# The Wisdom of the Crowd and Higher-Order Beliefs

Yi-Chun Chen${}^{\text{{a}}}$ ${}^{\text{{a}}}$Department of Economics and
Risk Management Institute, National University of Singapore
<EMAIL_ADDRESS>,  Manuel Mueller-Frank${}^{\text{{b}}}$
${}^{\text{{b}}}$IESE Business School, University of Navarra
<EMAIL_ADDRESS>and  Mallesh M. Pai${}^{\text{{c}}}$
${}^{\text{{c}}}$Department of Economics, Rice University
<EMAIL_ADDRESS>
###### Abstract.

The classic wisdom-of-the-crowd problem asks how a principal can “aggregate”
information about the unknown state of the world from agents without
understanding the information structure among them. We propose a new simple
procedure called _Population-Mean-Based Aggregation_ to achieve this goal. The
procedure only requires eliciting agents’ beliefs about the state, and also
eliciting some agents’ expectations of the average belief in the population.
We show that this procedure fully aggregates information: in an infinite
population, it always infers the true state of the world. The procedure can
accommodate correlations in agents’ information, misspecified beliefs, any
finite number of possible states of the world, and only requires very weak
assumptions on the information structure.

Keywords: aggregating beliefs, higher-order beliefs, wisdom of the crowd.

JEL Classification: D82, D83.

We thank Samuel Ho Cher Sien and Xinhan Zhang for excellent research
assistance. We thank Itai Arieli, Rahul Deb, Ben Golub, Tai-Wei Hu, Yanwei
Jia, Ran Smorodinsky, Tom Wilkening for helpful comments and discussions. Chen
gratefully acknowledges the financial support from the Singapore Ministry of
Education Academic Research Fund Tier 1. Mueller-Frank gratefully acknowledges
the financial support of the Spanish Ministry of Science, Innovation and
Universities (Ref. ECO2015-63711-P: MINECO/FEDER, UE) and the Department of
Economy and Knowledge of the Generalitat de Catalunya (Ref. 2017 SGR 1244).
Pai gratefully acknolwedges support from the NSF (CCF-1763349).

##  1\. Introduction

A long-held belief, starting from at least Aristotle if not earlier, is that
of the “wisdom of the crowd.” This refers to the idea that even though some
event may be uncertain, and the information each individual has may be noisy,
the “aggregate” of individual beliefs is accurate. Such aggregation obviously
has large social and private value, especially when it concerns important
social or economic events. As a consequence, there has been substantial effort
toward both improving existing institutions (e.g., polling), and designing and
implementing new methods (e.g., prediction markets).111There is also a large
and influential body of literature studies the possibility of such aggregation
within our current institutions, e.g., voting (see e.g., Austen-Smith and
Banks (1996), Feddersen and Pesendorfer (1997)), markets (e.g., Grossman and
Stiglitz (1980)) and social networks (see e.g., Golub and Jackson (2010),
Acemoglu, Dahleh, Lobel, and Ozdaglar (2011)). At a theoretical level, then,
it is important to understand the limits of such an exercise: without specific
assumptions on the nature of the information agents have, how can we aggregate
it, and how much can we learn from it? In short, how wise is the crowd, and
how can we tap into its wisdom?

To fix ideas, imagine a population of interest. There is an event whose
realization is uncertain. Individual agents each see a signal about the event
that is informative of the state. Agents also understand the informational
environment, i.e., they understand both what this signal implies for their own
beliefs, and the implied distribution of other agents’ beliefs. Even assuming
away the difficulties (both logistical and strategic) of eliciting the
relevant information from the population, the question we address in this
paper is: Does there exist a general procedure to aggregate individuals’
information without knowledge of the information structure among them?

One important insight from the existing literature is that even for the case
where agents’ signals are i.i.d. conditional on the state, knowing the first-
order beliefs of even an infinite set of agents is generally not sufficient to
learn the state.222See Prelec, Seung, and McCoy (2017) and Arieli, Babichenko,
and Smorodinsky (2017). The main takeaway from Arieli, Babichenko, and
Smorodinsky (2017) is that without further (very strong) assumptions about the
signal structure, the principal cannot learn the state of the world, even with
an infinite number of agents in the population: in the terminology of
econometrics, there is an “identification problem.”

Our answer to the posed question is founded on a simple procedure we name
_Population-Mean-Based Aggregation_ (herein PMBA). Our procedure involves
eliciting all agents’ beliefs about the uncertain event as well as eliciting
from some agents their expectation about the average of others’ beliefs. We
show that, under very weak conditions on the information structure generating
the agents’ signals, this procedure _fully aggregates_ the information among
agents in a large population. In other words, using information elicited from
agents, solely about agents’ own beliefs, and their expectation of the average
belief in the population (herein referred to as population average belief),
but with no other knowledge of the underlying information structure, it is
possible in theory to fully learn the underlying state. Of course, this idea
of eliciting and using higher-order beliefs is not new: In our context, we
note the elegant “Surprisingly Popular” (herein, SP) mechanism developed by
Prelec, Seung, and McCoy (2017) (henceforth PSM), who pioneered the use of
second-order beliefs for the purpose of information aggregation (their
mechanism works for a narrow set of information structures—we describe the
relation between the SP mechanism and our PMBA mechanism in more detail in
Section 5.1).

Our main result considers general environments with a finite number of states
of the world, and a large number of agents. We suppose that agents observe
signals according to an information structure that is _unknown to the
aggregation procedure_ , but understood among the agents themselves. We show
that, under general conditions, full aggregation can be obtained from the
population distribution of the first-order beliefs, and by eliciting, from as
many agents as there are states, agents’ expectations of the average belief of
the population. In particular, we explain how to more fruitfully use higher-
order beliefs when there are multiple states (indeed, PSM themselves point out
the difficulties of extending the SP mechanism when there are three or more
states, and we recap the issues with illustrative examples in Section 5.1).

The basic idea of our mechanism is intuitive and can be described verbally,
especially in the case of binary states. For simplicity, suppose every agent
sees a conditionally i.i.d. signal about the unknown state according to an
information structure that is known to them but unknown to the principal. Now
suppose the principal elicits, from each agent, their belief about the state.
By the law of large numbers, the population average belief must equal the
expected belief in that state.333This is an important point, see Remarks 3 and
4 below for further discussion. If the principal knows the expected belief in
each state, then they would be done: they could simply compare the elicited
average to the expected belief in each state.444If signals are informative, it
can be shown that these expected beliefs must be different in the two states.
However they do not know this mapping of the state to the expected
beliefs.555There is a trivial mechanism that simply elicits from some agent
the average belief in the population conditional on each state being the true
state of the world. We follow the literature, which has consistently only
considered eliciting expected population average beliefs, and ignore this
possibility. A possible justification is that there is substantial evidence in
practice that agents struggle with conditional assessments—see e.g., Charness
and Levin (2009), Esponda and Vespa (2014), or Martínez-Marquina, Niederle,
and Vespa (2019) for experimental evidence in different contexts, and Li
(2017) for a discussion. Suppose the principal elicited, from two agents with
different beliefs about the states, their expectation of the population
average belief. By the law of iterated expectations, each must report the
weighted average of the expected belief conditional on the state, weighted by
their (different) beliefs about the likelihood of the two states. This
presents the principal a simple system of two linear equations in two
unknowns; thus the expected beliefs in each state can be uniquely recovered
from the information elicited from these two agents. We show in what follows
that this basic idea can be adapted with more states and limited correlation
among agents. With an eye to real-world applications, we show how to adapt our
method when only eliciting “votes” from most agents, e.g., the state they
perceive to be most likely (this was also the main exercise considered in
Prelec, Seung, and McCoy (2017)). We also consider settings where agents
understand their own information, but misperceive the distribution of other
agents’ beliefs etc. (this idea of the misperception of others’ information is
related to the environments considered in Eyster and Rabin (2010), Bohren
(2016), and Frick, Iijima, and Ishii (2020)).

While our present paper is theoretical, the idea of eliciting agents’ higher-
order beliefs and using this information has been successfully implemented in
practice, both in the lab and in the field. There is now a substantial
literature describing implementations of various procedures that elicit and
use higher-order beliefs, including SP (we describe this and other related
literature in more detail in Section 5).

The plan for the rest of the paper is as follows. In Section 2, we introduce
the model and notation we use in this paper, and formally present our PMBA
mechanism. Section 2.2 covers the restricted case of binary states, since our
sufficient conditions (and the mechanism) are easier to describe in that
case—further that case is perhaps the leading candidate to be used in
practice. Section 2.3 gives the general mechanism for an arbitrary finite
number of states. Section 3.2 considers an extension to a setting where agents
have individually incorrect beliefs about the distribution of the beliefs of
others, and shows how to extend PMBA to such settings. It also discusses how
PMBA can be used to elicit beliefs truthfully. Section 4 attempts to
understand the full power and limitations of aggregation when eliciting
higher-order beliefs. Of course, the PMBA shows that with an infinite
population, access to agents’ first-order and (a few agents’) second-order
beliefs can suffice. So, we now consider the case of a finite population (in
particular, where signals are not conditional i.i.d.). We show that if the
principal can extract the full hierarchy of agents’ higher-order beliefs, they
can learn the “full information posterior”, i.e., the posterior of an
omniscient agent who shares the agents’ prior belief and sees all their
individual signals, despite the principal not knowing the prior and only
collecting information about posteriors. Conversely, we show that if the
principal only sees a finite prefix of the hierarchy (e.g., only the first $k$
beliefs) of a population, then there is an “identification problem,” i.e.,
there exist two different information structures that could both result in
this exact realization. In short, higher-order beliefs are not useful in the
case of a finite population unless the entire hierarchy can be elicited.
Section 5 discusses the related literature in further detail and concludes. In
particular, Section 5.1 studies the implications on the SP mechanism: we
demonstrate both how our analysis implies that the SP mechanism is more
generally applicable than previously understood, and also how our mechanism is
applicable when SP is not.

##  2\. Population-Mean-Based Aggregation

In this section, we introduce our core model; we describe our Population-Mean-
Based Aggregation procedure (herein PMBA) and prove its validity.

###  2.1. General Model and Notation

There is an infinite population of agents. To this end, let
$N=\\{1,2,\ldots\\}$ be a countably infinite set of agents, with $i$ denoting
a generic agent. There are a finite set of $L$ states of the world
$\Omega=\\{\omega_{1},\omega_{2}\ldots\omega_{L}\\}$. The space of feasible
beliefs over $\Omega$, denoted as $\Delta(\Omega)$, is the $L$ dimensional
simplex $\Delta^{L}$.

Each agent $i$ observes a signal in a set $S$ with the associated sigma field
$\mathcal{F}$. To allow for correlation, let us denote the common prior over
$\Omega\times S^{N}$ by $P$. Given that an agent observes some signal $s\in
S$, their posterior belief that the state is $\omega$ is given by
$P(\omega|s)$. Let us denote the posterior beliefs of agent $i$ by
$\mu_{i}\in\Delta^{L}$ (suppressing the dependence on the signal), and
$\tilde{\mu}_{i}$ as the associated random variable. We define

$\displaystyle\overline{\mu}_{i}(\omega)=\mathbb{E}[\tilde{\mu}_{i}|\omega].$

In words, $\overline{\mu}_{i}(\omega)$ is the expected belief of agent $i$ in
state $\omega$.

Finally, we define
$\overline{\mu}(\omega)=\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\overline{\mu}_{i}(\omega)$.666Note
that without further assumptions, it is possible that this limit might not
exist and/or might be different depending on the ordering of agents. We ignore
these possibilities. Simple sufficient conditions (e.g., conditional i.i.d.
signals) rule out these concerns with an infinite population, and they are
moot for any arbitrarily large finite population. This is the expected
population average belief in state $\omega$.

Given a vector $x\in\Delta(\Omega)$, we will let $x_{\omega}$ refer to the
component corresponding to $\omega$; i.e., $\mu_{i,\omega}$ is the belief of
agent $i$ that the state is $\omega$, $\tilde{\mu}_{i,\omega}$ is the random
variable of agent $i$’s belief that the state is $\omega$, etc.

###  2.2. Binary States

In this section, we consider the case of binary states, i.e., $L=2$. This
mechanism will form the core of our general analysis.

We let $P_{i,\omega}^{S}$ denote the induced marginal distribution of P over
signals for agent $i$ in state $\omega$. We also abuse notation so that
$\mu_{i}$ represents agent $i$’s belief that the state is $\omega_{1}$: since
there are only two states, their belief that the state is $\omega_{2}$ is
$1-\mu_{i}$.

######  Assumption 1.

We make the following assumptions on $P$:

  1. (1)

Imperfectly informed agents: for each agent $i$, we have that
$P^{S}_{i,\omega}$ and $P^{S}_{i,\omega^{\prime}}$ are mutually absolutely
continuous with respect to each other.

  2. (2)

Limited correlation: For every $\varepsilon>0$, there exists a finite
$n(\varepsilon)$ such that conditional on the state, for any agent $i$, all
but at most $n(\varepsilon)$ agents’ signals are $\varepsilon$-independent of
agent $i$’s signal. Formally:

$\displaystyle\forall\varepsilon>0,\exists n(\varepsilon)\in\mathbb{N}\text{
s.t. }\forall i,\exists N_{i}\subseteq N,|N_{i}|\leq n(\varepsilon),$
$\displaystyle\forall j\in N\backslash N_{i},\forall
E,E^{\prime}\in\mathcal{F},\forall\omega:|P(\tilde{s}_{i,\omega}\in
E,\tilde{s}_{j,\omega}\in
E^{\prime}|\omega)-P_{i,\omega}^{S}(E)P_{j,\omega}^{S}(E^{\prime})|\leq\varepsilon.$

This assumption may seem a little dense so some discussion is useful. The
first condition is simply to make the problem interesting/non-trivial: after
all, if an agent is perfectly informed then aggregation is trivial. Next, we
need an assumption regarding the correlations between agents’ signals.
Intuitively, assuming conditionally i.i.d. signals is likely overly strong and
does not capture the idea that agents’ information may be correlated. However,
we cannot allow for arbitrary correlation—for example, if all agents’
information is perfectly correlated, then effectively there is only one unique
opinion in the population and no further aggregation is possible. Our notion
allows for “enough independence,” thereby roughly guaranteeing that for any
agent, most other agents’ signals are approximately independent conditional on
the state. This ensures that the law of large numbers holds, so that the
population average belief is a deterministic function the of state, i.e., in
the notation of our model, in any state $\omega$ almost surely (a.s.) for any
realizations $\mu_{i}$ of beliefs of agents $i\in N$; we have that

$\hat{\mu}\equiv\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}\mu_{i}=\overline{\mu}(\omega).$

As we will see in what follows, our PMBA procedure requires that the
population average beliefs are different in the two states, i.e.,
$\overline{\mu}(\omega_{1})\neq\overline{\mu}(\omega_{2})$. This is a very
weak condition, but one that one may nevertheless worry about: in principle
different distributions of beliefs may nevertheless have the same average
expected belief. The following assumption delivers this: it requires that all
agents individually are at least minimally informed, formally defined with
respect to the total variation metric. This is also akin to a normalization—it
simply ensures that we do not have infinitely many arbitrarily uninformed
agents.

