#  Information Aggregation with Costly Information Acquisition111We would like
to thank Emir Kamenica, Anastasios Karantounias, Ehud Lehrer and participants
at Glasgow University, the MIMA Workshop in Macroeconomic Theory at Warwick,
the CEPR Workshop in Turin, Durham York Workshop in Economic Theory, the
Durham Economic Theory Conference, EWMES2023 in Manchester, and SAET2023 in
Paris. This research is funded under ESRC grant ES/V004425/1.

Spyros Galanis222Department of Economics, Durham University, Mill Hill Lane,
Durham, UK, DH1 3LB. Email<EMAIL_ADDRESS>
Sergei Mikhalishchev333Department of Economics, Durham University, Mill Hill
Lane, Durham, UK, DH1 3LB. Email<EMAIL_ADDRESS>(

( This draft: )

###### Abstract

We study information aggregation in a dynamic trading model with partially
informed traders. Ostrovsky (2012) showed that ‘separable’ securities
aggregate information in all equilibria, however, separability is not robust
to small changes in the traders’ private information. To remedy this problem,
we enhance the model by allowing traders to acquire signals with cost
$\kappa$, in every period. We show that ‘$\kappa$ separable securities’
aggregate information and, as the cost decreases, nearly all securities become
$\kappa$ separable, irrespective of the traders’ initial private information.
Moreover, the switch to $\kappa$ separability happens not gradually but
discontinuously, hence even a small decrease in costs can result in a security
aggregating information. Finally, even with myopic traders, cheaper
information may accelerate or decelerate information aggregation for all but
Arrow-Debreu securities.

JEL: C91, D82, D83, D84, G14, G41

Keywords: Information Aggregation, Information Acquisition, Financial Markets,
Prediction Markets.

##  1 Introduction

The question of whether financial markets reveal and aggregate the private
information of traders has been studied at least since Hayek (1945). Ostrovsky
(2012) provides a strong result, showing that information gets aggregated in
all Nash equilibria, as long as the traded securities are separable and
trading takes place for infinitely many periods. However, separability is not
robust to small changes in the composition of the market or to the traders’
information structure. As a result, a market designer who does not know who
participates or what is their information structure, cannot be sure that the
equilibrium price is a good predictor of the security’s value.

In this paper, we examine whether the ability to acquire costly signals during
trading can make markets more efficient at aggregating information. This
question becomes more relevant as the continuous improvements in information
technology have created an abundance of available information, which is now
cheaper than ever to acquire, analyze, and act upon.444For example, recent
advances in generative AI tools such as ChatGPT could add considerable value
for investors with information processing constraints (Kim et al., 2023) and
assist in picking stocks (Pelster and Val, 2023).

We use the dynamic trading model of Ostrovsky (2012) with infinitely many
periods and payoffs given by the Market Scoring Rule (MSR) (Hanson, 2003,
2007). We first characterize the class of securities which are always
separable, irrespective of who trades and what is their information structure.
This class is very small and uninformative, as it only contains the Arrow-
Debreu (A-D) security, which pays $a$ at some state and $b$ in all other
states, and the security that specifies three payoffs: the largest is paid in
one state of the world, the lowest in another, and the middle in all other
states.555These securities are not very informative because the A-D can only
predict whether one state has occurred or not, whereas the other security can
only predict whether two states have occurred or not. Note that combining more
than one A-D security to construct a composite and more informative security
will not solve this issue, because it will be non-separable for some
information structures. For any other security, there is a market (information
structure and prior) at which there is no information aggregation so that the
security’s price does not converge to its true value. This means that
information may not aggregate in ‘most’ markets.

Does the availability of cheap information alleviate this problem? To study
this question, we enhance the model of Ostrovsky (2012) by enabling traders to
buy a costly signal structure in each period, before trading. We allow for a
large class of information cost functions $\kappa$, including the Shannon
entropy.

We define the class of $\kappa$ separable securities and Theorem 6 shows that
they are necessary and sufficient for information aggregation, when the cost
of information is $\kappa$, thus generalising Ostrovsky (2012). As information
acquisition costs decrease, the securities that eventually become $\kappa$
separable, and therefore aggregate information in all equilibria and for all
information structures, have a very simple structure: they specify a different
payoff at each state. This class of securities with unique values is generic.
Hence, the main message of the paper is that the availability of cheap
information makes ‘most’ markets aggregate information.666This result is also
supported by empirical evidence. (Farboodi et al., 2022) show, using a
structural model, that as the value of a firm’s data grows, which is
equivalent in our model to a decrease in the cost of information acquisition,
then the information content of the price of the firm’s stock increases as
well.  Surprisingly, there is also a small class of securities, which Theorem
3 characterises, that never become $\kappa$ separable for all information
structures, and therefore may fail information aggregation, even when the cost
converges to zero. Such a security specifies the same payoff $d$ in two states
and, in two other states, it pays either higher or lower than $d$.

If we can decrease the cost of information as much as we want, do we even need
markets to aggregate information through prices? Each trader could buy the
necessary signals and then bid very close to the true value. We argue that
this intuition is incorrect, because markets become even more important in an
environment with information acquisition. As cost $\kappa$ decreases, a
security switches discontinuously from non-separable to $\kappa$ separable,
hence even a slight reduction can enable a market to aggregate information.
This is in contrast to the average of the traders’ opinions after receiving
the information, or a poll, because its predictive accuracy improves smoothly
as costs decrease. Hence, the availability of cheap information leverages the
value of the markets, enabling them to aggregate information long before the
cost goes to zero. Moreover, we show that a security is $\kappa$ separable if
and only if the market is more accurate than a poll, for all priors.

Finally, we examine whether information acquisition can make markets more
efficient by aggregating information faster. We show that this is not true.
Even in non-strategic environments, as long as the separable security is not
A-D, information aggregation can happen both faster and slower, depending on
the parameters. This implies that, as information becomes more affordable,
markets could become less efficient if the underlying security is not A-D.

