$ and consider
any _feasible_ vector distribution $\mathbf{V}^{\prime}$ with
$\mathbf{V}^{\prime}(a,b)>0$ for some $a\neq b$. Let $\mathbf{V}$ be a
distribution that is obtained from $\mathbf{V}^{\prime}$ by moving some
positive mass from $(a,b)$ to $(b,a)$. The distribution $\mathbf{V}$ becomes
vector infeasible because it violates the martingale condition for both
agents. However $\mathbf{P}(\mathbf{V})=\mathbf{P}(\mathbf{V}^{\prime})$
remains feasible. Another counterexample that does not violate the martingale
condition for any player is the following. Let $|\Omega|=2$ and $n=3$. The
distribution
$\mathbf{V}=\frac{1}{2}\delta_{(\frac{1}{3},\frac{1}{3},\frac{2}{3})}+\frac{1}{2}\delta_{(\frac{2}{3},\frac{2}{3},\frac{1}{3})}$
is vector infeasible because agents 1 and 2 are agreeing to disagree with
agent 3 in both points of the support of this distribution. However,
$\mathbf{P}(\mathbf{V})$, which is the distribution where with probability
$\frac{1}{2}$ two of the three agents have a posterior of $\frac{j}{3}$
(without specifying their identity) and the remaining agent has a posterior of
$\frac{3-j}{3}$ for $j=1,2$, is feasible by Corollary 1.  $\mathbf{V}$.

Therefore, the tools developed in this paper might be useful for determining
the vectorial infeasibility of $\mathbf{V}$ by checking our condition for the
feasibility of $\mathbf{P}(\mathbf{V})$. The following subsection demonstrates
that in sufficiently symmetric cases the feasibility of
$\mathbf{P}(\mathbf{V})$ serves as a necessary _and sufficient_ condition for
the vectorial feasibility of $\mathbf{V}$. In particular, using our results we
are able to deduce new insights about feasible and infeasible vector
distributions $\mathbf{V}$.

###  5.1 Private private information

A special case of distributions $\mathbf{V}$ whose vectorial feasibility has
been studied in (He et al., 2021; Arieli et al., 2021; Gutmann et al., 1991b)
is that of $|\Omega|=2$ and a _product_ distribution $V=\Pi_{i\in[n]}Q_{i}$
where $Q_{i}\in\Delta([0,1])$ is the distribution of the beliefs of agent $i$.
(He et al., 2021) refer to this class of distributions as _private private
information_ because the private information of agents other than $i$ does not
affect their beliefs about agent $i$’s belief. In any case, the belief of
every agent $j\neq i$ about agent $i$’s belief will be $Q_{i}$. For a
discussion on the attractiveness of this property see (He et al., 2021; Arieli
et al., 2021). A special case of private private information that has received
attention in the existing literature (see (He et al., 2021; Arieli et al.,
2021; Gutmann et al., 1991b)) is the _symmetric_ case where $V=Q^{n}$. The
characterization of the feasible and infeasible $V=Q^{n}$ is known for the
case of $n=2$ ((He et al., 2021)) but remains unclear for $n\geq 3$.

Consider now the case of a finitely supported $Q$. For the case of $V=Q^{n}$
the induced distribution over empirical distributions $\mathbf{P}(\mathbf{V})$
is the multinomial distribution. More formally, assume that $Q$ is supported
on $x_{1},...,x_{k}\in[0,1]$ and assigns a probability of $\alpha_{i}$ to
$x_{i}$. Let $\operatorname{MN}(n,Q)$ be the multinomial distribution that
draws $n$ samples from $\\{x_{i}\\}_{i\in[k]}$ according to $Q$. In the
standard definition of the multinomial distribution it is assumed that the
outcome is a vector of integers $(n_{1},...,n_{k})$ that sum up to $n$ and
indicates the number of times $x_{1},...,x_{k}$ have been chosen. In our case,
we refer to the outcome of $\operatorname{MN}(n,Q)$ as
$(\frac{n_{1}}{n},...,\frac{n_{k}}{n})$ and hence the outcome is an element of
$\Delta_{n}(\Delta(\Omega))$. The following proposition shows that in this
case the feasibility of $\mathbf{P}(\mathbf{V})$ is a necessary and sufficient
condition for the vectorial feasibility of $\mathbf{V}=Q^{n}$. In particular,
it provides a novel characterization of vectorial feasibility for this class
of distributions.