To this end, we need some more notation: let $G_{i,\omega}$ denote the induced
distribution of private beliefs of agent $i$ in state $\omega$. We will abuse
notation and also let it denote the cumulative distribution function of the
belief of agent $i$ in state $\omega$.

######  Assumption 2 (Minimal Information).

We assume that there exists $\delta>0$, such that for any agent $i$, the total
variation distance between $G_{i,\omega_{1}}$ and $G_{i,\omega_{2}}$ is
uniformly bounded below by $\delta$, i.e.,

$\exists\delta>0\text{ s.t. }\forall
i:\sup_{E\subseteq[0,1]}|G_{i,\omega_{1}}(E)-G_{i,\omega_{2}}(E)|\geq\delta.$

Procedure 1 Population-Mean-Based Aggregation, 2 states

  1. (1)

Elicit, from each agent $i\in N$, their belief $\mu_{i}$. Calculate
$\hat{\mu}$ defined as:

$\displaystyle\hat{\mu}=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}\mu_{i}$

  2. (2)

Select two agents, A and B, such that $\mu_{A}\neq\mu_{B}$. Elicit from each
of these agents their expectation of the average posterior beliefs in the
population, i.e., denoting $s_{i}$ as the signal seen by agent $i$, elicit for
each $i=A,B$:

$\displaystyle\alpha_{i}=\lim_{n\rightarrow\infty}\mathbb{E}\left[\frac{1}{n}\sum_{j=1}^{n}\tilde{\mu}_{j}\bigg{|}s_{i}\right].$

  3. (3)

Calculate $\overline{\mu}(\omega_{1}),\overline{\mu}(\omega_{2})$ as:

$\displaystyle\left(\begin{array}[]{c}\overline{\mu}(\omega_{1})\\\
\overline{\mu}(\omega_{2})\end{array}\right)=\left(\begin{array}[]{cc}\mu_{A}&1-\mu_{A}\\\
\mu_{B}&1-\mu_{B}\end{array}\right)^{-1}\left(\begin{array}[]{c}\alpha_{A}\\\
\alpha_{B}\end{array}\right)$ (7)

Note that the right-hand side is elicited from the agents directly; and since
$\mu_{A}\neq\mu_{B}$, the middle matrix is invertible.

  4. (4)

Recover the state by comparing $\hat{\mu}$ calculated in step $1$ with
$\overline{\mu}(\cdot)$ calculated above. The state of the world is $\omega$
s.t. $\hat{\mu}=\overline{\mu}(\omega).$

The first main result of our paper shows the power of the PMBA procedure as
described in Procedure 1.

######  Theorem 1.

Suppose the information structure among agents satisfies Assumptions 1 and 2.
Then the PMBA procedure (Procedure 1) correctly recovers the true state of the
world almost surely.

Before we discuss a sketch of the proof and some directions to generalize the
procedure, some remarks about the mechanism and the result are in order.

######  Remark 1.

Assumption 1 and 2 jointly are sufficient for PMBA to be valid, but it can be
shown that they are not necessary. A full characterization is easy to describe
but perhaps unsatisfying—as long as the information structure is such that the
law of large numbers holds, i.e., in any state $\omega$,
$\textrm{plim}_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\tilde{\mu}_{i}=\bar{\mu}(\omega)$;
and $\bar{\mu}\neq\bar{\mu^{\prime}}$, it will be clear from the proofs that
PMBA will successfully recover the true state of the world. Expressing this
(in particular, concentration) in terms of primitives, however, appears out of
reach.

######  Remark 2.

It is worth re-emphasizing that the PMBA mechanism does not require any
knowledge of the underlying information structure among agents—a different way
of interpreting our result therefore is that as long as these conditions are
satisfied, knowledge of the information structure among agents is not needed
to fully aggregate their information. That said, it is possible to construct
examples where these conditions are violated but a principal who knows the
information structure among agents can aggregate their information. For
example, one can construct settings, e.g., with two states, such that
$\bar{\mu}(\omega_{1})=\bar{\mu}(\omega_{2})$, but the population distribution
of posteriors in these two states are different. A Bayesian principal who
knows this would be able to perfectly aggregate agents’ information, but PMBA
would clearly fail in such a setting.777As a simple illustration, consider the
following example: there are two states, $\\{\omega_{1},\omega_{2}\\}$, and
two signals, $\\{s_{1},s_{2}\\}$. All agents’ prior is that either state is
equally likely. In state $\omega_{j}$, agent $i$ receives the “correct” signal
$s_{j}$ with probability $\frac{1}{2}+e^{-i}$, and with the remaining
probability receives the incorrect signal $s_{3-j}$ (this clearly violates
Assumption 2). PMBA will fail in this setting: the expected population average
belief in both states is $\frac{1}{2}$. However, by observation, a Bayesian
who knows the signal structure can learn from the agents’ beliefs.

######  Remark 3.

It is important, both conceptually and technically, to note that our
assumptions (in particular, Assumption 1 part (2)) imply that
$\bar{\mu}(\omega)$ is a degenerate random variable, i.e., a deterministic
function of the state. Note that our PMBA procedure would fail otherwise,
because it recovers $\mathbb{E}[\bar{\mu}(\omega)|\omega]$, which may not
equal the realization if $\bar{\mu}(\omega)$ is a non-degenerate random
variable. One can construct settings where PMBA would fail, but nevertheless a
Bayesian principal who knew the information structure among agents would learn
the true state of the world (for example, if $\bar{\mu}(\omega)$ is a non-
degenerate r.v. but has disjoint support across the states).

Conceptually then, our assumption puts additional structure on what we mean by
a “state of the world.” As an example, consider a setting where agents’
signals are exchangeable conditional on the state. In this case, DiFinetti’s
theorem tells us that the signals are i.i.d. conditional on state and an
additional latent random variable. Our assumption then rules out the existence
of additional latent random variables on which $\bar{\mu}(\omega)$ depends—the
description of the state must be rich enough so that $\bar{\mu}(\cdot)$ is a
deterministic function of $\omega$.

A formal proof is given in Appendix A. Here we want to discuss how the various
assumptions play a role in the result. As we pointed out earlier, Part (1) of
Assumption 1 is mainly a normalization. The key processing step in this
procedure is Step 3. By the law of iterated expectations, an agent $i$ with
beliefs $\mu_{i}$ will report, in step (2),

$\alpha_{i}=\mu_{i}\overline{\mu}(\omega_{1})+(1-\mu_{i})\overline{\mu}(\omega_{2}).$

Our procedure simply inverts this relationship to learn the unknown mapping
from $\omega$ to $\overline{\mu}(\omega)$, using (7).

Step (4) then recovers the state by comparing the calculated
$\overline{\mu}(\cdot)$ to the elicited $\hat{\mu}$. Part (2) of Assumption 1
ensures that an appropriate law of large numbers holds so that in state
$\omega$, $\hat{\mu}=\overline{\mu}(\omega)$, almost surely. Assumption 2 is
used to guarantee that
$\overline{\mu}(\omega_{1})\neq\overline{\mu}(\omega_{2})$. Therefore, almost
surely, the elicited $\hat{\mu}$ will match $\overline{\mu}(\omega)$ for a
unique state, and the procedure thus successfully concludes. Taken together,
therefore, these give us our result.

To be clear, the model and result of this section are explicitly written for a
common prior model where the agents perfectly understand the signal structure
among themselves, and only the principal does not. In applied contexts, this
kind of knowledge of each others’ information among agents may seem daunting
or even implausible.

To that end, we should first point out that our mechanism does not require
agents to have an exact understanding of the distribution of signals among
fellow agents. We can observe that it is sufficient for each agent to
perfectly understand the aggregate implications; that is to say, the average
population beliefs in each state, $\overline{\mu}(\cdot)$ since that is all
they need to know (other than their own beliefs) to calculate the information
that we elicit. This is a substantially lower informational requirement.
Indeed in the example in Section 3.1, we show how even less demanding
information may be elicited (e.g., asking most agents what state they think is
more likely).

However, even this may seem high—perhaps agents don’t perfectly understand
even the aggregate implications of the model—for example, each agent has noisy
information about the true $\overline{\mu}(\cdot)$, rather than exact
knowledge of it. In Section 3.2, we show how a modification of our mechanism
will continue to allow for population aggregation in such a setting.

###  2.3. $L>2$ States

We now consider the case of $L>2$ states. Intuitively, we generalize the
“matrix inversion” ideas of the previous subsection to propose a PMBA
procedure that works for $L$ states. However having more than $2$ states
introduces some difficulties: notably, the assumptions that guarantee that
this process works become slightly more demanding. Assumption 1 still applies
as written, observe that nothing in that assumption was specific to the case
of binary states.

However, a direct generalization of Assumption 2 to more than 2 states is
generally not sufficient to assure that the state-dependent averages are
unequal across states. So we assume the $L$ state generalization of its
implication directly.

######  Assumption 3.

We assume that $P$ is such that for any two states $\omega$ and
$\omega^{\prime}$,
$\overline{\mu}(\omega)\neq\overline{\mu}(\omega^{\prime}).$

Note that while Assumption 3 is not directly stated in terms of the primitives
of our model ($P$), it should be clear that this is a property that can be
easily satisfied.888For example, in the case of finite conditionally i.i.d
signals, this property is generically satisfied in the belief space, in the
sense of full measure.

Finally, in the $2-$state case, we are guaranteed the a.s. existence of two
agents who have different beliefs. The appropriate generalization turns out to
be the existence of $L$ agents whose beliefs constitute a full rank $L\times
L$ matrix so that the analog/generalization of (7) in step (3) can be carried
out. The following assumption guarantees this:

######  Assumption 4.

We assume that $P$ is such that for any agent $i$, the support of beliefs
$\textrm{supp}(\tilde{\mu}_{i})$ contains $L$ distinct points, such that those
$L$ beliefs, viewed as $L$-dimensional vectors, constitute a set of full rank.
Alternately and equivalently, we require that the convex-hull of the support
has an interior relative to $\Delta^{L}$.

One might wonder why such a condition was not required in the case of binary
states. The answer is that when $L=2$, this condition simply reduces to
requiring that the support of the agents’ beliefs has at least two points in
any state. Part (1) of Assumption 1, however, implies that the support of the
agents’ beliefs is the same across states. If this support was degenerate at a
single point, we would clearly violate Assumption 2. Note that with $L>2$,
there clearly exist settings that violate Assumption 4 despite satisfying
Assumptions 1 and 2; for instance, consider a setting where $L=3$ so that the
simplex is a triangle, but there are only two possible signals, so that there
only two induced beliefs.

We should note that this condition is weak in at least two ways. First, in the
standard genericity sense: a generically chosen set of feasible posteriors
will be full rank. Second, and perhaps more interestingly, even in settings
where there are fewer signals than states (e.g., agents can run only a unique
binary-outcome experiment whose outcome determines their posterior), as long
as the agents can acquire enough independent signals (e.g., conduct a
sufficient number of independent experiments), the set of feasible posteriors
is full rank. We formally state this claim below— the proof essentially
repurposes a lemma of Fu, Haghpanah, Hartline, and Kleinberg (2021).

######  Claim 1.

Suppose that signals are conditionally i.i.d. draws from a finite set $S$, and
Assumption 3 is satisfied. If each agent gets $L-1$ independent draws from the
distribution $P_{\omega}^{S}$, then Assumption 4 is satisfied with the signal
space $S^{L-1}$.

We are now in a position to describe our main procedure, Population-Mean-Based
Aggregation (PMBA) (Procedure 2 below) and state our main result for this
setting.

Procedure 2 Population-Mean-Based Aggregation, $L$ states

  1. (1)

Elicit, from each agent $i\in N$, their belief $\mu_{i}$. Calculate
$\hat{\mu}$ defined as:

$\displaystyle\hat{\mu}=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}\mu_{i}$

  2. (2)

Select $L$ agents $I=\\{i_{1},i_{2},\ldots,i_{L}\\}$ such that the $L\times L$
matrix $\bm{\mu}=(\mu_{i_{1}}^{T},\mu_{i_{2}}^{T},\ldots,\mu_{i_{L}}^{T})^{T}$
is full rank.

  3. (3)

Let $s_{i}$ be the signal seen by agent $i$. Elicit from each of these agents
their expectation of the average posterior beliefs in the population, i.e.,
elicit for each $i\in I$:

$\displaystyle\alpha_{i}=\lim_{n\rightarrow\infty}\mathbb{E}\left[\frac{1}{n}\sum_{j=1}^{n}\tilde{\mu}_{j}\bigg{|}s_{i}\right].$

Denote
$\bm{\alpha}=(\alpha_{i_{1}}^{T},\alpha_{i_{2}}^{T},\ldots,\alpha_{i_{L}}^{T})^{T}$.

  4. (4)

Solve

$\displaystyle\bar{\bm{\mu}}=\bm{\mu}^{-1}\bm{\alpha}$ (8)

Note that the right-hand side is elicited from agents directly and therefore
$\bar{\bm{\mu}}$ can be recovered. The state $\omega$ is the row of the
recovered $\bar{\bm{\mu}}$ that equals the calculated $\hat{\mu}$.

######  Theorem 2.

Suppose $L>2$ and the common prior $P$ satisfies Assumptions 1, 3, and 4. Then
the PMBA procedure (Procedure 2) recovers the true unknown state of nature
$\omega$ almost surely.

##  3\. Extensions

In this section, we show how to extend the basic PMBA to “more realistic”
settings. In Section 3.2, we show a variant of the procedure that works even
if the individual agents have potentially incorrect information about the
aggregate distribution of signals. In Section 3.1, we recognize that, in
practice, eliciting beliefs from agents is difficult—we show that a variant of
the procedure can aggregate information if we only elicit “simple” information
(e.g., report which state is thought more likely) from most agents, and elicit
more fine-grained information from only a small number of agents. Both
extensions are displayed for the case of $2$ states of the world.

###  3.1. Aggregating Guesses instead of Beliefs

In this section we discuss how our PMBA mechanism can be adjusted to
environments where not beliefs but rather finite guesses are elicited, e.g.,
which state is more likely rather than what is the probability of state
$\omega=1$ (this is also the original formulation in Prelec, Seung, and McCoy
(2017)—see Section 5.1). Consider a binary state space and an information
structure that satisfies Assumptions 1 and 2. The action-based PMBA procedure
is defined in Figure 3.