###  1.1 Literature

Our paper contributes to three strands of the literature. The first studies
the inefficiency of information acquisition and its effect on information
aggregation in markets.777See (Lim and Brooks, 2011) for a survey of the
empirical literature on market efficiency. Pavan et al. (2022) show that
traders acquire and use information inefficiently. Moreover, as the cost of
information declines, traders over-invest in information acquisition and trade
too much on their private information. Several experimental studies support
these results and find that traders tend to over-acquire information. In
addition, while information acquisition is positively correlated with market
efficiency, market prices do not aggregate all private information (Kraemer et
al., 2006; Page and Siemroth, 2017, 2021; Corgnet et al., 2022). Mele and
Sangiorgi (2015) analyze costly information acquisition in asset markets with
ambiguity-averse traders and show that when uncertainty is high enough,
information acquisition decisions become strategic complements and lead to
multiple equilibria. Our paper complements and differs from this literature.
We find that, as the cost of information acquisition decreases, the number of
securities (and therefore markets) that aggregate information increases.
However, some securities are never able to aggregate information, even if the
cost is almost zero. Finally, information aggregation can be delayed when
information acquisition is cheap, thus introducing another element of
inefficiency.

The second strand looks at the information aggregation properties of financial
and, in particular, prediction markets.888See Wolfers and Zitzewitz (2004) for
an early overview of the literature. DeMarzo and Skiadas (1998, 1999) first
introduced the notion of separable securities. Ostrovsky (2012) and Chen et
al. (2012) show that in a market with dynamically consistent traders,
separable securities are both necessary and sufficient for information
aggregation. Dimitrov and Sami (2008) and Chen et al. (2010) examine
information aggregation by varying the assumptions regarding the traders’
information structure. Galanis et al. (2024) study information aggregation
with ambiguity-averse traders, whereas Galanis and Kotronis (2021) allow for
boundedly rational traders who are unaware of some
contingencies.999Unawareness and ambiguity aversion generate dynamic
inconsistency and negative value of information, which are partly responsible
for no information aggregation. See (Galanis (2011, 2013)) for a model of
unawareness and Galanis (2021) for a connection between dynamic inconsistency
and the negative value of information. We contribute to this literature by
allowing traders to acquire costly signals at every period and we characterize
the $\kappa$ separable securities which aggregate information.

Finally, the paper contributes to the growing literature on the implications
of rational inattention, originated by Sims (2003). We build on the results of
(Denti, 2022; Caplin et al., 2019; Matějka and McKay, 2015; Caplin and Dean,
2015) to characterize the traders’ optimal behavior in a game with infinitely
many periods, where traders have posterior-separable cost functions and can
buy signals in every period. We show that, in any Nash equilibrium, almost any
security aggregates information for a sufficiently small marginal cost of
information.101010Atakan and Ekmekci (2023) show that common value auctions
with uninformed bidders who can acquire costly signals aggregate information
as long as the minimum cost-accuracy ratio is equal to zero. See also
Maćkowiak et al. (2023) for a recent survey of the literature on rational
inattention.

We conclude by motivating our choice of the MSR model. First, in the MSR
model, there are no noise traders and no strategic market makers, hence the
issue of information aggregation is not intertwined with that of information
revelation, as in Kyle (1985). Unlike the MSR, in Kyle (1985) it is not always
the case that the price will converge to the true value of the security, even
if there is only one trader and therefore information aggregation is achieved
by default. Second, a prediction market with the MSR can be reinterpreted as
an inventory-based market with a market maker who continuously adjusts the
price of the securities depending on the orders she receives.111111See
Ostrovsky (2012) and Galanis et al. (2024) for examples. The advantage of the
MSR over more well-known market mechanisms, such as the continuous double
auction, is that an agent can make her prediction/trade without waiting for
another agent to take the opposite side, or submit a limit order and wait for
it to be filled. This feature makes it an attractive mechanism for markets
with relatively few participants who do not trade daily, or in markets with
automated market makers.121212Automated market makers are widely used in
Decentralized Finance, see Schlegel et al. (2022) for an axiomatization of the
logarithmic MSR. MSR-based prediction markets have been used widely, for
example, by firms such as Ford, Google, General Electric, and Chevron (see
Ostrovsky (2012), Cowgill and Zitzewitz (2015)) as well as governments, for
example, in the UK and the Czech Republic (The Economist (2021)).

The paper adheres to the following plan. Section 2 describes the model.
Section 3 shows that the class of always separable securities is very small
and uninformative, thus prompting Section 4, where we include our main results
in an environment with information acquisition. Section 6 concludes.

##  2 The Model

###  2.1 Preliminaries

Uncertainty is described by a finite state space
$\Omega=\\{\omega_{1},...,\omega_{l}\\}$ and the set of traders is denoted
$I=\\{1,\ldots,n\\}$. Trader $i$’s initial private information is represented
by partition $\Pi_{i}$ of $\Omega$. Let $\Pi_{i}(\omega)$ be a partition
element of $\Pi_{i}$ that contains $\omega$, so that
$\omega\in\Pi_{i}(\omega)\in\Pi_{i}$. When the true state is
$\omega\in\Omega$, Trader $i$ considers all states in
$\Pi_{i}(\omega)\subseteq\Omega$ to be possible. We assume that the join (the
coarsest common refinement) of partitions $\Pi=\\{\Pi_{1},\ldots\Pi_{n}\\}$
consists of singleton sets so that $\underset{i\in
I}{\bigcap}\Pi_{i}(\omega)=\omega$ for all $\omega\in\Omega$, which means that
the traders’ pooled information always reveals the true state.131313This
assumption is also made by Ostrovsky (2012) and it is without loss of
generality because if the conjunction of the traders’ private information does
not reveal the state, we cannot expect that trading the security will reveal
it. This implies that, for any two states $\omega_{1}\neq\omega_{2}$, there
exists Trader $i$ such that $\Pi_{i}(\omega_{1})\neq\Pi_{i}(\omega_{2})$. Let
${\cal P}$ be the collection of all information structures $\Pi$ where
$\Omega$ has at least three states and $\underset{i\in
I}{\bigcap}\Pi_{i}(\omega)=\\{\omega\\}$ for all $\omega\in\Omega$. Traders
have a full-support common prior $\mu_{0}$ over $\Omega$ and they are risk-
neutral.