######  Proposition 3.

For a binary state space $\Omega$ and $Q\in\Delta([0,1])$ the product
distribution $\mathbf{V}=Q^{n}$ for $n$ agents is vector feasible if and only
if the multinomial distribution $\operatorname{MN}(n,Q)$ is a mean-preserving
spread of777We recall that $Q^{\omega}\in\Delta([0,1])$ denotes the
distribution of beliefs conditional on state $\omega$ in the case where the
ex-ante distribution of beliefs is $Q$. $(1-\mu)Q^{\omega=0}+\mu
Q^{\omega=1}$.

###### Proof.

Denote $\mathbf{P}=\mathbf{P}(\mathbf{V})=\operatorname{MN}(n,Q)$. Note that
$\tilde{\mathbf{P}}=Q$. By Theorem 1, $\mathbf{P}$ is feasible if and only if
$\mathbf{P}$ is a mean-preserving spread of $(1-\mu)Q^{\omega=0}+\mu
Q^{\omega=1}$.

If $\mathbf{P}=\mathbf{P}(\mathbf{V})$ is infeasible then $\mathbf{V}$ is not
vector feasible, as discussed above.

If $\mathbf{P}$ is feasible, we observe that $\mathbf{V}$ can be viewed as the
_unique_ distribution $\mathbf{W}$ with the following two properties:

  1. 1.

$\mathbf{P}(\mathbf{W})=\mathbf{P}$.

  2. 2.

$\mathbf{W}$ assigns an equal probability to every two vectors
$(y_{1},...,y_{n}),(y^{\prime}_{1},...,y^{\prime}_{n})\in\\{x_{1},...,x_{k}\\}^{n}$
with the same empirical distributions of beliefs.

It is easy to check that our construction of the information structure in the
proof of Theorem 1 induces a distribution over vectors of beliefs that
satisfies these two properties and hence this information structure implements
$\mathbf{V}$. Thus, $\mathbf{V}$ is feasible. ∎

Determining whether $\operatorname{MN}(n,Q)$ is a mean-preserving spread of
$(1-\mu)Q_{0}+\mu Q_{1}$ might not be an easy task in the case where $k$
(i.e., the support size of $Q$) is large. This is because
$\operatorname{MN}(n,Q)$ is a distribution over elements in $\Delta([k])$, and
in high dimensions the mean-preserving condition might not be easy to check.
However, for $k=2$ the multinomial distribution becomes simply a binomial
distribution and the mean-preserving condition is easily verifiable by Lemma
1. Thus, for $Q$ with a binary support we get the following condition for the
vectorial feasibility of $\mathbf{V}=Q^{n}$.

######  Corollary 3.

Let $Q\in\Delta(\\{a,b\\})$ where $0<a<\mu<b<1$. The distribution
$\mathbf{V}=Q^{n}$ is vector feasible (for $n$ agents) if and only if888We
adopt here our previous convention for the multinomial distribution, and
assume that the Binomial distribution receives values in
$\\{0,\frac{1}{n},...,1\\}$ and not in $\\{0,1,...,n\\}$.
$\mathbb{E}[\operatorname{Bin}(n,\frac{\mu-a}{b-a})|_{\mu}]\leq\frac{(\mu-a)b}{(b-a)\mu}$.