Procedure 3 Action-based PMBA, 2 states

  1. (1)

Elicit from each agent $i\in N$, the state $\omega\in{0,1}$ they consider more
likely, $a_{i}\in{0,1}$. If their belief equals 0.5, have them report
$a_{i}=1$. Calculate $\hat{\alpha}$ defined as:

$\displaystyle\hat{\alpha}=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}$

  2. (2)

Select $2$ agents, A and B, such that $a_{A}\neq a_{B}$. Elicit from each of
these agents $i=A,B$ their belief $\mu_{i}$ and their expectation of the
average report in the population, i.e., denoting $s_{i}$ as the signal seen by
agent $i$, elicit for each $i=A,B$:

$\displaystyle\alpha_{i}=\lim_{n\rightarrow\infty}\mathbb{E}\left[\frac{1}{n}\sum_{j=1}^{n}a_{j}\bigg{|}s_{i}\right].$

  3. (3)

Calculate $\bar{\alpha}(\omega_{1}),\bar{\alpha}(\omega_{2})$ as:

$\displaystyle\left(\begin{array}[]{c}\bar{\alpha}(\omega_{1})\\\
\bar{\alpha}(\omega_{2})\end{array}\right)=\left(\begin{array}[]{cc}\mu_{A}&1-\mu_{A}\\\
\mu_{B}&1-\mu_{B}\end{array}\right)^{-1}\left(\begin{array}[]{c}\alpha_{A}\\\
\alpha_{B}\end{array}\right)$ (15)

Note that the right-hand side is elicited from agents directly; and since
$\alpha_{A}\neq\alpha_{B}$, the middle matrix is invertible.

  4. (4)

Recover the state by comparing $\hat{\alpha}$ calculated in step $1$ with
$\bar{\alpha}(\cdot)$ calculated above. The state of the world is $\omega$
s.t. $\hat{a}=\bar{a}(\omega).$

For the action based PMBA procedure to be viable we require the existence of
two agents $A,B$ with differing first-order guesses, $a_{A}\neq a_{B}$.
Assumptions 1 and 2 are not sufficient to assure this. The following
assumption, however, guarantees the existence of two such agents.

######  Assumption 5 (No Herding).

We assume that there exists $\delta>0$, such that for any agent $i$ and any
state $\omega_{j}=\omega_{1},\omega_{2}$, we have that the probability agent
$i$ assigns to state $\omega_{j}$ conditional on their signal $s_{i}$ is
larger than half with a probability of at least $\delta$.

We can now present our result on aggregation in this environment.

######  Proposition 1.

Consider a binary state space. Suppose that the information structure
satisfies Assumptions 1, 2, and 5. Then the _Action Based PMBA_ procedure
(Procedure 3) correctly recovers the true state of the world a.s.

The result follows along the same lines as Theorem 1.

###  3.2. Aggregation in Misspecified Information Settings

In what follows, we show that even if the agents have limited and individually
incorrect knowledge about the aggregate information structure, nevertheless
similar ideas can be used to aggregate population information. That is,
eliciting first-order beliefs from all, and a second-order statistic from a
subgroup of agents will be sufficient to learn the state. In the interests of
brevity, we consider the case of two states; the extension to $L$ states is
analogous to the previous section.

We maintain the assumptions on the true signal generating process $P$ as in
Section 2.2, i.e., Assumptions 1 and 2. Instead of assuming common knowledge
of $P$, we assume that the agents’ information about $P$ is misspecified—agent
$i$ believes that the state-dependent population averages are
$(\alpha_{i}^{\omega_{1}},\alpha^{\omega_{2}}_{i})\in[0,1]^{2}$. Note that
each agent understands how to interpret their own signal correctly, i.e., the
misspecification is purely about knowledge of the population averages. We make
the following assumption on the relation between agents’ beliefs and the true
state-dependent population means.

######  Assumption 6.

We make the following assumption regarding the agents’ knowledge about $P$:

For each agent $i$, and each state $\omega\in\Omega$ we have that:

$\alpha_{i}^{\omega}=\overline{\mu}(\omega)+\zeta_{i}^{\omega}$

where, conditional on state $\omega$, $\zeta_{i}^{\omega}$ has mean 0, and
further is independent of the agent’s signal and other agents’ $\zeta$’s,
i.e., $\zeta_{i}^{\omega}\perp s_{i}$, $\forall i\neq j$,
$\zeta_{i}^{\omega}\perp\zeta_{j}^{\omega}$. Further, we assume that
$\alpha_{i}^{\omega_{1}}$ and $\alpha_{i}^{\omega_{2}}$ have disjoint support.

Let us briefly discuss Assumption 6. One implication of this is that each
agent understands that conditional on the state their expectation of the
population mean is independent of their first-order belief. The assumption
that supports are disjoint is necessary keeping in mind that agents are fully
Bayesian in their inferences, and they simply have misspecified knowledge. To
be precise, this assumption avoids that their expectation for the population
average in state $\omega_{1}$ is larger than in state $\omega_{2}$ which would
contradict Bayes rationality (see Lemma 2 for further details).

Procedure 4 Limited Information Population-Mean-Based Aggregation

  1. (1)

Elicit, from each agent $i$, their belief $\mu_{i}$. Calculate $\hat{\mu}$
defined as:

$\displaystyle\hat{\mu}=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}\mu_{i}$

  2. (2)

Elicit from all agents their interim expectation of the average posterior
belief over all agents—denote this as $\alpha_{i}$ for agent $i$.

Let group $A$ be the set of agents $i$ such that $\mu_{i}\leq\hat{\mu}$, and
$B$ all the other agents. Calculate the average of the $\alpha_{i}$’s in each
group, $A,B$. For example, without loss, rename agents so that
$A=\\{1,2,3,\ldots\\}$. Then define

$\displaystyle\alpha_{A}=\lim_{n\to\infty}\frac{1}{n}\sum_{i\leq
n}\alpha_{i},$ $\displaystyle\mu_{A}=\lim_{n\to\infty}\frac{1}{n}\sum_{i\leq
n}\mu_{i}.$

Calculate $\mu_{B},\alpha_{B}$ analogously.

  3. (3)

Calculate $\overline{\mu}(\omega)$ as

$\displaystyle\left(\begin{array}[]{c}\overline{\mu}(\omega_{1})\\\
\overline{\mu}(\omega_{2})\end{array}\right)=\left(\begin{array}[]{cc}\mu_{A}&1-\mu_{A}\\\
\mu_{B}&1-\mu_{B}\end{array}\right)^{-1}\left(\begin{array}[]{c}\alpha_{A}\\\
\alpha_{B}\end{array}\right)$ (22)

Recover the state by comparing $\hat{\mu}$ calculated in step $1$ with
$\overline{\mu}(\omega),\omega\in\Omega$ calculated above.

We can now state our aggregation result in the limited information setting.

######  Theorem 3.

Suppose the true signal distribution process satisfies Assumptions 1 and 2,
and that the agents’ knowledge regarding the information structure satisfies
Assumption 6. Then, _Limited Information Population- Mean-Based Aggregation_
(Procedure 4) correctly recovers the true state of the world almost surely.

Theorem 3 establishes that indeed the conceptual idea of second-order
elicitation successfully overcomes the limited understanding of agents’
regarding the private belief distributions of their peers. Compared to the
PMBA aggregation under complete knowledge of the information structure, the
two infinite groups $A$ and $B$ play the roles of the two agents in the PMBA.

Thus moving to a more realistic assumption of an incomplete and individually
incorrect understanding of the information structure necessitates the
elicitation of the expected population averages from large groups: we are
relying on the crowd for both first- and second-order statistics. The main
issue is that individual agents’ understanding of the population averages is
incorrect. However, by the law of large numbers we can appropriately average
these individual agents’ reports over two groups, $A$ and $B$. The average
first-order belief, and average report of population average, of agents in
each group, will serve as the analogs of the two individuals in the baseline
PMBA procedure (Procedure 1).

The basic intuition is that even though individual agents are misinformed
about the distribution of thier fellow agents’ beliefs, these errors are not
systematic. One can show, therefore, that, the constructed
$\alpha_{A},\alpha_{B}$ for these groups $A$ and $B$ would be the same as
those of agents with beliefs $\mu_{A},\mu_{B}$ respectively who also were
correctly informed about the distribution of agents’ beliefs. Assumption 6
makes this possible.

###  3.3. Incentivizing Truthful Reporting

In our analysis so far we abstracted away from incentive issues, assuming
truth-telling. We now discuss how truthful elicitation and aggregation can
simultaneously be achieved via a slightly adjusted mechanism that includes
payments which ensure truthful reporting is a (strict) Bayes Nash Equilibrium.

Recall that when there is a verifiable state that is or will be known to the
principal, the principal can truthfully elicit agents’ beliefs about the state
by paying them using a proper scoring rule. A proper scoring rule pays agents
as a function of the reported beliefs and the true state so that for any
agent, truthful reporting of beliefs is a dominant strategy. For example,
classic scoring rules such as the Brier score (Brier, 1950) or the logarithmic
scoring rule make truthful reporting a dominant strategy for expected utility
maximizing agents.999There is a large literature proposing more robust scoring
rules for various settings, see e.g., Karni (2009) for an example.

When the state is not verifiable/observable to the principal, proper scoring
rules are not applicable. However, the principal may nevertheless be
interested in ensuring that agents are incentivized to report their beliefs
truthfully. A leading example from Prelec (2004) is the case of subjective
surveys/evaluations of a new product. In their example, the principal (e.g., a
marketer) wishes to incentivize truthful reporting by agents, but there is no
“ground truth” on which to base the rewards. The basic idea of Bayesian Truth
Serum of Prelec (2004) is that agents’ beliefs also have implications on their
second-order beliefs (i.e., how they think the population will evaluate them).
The population averages are indeed verifiable, and therefore can be used to
incentivize agents. A similar idea works here, we describe it loosely for
completeness:

  1. (1)

Elicit beliefs and second-order beliefs from agents as prescribed in PMBA to
learn the true state $\omega$ (which variant of PMBA is used, and what is
elicited, depends on the setting).

  2. (2)

For the agents who were asked for second-order beliefs, pay them based on a
proper scoring rule as a function of their expected population average
beliefs, and the true calculated population average beliefs.

  3. (3)

For all agents, pay them using a proper scoring rule as a function of their
beliefs and the state $\omega$ recovered by PMBA.

Observe that under the maintained assumptions of correctness of the
corresponding PMBA mechanism, truthful reporting will be a perfect Bayesian
equilibrium among agents. To understand why, first note that since there is a
large population, the average beliefs are effectively observed: even if a
single agent misreports, this does not affect the average. Therefore,
rewarding the two agents with a proper scoring rule based on their expected
population average beliefs is feasible and will incentivize truthful
reporting. These agents have no incentive to misreport their first-order
beliefs because there is no associated payment. Note that given this, the true
underlying state can be correctly calculated. A proper scoring rule against
this recovered state ensures the truthful reporting of first-order beliefs for
all the agents. Finally, there is no “compound” incentive for an agent who
reports both first- and second-order beliefs to misreport their beliefs: their
incentive to report their second-order beliefs truthfully was already argued,
and misreporting their first-order beliefs will cause the PMBA to fail and
therefore will void payments.

##  4\. Aggregation with Finite Populations

So far, we have discussed the possibility of aggregating information with an
infinite population. In practice, of course, populations are finite. The
procedures given above will correctly identify the true state with high
probability in large populations. For instance, in our baseline model, even
with a large but finite population, the average belief in the population will
concentrate around the expected belief conditional on the true state. As long
as the expected beliefs conditional on state are sufficiently different across
the states, an appropriately modified procedure will recover the true state of
the world with high probability.

Nevertheless, at a theoretical level, one may wonder if, with a finite
population, there is any value to eliciting higher-order beliefs, ignoring the
difficulties of such elicitation in practice. The benchmark we use to assess
this is one of a “full information posterior,” i.e., can we, without knowledge
of the underlying information structure $P$, and eliciting solely agents
beliefs (and higher-order beliefs), nevertheless reach the same posterior
beliefs as an omniscient agent who knew $P$ and directly observed all the
agents’ signals? Our previous results answered this in the positive and showed
that for the case of an infinite population (under the maintained
assumptions), the output of the PMBA could identify the degenerate belief on
the true state.

In this section, we answer these questions primarily in the negative. First we
show that with a finite population, and with elicitation of the entire
hierarchy of beliefs, an agnostic mechanism can learn the prior and signals of
each agent. This result is essentially a straightforward corollary of the
results of Mertens, Sorin, and Zamir (2015). However, elicitation of the
entire hierarchy is obviously impractical in real-world applications.

By contrast, we show that, for elicitation up to any finite-order of beliefs,
there is an identification problem: there exist information structures where
the exact same finite hierarchy can be realized among agents in both states.

###  4.1. Eliciting the entire hierarchy of beliefs

Consider a finite set of agents $\left\\{1,2,...,N\right\\}$ with a common
prior $P$ defined on a finite set $\Omega\times S$ (where
$S=\times_{i=1}^{N}S_{i}$ is the set of their signal profiles). The aggregator
does not know $P$ but can ask the agents to report their higher-order beliefs.
In this section, we show that the aggregator can effectively elicit the full
information posterior, provided that they can ask the agents to report their
entire hierarchy of beliefs and that each hierarchy of beliefs uniquely
identifies a signal of an agent.

To this end, we recall the standard formulation of higher-order beliefs by
Mertens and Zamir (1985). Denote by $s_{i}^{k}$ the $k$th-order belief of
agent $i$ over $\Omega$. For instance,
$s_{i}^{1}=$marg${}_{\Omega}P\left(\cdot|s_{i}\right)$,

$s_{i}^{2}\left(\omega,s_{-i}^{1}\right)=P\left(\left\\{\left(\omega^{\prime},s_{-i}^{\prime}\right):\omega^{\prime}=\omega\text{
and }s_{-i}^{\prime 1}=s_{-i}^{1}\right\\}|s_{i}\right)\text{,}$

and so on. Denote by $\tilde{s}_{i}$ the hierarchy of beliefs induced by
signal $s_{i}$, i.e., $\tilde{s}_{i}=\left(s_{i}^{1},s_{i}^{2},...\right)$.
Let $\tilde{S}_{i}$ be the set of hierarchies of beliefs induced from $S_{i}$
and $\tilde{P}$ the distribution induced by $P$ on $\Omega\times\tilde{S}$.
Moreover, by Mertens and Zamir (1985), each $\tilde{s}_{i}$ induces a belief
$\tilde{\pi}_{i}\left(\tilde{s}_{i}\right)$ over $\Omega\times\tilde{S}_{-i}$,
where $\tilde{S}_{-i}=\times_{j\neq i}\tilde{S}_{j}$.