###  2.2 Trading environment

Trading is organized as follows. At time $t_{0}=0$, nature selects a state
$\omega^{*}\in\Omega$ and the uninformed market maker makes a prediction
$y_{0}$ about the value of security $X:\Omega\rightarrow{\mathbb{R}}$. At time
$t_{1}>t_{0}$, Trader 1 makes a revised prediction $y_{1}$, at $t_{2}>t_{1}$
trader 2 makes his prediction, and so on. At time $t_{n+1}>t_{n}$, Trader 1
makes another prediction $y_{n+1}$, and the whole process repeats until time
$t_{\infty}\equiv\lim_{k\to\infty}t_{k}=1$. All predictions are observed by
all traders. Each prediction $y_{k}$ is required to be within the set
$[\underset{\omega\in\Omega}{\min}X(\omega),\underset{\omega\in\Omega}{\max}X(\omega)]$.
At some time $t^{*}>1$ the true value $x^{*}=X(\omega^{*})$ of the security is
revealed.

The traders’ payoffs are computed using a scoring rule, $s(y,x^{*})$, where
$x^{*}$ is the true value of the security and $y$ is a prediction. A scoring
rule is proper if, for any probability measure $p$ and any random variable
$X$, the expectation of $s$ is maximized at $y=E_{p}[X]$. It is strictly
proper if $y$ is unique. We focus on continuous strictly proper scoring rules.
Examples are the quadratic, where $s(y,x)=-(x-y)^{2}$, and the logarithmic,
where $s(y,x)=(x-a)ln(y-a)+(b-x)ln(b-y)$ with
$a<\underset{\omega\in\Omega}{\min}X(\omega),b>\underset{\omega\in\Omega}{\max}X(\omega)$.

Under the market scoring rule (MSR) (McKelvey and Page (1990), Hanson (2003,
2007)), a trader is paid for each revision he makes. In particular, his
payoff, from announcing $y_{n}$ at $t_{n}$, is
$s(y_{n},x^{*})-s(y_{n-1},x^{*})$, where $y_{n-1}$ is the previous
announcement and $x^{*}$ is the true value of the security. For all proper
scoring rules, as $E_{q}[X]$ converges to $X(\omega)$, $s(E_{q}[X],X(\omega))$
converges to 0. Moreover, if $y_{n-1}$ is further away from $X(\omega)$ than
$E_{q}(X)$ is from $X(\omega)$, then
$s(E_{q}[X],X(\omega))-s(y_{n-1},X(\omega))$ is strictly positive. We then say
that the trader “buys out” the previous trader’s prediction. If he repeats the
previous announcement, his period payoff is zero. We say that prior $\mu$ is
non-degenerate given security $X$ if it does not assign probability 1 to a
unique value of $X$.

###  2.3 Two trading settings

We examine trading in two settings. In the myopic, or non-strategic, setting,
each trader does not care about future payoffs when acquiring information and
making an announcement. We denote this setting by
$\Gamma^{M}(\Omega,I,\Pi,X,y_{0},\mu,\underline{y},\overline{y},s)$, where $I$
is the set of $n$ players, $s$ is a strictly proper scoring rule, $y_{0}$ is
the market maker’s initial announcement at time $t_{0}$, $\mu$ is the common
prior, $[\underline{y},\overline{y}]$ is the set of possible announcements.

The strategic setting is studied in Section 4.4. Following Dimitrov and Sami
(2008), we focus on the discounted MSR, which specifies that the payment at
$t_{k}$ is $\beta^{k}(s(y_{t_{k}},x^{*})-s(y_{t_{k-1}},x^{*}))$, where
$0<\beta\leq 1$ is the common discount factor. The total payoff of each trader
is the sum of all payments for revisions. We denote this setting by
$\Gamma^{S}(\Omega,I,\Pi,X,y_{0},\mu,\underline{y},\overline{y},s,\beta)$. In
Section 4, we will add function $h$ in the specification of the game, which
denotes the cost of statistical experiments.

###  2.4 Information aggregation

We say that information is aggregated if the traders’ predictions converge to
the intrinsic value $X(\omega)$ of security $X$, for all $\omega\in\Omega$.
For every $\omega\in\Omega$, let $y_{k}(\omega)$ be the announcement of the
trader who moves in period $t_{k}$. The announcement $y_{k}(\omega)$ depends
on $\omega$ because traders have different private information across states.
Because $\\{y_{k}\\}_{k=1}^{\infty}$ is a sequence of random variables, we
need a probabilistic version of convergence.

######  Definition 1.

Under a profile of strategies in $\Gamma^{M}$ or $\Gamma^{S}$, information
aggregates if sequence $\\{y_{k}\\}_{k=1}^{\infty}$ converges in probability
to random variable $X$.

###  2.5 Example

Consider the following Example, which illustrates aspects of our model.

######  Example 1.

The state space is $\Omega=\\{\omega_{1},\omega_{2},\omega_{3},\omega_{4}\\}$,
the security is $X=(0,\frac{10}{7},0,1)$, and there are two myopic traders
with common prior $\mu=(\frac{1}{4},\frac{1}{6},\frac{1}{4},\frac{1}{3})$.
Trader 1’s partition is
$\\{\\{\omega_{1},\omega_{2}\\},\\{\omega_{3},\omega_{4}\\}\\}$ and Trader 2’s
is $\\{\\{\omega_{1},\omega_{4}\\},\\{\omega_{2},\omega_{3}\\}\\}$.

As traders are myopic, under the MSR they take turns in announcing their
expected value of $X$. Suppose that the true state is $\omega_{1}$ and traders
cannot acquire costly signals. In period 1, Trader 1 with conditional beliefs
$(\frac{3}{5},\frac{2}{5},0,0)$ will announce $\frac{4}{7}$. The same
announcement would be made by Trader 1 in all states and, thus, no information
is transmitted to Trader 2. In period 2, Trader 2 also makes the same
announcement at $\omega_{1}$ and, furthermore, the same announcement would be
made in all states. Hence, no information is transmitted to Trader 1. Because
the two traders agree on the announcement, there is no information updating
and the process ends. We say that there is no information aggregation at
$\omega_{1}$ because the final announcement $\frac{4}{7}$ is not equal to the
intrinsic value of $X$ at $\omega_{1}$, which is 0.141414Information
aggregation fails in the first round in this example and no-one updates from
$\mu$. In general, traders could start from a different common prior
$\mu^{\prime}$, and after several rounds of updating they could update to some
other posterior $\mu^{\prime\prime}$, at which there is no information
aggregation.