That is, the corollary states that in order to determine the feasibility of
$\mathbf{V}=Q^{n}$ one should calculate the expectation of the $\mu$-quantile
of the Binomial distribution. If it exceeds some constant, then the
distribution is infeasible. Otherwise it is feasible. Below is a case where we
can provide an explicit formula for the feasibility for every number of
agents.

######  Example 2.

$n$$a$2345678910110.50.250.4 Figure 1: The values of $a$ for which the product
distribution $Q^{n}$ is feasible as a function of the number of agents $n$.

Let $\mu=\frac{1}{2}$ and let
$Q=\frac{1}{2}\delta_{a}+\frac{1}{2}\delta_{1-a}$ for $0<a<\frac{1}{2}$. For
$n=2m$ and for $n=2m+1$, the distribution $\mathbf{V}=Q^{n}$ is feasible if
and only if $a\geq\frac{1}{2}-\binom{2m}{m}2^{-2m-1}$ (see Figure 1). This
observation is a consequence of Corollary 3, where the expectation of the
$\frac{1}{2}$-quantile of the $\operatorname{Bin}(n,\frac{1}{2})$ distribution
can be explicitly computed and is provided in the following lemma.

######  Lemma 2.

For every $m\in\mathbb{N}$ we have

$\mathbb{E}[\operatorname{Bin}(2m,p)|_{\frac{1}{2}}]=\mathbb{E}[\operatorname{Bin}(2m+1,p)|_{\frac{1}{2}}]=\frac{1}{2}-\binom{2m}{m}2^{-2m-1}.$

The proof of this lemma is relegated to Appendix B.

Asymptotically, for large population of $n$ agents, the threshold value of $a$
for which $V=Q^{n}$ is feasible is
$a\geq\frac{1}{2}-\binom{2m}{m}2^{-2m-1}\approx\frac{1}{2}-\frac{1}{\sqrt{2\pi
n}}$. Another interesting phenomenon is that the threshold value of $a$ is
identical for an even $n$ and for $n+1$.

##  6 Private Bayesian Persuasion of a Homogeneous Population

We next consider an application of our result to private Bayesian persuasion.
As in (Arieli and Babichenko, 2019), consider a sender who is a marketer that
tries to sell a product to a group of agents. Consider a binary state space
$\Omega=\\{0,1\\}$ with a common prior $\mu$. The agents’ action set is
$A=\\{0,1\\}$, where action $a=1$ represents a decision to buy the product and
action $a=0$ represents a decision not to buy the product. Playing $a=0$
yields a utility of $0$. The state represents the quality of the product. In
$\omega=1$ the product is of good quality and worth $1$ to the agents while in
state $\omega=0$ it is worth $-\frac{\tau}{1-\tau}$ to the agent, where
$\tau\in(0,1)$. Thus, at a posterior of $\tau$ the agents are indifferent
between buying and not buying the product. We assume that the agents share no
payoff externalities. Let $u:[0,1]\rightarrow\real$ be the utility of the
sender, where $u(x)$ represents her utility from persuading a fraction $x$ of
agents to take action 1. We assume that $u$ is non-decreasing and that
$\mu<\tau$ (otherwise the policy that reveals no information would be
optimal). We denote by $V_{n}$ the optimal utility the sender can achieve in
an interaction with $n$ agents.

(Arieli and Babichenko, 2019; Babichenko and Barman, 2016) study a
heterogeneous variant of this setting, where each agent possesses an
individual threshold $\tau_{i}\in(0,1)$. In the heterogeneous case a clean
formula for $V_{n}$ has been deduced only in the case where $u$ is concave;
see (Arieli and Babichenko, 2019).999A polynomial algorithm for computing
$V_{n}$ has been deduced in (Babichenko and Barman, 2016). Here we observe
that the homogeneous population variant of the problem can be solved using our
main result (Theorem 1) and a clean formula for $V_{n}$ can be provided.

0$\frac{1}{n}$$\frac{2}{n}$1 Figure 2: Concavification over a grid. The
function $u$ is displayed in black. The concavification over the grid
$\operatorname{cav}_{n}(u)$ is displayed in blue.