######  Theorem 4.

Suppose that $\tilde{\mu}$ has a finite support and the agents report
$\tilde{s}=\left(\tilde{s}_{i}\right)_{i}$. Then there exists a procedure
which recovers the “pooled information” posterior on the states, i.e.,
$\tilde{P}(\cdot|\tilde{s}).$

###  4.2. Eliciting Higher-Order Beliefs up to a Finite Order $m$

We now explain that the above cannot be done if the aggregator only knows the
reported beliefs up to order $m$. Our argument is adapted from the leading
example of Lipman (2003).

We adopt his notation here for ease of comparison. Suppose there are two
players and two states of nature $\left\\{\omega_{1},\omega_{2}\right\\}$ as
in our paper. The construction is easier to explain in terms of the standard
partitional model of knowledge.

The model below considers $8$ extended states:
$\\{\left(\sigma_{l},k\right):l\in\\{1,2\\}\text{ and }k\in\\{1,2,3,4\\}\\}$.
Interpret state $(\sigma_{l},k)$ in this model as corresponding to
$\omega_{l}$ realized for $l=1,2$ and any $k$. In this example, Lipman
considers the hierarchy of belief induced by common knowledge that player $1$
assigns probability $2/3$ to $\omega_{1}$ and player $2$ assigns probability
$1/3$ to $\omega_{1}$. Such a type does not admit a common prior. However,
Lipman shows that the model below, which has a uniform common prior, admits a
state $\left(\sigma_{1},1\right)$ where the players have the same belief as
that of the common knowledge type described above, up to any finite order $m$.

The prior for the players is the uniform distribution over the extended
states, i.e.:

| $\left(\sigma_{1},4\right)$ | $\left(\sigma_{1},3\right)$ | $\left(\sigma_{2},2\right)$ | $\left(\sigma_{1},1\right)$ | $\left(\sigma_{2},1\right)$ | $\left(\sigma_{1},2\right)$ | $\left(\sigma_{2},3\right)$ | $\left(\sigma_{2},4\right)$
---|---|---|---|---|---|---|---|---
$\mu$ | $\frac{1}{8}$ | $\frac{1}{8}$ | $\frac{1}{8}$ | $\frac{1}{8}$ | $\frac{1}{8}$ | $\frac{1}{8}$ | $\frac{1}{8}$ | $\frac{1}{8}$

Agents’ information is identified with the following partitions:

$\displaystyle\Pi_{1}$ $\displaystyle=$
$\displaystyle\left\\{\left\\{\left(\sigma_{1},4\right),\left(\sigma_{1},3\right),\left(\sigma_{2},2\right)\right\\}\left\\{\left(\sigma_{1},1\right),\left(\sigma_{2},1\right),\left(\sigma_{1},2\right)\right\\}\text{
}\left\\{\left(\sigma_{2},3\right),\left(\sigma_{2},4\right)\right\\}\right\\}$
$\displaystyle\Pi_{2}$ $\displaystyle=$
$\displaystyle\left\\{\left\\{\left(\sigma_{1},4\right),\left(\sigma_{1},3\right)\right\\}\left\\{\left(\sigma_{2},2\right),\left(\sigma_{1},1\right),\left(\sigma_{2},1\right)\right\\}\left\\{\left(\sigma_{1},2\right),\left(\sigma_{2},3\right),\left(\sigma_{2},4\right)\right\\}\right\\}\text{.}$

Note that each $\Pi_{i}\left(\sigma^{\prime},k\right)$ is identified with a
signal of player $i$. Observe that player 1 assigns probability one to
$\omega_{2}$ at $\Pi_{1}\left(\sigma_{2},4\right)$ and player $2$ assigns
probability one to $\omega_{1}$ at $\Pi_{2}\left(\sigma_{1},4\right)$. Hence,
each partition cell of each player induces a different second-order belief. In
particular,
$\left(\Pi_{1}\left(\sigma_{1},1\right),\Pi_{2}\left(\sigma_{1},1\right)\right)$
is the only partition profile at which the second-order beliefs of both
players are identical to $t$. Moreover, conditional on the reported second-
order belief at
$\left(\Pi_{1}\left(\sigma_{1},1\right),\Pi_{2}\left(\sigma_{1},1\right)\right)$
(denoted as
$\left(\Pi_{1}\left(\sigma_{1},1\right),\Pi_{2}\left(\sigma_{1},1\right)\right)^{2}$),
we have

$\mu\left(\omega_{1}|\left(\Pi_{1}\left(\sigma_{1},1\right),\Pi_{2}\left(\sigma_{1},1\right)\right)\right)=\frac{1}{2}\text{.}$

Now consider another model modified from the previous one by adding one
additional state with the following prior:

| $\left(\sigma_{1},4\right)^{\prime}$ | $\left(\sigma_{1},3\right)^{\prime}$ | $\left(\sigma_{2},2\right)^{\prime}$ | $\left(\sigma_{1},1\right)^{\prime}$ | $\left(\sigma_{1},1\right)$ | $\left(\sigma_{2},1\right)$ | $\left(\sigma_{1},2\right)$ | $\left(\sigma_{2},3\right)$ | $\left(\sigma_{2},4\right)$
---|---|---|---|---|---|---|---|---|---
$\mu^{\prime}$ | $\frac{1}{20}$ | $\frac{1}{20}$ | $\frac{1}{10}$ | $\frac{1}{10}$ | $0$ | $\frac{1}{10}$ | $\frac{1}{5}$ | $\frac{1}{5}$ | $\frac{1}{5}$

where ′ is to indicate that a state is to the left of
$\left(\sigma_{1},1\right)$ and it will be useful in generalizing the idea to
eliciting $m$ orders of beliefs, for $m\geq 3$ in what follows. Agents’
information is now given by the partitions:

$\displaystyle\Pi_{1}^{\prime}$ $\displaystyle=$
$\displaystyle\left\\{\left\\{\left(\sigma_{1},4\right)^{\prime},\left(\sigma_{1},3\right)^{\prime},\left(\sigma_{2},2\right)^{\prime},\left(\sigma_{1},1\right)^{\prime}\right\\},\left\\{\left(\sigma_{1},1\right),\left(\sigma_{2},1\right),\left(\sigma_{1},2\right)\right\\}\text{
}\left\\{\left(\sigma_{2},3\right),\left(\sigma_{2},4\right)\right\\}\right\\}\text{;}$
$\displaystyle\Pi_{2}^{\prime}$ $\displaystyle=$
$\displaystyle\left\\{\left\\{\left(\sigma_{1},4\right)^{\prime},\left(\sigma_{1},3\right)^{\prime}\right\\},\left\\{\left(\sigma_{2},2\right)^{\prime},\left(\sigma_{1},1\right)^{\prime},\left(\sigma_{1},1\right),\left(\sigma_{2},1\right)\right\\},\left\\{\left(\sigma_{1},2\right),\left(\sigma_{2},3\right),\left(\sigma_{2},4\right)\right\\}\right\\}\text{.}$

Observe now that, conditional on the reported second-order belief at
$\left(\Pi_{1}^{\prime}\left(\sigma_{1},1\right),\Pi_{2}^{\prime}\left(\sigma_{1},1\right)\right)$
(denoted as
$\left(\Pi_{1}^{\prime}\left(\sigma_{1},1\right),\Pi_{2}^{\prime}\left(\sigma_{1},1\right)\right)^{2}$),
we have

$\mu^{\prime}\left(\omega_{1}|\left(\Pi_{1}^{\prime}\left(\sigma_{1},1\right),\Pi_{2}^{\prime}\left(\sigma_{1},1\right)\right)^{2}\right)=0\text{.}$

Second, observe that

$\left(\Pi_{1}^{\prime}\left(\sigma_{1},1\right),\Pi_{2}^{\prime}\left(\sigma_{1},1\right)\right)^{2}=\left(\Pi_{1}\left(\sigma_{1},1\right),\Pi_{2}\left(\sigma_{1},1\right)\right)^{2}\text{.}$

This means that the reported second-order beliefs at
$\left(\Pi_{1}^{\prime}\left(\sigma_{1},1\right),\Pi_{2}^{\prime}\left(\sigma_{1},1\right)\right)$
under $\mu^{\prime}$ are the same as at
$\left(\Pi_{1}\left(\sigma_{1},1\right),\Pi_{2}\left(\sigma_{1},1\right)\right)$
under $\mu$. The basic idea is that in $\mu^{\prime}$, we “shift” the
probability which $\mu$ assigns to $\left(\sigma_{1},1\right)$ to the
additional state $\left(\sigma_{1},1\right)^{\prime}$. This additional state
helps us preserve the first-order belief of player $2$ at
$\Pi_{2}^{\prime}\left(\sigma_{1},1\right)$ at $1/3$, while we decrease the
probability $\left(\sigma_{1},4\right)$ to $0$ to preserve the first-order
belief of player $1$ at
$\Pi_{1}^{\prime}\left(\left(\sigma_{1},4\right)^{\prime}\right)$. This takes
care of the states to the left of $\left(\sigma_{1},1\right)$. For states to
the right, we double the probability of $\left(\sigma_{2},2\right)$,
$\left(\sigma_{2},3\right)$, and $\left(\sigma_{2},4\right)$, so that the
first-order belief of player $1$ at
$\Pi_{1}^{\prime}\left(\sigma_{1},1\right)$ and the first-order belief of
player $2$ at $\Pi^{\prime}$ are also preserved. Therefore, just eliciting the
first two orders of beliefs, we cannot distinguish between the model
corresponding to $\mu$ (under which the posterior would be that both states
are equally likely), and $\mu^{\prime}$ (under which the posterior would be
that the state is $\omega_{2}$ for sure).

The construction for $m>2$ is similar but more involved. The details are in
Appendix C.1.

##  5\. Related Literature and Conclusions

In what follows, we discuss the related literature more closely (with the
benefit of being able to explain some of the ideas more formally with our
notation). We first discuss the SP mechanism of Prelec, Seung, and McCoy
(2017) in some detail. Then we discuss other related literature and
approaches. Finally, we conclude.

###  5.1. The Surprisingly Popular Mechanism

The original SP mechanism considers the case analogous to what we consider in
Section 3.1—eliciting from each agent a “guess” or “vote” of the state they
consider most likely given their beliefs. It also considers the case of
directly eliciting beliefs (see Section 1.4 of the Supplementary Information
Section of Prelec, Seung, and McCoy (2017)). We focus on that for ease of
comparison to our core mechanism of PMBA. Analogous comments would apply,
_mutatis mutandis_ , to the case of eliciting votes.

We formally define the mechanism in our general environment, generalize the
existing results, and highlight its limitation relative to PMBA. To start,
consider the case of binary states and let the common prior P among agents
satisfy Assumption 1 and Assumption 2. In our notation, the SP mechanism can
be described as follows:

Procedure 5 Surprisingly Popular, 2 states

  1. (1)

Elicit from each agent $i\in N$, their belief $\mu_{i}$. Calculate $\hat{\mu}$
defined as:

$\displaystyle\hat{\mu}=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}\mu_{i}$

  2. (2)

Select one agent, $i$, and elicit their expectation of the average posterior
beliefs of the population, i.e., denoting $s_{i}$ as the signal seen by agent
$i$:

$\displaystyle\alpha_{i}=\lim_{n\rightarrow\infty}\mathbb{E}\left[\frac{1}{n}\sum_{j=1}^{n}\tilde{\mu}_{j}\bigg{|}s_{i}\right].$

  3. (3)

Compare the realized population average belief $\hat{\mu}$ with agent $i$’s
expectation $\alpha_{i}$. If $\hat{\mu}<\alpha_{i}$, the state of the world is
$\omega=1$, otherwise it is $\omega=0$.

The SP procedure is arguably even simpler than our PBMA procedure. Only one
agent is required to report their expectation of the population average and no
calculation is required. Nevertheless, SP can recover the truth in the binary
state setting as the following result shows.

######  Proposition 2.

Consider a binary state space. Suppose that the information structure among
agents satisfies Assumptions 1 and 2. Then the _SP_ procedure (Procedure 5)
correctly recovers the true state of the world.

Note that this result provides a stronger theoretical guarantee of the
applicability of the SP procedure: the existing results about SP require
significantly stronger assumptions on primitives. We omit a formal proof. At a
high-level, this proceeds straightforwardly from Lemma 2: In a two-state
setting (with two or more signals), for any agent $i$, the induced
distribution of beliefs on $\omega$ at state $\omega$ first-order
stochastically dominates the induced distribution of the belief on $\omega$ at
state $\omega^{\prime}\neq\omega$. Therefore the population average belief of
state $\omega$ in $\omega$ must be higher than the population average belief
of state $\omega$ in state $\omega^{\prime}$. An agent offering their expected
population average belief must be submitting a convex combination of these
two. Therefore if the true state is indeed $\omega$, it will be “surprisingly
popular” relative to this expected belief.

In summary, the SP procedure is extremely powerful in aggregating beliefs in
the binary state setting. However, it is not applicable beyond two states
without strong additional assumptions. The reason is that, in general, more
than two states can be “surprisingly popular”, leaving an indeterminacy in how
to define the mechanism. Even when defining the SP procedure as e.g., picking
the state that surpassed by most its expectation, the procedure does not
achieve aggregation. This issue is discussed in Section 1.3 of the
Supplementary Information of Prelec, Seung, and McCoy (2017). The authors
impose an additional assumption (see e.g., Theorem 3), one of “diagonal
dominance,” in which roughly the posterior of an agent who thinks state
$\omega$ is most likely must also place a higher posterior on $\omega$ than
any voter voting for $\omega^{\prime}\neq\omega$ places on $\omega$. Simple
counter examples show that this assumption is necessary. We exhibit one below.

######  Example 1.

Suppose there are three states
$\Omega=\\{\omega_{1},\omega_{2},\omega_{3}\\}$, and three signals
$S=\\{s_{1},s_{2},s_{3}\\}$. All agents have the initial uniform prior. Agents
have a uniform prior over signals and receive conditionally i.i.d. signals,
conditional on the state, so that their posteriors can be described as:

$\displaystyle\begin{array}[]{c|ccc}&P(\omega_{1}|\cdot)&P(\omega_{2}|\cdot)&P(\omega_{3}|\cdot)\\\
\hline\cr s_{1}&0.4&0.21&0.39\\\ s_{2}&0.45&0.54&0.01\\\
s_{3}&0.44&0.06&0.5\end{array}$

i.e., the $i^{\text{th}}$ row and $j^{\text{th}}$ column is the posterior the
agent places on $\omega_{j}$ upon signal $s_{i}$, $P(\omega_{j}|s_{i})$. Note
that this violates the assumption of “diagonal dominance”—
$P(\omega_{1}|s_{2})>P(\omega_{1}|s_{1})$, i.e., an agent who sees $s_{2}$
(or, indeed $s_{3}$) places a higher posterior on $\omega_{1}$ than an agent
who sees signal $s_{1}$. However note that an agent who sees signal $s_{i}$
believes that $\omega_{i}$ is the most likely state.