Suppose now that traders are able to acquire a noisy signal about the value of
the security before making their announcement. For instance, Trader 1 can
acquire a statistical experiment ${\cal R}$ that generates a signal $z$, given
each state, with probabilities ${\cal R}_{\omega_{1}}(z)=1$ and ${\cal
R}_{\omega_{2}}(z)={\cal R}_{\omega_{3}}(z)={\cal R}_{\omega_{4}}(z)=0.5$.

At state $\omega_{1}$, Trader 1’s prior belief is
$(\frac{3}{5},\frac{2}{5},0,0)$ and, after receiving signal $z$, his posterior
belief is $(\frac{3}{4},\frac{1}{4},0,0)$. He then announces the expected
value of $X$, which is $\frac{5}{14}$. Trader 2 considers states $\omega_{1}$
and $\omega_{4}$ to be possible. The public announcement of $\frac{5}{14}$
reveals to Trader 2 that the true state is $\omega_{1}$. The reason is that,
irrespective of whether Trader 1 received signal $z$ or not, his posteriors at
$\omega_{4}$ are $(0,0,\frac{3}{7},\frac{4}{7})$ and he would have announced
$\frac{4}{7}$. As a result, Trader 2 announces 0 in the second round and the
game ends. Note that the final price is equal to the intrinsic value of $X$ at
$\omega_{1}$, hence information aggregation does occur. In summary, the
ability to acquire extra signals transforms $X$ from non-separable to
separable, enabling information aggregation.

We make two observations. First, we have abstracted from the cost of acquiring
an experiment. Each trader will acquire the signal structure only if the
expected benefit from making a better prediction outweighs the cost. See
Section 4 for the formal treatment. Second, Trader 2 free-rides on Trader 1
buying the signal. By moving the price from $\frac{5}{14}$ to $0$, he books a
profit, without paying the cost of a signal. This example illustrates that the
ability to acquire information can turn a non-separable security into a
$\kappa$ separable.

Security $X$ is $\kappa$ separable if whenever there is agreement about the
expected value of $X$, given a prior that does not put probability 1 to only
one value of $X$, then at least one (myopic) trader finds it profitable to
acquire information. It is therefore a generalization of separability, which
requires that such a prior does not exist. In Section 4.3, we formally define
the notion of $\kappa$ separability and Theorem 6 shows that this class of
securities characterizes information aggregation when the cost is $\kappa$. A
natural question is whether all securities eventually become $\kappa$
separable, for sufficiently low cost of information. Surprisingly, Theorem 3
shows that there is a very small class of securities that never become
$\kappa$ separable, even if the cost is negligible but strictly positive. We,
therefore, have a discontinuity as costs converge to zero.

##  3 No Information Acquisition

In an environment without information acquisition, Ostrovsky (2012) showed
that separable securities are necessary and sufficient for aggregating
information. In this section, we first define the class of separable
securities. We then show that the class of always separable securities, which
are separable for all information structures, is small and uninformative.

###  3.1 Separable Securities

Consider the following example, which appears in similar form in Geanakoplos
and Polemarchakis (1982) and in Example 1 of Ostrovsky (2012).

######  Example 2.

The state space is $\Omega=\\{\omega_{1},\omega_{2},\omega_{3},\omega_{4}\\}$,
the security is $X=(0,1,0,1)$, and there are two traders with common prior
$\mu=(1/4,1/4,1/4,1/4)$. Trader 1’s partition is
$\\{\\{\omega_{1},\omega_{2}\\},\\{\omega_{3},\omega_{4}\\}\\}$ and Trader 2’s
is $\\{\\{\omega_{1},\omega_{4}\\},\\{\omega_{2},\omega_{3}\\}\\}$.

At all states, it is common knowledge that both traders agree that the
expected value of $X$ is 0.5, hence there can be no more learning from further
announcements. However, it is also common knowledge that the intrinsic value
of $X$ is not 0.5; it is either 0 or 1, which implies that there is no
information aggregation. When both these conditions are satisfied for some
prior and some partitions, the security is non-separable.

######  Definition 2.

A security $X$ is called non-separable under information structure $\Pi$ if
there exists probability $\mu$ and value $v\in\mathbb{R}$ such that:

  * $(i)$

$X(\omega)\neq v$ for some $\omega\in Supp(\mu)$,

  * $(ii)$

$E_{\mu}[X|\Pi_{i}(\omega)]=v$ for all $i=1,...,n$ and $\omega\in Supp(\mu)$.

We then say that security $X$ is non-separable at $\mu$. Otherwise, it is
called separable.

A security $X$ is non-separable if, for some prior $\mu$ and at all states in
its support, all traders’ expected value of $X$ is $v$, yet there is
uncertainty about the value of $X$. If for any prior at least one condition is
violated, then the security is separable. Note that separability (and non-
separability) is a property that depends on the information structure $\Pi$.
For a different $\Pi$, a security may switch from separable to non-separable
and vice-versa.

######  Theorem (Ostrovsky (2012)).

Fix information structure $\Pi$. Then:

  * •

If security $X$ is separable under $\Pi$, then in any Nash equilibrium of game
$\Gamma^{S}(\Omega,I,\Pi,X,y_{0},\mu,\underline{y},\overline{y},s,\beta)$,
information gets aggregated.

  * •

If security $X$ is non-separable under $\Pi$, then there exists prior $\mu$
such that for all $s$, $y_{0}$, $\underline{y}$, $\overline{y}$, and $\beta$,
there exists a Perfect Bayesian equilibrium of the corresponding game
$\Gamma^{S}$ in which information does not get aggregated.

This is a powerful result because it applies to all equilibria and
irrespective of the market power of traders. Separable securities can be very
useful for a market designer because they always aggregate information, hence
the price can predict whether an event has occurred or not. However,
separability depends on the information structure $\Pi$. If the market maker
does not know $\Pi$, he cannot be certain that the equilibrium price is a good
predictor of the intrinsic value of the security.

###  3.2 Always Separable Securities

The dependence of separability on $\Pi$ is not a problem if the security is
separable for all information structures. In this section, we characterize the
securities that have this property, so that information aggregation does not
depend on who trades or what is their private information.