We denote by $\operatorname{cav}_{n}(u):[0,1]\to\real$ the concavification of
$u$ when $u$ is restricted to the $\frac{1}{n}$-grid points
$\\{0,\frac{1}{n},\frac{2}{n},...,1\\}$ only (see Figure 2). Namely,
$\operatorname{cav}_{n}(u)$ is the minimial concave function that satisfies
$\operatorname{cav}_{n}(u)(x)\geq u(x)$ for every
$x\in\\{0,\frac{1}{n},\frac{2}{n},...,1\\}$. Equivalently, for every
$y\in[0,1]$ we define

$\displaystyle\operatorname{cav}_{n}(u)(y)=\max_{Q\in\Delta_{n}(\\{0,1\\}),\mathbb{E}[Q]=y}\mathbb{E}_{x\sim
Q}[u(x)].$ (7)

The characterization of the sender’s optimal value is as follows.

######  Proposition 4.

The optimal utility for the sender is

$V_{n}=\mu
u(1)+(1-\mu)\operatorname{cav}_{n}(u)(\frac{\mu(1-\tau)}{\tau(1-\mu)}).$

The formula for the value can be interpreted as follows. In state $\omega=1$
all agents take action 1 and the utility is $u(1)$. In state $\omega=0$ an
expected fraction of $\frac{\mu(1-\tau)}{\tau(1-\mu)}$ agents adopt the
product and the sender is allowed to optimize over all the distributions with
expectation $\frac{\mu(1-\tau)}{\tau(1-\mu)}$ in order to achieve
$\operatorname{cav}_{n}(u)(\frac{\mu(1-\tau)}{\tau(1-\mu)})$.

Note that the concavification in Proposition 4 differs from the classic
characterization of the value via concavification ((Kamenica and Gentzkow,
2011)). The standard characterization applies concavification to a function
whose domain is that of the posterior beliefs. Our characterization (for this
special case) applies concavification to a function whose domain is that of a
fraction of the buyers.

Proposition 4 also provides an asymptotic characterization of the value for
large populations. Denote $V^{*}=\lim_{n\to\infty}V_{n}$ and denote by
$\operatorname{cav}(u)$ the standard notion of concavification (without
restricting it to a grid). By Proposition 4 we have

$V^{*}=\mu u(1)+(1-\mu)\operatorname{cav}(u)(\frac{\mu(1-\tau)}{\tau(1-\mu)})$

because $\operatorname{cav}_{n}(u)$ point-wise converges to
$\operatorname{cav}(u)$ as $n\to\infty$ for non-decreasing functions $u$.

###### Proof of Proposition 4.

The idea is to utilize Corollary 1. To do so, we should be able to identify
the possible posteriors that agents could have in an optimal policy, and to
prove that there are only two possible values for these posteriors. To this
end, we utilize the following observation from (Arieli and Babichenko, 2019).

######  Lemma 3 ((Arieli and Babichenko, 2019)).

There exists an optimal policy with the following properties:

  * •

The policy uses binary signals $S_{i}=\\{s^{i}_{0},s^{i}_{1}\\}$ for each
agent $i$, and each agent takes action $1$ if and only if she receives the
signal $s^{i}_{1}$.

  * •

In state $\omega=1$ signal $s^{i}_{1}$ is sent to all agents with probability
$1$.

  * •

In state $\omega=0$ each agent $i$ gets the signal $s^{i}_{1}$ with
probability101010The Lemma does not indicate on the correlation of the
$s^{i}_{1}$ signals among the agents conditional on state $\omega=0$.
$\frac{\mu(1-\tau)}{\tau(1-\mu)}$.