To be clear, note that this can be achieved by the following distribution of
signals conditional on the state (numbers rounded):

$\displaystyle\begin{array}[]{c|ccc}&P(\cdot|\omega_{1})&P(\cdot|\omega_{2})&P(\cdot|\omega_{3})\\\
\hline\cr s_{1}&0.31&0.259&0.433\\\ s_{2}&0.349&0.667&0.011\\\
s_{3}&0.341&0.074&0.556\end{array}$

This results, by mechanical calculation, on the following average beliefs in
the population:

$\displaystyle\begin{array}[]{c|ccc}&\overline{\mu}(\omega_{1})&\overline{\mu}(\omega_{2})&\overline{\mu}(\omega_{3})\\\
\hline\cr\omega_{1}&0.431&0.436&0.422\\\ \omega_{2}&0.274&0.419&0.130\\\
\omega_{3}&0.295&0.145&0.447\end{array}$

here the $i^{\text{th}}$ row and and $j^{\text{th}}$ column represent the
average belief in the population that the state is $\omega_{i}$ when the true
state is $\omega_{j}$.

Note that the resulting expected population average belief is:

$\displaystyle\begin{array}[]{c|ccc}&\alpha(s_{1})&\alpha(s_{2})&\alpha(s_{3})\\\
\hline\cr\omega_{1}&0.429&0.434&0.427\\\ \omega_{2}&0.248&0.351&0.211\\\
\omega_{3}&0.323&0.215&0.362\end{array}$

i.e., the column labeled $\alpha(s_{i})$ is the expected population average
belief of an agent seeing signal $s_{i}$.

More generally, we can summarize the set of “surprisingly popular” state(s) in
each case as:

$\displaystyle\begin{array}[]{c|ccc}&\omega_{1}&\omega_{2}&\omega_{3}\\\
\hline\cr\alpha(s_{1})&\omega_{1},\underline{\omega_{2}}&\omega_{1},\underline{\omega_{2}}&\omega_{3}\\\
\alpha(s_{2})&\omega_{3}&\omega_{1},\underline{\omega_{2}}&\omega_{3}\\\
\alpha(s_{3})&\omega_{1},\underline{\omega_{2}}&\omega_{1},\underline{\omega_{2}}&\omega_{3}\end{array}$

i.e., the entry in row $i$ corresponding to $\alpha(s_{i})$ and column $j$
corresponding to state $\omega_{j}$ is the set of states that are surprisingly
popular, in true state $\omega_{j}$, relative to the expected population
average beliefs of an agent who received signal $s_{i}$. The underlined entry,
if there are multiple, is the one which is surprising by the largest
magnitude. For example, relative to an agent seeing $s_{1}$ in true state
$\omega_{1}$, both $\omega_{1}$ and $\omega_{2}$ are surprisingly popular, but
$\omega_{2}$ is the most surprisingly popular:
$(0.274-0.248=0.028>0.02=0.431-0.429)$.

Note that in our example, the “most surprisingly popular” mechanism fails to
identify the true state when the state is $\omega_{1}$, regardless of the
signal of who is polled for their expectation about the population average
beliefs. Indeed, if an agent who saw signal $s_{2}$ is polled, the true state
is not even in the set of surprisingly popular states. Note also the failure
of first-order stochastic dominance of population average beliefs in this
3-state example—as we argued earlier, that was key to why SP works with binary
states, but this can be violated with three or more states.

Another closely related approach proposed is that of “prediction normalized
votes,” each agent votes for the state they believe is more likely (which, in
this example, was the signal they saw, i.e., an agent votes for state
$\omega_{i}$ if they see signal $s_{i}$). The fraction of votes each state
$\omega_{j}$ receives is normalized by the sum, over all states $\omega_{k}$,
the ratio of predicted vote fraction for $\omega_{k}$ by a $\omega_{j}$ voter
to the predicted vote fraction for $\omega_{j}$ by an $\omega_{k}$ voter. A
simple calculation shows that given the specific numerical assumption we
constructed, $\omega_{2}$ will have a higher prediction normalized vote than
$\omega_{1}$ when the true state is $\omega_{1}$.

######  Remark 4.

One possibility a reader may consider feasible is, in a setting with $L>2$
states $\\{\omega_{1},\ldots,\omega_{L}\\}$, to run SP sequentially. That is
to say first elicit beliefs corresponding to states $\omega_{1}$ and its
complement (states $\\{\omega_{2},\ldots,\omega_{L}\\}$), and run SP on this
“binary state” environment. However, note that in this case, the realized
population average belief conditional on the second “state”
($\\{\omega_{2},\ldots,\omega_{L}\\}$) is not deterministic as a function of
this “state” (i.e., it depends on which of the states $\omega_{1}$ thru
$\omega_{L}$ actually realized). As a result SP does not apply—indeed, the
binary state PMBA cannot be used sequentially for similar reasons, recall
Remark 3.

###  5.2. Other Approaches and Related Literature

The literature on aggregation of information in a population is perhaps far
too vast to comprehensively cite. Multiple strands attempt to understand, in a
variety of settings, when and how existing institutions aggregate dispersed
information in strategic settings. These include studies on information
aggregation in voting, a vast literature with its roots in Condorcet, but more
recently studied formally starting from Feddersen and Pesendorfer (1997);
aggregation in auctions (see e.g., Milgrom and Weber (1982) and subsequent
literature) and of course information aggregation in markets (see e.g.,
Grossman and Stiglitz (1980)).

We limit attention here explicitly to work that studies the design/
possibility of aggregation mechanisms/ algorithms, or institutions that solely
exist to aggregate information.

A key idea in the former space (indeed, in a sense an idea that is also at the
heart of Prelec, Seung, and McCoy (2017)) is the Bayesian Truth Serum (Prelec
(2004)— this is a procedure to truthfully elicit subjective information from
agents by rewarding them as a function of others’ reports. This paper
pioneered the idea that agents’ higher-order beliefs could be used to
incentivize them successfully even if the planner/ designer did not know the
prior among agents. A literature studying this followed. Prelec and Seung
(2006) showed how to modify agents’ reports to better aggregate them (akin to
the prediction normalized voting rule we described in Example 1). Several
extensions to the BTS have been proposed— see e.g., Cvitanić, Prelec, Riley,
and Tereick (2019), Witkowski and Parkes (2012), Radanovic and Faltings
(2013), and Radanovic and Faltings (2014). The latter three papers attempt to
make the basic Bayesian Truth Serum robust, i.e., to ensure that it has good
properties even in finite populations, and to ensure that participation is
individually rational etc.

These ideas have also been successfully employed in real-world applications:
see e.g., Shaw, Horton, and Chen (2011) for an application on Mechanical Turk,
or Rigol, Hussam, and Roth (2021), who use robust BTS to identify high-ability
micro-entrepreneurs by surveying peers. There is also a growing literature
attempting to elicit and use higher-order beliefs to improve aggregation—see
e.g. recent works such as Palley and Soll (2019), Palley and Satopää (2020).
Some of these works consider modifications of the SP algorithm that improve
performance in experiments, e.g., Wilkening, Martinie, and Howe (2021) or
Martinie, Wilkening, and Howe (2020).

On the latter, there is an important literature which has attempted to
understand the performance of prediction markets: see e.g., Wolfers and
Zitzewitz (2004) for an early piece summarizing the issues and Baillon (2017)
for an alternate design. Wolfers and Zitzewitz (2006) study when the
prediction market price can be interpreted as the average of traders’ beliefs,
while Ottaviani and Sørensen (2015) study when the market price underreacts to
new information. Dai, Jia, and Kou (2020) show how to infer the state of the
world from prediction market trading data using these theoretical ideas. There
is also a large literature on pari-mutuel markets, and on adapting them to
aggregate information—see e.g., Pennock (2004).

On a technical level, there is a connection between our approach in this paper
and the work of Libgober (2021). He considers a hypothetical setting where a
single agent conducts an unknown Blackwell experiment. The principal does not
know the agent’s prior, but can elicit, after the experiment, their posterior
and their contingent hypothetical beliefs (their posteriors contingent on
seeing any signal). He shows that the original problem can be recovered from
these contingent hypothetical beliefs. While the applications are of course
very different, at a technical level, our “expected population average
beliefs” serve a similar role to the “contingent hypothetical beliefs” in his
paper. On a related theme, there are the works of Samet (1998a), Samet
(1998b), and Golub and Morris (2017). These papers are interested in interim
characterizations of the common prior, i.e., understanding whether or not
agents have a common prior from the properties of their interim
beliefs/higher-order beliefs.

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##  Appendix A Proofs from Section 2.2

First we prove a general property:

######  Lemma 1.

Suppose Part (2) of Assumption 1 is satisfied. Then for any state $\omega$,

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{\infty}\tilde{\mu}_{i}\to_{P}\overline{\mu}(\omega).$

  * Proof.

The result is established in two steps. For any state $\omega$ below, we show
that agents’ beliefs that the state is $\omega^{\prime}\in\Omega$ satisfy
vanishing correlation. As beliefs lie in $[0,1]$ and therefore have bounded
variance, we can apply the Bernstein law of large numbers. Together, these
imply that when the state of the world is $\omega;$ for any state
$\omega^{\prime}\in\Omega$, we have that:

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{\infty}\tilde{\mu}_{i,\omega^{\prime}}\to_{P}\overline{\mu}_{\omega^{\prime}}(\omega).$

Since there are only a finite set of states, the main result follows.

First, let us prove our claim of vanishing correlations:

######  Claim 2.

Conditioned on state $\omega\in\Omega$, the sequence of (random) beliefs
$\left\langle\tilde{\mu}_{i,\omega^{\prime}}\right\rangle_{i=1}^{\infty}$ that
the state of the world is any $\omega^{\prime}\in\Omega$ satisfies vanishing
correlation, i.e.,

$\lim_{|i-j|\to\infty}\textrm{cov}(\tilde{\mu}_{i,\omega^{\prime}},\tilde{\mu}_{j,\omega^{\prime}}|\omega)=0.$

  * Proof of Claim.

Consider agents $i$ and $j$ such that $\varepsilon$-independence among signals
holds, i.e., for all events $E,E^{\prime}\in\mathcal{F}$ we have

$|P\left[\tilde{s}_{i}\in E,\tilde{s}_{j}\in
E^{\prime}|\omega\right]-P_{i,\omega}^{S}(E)P_{j,\omega}^{S}(E^{\prime})|\leq\varepsilon.$

Let $f_{i}$ denote the conditional probability function that assigns Bayesian
beliefs of state $\omega^{\prime}$ to signals for agent $i$. By measurability
of $f_{i}$’s we have that every event $S,S^{\prime}\subseteq[0,1]$ has a
corresponding event $f^{-1}_{i}(S)$ and $f^{-1}_{j}(S^{\prime})$ such that

$\displaystyle\left|\Pr\left[\tilde{\mu}_{i,\omega^{\prime}}\in
S,\tilde{\mu}_{j,\omega^{\prime}}\in
S^{\prime}\left|\omega\right.\right]-\Pr\left[\tilde{\mu}_{i,\omega^{\prime}}\in
S\left|\omega\right.\right]\Pr\left[\tilde{\mu}_{j,\omega^{\prime}}\in
S^{\prime}\left|\omega\right.\right]\right|$ $\displaystyle=$
$\displaystyle\left|P\left[\tilde{s}_{i}\in f^{-1}_{i}(S),\tilde{s}_{j}\in
f^{-1}_{j}(S^{\prime})\left|\omega\right.\right]-P^{S}_{i,\omega}(f_{i}^{-1}(S))P^{S}_{j,\omega}(f^{-1}_{j}(S^{\prime}))\right|$

establishing approximate independence of beliefs, conditional on the state.

To establish state dependent vanishing correlation among beliefs, consider the
random beliefs
$\tilde{\mu}_{i,\omega^{\prime}},\tilde{\mu}_{j,\omega^{\prime}}$. For any
number $n>1$, let $G_{n}$ denote a partition of $[0,1]$ into $n+1$ equal
intervals, i.e., $G^{k}_{n}=[\frac{k-1}{n},\frac{k}{n})$ for $k=1$ thru $n-1$;
$G^{n}_{n}=[\frac{n-1}{n},1]$.

Note that:

$\displaystyle\mathbb{E}[\tilde{\mu}_{i,\omega^{\prime}}|\omega]\leq\sum_{k=1}^{n}\frac{k}{n}\Pr(\tilde{\mu}_{i,\omega^{\prime}}\in
G^{k}_{n}|\omega)\equiv U_{i}^{n},$
$\displaystyle\mathbb{E}[\tilde{\mu}_{j,\omega^{\prime}}|\omega]\leq\sum_{k=1}^{n}\frac{k}{n}\Pr(\tilde{\mu}_{j,\omega^{\prime}}\in
G^{k}_{n}|\omega)\equiv U_{j}^{n},$
$\displaystyle\mathbb{E}[\tilde{\mu}_{i,\omega^{\prime}}\tilde{\mu}_{j,\omega^{\prime}}|\omega]\leq\sum_{l=1}^{n}\sum_{k=1}^{n}\frac{lk}{n}\Pr(\tilde{\mu}_{i,\omega^{\prime}}\in
G^{l}_{n},\tilde{\mu}_{j,\omega^{\prime}}\in G^{k}_{n}|\omega)\equiv
U_{ij}^{n}.$

Since $\tilde{\mu}_{i,\omega^{\prime}}$ and
$\tilde{\mu}_{j,\tilde{\mu}^{\prime}}$ satisfy $\varepsilon$-independence, we
have that

$\displaystyle|U_{ij}^{n}-U_{i}^{n}U_{j}^{n}|\leq\varepsilon.$

However, as $n$ goes to infinity, by observation,
$U^{n}_{i}\to\mathbb{E}[\tilde{\mu}_{i,\omega^{\prime}}|\omega]$,
$U^{n}_{j}\to\mathbb{E}[\tilde{\mu}_{j,\omega^{\prime}}|\omega]$ and
$U^{n}_{ij}\to\mathbb{E}[\tilde{\mu}_{i,\omega^{\prime}}\tilde{\mu}_{j,\omega^{\prime}}|\omega]$.
Therefore
$\text{cov}(\tilde{\mu}_{i,\omega^{\prime}},\tilde{\mu}_{j,\omega^{\prime}}|\omega)\leq\varepsilon$.