Unfortunately, the following Proposition shows that this class is very small
and uninformative. It consists of just three types of securities. The first is
the constant, which pays the same at all states. The second is the Arrow-
Debreu (A-D), which pays $a$ at some state $\omega$ and $b$ at all other
states. The third pays $a$ at some state $\omega$, $d$ at $\omega^{\prime}$,
and $b$ at all other states, where $a<b<d$.

proposition The only non-constant securities that are separable for all
information structures in ${\cal P}$ are the A-D and the security that is of
the following form. There are values $a<b<d$ such that $X(\omega_{a})=a$ and
$X(\omega_{d})=d$ for two states $\omega_{a},\omega_{d}$, and $X(\omega)=b$
for all $\omega\neq\omega_{a},\omega_{d}$.

We start by restating a useful characterization of separable securities by
Ostrovsky (2012). It specifies that $X$ is separable if and only if, for any
possible announcement $v$, we can find numbers $\lambda_{i}(\Pi_{i}(\omega))$,
for each $i$ and $\omega$, such that the sum over all traders has the same
sign as the difference of $X(\omega)-v$. Intuitively, for any $v$ and at each
$\omega$, all traders “vote” and the sign of the sum of the votes has to agree
with the sign of the difference between the value of the security and $v$.

######  Proposition 1 (Ostrovsky (2012)).

Security $X$ is separable under partition structure $\Pi$ if and only if, for
every $v\in{\mathbb{R}}$, there exist functions
$\lambda_{i}:\Pi_{i}\rightarrow{\mathbb{R}}$ for $i=1,\ldots,n$ such that, for
every state $\omega$ with $X(\omega)\neq v$,

$(X(\omega)-v)\underset{i\in I}{\sum}\lambda_{i}(\Pi_{i}(\omega))>0.$

If $\Omega$ has up to three states, then all securities are of the two types
that we have described or the uninteresting case of a constant security.
Hence, without loss of generality, we fix a state space $\Omega$ with at least
four states and a security $X$. If $X$ is constant, it is trivially separable.
Ostrovsky (2012) shows that an A-D security is always separable.

We now show that $X$ is always separable if it is of the following form.
Suppose there are $a<b<c$ such that $X(\omega_{a})=a$ and $X(\omega_{c})=c$
for two states $\omega_{a},\omega_{c}$, whereas $X(\omega)=b$ for all
$\omega\neq\omega_{a},\omega_{c}$.151515Our proof for this type of security
also applies to an A-D security. Ostrovsky (2012) used Corollary 1 to show
that an A-D security is always separable, however, we cannot use it for this
type of security. Using Proposition 1, we need to show that for every
$v\in{\mathbb{R}}$, there exist functions
$\lambda_{i}:\Pi_{i}\rightarrow{\mathbb{R}}$ for $i=1,\ldots,n$ such that, for
every state $\omega$ with $X(\omega)\neq v$,

$(X(\omega)-v)\underset{i\in I}{\sum}\lambda_{i}(\Pi_{i}(\omega))>0.$ (1)

If $v\leq a$, then condition (1) is satisfied by setting
$\lambda_{i}(\Pi_{i}(\omega))=1$ for all $i\in I$ and $\omega\in\Omega$.
Similarly, if $v\geq c$, we set $\lambda_{i}(\Pi_{i}(\omega))=-1$ for all
$i\in I$ and $\omega\in\Omega$. Suppose that $a<v\leq b<c$. For all $i\in I$,
set $\lambda_{i}(\Pi_{i}(\omega_{a}))=-1$ and (1) is satisfied for
$\omega_{a}$. For all $\omega$ with $\Pi_{i}(\omega)\neq\Pi_{i}(\omega_{a})$,
set $\lambda_{i}(\Pi_{i}(\omega))=k$, where $k=|I|$ is the number of agents.
Because of our assumption that the join of all partitions consists of
singleton sets, we have that for each $\omega\neq\omega_{a}$, there exists $i$
such that $\Pi_{i}(\omega)\neq\Pi_{i}(\omega_{a})$. This implies that if
$X(\omega)-v>0$, we also have $\underset{i\in
I}{\sum}\lambda_{i}(\Pi_{i}(\omega))\geq k-(k-1)>0$ and (1) is satisfied for
$\omega$. Using a symmetric argument, we can show that (1) is satisfied for
$a<b\leq v<c$, by setting $\lambda_{i}(\Pi_{i}(\omega_{c}))=1$ and
$\lambda_{i}(\Pi_{i}(\omega))=-k$ for $\omega$ with
$\Pi_{i}(\omega)\neq\Pi_{i}(\omega_{c})$, for all $i\in I$. By applying
Proposition 1, security $X$ is always separable.

Suppose that $X$ is not of the three aforementioned types. Then, we can find
four distinct states where $X$ assigns values $a\leq b<c\leq d$. For
simplicity, we refer to the state with value $a$ as state $a$ and similarly
for $b,c$, and $d$.

We will show that $X$ is non-separable for an information structure in ${\cal
P}$ with two agents. The partition of Trader 1 is $\\{\\{a,d\\},\\{b,c\\}\\}$
for these four states, whereas for any other state, we have
$\Pi_{1}(\omega)=\\{\omega\\}$. For Trader 2 it is $\\{\\{a,c\\},\\{b,d\\}\\}$
and for any other state we have $\Pi_{2}(\omega)=\\{\omega\\}$. Hence, the
information structure is in ${\cal P}$.

To show that $X$ is non-separable, it is enough to find a prior $p$ with
support on $\\{a,b,c,d\\}$ such that, for some $v$,

  * $(i)$

$X(\omega)\neq v$ for some $\omega\in Supp(p)$,

  * $(ii)$

$E_{p}[X|\Pi_{i}(\omega)]=v$ for all $i=1,2$ and $\omega\in Supp(p)$.