By Lemma 3 we deduce that the only two possible posteriors for an agent are
$0$ and $\tau$. Now we can use Corollary 1, which states that
$\mathbf{P}\in\Delta(\\{0,\frac{1}{n},\frac{2}{n},...,1\\})$ is feasible if
and only if $\mathbf{P}$ is a mean-preserving spread of
$\mu\delta_{1}+(1-\mu)\delta_{\frac{\mu(1-\tau)}{\tau(1-\mu)}}$, where
$\mathbf{P}$ is the number of agents with posterior $\tau$. This set of mean-
preserving spreads consists of all $\mu\delta_{1}+(1-\mu)Q$ where
$Q\in\Delta(\\{0,\frac{1}{n},\frac{2}{n},...,1\\})$ satisfy
$\mathbb{E}[Q]=\frac{\mu(1-\tau)}{\tau(1-\mu)}$ and the utility of the sender
for each such mean-preserving spread is $\mu u(1)+(1-\mu)\mathbb{E}_{x\sim
Q}[u(x)]$, which is precisely the concavification formula of Equation (7). ∎

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##  Appendix A Extending the Proof of Theorem 1 to infinitely supported
distributions

We have proved in Section 3.3 that a finitely supported feasible $\mathbf{P}$
must be a mean-preserving spread of $\mathbf{P}^{\prime}$. Here we extend this
result to arbitrary feasible distributions $\mathbf{P}$.

Let $G$ be an information structure $G$ for $n$ agents where
$\\{S_{i}\\}_{i=1}^{n}$ are (possibly infinite) measurable sets. Note that the
measure $\mathbf{P}_{G}$ that is induced by $G$ can be weakly approximated
using $\mathbf{P}_{G^{\prime}}$ when $G^{\prime}$ is a finite information
structure. To do this, we use the following standard trick. First we partition
$\Delta(\Omega)$ using a finite partition $\mathcal{D}_{\delta}$ such that
each element $D\in\mathcal{D}_{\delta}$ is a convex set and has a diameter at
most $\delta$. We define the new signal space
$S^{\prime}_{i}=\mathcal{D}_{\delta}$ for every agent $i$. We then couple
together $G$ and $G^{\prime}$ as follows. We draw a signal profile according
to $G$ and tell each agent $i$ only of the partition element in
$\mathcal{D}_{\delta}$ in which the posterior $\mathbf{P}(\omega=1|s_{i})$
lies. It is easy to see that $\mathbf{P}_{G^{\prime}}$ weakly converges to
$\mathbf{P}$ as $\delta$ goes to $0$.

It now remains to show that if
$\\{\mathbf{P}_{k}\\}_{k}\subset\Delta(\Delta_{n}(\Delta(\Omega)))$ is a
sequence of measures with finite support that weakly converges to $\mathbf{P}$
such that $\mathbf{P}_{k}$ is a mean-preserving spread of
$\mathbf{P}^{\prime}$ for every $k$, then $\mathbf{P}$ also satisfies the this
condition.

Note that
$\mathbf{P}_{k}=\sum_{\omega\in\Omega}\mu_{\omega}\mathbf{P}^{\omega}_{k}$.
Since $\Delta(\Delta_{n}(\Delta(\Omega)))$ is a weakly compact space we can
assume (by taking subsequences if necessary) that all sequences
$\\{\mathbf{P}^{\omega}_{k}\\}_{k}$ weakly converge to a measure
$\mathbf{P}^{\omega}$ for every state $\omega\in\Omega$. Therefore,
$\mathbf{P}=\sum_{\omega\in\Omega}\mu_{\omega}\mathbf{P}_{\omega}$. We further
note that taking the expected measure is a weakly continuous operator from
$\Delta(\Delta_{n}(\Delta(\Omega)))$ to $\Delta(\Delta(\Omega))$. Therefore,
the sequences $\\{\tilde{\mathbf{P}}_{k}\\}_{k}$ and
$\\{\tilde{\mathbf{P}}^{\omega}_{k}\\}_{k}$ converge to $\tilde{\mathbf{P}}$
and $\tilde{\mathbf{P}}^{\omega}$ respectively for every $\omega$. Therefore
$\mathbf{P}$ is a mean-preserving spread of
$\mathbf{P}^{\prime}=\sum_{\omega\in\Omega}\mu_{\omega}\delta_{\tilde{\mathbf{P}}^{\omega}}$
as desired.