Now, by part (2) of Assumption 1, we know that for any $i$, and
$\varepsilon>0$, $\exists n(\varepsilon)$ large enough so that if $j-i>n$ so
that we have that conditional on state, $\tilde{\mu}_{i}$ and
$\tilde{\mu}_{j}$ are $\varepsilon$-independent. The result trivially follows.
∎

Now, recall the Bernstein law of large numbers (See e.g 3.3.2 of Shiryaev
(2012)):

######  Theorem 5.

Let $\tilde{X}_{i}$, $i=1,2,3\ldots$ be a sequence of real valued random
variables such that (1) $\mathbb{E}[\tilde{X}_{i}]$ is finite for each $i$,
(2) there exists $K$ finite such that $\mathbb{V}[\tilde{X}_{i}]\leq K$ for
all $i$, and (3) $\lim_{|i-j|\rightarrow\infty}\text{cov}(X_{i},X_{j})=0$.
Then

$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\left(\tilde{X}_{i}-\mathbb{E}[X_{i}]\right)\to_{P}0.$

Observe that since $\tilde{\mu}_{i,\omega^{\prime}}$ is a belief, it lies
between $[0,1]$ and so the first two conditions of the Theorem are verified.
Claim 2 verifies the third condition conditional on any true state $\omega$.
Therefore we have the desired result for any $\omega^{\prime}\in\Omega$,
conditional on true state $\omega.$ Since the set of states is finite, we have
convergence in probability of the entire vector. ∎

######  Lemma 2.

Suppose $L=2$. Suppose further that part (1) of Assumption 1 holds and
$P_{i,\omega_{1}}^{S}\neq P_{i,\omega_{2}}^{S}$. Then following statements
hold:

  1. (1)

$\frac{\hat{G}_{i,\omega_{1}}(r)}{\hat{G}_{i,\omega_{2}}(r)}$ is non-
increasing in $r\,$

  2. (2)

$\frac{\hat{G}_{i,\omega_{1}}(r)}{\hat{G}_{i,\omega_{2}}(r)}>1$ for all
$r\in\left(\beta,\bar{\beta}\right)$ where
$\left[\underline{\beta},\bar{\beta}\right]$ denote the convex hull of the
support of private beliefs.101010Since signal generating measures are
absolutely continuous each other, the support is identical across states.

  3. (3)

$\overline{\mu}_{i}(\omega_{1})<\overline{\mu}_{i}(\omega_{2}).$

  * Proof.

By Bayes’ rule, we know that

$r=\frac{dG_{i,\omega_{2}}\left(r\right)P\left(\omega_{2}\right)}{dG_{i,\omega_{1}}\left(r\right)P\left(\omega_{1}\right)+dG_{i,\omega_{2}}\left(r\right)P\left(\omega_{2}\right)}\text{.}$
(23)

We define $\alpha=P(\omega_{2})/P(\omega_{1})$ as the relative likelihood of
the states according to the prior $P.$ Since no signal is completely
informative, i.e., $r\notin\left\\{0,1\right\\}$, we can rewrite (23) as

$\frac{dG_{i,\omega_{1}}}{dG_{i,\omega_{2}}}\left(r\right)=\left(\frac{1-r}{r}\right)\alpha\text{.}$
(24)

Hence, for any $r\in\left(0,1\right)$, we have

$G_{i,\omega_{1}}\left(r\right)=\int_{0}^{r}dG_{i,\omega_{1}}\left(x\right)=\int_{0}^{r}\left(\frac{1-x}{x}\right)\alpha
dG_{i,\omega_{2}}\left(x\right)\geq\left(\frac{1-r}{r}\right)\alpha
G_{i,\omega_{2}}\left(r\right)\text{.}$ (25)

#### Part (1)

Pick any $r^{\prime},r$ s.t. $r^{\prime}>r$.Observe that:

$\displaystyle\frac{G_{i,\omega_{1}}(r)}{G_{i,\omega_{2}}(r)}-\frac{G_{i,\omega_{1}}(r^{\prime})}{G_{i,\omega_{2}}(r^{\prime})},$
$\displaystyle=$
$\displaystyle\frac{G_{i,\omega_{1}}(r)G_{i,\omega_{2}}(r^{\prime})-G_{i,\omega_{1}}(r^{\prime})G_{i,\omega_{2}}(r)}{G_{i,\omega_{2}}(r)G_{i,\omega_{2}}(r^{\prime})}.$
Define $\Delta_{i,\omega}=G_{i,\omega}(r^{\prime})-G_{i,\omega}(r)$. We then
have: $\displaystyle=$
$\displaystyle\frac{G_{i,\omega_{1}}(r)\Delta_{i,\omega_{2}}-\Delta_{i,\omega_{1}}G_{i,\omega_{2}}(r)}{G_{i,\omega_{2}}(r)G_{i,\omega_{2}}(r^{\prime})},$
Observe that
$\Delta_{i,\omega_{1}}=\int_{r}^{r^{\prime}}dG_{i,\omega_{1}}(x)=\int_{r}^{r^{\prime}}\left(\frac{1-x}{x}\right)\alpha
dG_{i,\omega_{2}}\left(x\right)\leq\left(\frac{1-r}{r}\right)\alpha\Delta_{i,\omega_{2}}$.
Substituting in, $\displaystyle\geq$
$\displaystyle\frac{G_{i,\omega_{1}}(r)\Delta_{i,\omega_{2}}-\frac{1-r}{r}\alpha\Delta_{i,\omega_{2}}G_{i,\omega_{2}}(r)}{G_{i,\omega_{2}}(r)G_{i,\omega_{2}}(r^{\prime})},$
$\displaystyle\geq$ $\displaystyle 0,$

where the last inequality follows by substituting in $G_{i,\omega_{1}}(r)$ in
the numerator from (25).

#### Part (2)

It follows from (25) that

$G_{i,\omega_{1}}\left(r\right)\geq G_{i,\omega_{2}}\left(r\right)\text{ if
}\left(\frac{1-r}{r}\right)\alpha\geq 1\text{.}$ (26)

If $\left(\frac{1-r}{r}\right)\alpha<1$, then

$\displaystyle 1-G_{i,\omega_{1}}\left(r\right)$ $\displaystyle=$
$\displaystyle\int_{r}^{1}dG_{i,\omega_{1}}\left(x\right)$ (27)
$\displaystyle=$ $\displaystyle\int_{r}^{1}\left(\frac{1-x}{x}\right)\alpha
dG_{i,\omega_{2}}\left(x\right)$ $\displaystyle\leq$
$\displaystyle\left(\frac{1-r}{r}\right)\alpha\left(1-G_{i,\omega_{2}}\left(r\right)\right)$
$\displaystyle\leq$ $\displaystyle 1-G_{i,\omega_{2}}\left(r\right)\text{.}$

Combining (26) and (27), we obtain that

$\frac{G_{i,\omega_{1}}\left(r\right)}{G_{i,\omega_{2}}\left(r\right)}\geq
1\text{.}$ (28)

To prove
$\frac{G_{i,\omega_{1}}\left(r\right)}{G_{i,\omega_{2}}\left(r\right)}>1$ for
any $r\in\left(\underline{\beta},\overline{\beta}\right)$, suppose to the
contrary that
$\frac{G_{i,\omega_{1}}\left(r\right)}{G_{i,\omega_{2}}\left(r\right)}=1$ for
some $r\in\left(\underline{\beta},\overline{\beta}\right)$. We derive
contradiction in each of the following two cases

Case 1:

$\left(\frac{1-r}{r}\right)\alpha<1$. Then,

$\displaystyle G_{i,\omega_{1}}\left(1\right)$ $\displaystyle=$ $\displaystyle
G_{i,\omega_{1}}\left(r\right)+\int_{r}^{1}dG_{i,\omega_{1}}\left(x\right)$
$\displaystyle=$ $\displaystyle
G_{i,\omega_{2}}\left(r\right)+\int_{r}^{1}dG_{i,\omega_{1}}\left(x\right)$
$\displaystyle=$ $\displaystyle
G_{i,\omega_{2}}\left(r\right)+\int_{r}^{1}\left(\frac{1-x}{x}\right)\alpha
dG_{i,\omega_{2}}\left(x\right)$ $\displaystyle\leq$ $\displaystyle
G_{i,\omega_{2}}\left(r\right)+\left(\frac{1-r}{r}\right)\alpha\int_{r}^{1}dG_{i,\omega_{2}}\left(x\right)$
$\displaystyle=$ $\displaystyle
G_{i,\omega_{2}}\left(r\right)+\left(\frac{1-r}{r}\right)\alpha\left[1-G_{i,\omega_{2}}\left(r\right)\right]$

which yields a contradiction unless $G_{i,\omega_{2}}\left(r\right)=1$.
However, $G_{i,\omega_{2}}\left(r\right)=1$ implies $r\geq\overline{\beta}$
which contradicts $r\in\left(\underline{\beta},\overline{\beta}\right)$.

Case 2:

$\left(\frac{1-r}{r}\right)\alpha\geq 1$ . Then, since
$\frac{G_{i,\omega_{1}}\left(r\right)}{G_{i,\omega_{2}}\left(r\right)}=1$ and
$\frac{G_{i,\omega_{1}}\left(r\right)}{G_{i,\omega_{2}}\left(r\right)}$ is
non-increasing in $r$, together with (28) implies that
$\frac{G_{i,\omega_{1}}\left(r^{\prime}\right)}{G_{i,\omega_{2}}\left(r^{\prime}\right)}=1$
for all $r^{\prime}\geq r$. Pick $r^{\prime}>\frac{\alpha}{1+\alpha}$ (note
that since $\frac{\alpha}{1+\alpha}$ is just the prior belief that
$\omega_{1}=1$, it must be in the interior of the support of beliefs
$[\underline{\beta},\bar{\beta}]$—this leads to the contradiction in Case 1.

#### Part (3)

This follows directly from the first-order stochastic dominance relationship
between $G_{i,\omega_{1}}$ and $G_{i,\omega_{2}}$ in established part (2). ∎

######  Lemma 3.

Suppose $L=2$ and the common prior $P$ satisfies both Assumptions 1 and 2.
Then, there exists an $\varepsilon>0$ such that the
$\overline{\mu}_{i}(\omega_{1})+\varepsilon<\overline{\mu}_{i}(\omega_{2})$
for all $i\in N.$

  * Proof.

Fix any agent $i$ and let the convex hull of the common support of its
posteriors be $[\underline{\beta},\overline{\beta}]$. First observe that the
prior belief about the state is $\frac{\alpha}{1+\alpha}$. Therefore it must
be that $\underline{\beta}<\frac{\alpha}{1+\alpha}<\overline{\beta}$ (if
exactly one of these is an equality we violate the condition that the expected
posterior is the prior, if both are equalities then the signal is
uninformative violating Assumption 2.

######  Claim 3.

For $\delta>0$ if the total variation distance between $G_{i,\omega_{1}}$ and
$G_{i,\omega_{2}}$ is larger than $\delta$, there exists $\delta^{\prime}>0$
such that
$\underline{\beta}+\delta^{\prime}<\frac{\alpha}{1+\alpha}<\overline{\beta}-\delta^{\prime}$,
and further
$\max\left\\{G_{i,\omega_{1}}(\frac{\alpha}{1+\alpha}-\delta^{\prime}),(1-G_{i,\omega_{2}}(\frac{\alpha}{1+\alpha}+\delta^{\prime})\right\\}\geq\delta^{\prime}$.

  * Proof.

To see this observe that we showed in the proof of Lemma 2 that for any
$r\not\in\\{0,1\\}$, we have

$\frac{dG_{i,\omega_{1}}}{dG_{i,\omega_{2}}}\left(r\right)=\left(\frac{1-r}{r}\right)\alpha\text{.}$

Therefore for $r$ arbitrarily close to $\frac{\alpha}{1+\alpha}$,
$\frac{dG_{i,\omega_{1}}}{dG_{i,\omega_{2}}}\left(r\right)$ is arbitrarily
close to $1$. Further, by the fact that $G_{i,\omega_{2}}$ first order
stochastically dominates $G_{i,\omega_{1}}$, we have
$\delta^{\prime}>G_{i,\omega_{1}}(\frac{\alpha}{1+\alpha}-\delta^{\prime})>G_{i,\omega_{2}}(\frac{\alpha}{1+\alpha}-\delta^{\prime})$,
and similarly
$\delta^{\prime}>(1-G_{i,\omega_{2}}(\frac{\alpha}{1+\alpha}+\delta^{\prime})>(1-G_{i,\omega_{1}}(\frac{\alpha}{1+\alpha}+\delta^{\prime})$

Therefore if the claim is not satisfied, both measures are mostly supported on
$[\frac{\alpha}{1+\alpha}-\delta^{\prime},\frac{\alpha}{1+\alpha}+\delta^{\prime}]$
the total variation distance is $\leq 2\frac{1+\alpha}{\alpha}\delta^{\prime
2}+2\delta^{\prime}$. Therefore the claim follows for e.g.,
$\delta^{\prime}=\frac{1}{3}\delta$. ∎

Now observe that
$\overline{\mu}_{i}(\omega)=\int_{0}^{1}(1-G_{i,\omega}(r))dr$. Therefore

$\displaystyle\overline{\mu}_{i}(\omega_{2})-\overline{\mu}(\omega_{1})$
$\displaystyle=$
$\displaystyle\int_{0}^{1}(G_{i,\omega_{1}}(r)-G_{i,\omega_{2}}(r))dr$ We
showed in Lemma 2 that $G_{i,\omega_{2}}$ first order stochastically dominates
$G_{i,\omega_{1}}$. Therefore we have, for $\delta^{\prime}$ that satisfies
the statement of the claim above, $\displaystyle\geq$
$\displaystyle\int_{\frac{\alpha}{1+\alpha}-\delta^{\prime}}^{\frac{\alpha}{1+\alpha}}(G_{i,\omega_{1}}(r)-G_{i,\omega_{2}}(r))dr$
$\displaystyle=$
$\displaystyle\delta^{\prime}(G_{i,\omega_{1}}(\frac{\alpha}{1+\alpha}-\delta^{\prime})-G_{i,\omega_{2}}(\frac{\alpha}{1+\alpha}-\delta^{\prime}))$
$\displaystyle+\int_{\frac{\alpha}{1+\alpha}-\delta^{\prime}}^{\frac{\alpha}{1+\alpha}}\left(\int_{\frac{\alpha}{1+\alpha}-\delta^{\prime}}^{r}(dG_{i,\omega_{1}}(x)-dG_{i,\omega_{2}}(x))\right)dr$
We know from (24) that $dG_{i,\omega_{1}}(r)>dG_{i,\omega_{2}}(r)$ for
$r<\frac{\alpha}{1+\alpha}$. So we have, $\displaystyle\geq$
$\displaystyle\delta^{\prime}(G_{i,\omega_{1}}(\frac{\alpha}{1+\alpha}-\delta^{\prime})-G_{i,\omega_{2}}(\frac{\alpha}{1+\alpha}-\delta^{\prime})).$