Let $p_{1}$ be 1’s probability of state $a$ conditional on $\\{a,d\\}$,
whereas $q_{1}$ is 1’s probability of state $b$ conditional on $\\{b,c\\}$.
Let $p_{2}$ be 2’s probability of state $a$ conditional on $\\{a,c\\}$,
whereas $q_{2}$ is 2’s probability of state $b$ conditional on $\\{b,d\\}$.
Condition (ii) then translates to the following equations

$\begin{cases}ap_{1}+d(1-p_{1})=bq_{1}+c(1-q_{1})\\\
ap_{1}+d(1-p_{1})=ap_{2}+c(1-p_{2})\\\ ap_{1}+d(1-p_{1})=bq_{2}+d(1-q_{2})\\\
bq_{1}+c(1-q_{1})=ap_{2}+c(1-p_{2})\\\ ap_{1}+d(1-p_{1})=bq_{2}+d(1-q_{2})\\\
ap_{2}+c(1-p_{2})=bq_{2}+d(1-q_{2})\end{cases}$ (2)

The posteriors of the two agents can be derived by a common prior $p$ if the
following conditions hold:

$\begin{cases}xp_{1}=yp_{2}\\\ (1-x)q_{1}=(1-y)q_{2}\\\
(1-x)(1-q_{1})=y(1-p_{2})\\\ x(1-p_{1})=(1-y)(1-q_{2})\end{cases}$ (3)

where $x$ is the prior probability of $(a,d)$ and $y$ is the prior probability
of $(a,c)$.

When $a\leq b<c\leq d$, the system (2 \- 3) has the following solution:

$\begin{split}q_{1}=\frac{a-c}{a+b-c-d},\\\ p_{1}=\frac{b-d}{a+b-c-d},\\\
p_{2}=\frac{b-c}{a+b-c-d},\\\ q_{2}=\frac{a-d}{a+b-c-d},\\\ x=p_{2},\\\
y=p_{1}.\end{split}$

These posteriors uniquely define the respective prior probabilities.

All three types of securities are ‘uninformative’. Even when there is
information aggregation and the price of $X$ always converges to the true
value $X(\omega)$ at state $\omega$, this only reveals that either $\omega$,
$\omega^{\prime}$, or neither, have occurred. In other words, the price of $X$
does not reveal information about most events in $\Omega$. In contrast, the
most informative security $X$ pays differently across all states. If there is
information aggregation, then $X$ reveals whether any event in $\Omega$ has
occurred or not. However, because $X$ is not always separable, we know that
for some information structure $\Pi$, information does not aggregate.

##  4 Information Acquisition

In this section, we show how the problem of non-separability and no
information aggregation can be alleviated if we enhance the model of Ostrovsky
(2012), by allowing traders to acquire information in every period where they
make an announcement. We define a new class of securities, called $\kappa$
separable, where $\kappa$ is the cost of acquiring information. Our first main
result is Theorem 3, which characterizes the securities that are $\kappa$ non-
separable for some information structure $\Pi$ but for all $\kappa$. Our
second main result, Theorem 6, shows that $\kappa$ separable securities
characterize information aggregation, in both strategic and non-strategic
settings, thus generalizing Ostrovsky (2012) in an environment with
information acquisition.

###  4.1 Cost of Information

We first formalise the cost of acquiring information. For ease of exposition,
many of the technical details are relegated to Section A.1 in the Appendix.

Let $T$ be a Polish space of possible signals with Borel $\sigma$-algebra
${\cal T}$. Let $\Delta(T)$ be the set of Borel probabilities on $T$.
Following Blackwell (1951), we model the acquisition of information using
statistical experiments. A statistical experiment is a function from states
into probabilities on signals, ${\cal R}:\Omega\rightarrow\Delta(T)$. An
experiment is bounded if it does not definitely rule out any state. Let ${\cal
E}$ be the collection of all experiments and ${\cal E}_{b}$ be the collection
of all bounded experiments.

Given a prior belief $\mu\in\Delta(\Omega)$, a statistical experiment ${\cal
R}$ induces via Bayesian updating a probability distribution over posteriors,
or random posterior, $B(\mu,{\cal R})=Q\in\Delta(\Delta(\Omega))$. Let
$Q(\gamma)$ be the probability of posterior $\gamma\in\Delta(\Omega)$, ${\cal
Q}(\mu)=\\{B(\mu,{\cal R}):{\cal R}\in{\cal E}\\}$ be the set of all random
posteriors that can be generated by some experiment ${\cal R}\in{\cal E}$, and
${\cal Q}=\underset{\mu\in\Delta(\Omega)}{\bigcup}{\cal Q}(\mu)$. Note that
$Q\in{\cal Q}(\mu)$ if and only if $\underset{\Delta(\Omega)}{\int}\gamma
Q(d\gamma)=\mu$.

A cost on experiments is a map $h:\Delta(\Omega)\times{\cal
E}\rightarrow[0,\infty]$, where $h(\mu,\cdot)$ is Borel measurable for each
prior $\mu\in\Delta(\Omega)$. The cost $h$ generates a cost structure
$\kappa=(K,c)$ on random posteriors, where $c>0$ is the unit cost of
information and $K:{\cal F}\rightarrow[0,\infty]$ maps elements of ${\cal
F}=\\{(\mu,Q):\mu\in\Delta(\Omega),Q\in{\cal Q}(\mu)\\}$, the set of all
priors $\mu$ and all posterior distributions $Q$ that can be generated by some
experiment ${\cal R}\in{\cal E}$, to the extended real line.

For Theorem 3 and Proposition 2, we assume that it is prohibitively costly to
acquire an unbounded experiment.

######  Assumption 1.

If ${\cal R}\notin{\cal E}_{b}$, then $h(\mu,{\cal R})=\infty$.

If all unbounded experiments have infinite cost, then generating a random
posterior with support on a posterior $\delta_{\omega}$, which assigns
probability 0 to a state $\omega\in Supp(\mu)$, is infinitely costly.

A widely used class of functions is the posterior-separable cost functions
(Caplin et al. (2022)).

######  Definition 3.

A cost of information function $K$ is posterior-separable if, given
$\mu\in\Delta(\Omega)$ and any Bayes-consistent posteriors $Q\in{\cal
Q}(\mu)$,

$K(\mu,Q)=\underset{\gamma\in Supp(Q)}{\sum}Q(\gamma)W_{\mu}(\gamma)$

for some function $W_{\mu}:\Delta(Supp(\mu))\rightarrow\overline{\mathbb{R}}$
which is strictly convex and continuous in $\gamma$, $T_{\mu}(\gamma)<\infty$
on $int\Delta(Supp(\mu))$ and $W_{\mu}(\gamma)\geq 0$.