Furthermore, since for every $k$ and $\omega\in\Omega$,

$\mathrm{d}\tilde{\mathbf{P}}^{\omega}_{k}(x)=\frac{1}{\mu_{\omega}}x_{\omega}\mathrm{d}\tilde{\mathbf{P}}_{k}(x),$

it follows that in the limit for every $\omega\in\Omega$,

$\mathrm{d}\tilde{\mathbf{P}}^{\omega}(x)=\frac{1}{\mu_{\omega}}x_{\omega}\mathrm{d}\tilde{\mathbf{P}}(x),$

as desired.

##  Appendix B Proof of Lemma 2

The distribution $\operatorname{Bin}(2m+1,\frac{1}{2})|_{\frac{1}{2}}$ assigns
a probability of $\binom{2m+1}{k}2^{-2m}$ to $\frac{k}{2m+1}$ for every $k\leq
m$, hence its expectation is given by

$\displaystyle\begin{aligned}
\mathbb{E}[\operatorname{Bin}(2m+1,\frac{1}{2})|_{\frac{1}{2}}]&=\sum_{k=1}^{m}\binom{2m+1}{k}2^{-2m}\frac{k}{2m+1}=\frac{2^{-2m}}{2m+1}\sum_{k=1}^{m}\binom{2m+1}{k}k\\\
&=\frac{2^{-2m}}{2m+1}(2m+1)\sum_{k=0}^{m-1}\binom{2m}{k}=2^{-2m}\frac{1}{2}\left(2^{2m}-\binom{2m}{m}\right)\\\
&=\frac{1}{2}-\binom{2m}{m}2^{-2m-1}\end{aligned}$ (8)

where the third equality follows from the fact that both expressions
$\sum_{k=1}^{m}\binom{2m+1}{k}k$ and $(2m+1)\sum_{k=0}^{m-1}\binom{2m}{k}$
count the number of possible choices of a subset of at most $k$ people and
their leader out of a group of $2m+1$ people. The first expression first
chooses the group and thereafter the leader. The second expression first
chooses the leader and thereafter complements it with a subset.

The distribution $\operatorname{Bin}(2m,\frac{1}{2})|_{\frac{1}{2}}$ assigns a
probability of $\binom{2m}{k}2^{-2m+1}$ to $\frac{k}{2m}$ for every $k<m$ and
assigns a probability of $\binom{2m}{m}2^{-2m}$ to $\frac{1}{2}$, and hence
its expectation is given by

$\displaystyle\mathbb{E}[\operatorname{Bin}(2m,\frac{1}{2})|_{\frac{1}{2}}]$
$\displaystyle=\sum_{k=1}^{m-1}\binom{2m}{k}2^{-2m+1}\frac{k}{2m}+\binom{2m}{m}2^{-2m-1}$
$\displaystyle=\frac{2^{-2m+1}}{2m}\sum_{k=1}^{m-1}\binom{2m}{k}k+\binom{2m}{m}2^{-2m-1}$
$\displaystyle=\frac{2^{-2m+1}}{2m}2m\sum_{k=0}^{m-2}\binom{2m-1}{k}+\binom{2m}{m}2^{-2m-1}$
$\displaystyle=2^{-2m+1}\frac{1}{2}\left(2^{2m-1}-2\binom{2m-1}{m-1}\right)+\binom{2m}{m}2^{-2m-1}$
$\displaystyle=\frac{1}{2}-2^{-2m+1}\binom{2m-1}{m-1}+\binom{2m}{m}2^{-2m-1}$
$\displaystyle=\frac{1}{2}-\binom{2m}{m}2^{-2m-1}$

where the argument for the third equality is similar to the third equality in
Equation (8).