Now observe that by (25) we have

$\displaystyle G_{i,\omega_{1}}(r)\geq\frac{1-r}{r}\alpha
G_{i,\omega_{2}}(r),$ $\displaystyle\implies$ $\displaystyle
G_{i,\omega_{1}}(r)-G_{i,\omega_{2}}(r)\geq\left(\frac{1-r}{r}\alpha-1\right)G_{i,\omega_{2}}(r).$
For $r=\frac{\alpha}{1+\alpha}-\delta^{\prime}$ we have
$\displaystyle\implies$ $\displaystyle
G_{i,\omega_{1}}(\frac{\alpha}{1+\alpha}-\delta^{\prime})-G_{i,\omega_{2}}(\frac{\alpha}{1+\alpha}-\delta^{\prime})\geq\frac{\delta^{\prime}(\alpha+1)^{2}}{\alpha-(\alpha+1)\delta^{\prime}}G_{i,\omega_{2}}(\frac{\alpha}{1+\alpha}-\delta^{\prime}).$
However we also have by the statement of the claim that $\displaystyle
G_{i,\omega_{1}}(\frac{\alpha}{1+\alpha}-\delta^{\prime})\geq\delta^{\prime},$
$\displaystyle\implies$ $\displaystyle
G_{i,\omega_{1}}(\frac{\alpha}{1+\alpha}-\delta^{\prime})-G_{i,\omega_{2}}(\frac{\alpha}{1+\alpha}-\delta^{\prime})\geq\delta^{\prime}-G_{i,\omega_{2}}(\frac{\alpha}{1+\alpha}-\delta^{\prime}).$
Combining, we have that
$\displaystyle\geq\max\\{\delta^{\prime}-G_{i,\omega_{2}}(\frac{\alpha}{1+\alpha}-\delta^{\prime}),\frac{\delta^{\prime}(\alpha+1)^{2}}{\alpha-(\alpha+1)\delta^{\prime}}G_{i,\omega_{2}}(\frac{\alpha}{1+\alpha}-\delta^{\prime})\\}$

Note that the first term is decreasing in
$G_{i,\omega_{2}}(\frac{\alpha}{1+\alpha}-\delta^{\prime})$, while the second
is increasing. Therefore, the minimum is achieved at

$\displaystyle\delta^{\prime}-G_{i,\omega_{2}}(\frac{\alpha}{1+\alpha}-\delta^{\prime})=\frac{\delta^{\prime}(\alpha+1)^{2}}{\alpha-(\alpha+1)\delta^{\prime}}G_{i,\omega_{2}}(\frac{\alpha}{1+\alpha}-\delta^{\prime})$
$\displaystyle\implies$ $\displaystyle
G_{i,\omega_{2}}(\frac{\alpha}{1+\alpha}-\delta^{\prime})=\frac{\alpha-(\alpha+1)\delta^{\prime}}{\alpha+\alpha\delta^{\prime}(\alpha+1)}\delta^{\prime}$

Substituting in we have that,

$\displaystyle
G_{i,\omega_{1}}(\frac{\alpha}{1+\alpha}-\delta^{\prime})-G_{i,\omega_{2}}(\frac{\alpha}{1+\alpha}-\delta^{\prime})\geq\frac{\delta^{\prime}(\alpha+1)^{2}}{\alpha+\alpha\delta^{\prime}(\alpha+1)}\delta^{\prime},$

and therefore we have that
$\overline{\mu}_{i}(\omega_{2})-\overline{\mu}_{i}(\omega_{1})\geq\varepsilon$
for
$\varepsilon=\frac{\delta^{\prime}(\alpha+1)^{2}}{\alpha+\alpha\delta^{\prime}(\alpha+1)}\delta^{\prime
2}$. ∎

We are now finally in a position to give the proof of Theorem 1.

See 1

  * Proof of Theorem 1.

We prove this by analyzing each of the steps of Procedure 1.

    1. (1)

By Lemma 1, the law of large numbers holds and when the realizes state is
$\omega$,

$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\tilde{\mu}_{i}=\overline{\mu}(\omega).$

Therefore the population mean of the elicited beliefs $\hat{\mu}$ in Step (1)
of the procedure equals $\overline{\mu}(\omega)$ in state $\omega.$

    2. (2)

By part (1) of Assumption 1, we know that the beliefs of any agent $i$ have a
common support across the states. Further, by Assumption 2, this support must
have at least 2 distinct points in it. Combining this with the fact that
signals are approximately independent, there must exist at least two agents
with different beliefs almost surely. So step 2 can find 2 such agents.
Further, in step 2, each agent $i=A,B$ will report, by the law of total
probability:
$\alpha_{i}=\mu_{i}\overline{\mu}(\omega_{1})+(1-\mu_{i})\overline{\mu}(\omega_{2}).$

    3. (3)

Since $\mu_{A}\neq\mu_{B}$, equation 7 can be solved.

    4. (4)

By Lemma 3, there is a $\varepsilon>0$ such that for all $i$
$\overline{\mu}_{i}(\omega_{1})<\overline{\mu}_{i}(\omega_{2})-\varepsilon$.
Therefore, the expected poptulation average beliefs,
$\overline{\mu}(\omega_{1})<\overline{\mu}(\omega_{2})-\varepsilon$, i.e.,
$\overline{\mu}(\omega_{1})\neq\overline{\mu}(\omega_{2}).$

Therefore comparing the elicited $\hat{\mu}$ in step (1) with the recovered
$(\overline{\mu}(\omega_{1}),\overline{\mu}(\omega_{2}))$ must reveal the true
state. ∎

See 2

  * Proof.

We prove this by analyzing each of the steps of Procedure 2 analogously to our
proof of Theorem 1.

    1. (1)

By Lemma 1, the law of large numbers holds and when the realizes state is
$\omega$,

$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\tilde{\mu}_{i}=\overline{\mu}(\omega).$

Therefore the population mean of the elicited beliefs $\hat{\mu}$ in Step (1)
of the procedure equals $\overline{\mu}(\omega)$ in state $\omega.$

    2. (2)

By part (1) of Assumption 1, we know that the beliefs of any agent $i$ have a
common support across the states. Further, by Assumption 4, this support must
has an interior relative to $\Delta^{L}$. Combining this with the fact that
signals are approximately independent, there must exist at least $L$ agents
whose beliefs form a full rank matrix a.s.. So step 2 can find $L$ such
agents. Further, in step 2, each agent $i\in I$ will report, by the law of
total probability: $\alpha_{i}=\mu_{i}^{T}\bar{\bm{\mu}}.$

    3. (3)

Since the set $I$ of agents is selected so that $\bm{\mu}$ is full-rank,
equation 8 can be solved.

    4. (4)

By Assumption 3, $\overline{\mu}(\omega)\neq\overline{\mu}(\omega^{\prime})$
for any $\omega,\omega^{\prime}\in\Omega$.

Therefore comparing the elicited $\hat{\mu}$ in step (1) with the recovered
$\bar{\bm{\mu}}$ must reveal the true state. ∎

See 1

  * Proof.

At each $\omega$, the distribution of the $L-1$ signals can be summarized by
the outer product

$\left(\otimes P_{\omega}^{S}\right)^{L-1}=\underset{L-1\text{
times}}{\underbrace{P_{\omega}^{S}\otimes\cdots\otimes
P_{\omega}^{S}}}\text{.}$

Since Assumption 3 is satisfied, the dimension of the linear space spanned by
$\left\\{P_{\omega}^{S}\right\\}_{\omega\in\Omega}$ is at least $2$. Hence, by
Lemma 4 by Fu, Haghpanah, Hartline, and Kleinberg
(2021),$\left\\{\left(\otimes
P_{\omega}^{S}\right)^{L-1}\right\\}_{\omega\in\Omega}$ are linearly
independent and hence $\left\\{\left(\otimes
P_{\omega}^{S}\right)^{L-1}P\left(\omega\right)\right\\}_{\omega\in\Omega}$
are also linearly independent ($P\left(\omega\right)$ denotes the prior
probability of $\omega$). Identify $\left\\{\left(\otimes
P_{\omega}^{S}\right)^{L-1}P\left(\omega\right)\right\\}_{\omega\in\Omega}$
with a $\left|S\right|^{L-1}\times L$ matrix which we denote by $M$. Hence,
rank$\left(M\right)=L$. For $\mathbf{s=}\left(s^{1},...,s^{L-1}\right)$ and
$\omega\in\Omega$, observe that
$M_{\mathbf{s},\omega}=P\left(\mathbf{s},\omega\right)$. Since
rank$\left(M\right)=L$, the conditional probability matrix
$\left\\{P\left(\omega|\mathbf{s}\right)\right\\}$ also has rank $L$. Hence,
Assumption 4 is satisfied with the signal space $S^{L-1}$. ∎

##  Appendix B Proofs from Section 3

We begin with a helpful lemma. First, let us first introduce some notation.
Recall that $N=\left\\{1,2,3,...\right\\}$ is the infinite set of agents. Let
$N_{n}$ be the set of the first $n$ agents in $N,$

$N_{n}=\left\\{1,...,n\right\\}.$

######  Lemma 4.

Let $\left\langle\zeta_{i}\right\rangle_{i\in N}$ be a sequence of independent
random variables $E[\zeta_{i}]=0$ and let $\left\langle
x_{i}\right\rangle_{i\in N}$ be a sequence of i.i.d. indicator random
variables distributed according to $x$ with $E[x]>0$ such that $\zeta_{i}$ and
$x_{i}$ are independent for all $i\in N$. Let $N_{n}^{1}\subseteq N_{n}$ be
the random subset of agents with $x_{i}=1$. Consider the random variable

$\displaystyle\rho_{n}=\begin{cases}\frac{1}{|N^{1}_{n}|}\sum_{i\in
N_{n}^{1}}\zeta_{i}&\text{ if }|N_{n}^{1}|>0,\\\
c&\text{otherwise.}\end{cases}$

Where $c\in\mathbb{R}$. Then $\rho_{n}\rightarrow 0$ almost surely.

  * Proof.

Note that since $E[x]>0$, $|N^{1}_{n}|\to\infty$ almost surely as
$n\to\infty$. Fix any realization of $x_{i}$’s such that $N^{1}_{n}\to\infty$.
Consider the subsequence of $i^{\prime}s$ such that $x_{i}=1$. Since
$\zeta_{i}$’s are independent of $x_{i}$’s by assumption, the limit average of
$\zeta_{i}$’s on this subsequence must converge to the limit average of
$E[\zeta_{i}]$ by the strong law of large numbers. Since $E[\zeta_{i}]=0$ by
assumption, we have that

$\displaystyle\lim_{n\to\infty}\frac{1}{|N^{1}_{n}|}\sum_{i\in
N_{n}^{1}}\zeta_{i}=0,$

for this realization of $x_{i}$’s. Since such realizations (i.e., ones where
$|N_{n}^{1}|\to\infty$) occur almost surely, we have that $\rho_{n}\to 0$ a.s.
as desired. ∎

We are now in a position to provide a proof of the main theorem, Theorem 3
(restated below for the reader’s convenience). See 3

  * Proof.

Consider a realization of the first order reports. Let groups $A$ and $B$ be
as defined in Procedure 4. By our maintained assumptions, each of these groups
must have an infinite number of members almost surely. Part (1) of Assumption
1 implies that the agents’ beliefs must lie in $(0,1)$. Therefore the
distribution of beliefs conditional on state must have at least two points in
their support. Part (2) implies that (most) agents’ beliefs are approximately
independent conditional on state. The result follows.

For agent $i\in A$, let $\alpha_{i}$ denote her expectation of the population
mean. Given her first order belief $\mu_{i}$ and pair of state-dependent
population beliefs $(\alpha_{i}^{0},\alpha_{i}^{1})$ she reports

$\displaystyle\alpha_{i}=\mu_{i}\alpha_{i}^{1}+(1-\mu_{i})\alpha_{i}^{2}.$

By Assumption 6 this can be rewritten as

$\displaystyle\alpha_{i}=\mu_{i}\overline{\mu}(\omega_{1})+(1-\mu_{i})\overline{\mu}(\omega_{2})+\left(\mu_{i}\zeta_{i}^{\omega_{1}}+(1-\mu_{i})\zeta_{i}^{\omega_{2}}\right).$

Denote by $A^{l}$ the set of first $l$ agents in $A$ according to the order on
$N$. Consider the average over the reports of the agents in $A^{l}$

$\displaystyle\frac{1}{l}\overset{}{\sum_{i\in A^{l}}}\alpha_{i}$
$\displaystyle=\frac{1}{l}\overset{}{\sum_{i\in
A^{l}}}\left((1-\mu_{i})\alpha_{i}^{1}+\mu_{i}\alpha_{i}^{2}\right),$
$\displaystyle=\frac{1}{l}\overset{}{\sum_{i\in
A^{l}}}\left(\mu_{i}\overline{\mu}(\omega_{1})+(1-\mu_{i})\overline{\mu}(\omega_{2})+\left(\mu_{i}\zeta_{i}^{\omega_{1}}+(1-\mu_{i})\zeta_{i}^{\omega_{2}}\right)\right)$

By the conclusion of Lemma 4, we therefore have that

$\displaystyle\underset{l\rightarrow\infty}{\lim}\frac{1}{l}\overset{}{\sum_{i\in
A^{l}}}\alpha_{i}$
$\displaystyle=\underset{l\rightarrow\infty}{\lim}\frac{1}{l}\overset{}{\sum_{i\in
A^{l}}}\left(\mu_{i}\overline{\mu}(\omega_{1})+(1-\mu_{i})\overline{\mu}(\omega_{2})\right)$

Therefore, we have that

$\displaystyle\alpha_{A}\equiv\underset{l\rightarrow\infty}{\lim}\frac{1}{l}\overset{}{\sum_{i\in
A^{l}}}\alpha_{i}$
$\displaystyle=\mu_{A}\overline{\mu}(\omega_{1})+(1-\mu_{A})\overline{\mu}(\omega_{2}).$

Similarly, for the agents in $B$ we have that

$\displaystyle\alpha_{B}\equiv\underset{l\rightarrow\infty}{\lim}\frac{1}{l}\overset{}{\sum_{i\in
B^{l}}}\alpha_{i}$
$\displaystyle=\mu_{B}\overline{\mu}(\omega_{1})+(1-\mu_{B})\overline{\mu}(\omega_{2}),$

where $\mu_{A}$ and $\mu_{B}$ are as defined in Procedure 4. As before, we can
recover $\overline{\mu}(\omega_{1}),\overline{\mu}(\omega_{2})$ as in (22):

$\displaystyle\left(\begin{array}[]{c}\overline{\mu}(\omega_{1})\\\
\overline{\mu}(\omega_{2})\end{array}\right)=\left(\begin{array}[]{cc}\mu_{A}&1-\mu_{A}\\\
\mu_{B}&1-\mu_{B}\end{array}\right)^{-1}\left(\begin{array}[]{c}\alpha_{A}\\\
\alpha_{B}\end{array}\right).$

Note that $\alpha_{A},\alpha_{B}$ are directly elicited, while
$\mu_{A}\neq\mu_{B}$ by construction implies the matrix is invertible. By
Lemma 1) we have that

$\underset{n\rightarrow\infty}{\lim}\frac{1}{n}\sum\limits_{i=1}^{n}\mu_{i}=\overline{\mu}(\omega)$

almost surely, and further that
$\overline{\mu}(\omega_{1})\neq\overline{\mu}(\omega_{2})$. Therefore by
comparing the realized average of elicited beliefs, to the
$\overline{\mu}(\cdot)$ recovered from (22), we can almost surely recover the
true state of the world, $\omega$, as desired. ∎

##  Appendix C Proofs from Section 4

See 4

  * Proof.