An example of a posterior-separable cost function that satisfies Assumption 1
is the Shannon cost function,

$K^{S}(\mu,Q)=\underset{\gamma\in Supp(Q)}{\sum}Q(\gamma)\underset{\omega\in
Supp(\gamma)}{\sum}\gamma(\omega)\ln\gamma(\omega)-\underset{\omega\in
Supp(\mu)}{\sum}\mu(\omega)\ln\mu(\omega).$

We have described two ways of representing the cost of information
acquisition. The first is in terms of costly statistical experiments, whereas
the second is in terms of costly random posteriors. There is a growing
literature that examines the connection between the two approaches. Bloedel
and Zhong (2020) and Hébert and Woodford (2021, 2023) show how costly
statistical experiments, with a function $h$ that depends both on the agent’s
prior and the experiment, can generate uniformly posterior-separable cost
functions for random posteriors. Denti et al. (2022), however, restrict $h$ to
depend only on the experiment and show that uniformly posterior-separable cost
functions cannot be generated.

We do not take a stance on which representation is the most suitable. Our
results do not require any specific functional forms, such as posterior
separability, however, we adopt this framework for most of the paper because
it is the most well-known. When we define the game in Section 4.4, we use the
standard framework of costly statistical experiments with function $h$.

###  4.2 The myopic problem

Suppose that it is Trader $i$’s at time $t$ turn to make an announcement.
Having observed all previous announcements and using the public information
that is revealed, an outside observer updates the common prior $\mu_{0}$ to a
belief $\mu$ over $\Omega$. If the true state is $\omega$, then Trader $i$’s
private information is $\Pi_{i}(\omega)$ and his posterior belief is the
Bayesian update of $\mu$, denoted $\mu_{\Pi_{i}(\omega)}$. In other words, he
updates using both his private information and the public information revealed
by previous announcements.

His myopic problem consists of buying a random posterior $Q\in{\cal
Q}(\mu_{\Pi_{i}(\omega)})$ at cost $cK(\mu_{\Pi_{i}(\omega)},Q)$, so that when
his posterior beliefs are $\gamma\in Supp(Q)$, he optimally announces
$E_{\gamma}[X]$ because the scoring rule is proper. The optimal $Q$ solves the
problem

$\underset{Q\in{\cal
Q}(\mu_{\Pi_{i}(\omega)})}{\sup}\Bigg{(}\underset{\gamma\in
Supp(Q)}{\sum}Q(\gamma)\underset{\omega^{\prime}\in\Omega}{\sum}\gamma(\omega^{\prime})\Big{[}s(E_{\gamma}[X],X(\omega^{\prime}))-s(z,X(\omega^{\prime}))\Big{]}-cK(\mu_{\Pi_{i}(\omega)},Q)\Bigg{)},$
(4)

where $c>0$ is the unit cost of information and $z$ is the previous
announcement.

In the standard model of Ostrovsky (2012), without information acquisition,
the previous announcement $z$ does not influence the myopic best announcement
because the scoring rule is proper. The same is true here, where we allow for
information acquisition.171717Interestingly, with ambiguity averse
preferences, the myopic best depends on the previous announcement, as shown in
Galanis et al. (2024). To see this, note that we can rewrite the expression in
the parenthesis as

$\underset{\gamma\in
Supp(Q)}{\sum}Q(\gamma)\underset{\omega^{\prime}\in\Omega}{\sum}\gamma(\omega^{\prime})\Big{[}s(E_{\gamma}[X],X(\omega^{\prime})))\Big{]}-\underset{\omega^{\prime}\in
Supp(\mu)}{\sum}\mu(\omega^{\prime})s(z,X(\omega^{\prime}))-cK(\mu_{\Pi_{i}(\omega)},Q),$

hence the previous announcement $z$ does not influence the choice of $Q$ or
the announcement.

Note that the trader can always receive a payoff of 0 by just repeating the
previous announcement $z$ and not acquiring any new information. If his payoff
at (4) is smaller or equal to 0 for all $Q\in{\cal Q}(\mu_{\Pi_{i}(\omega)})$,
we say that he does not prefer to acquire information at $\omega$. If, given a
common prior $\mu$, no Trader $i$ prefers to acquire information at any state
$\omega\in Supp(\mu)$, we say that there is no information acquisition for
security $X$.

######  Definition 4.

There is no information acquisition for security $X$, given prior $\mu$ and
previous announcement $z$, cost structure $\kappa=(K,c)$ and information
structure $\Pi\in{\cal P}$, if for all states $\omega\in Supp(\mu)$, all
traders $i$, and all $Q\in{\cal Q}(\mu_{\Pi_{i}(\omega)})$,

$\underset{\gamma\in
Supp(Q)}{\sum}Q(\gamma)\underset{\omega^{\prime}\in\Omega}{\sum}\gamma(\omega^{\prime})\Big{[}s(E_{\gamma}[X],X(\omega^{\prime}))-s(z,X(\omega^{\prime}))\Big{]}-cK(\mu_{\Pi_{i}(\omega)},Q)\leq
0.$

Otherwise, there is information acquisition.

###  4.3 $\kappa$ Separable Securities

The result of Proposition 3.2, that very few securities are always separable,
indicates that ‘most’ markets may not aggregate information. A natural
question is whether the possibility of acquiring information fixes this
problem. Intuitively, a non-separable security may still aggregate information
if, whenever everyone makes the same announcement, at least one trader finds
it profitable to acquire more information, thus changing his posterior and his
announcement.

In this section, we define the class of $\kappa$ separable securities, where
$\kappa=(K,c)$ is the cost of information acquisition. In the next section, we
show that they are necessary and sufficient for information aggregation, thus
generalising the main result of Ostrovsky (2012).

We say that security $X$ is $\kappa$ non-separable given an information
structure $\Pi$ and a cost structure $\kappa$ if there is a prior $\mu$ such
that no one is acquiring any information, yet the security is non-separable at
$\mu$.

######  Definition 5.

Security $X$ is $\kappa$ non-separable given an information structure
$\Pi\in{\cal P}$ and cost structure $\kappa\in{\cal K}$, if there exists prior
$\mu$ such that

  * •

Security $X$ is non-separable at $\mu$,

  * •

There is no information acquisition for $X$ given $\mu$.

Otherwise, security X is $\kappa$ separable.