Assume that $\text{marg}_{\tilde{S}}\tilde{\mu}\left(\tilde{s}\right)>0$. It
follows from Theorem III.2.7 of Mertens, Sorin, and Zamir (2015) that the
aggregator can derive:

    1. (1)

The set $E\left(\tilde{s}\right)$ which is the smallest set
$Y\subseteq\Omega\times\tilde{S}$ of state-hierarchy profiles satisfying

      1. (a)

$\left(\omega,\tilde{s}\right)\in Y$ for some $\omega\in\Omega$, and,

      2. (b)

for each $\left(\omega,\tilde{t}\right)\in Y$, we have
$\left\\{\tilde{t}_{i}\right\\}\times$supp
$\tilde{\pi}_{i}\left(\tilde{t}_{i}\right)\subseteq Y$.

    2. (2)

The unique consistent probability $\tilde{\mu}$ on $E\left(\tilde{s}\right)$.

With (1) and (2), the aggregator can compute
$\tilde{\mu}\left(\cdot|\tilde{s}\right)$. We recap and illustrate both (1)
and (2) for the ease of reference.

To construct (1), define
$C_{i}^{1}\left(\omega,\tilde{t}\right)=\left\\{\tilde{t}_{i}\right\\}\times$supp$\tilde{\pi}_{i}\left(\tilde{t}_{i}\right)$
for each $\left(\omega,\tilde{t}\right)\in\Omega\times\tilde{S}$; and
inductively, for every $l\geq 1$ and $\tilde{t}\in\tilde{S}$, define

$C_{i}^{l+1}\left(\omega,\tilde{t}\right)=C_{i}^{l}\left(\omega,\tilde{t}\right)\bigcup\bigcup\limits_{\left(\omega^{\prime},\tilde{t}^{\prime}\right)\in
C_{i}^{l}\left(\omega,\tilde{t}\right)}\bigcup\limits_{j=1}^{N}C_{j}^{1}\left(\omega^{\prime},\tilde{t}^{\prime}\right)\text{.}$

Then, let

$C_{i}\left(\omega,\tilde{t}\right)=\bigcup\limits_{l=1}^{\infty}C_{i}^{l}\left(\omega,\tilde{t}\right)\text{.}$

These include $\left(\omega,\tilde{t}\right)$, the state-hierarchy profiles
which agent $i$ regards as possible (i.e.,
$C_{i}^{1}\left(\omega,\tilde{t}\right)$) at $\left(\omega,\tilde{t}\right)$,
the state-hierarchy profiles which some agent regards at some state-hierarchy
profile in $C_{i}^{1}\left(\omega,\tilde{t}\right)$ (i.e.,
$C_{i}^{2}\left(\omega,\tilde{t}\right)$), and so on.

Consider any $\omega^{\ast}$ such that
$\tilde{\mu}\left(\omega^{\ast},\tilde{s}\right)>0$. Note that
$C_{i}\left(\omega^{\ast},\tilde{s}\right)$ satisfies properties (a) and (b)
above by construction and hence
$C_{i}\left(\omega^{\ast},\tilde{s}\right)\supseteq
E\left(\tilde{s}\right)$.111111Property (b) is by construction and property
(a) follows from the assumption that
$\tilde{\mu}\left(\omega^{\ast},\tilde{s}\right)>0$. Also we can argue that
$C_{i}\left(\omega^{*},\tilde{s}\right)\subseteq E\left(\tilde{s}\right)$
inductively.

For (2), it follows from Bayes’ rule that for every
$\left(\omega^{\prime},\tilde{t}^{\prime}\right)\in
C_{i}^{1}\left(\omega,\tilde{t}\right)$ and
$\tilde{\mu}\left(\omega,\tilde{t}\right)>0$, we must have
$\tilde{t}_{i}=\tilde{t}_{i}^{\prime}$ and hence

$\frac{\tilde{\mu}\left(\omega^{\prime},\tilde{t}^{\prime}\right)}{\tilde{\mu}\left(\omega,\tilde{t}\right)}=\frac{\tilde{\pi}_{i}\left(\tilde{t}_{i}^{\prime}\right)\left(\omega^{\prime},\tilde{t}^{\prime}\right)\tilde{\mu}_{i}\left(\tilde{t}_{i}^{\prime}\right)}{\tilde{\pi}_{i}\left(\tilde{t}_{i}\right)\left(\omega,\tilde{t}_{-i}\right)\tilde{\mu}_{i}\left(\tilde{t}_{i}\right)}=\frac{\tilde{\pi}_{i}\left(\tilde{t}_{i}^{\prime}\right)\left(\omega^{\prime},\tilde{t}^{\prime}\right)}{\tilde{\pi}_{i}\left(\tilde{t}_{i}\right)\left(\omega,\tilde{t}_{-i}\right)}>0\text{.}$

The aggregator knows $\tilde{\pi}_{i}\left(\tilde{t}_{i}\right)$ and hence can
express $\tilde{\mu}\left(\omega^{\prime},\tilde{t}^{\prime}\right)$ as a
multiple of $\tilde{\mu}\left(\omega,\tilde{t}\right)$. Inductively, for every
$\left(\omega^{\prime},\tilde{t}^{\prime}\right)\in
C_{i}^{l}\left(\omega,\tilde{t}\right)$,
$\tilde{\mu}\left(\omega^{\prime},\tilde{t}^{\prime}\right)$ can also be
expressed as a multiple of $\tilde{\mu}\left(\omega,\tilde{t}\right)$.

Hence, $\tilde{\mu}$ is uniquely determined on
$E\left(\tilde{s}\right)=C_{i}\left(\omega^{*},\tilde{s}\right)$ since
$\tilde{\mu}\left(\omega^{\ast},\tilde{s}\right)>0$.
$\tilde{\mu}(\cdot|\tilde{s})$ can thus be calculated. ∎

###  C.1. Construction for $m>2$

Similarly, we can generalize the construction in Section 4.2 to the case with
$m\geq 3$ as follows:

$\displaystyle\Pi_{1}=\left\\{\left\\{\left\\{\left(\sigma_{1},2k-1\right),\left(\sigma_{1},2k\right),\left(\sigma_{2},k\right)\right\\}:k=1,...,2^{m-1}\right\\},\left\\{\left(\sigma_{2},2^{m-1}+1\right),....,\left(\sigma_{2},2^{m}\right)\right\\}\right\\}\text{;}$
$\displaystyle\Pi_{2}=\left\\{\left\\{\left\\{\left(\sigma_{2},2k-1\right),\left(\sigma_{2},2k\right),\left(\sigma_{1},k\right)\right\\}:k=1,...,2^{m-1}\right\\},\left\\{\left(\sigma_{1},2^{m-1}+1\right),....,\left(\sigma_{1},2^{m}\right)\right\\}\right\\}\text{.}$
$\mu\left(\sigma_{l},k\right)=\frac{1}{2^{m+1}},\forall l=1,2,\forall
k=1,2,...,2^{m}\text{.}$

Without loss of generality, assume that $m\geq 3$ is odd. Construct the new
model as follows:

$\Pi^{\prime}_{1}=\left\\{\begin{array}[]{c}\left\\{\left(\sigma_{1},1\right),\left(\sigma_{2},1\right),\left(\sigma_{1},2\right)\right\\},\left\\{\left(\sigma_{1},1\right)^{\prime},\left(\sigma_{2},2\right)^{\prime},\left(\sigma_{1},3\right)^{\prime},\left(\sigma_{1},4\right)^{\prime}\right\\},\\\
\left\\{\left\\{\left(\sigma_{1},2k-1\right)^{\prime},\left(\sigma_{1},2k\right)^{\prime},\left(\sigma_{2},k\right)^{\prime}\right\\}:k=2^{n-1}+1,...,2^{n}\text{
and }n=3,5,...,m-2\right\\},\\\
\left\\{\left\\{\left(\sigma_{1},2k-1\right),\left(\sigma_{1},2k\right),\left(\sigma_{2},k\right)\right\\}:k=2^{n-1}+1,...,2^{n}\text{
and }n=2,4,...,m-1\right\\},\\\
\left\\{\left(\sigma_{2},2^{m-1}+1\right)^{\prime},....,\left(\sigma_{2},2^{m}\right)^{\prime}\right\\}\end{array}\right\\}\text{;}$
$\Pi^{\prime}_{2}=\left\\{\begin{array}[]{c}\left\\{\left(\sigma_{1},1\right),\left(\sigma_{2},1\right),\left(\sigma_{1},1\right)^{\prime},\left(\sigma_{2},2\right)^{\prime}\right\\},\\\
\left\\{\left\\{\left(\sigma_{2},2k-1\right)^{\prime},\left(\sigma_{2},2k\right)^{\prime},\left(\sigma_{1},k\right)^{\prime}\right\\}:k=2^{n-1}+1,...,2^{n}\text{
and }n=2,4,...,m-1\right\\},\\\
\left\\{\left\\{\left(\sigma_{2},2k-1\right),\left(\sigma_{2},2k\right),\left(\sigma_{1},k\right)\right\\}:k=2^{n-1}+1,...,2^{n}\text{
and }n=1,3,...,m-2\right\\},\\\
\left\\{\left(\sigma_{1},2^{m-1}+1\right),....,\left(\sigma_{1},2^{m}\right)\right\\}\end{array}\right\\}\text{.}$

As in the case of $m=2$, we start from the partition cells
$\left\\{\left(\sigma_{1},1\right),\left(\sigma_{2},1\right),\left(\sigma_{1},2\right)\right\\}$
and
$\left\\{\left(\sigma_{1},1\right),\left(\sigma_{2},1\right),\left(\sigma_{1},1\right)^{\prime},\left(\sigma_{2},2\right)^{\prime}\right\\}$
which contain $\left(\sigma_{1},1\right)$. The states with ′ are those “to the
left” of $\left(\sigma_{1},1\right)$, whereas the states without ′ are like
those “to the right” of $\left(\sigma_{1},1\right)$. We can then mimic the
idea for $m=2$ to solve for a prior $\mu^{\prime}$ with the desired properties
as follows:

  1. (1)

Again, for $x>0$, set

$\mu^{\prime}\left(\sigma_{1},1\right)=0\text{ and
}\mu^{\prime}\left(\sigma_{2},1\right)=\mu^{\prime}\left(\sigma_{1},1\right)^{\prime}=\mu^{\prime}\left(\sigma_{2},2\right)^{\prime}=x\text{.}$

  2. (2)

The number of states with ′ excluding $\left(\sigma_{1},1\right)^{\prime}$ and
$\left(\sigma_{2},2\right)^{\prime}$ (i.e., states in
$\Pi_{2}\left(\sigma_{1},k\right)^{\prime}\,\ $where $k=2^{n-1}+1,...,2^{n}$
and $n=2,4,...,m-1$) is :

$\displaystyle y\equiv 3\times\left(2^{1}+2^{3}+\cdots+2^{m-2}\right),$ i.e.,
$\displaystyle y=2^{m}-2.$

We assign probability $\tfrac{x}{2}$ to each of these “left” states.

  3. (3)

The number of states without ′ excluding $\left(\sigma_{1},1\right)$ and
$\left(\sigma_{2},1\right)$ (i.e., $\left(\sigma_{2},2\right)$ and states in
$\Pi_{1}\left(\sigma_{2},k\right)$ where $k=2^{n-1}+1,...,2^{n}$ and
$n=2,4,...,m-1$) is:

$1+3\times\left(2^{1}+2^{3}+\cdots+2^{m-2}\right)=y+1\text{.}$

We assign probability $2x$ to each of the “right” states.

  4. (4)

Hence for the total probability to sum to $1$, we must have:

$\displaystyle 3x+y\frac{x}{2}+(y+1)2x=1,$ $\displaystyle\implies$
$\displaystyle x=\frac{2}{10+5y},$ $\displaystyle\implies$ $\displaystyle
x=\frac{1}{5\times 2^{m}}\text{.}$

In summary, observe that as desired, we have that

$\displaystyle\mu\left(\omega_{1}|\left(\Pi_{1}\left(\sigma_{1},1\right),\Pi_{2}\left(\sigma_{1},1\right)\right)^{m}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{2}\text{.}$
$\displaystyle\mu^{\prime}\left(\omega_{1}|\left(\Pi_{1}^{\prime}\left(\sigma_{1},1\right),\Pi_{2}^{\prime}\left(\sigma_{1},1\right)\right)^{m}\right)$
$\displaystyle=$ $\displaystyle 0\text{.}$
$\displaystyle\left(\Pi_{1}^{\prime}\left(\sigma_{1},1\right),\Pi_{2}^{\prime}\left(\sigma_{1},1\right)\right)^{m}$
$\displaystyle=$
$\displaystyle\left(\Pi_{1}\left(\sigma_{1},1\right),\Pi_{2}\left(\sigma_{1},1\right)\right)^{m}\text{.}$

Clearly, we can flip left and right to construct another model
$\Pi_{1}^{\prime\prime}$ and $\Pi_{2}^{\prime\prime}$ with

$\displaystyle\mu^{\prime}\left(\omega_{1}|\left(\Pi_{1}^{\prime\prime}\left(\sigma_{1},1\right),\Pi_{2}^{\prime\prime}\left(\sigma_{1},1\right)\right)^{m}\right)$
$\displaystyle=$ $\displaystyle 1\text{;}$
$\displaystyle\left(\Pi_{1}^{\prime\prime}\left(\sigma_{1},1\right),\Pi_{2}^{\prime\prime}\left(\sigma_{1},1\right)\right)^{m}$
$\displaystyle=$
$\displaystyle\left(\Pi_{1}\left(\sigma_{1},1\right),\Pi_{2}\left(\sigma_{1},1\right)\right)^{m}\text{.}$

The construction here makes use of the feature that all higher-order beliefs
of the common knowledge hierarchy $t$ are degenerate. More precisely, this
feature ensures that in “shifting” out the probability which $\mu$ assigns to
$\left(\sigma_{1},1\right)$, we can preserve the higher-order beliefs of
$\Pi_{1}\left(\sigma_{1},1\right)$ and $\Pi_{2}\left(\sigma_{1},1\right)$ as
long as we can preserve the first-order belief at every partition cell except
for $\Pi_{1}\left(\sigma_{2},2^{m}\right)$ and
$\Pi_{2}\left(\sigma_{1},2^{m}\right)^{\prime}$.