Recall that a non-separable security is non-separable for at least one $\mu$,
whereas a separable security is separable for all $\mu$. If security $X$ is
$\kappa$ separable, then for each $\mu$ there are two cases. Either $X$ is
separable at $\mu$, or $X$ is non-separable at $\mu$ but there is information
acquisition.

We make the following remarks. First, separability implies $\kappa$
separability, for all $\kappa$, because there does not exist a prior at which
the security is non-separable. Second, non-separability implies $\kappa$ non-
separability for some $\kappa$, because for a high enough marginal cost $c$,
no trader will acquire any information.

######  Remark 1.

If security $X$ is separable given an information structure $\Pi\in{\cal P}$,
then it is also $\kappa$ separable for any cost structure $\kappa\in{\cal K}$.

######  Remark 2.

If security $X$ is non-separable given an information structure $\Pi\in{\cal
P}$, then for any cost function $K$ it is $\kappa=(K,c)$ non-separable for
some marginal cost of information $c$.

Note that if $X$ is $\kappa=(K,c)$ non-separable, then there is a $\mu$ for
which there is no information acquisition and $X$ is non-separable at $\mu$.
If we increase the marginal cost to $c^{\prime}>c$, then there is still no
information acquisition for $\mu$, hence $X$ is
$\kappa^{\prime}=(K,c^{\prime})$ non-separable. Conversely, if $X$ is
$\kappa=(K,c)$ separable, it is separable for all $\mu$ for which there is no
information acquisition. If we decrease the marginal cost to $c^{\prime}<c$,
then the set of priors for which there is no information acquisition will
shrink and therefore $X$ will be $\kappa^{\prime}=(K,c^{\prime})$ separable
for all $c^{\prime}<c$. We, therefore, have the following remark.

######  Remark 3.

If $X$ is $\kappa=(K,c)$ non-separable (separable), then it is
$\kappa^{\prime}=(K,c^{\prime})$ non-separable (separable) for all
$c^{\prime}>c$ ($c^{\prime}<c$).

It would be natural to expect that, for a sufficiently low marginal cost $c$,
any security eventually becomes $\kappa$ separable. But this is not true.
Recall Example 2 with state space
$\Omega=\\{\omega_{1},\omega_{2},\omega_{3},\omega_{4}\\}$, security
$X=(0,1,0,1)$, and two traders with the following information structure.
Trader 1’s partition is
$\\{\\{\omega_{1},\omega_{2}\\},\\{\omega_{3},\omega_{4}\\}\\}$ and Trader 2’s
is $\\{\\{\omega_{1},\omega_{4}\\},\\{\omega_{2},\omega_{3}\\}\\}$. With a
uniform prior, all traders would agree that the expected value of $X$ is 0.5,
at all states, hence the security is non-separable.

Consider now prior $\mu=(\frac{1-2m}{2},m,\frac{1-2m}{2},m)$, where $0<m<0.5$.
Given any $m$, each trader’s expected value of $X$ at all states is $2m$. As
$c$ converges to 0, we can always specify $m$ that is sufficiently close to
$0.5$, so that no trader is willing to buy a signal, because this will move
their expected value of $X$ from $1-2\epsilon$ to $1-\epsilon$, for small
enough $\epsilon>0$, hence the expected benefit is lower than $c$. Because
there is no information acquisition and all traders agree on the expected
value, security $X$ is $\kappa=(K,c)$ non-separable for all $c>0$.

Note that this result requires Assumption 1, which specifies that it is
impossible to buy an experiment that reveals the true state with
certainty.181818It is interesting to note that the incompatibility of the
$(0,1,0,1)$ vector with information aggregation is met in other settings as
well. In the model of DeGroot (1974), agents update their beliefs naively, by
looking at their immediate neighbours, according to a fixed network. If the
network is represented by a periodic matrix, such as $A=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}$, then beliefs do not converge (Golub and Jackson (2010)).
Moreover, all traders agree, at some state $\omega$, that at least two values
are possible, 0 and 1. Proposition 2 in the Appendix shows that this condition
is necessary for a security to be $\kappa$ non-separable for all $\kappa$ and
a specific information structure. Additionally, it specifies a second,
independent condition, which is an extension of Theorem 7 in Ostrovsky (2012),
a dual characterisation of separability.

However, what if we do not know the information structure of traders? The
following Theorem shows that security $X$ is $\kappa$ non-separable given some
$\Pi$ and for all $\kappa$, if and only if there are four states at which $X$
pays $(a,d,b,d)$, where either $a,b<d$ or $a,b>d$.

theorem Suppose that $\Omega$ has at least four states. Under Assumption 1,
the following are equivalent:

  * •

$X$ is $\kappa$ non-separable given some $\Pi$ and for all $\kappa$,

  * •

$X$ pays $(a,b,d,d)$ in four states, where either $a,b<d$ or $a,b>d$.

Suppose $\Omega$ has at least four states and let
$E=\\{\omega_{1},\omega_{2},\omega_{3},\omega_{4}\\}$, where $X$ pays
$(a,d,b,d)$, with either $a,b<d$ or $a,b>d$. What $X$ pays outside $E$ is
irrelevant, because we will only consider priors that have full support on
$E$, in order to show that $X$ is $\kappa$ non-separable for all $\kappa$.
Consider two traders with the following information structure on $E$. Trader
1’s partition is
$\\{\\{\omega_{1},\omega_{2}\\},\\{\omega_{3},\omega_{4}\\}\\}$ and Trader 2’s
is $\\{\\{\omega_{1},\omega_{4}\\},\\{\omega_{2},\omega_{3}\\}\\}$. We need to
show that for any $\kappa=(K,c)$, $X$ is $\kappa$ non-separable. It is enough
to show that, as the marginal cost $c$ converges to 0, there is always a
common prior $\mu$ on $E$ such that no trader acquires any information and $X$
is non-separable at $\mu$.

Consider prior $\mu=(p,m,1-p-2m,m)$, where $0<m<0.5$. We specify $p$ such
that, given any $m$, each trader’s expected value of $X$ is the same at all
states. In the following equation, the left hand-side computes Trader 1’s
expected value of $X$ at $\\{\omega_{1},\omega_{2}\\}$ and Trader 2’s at
$\\{\omega_{1},\omega_{4}\\}$, whereas the right hand-side computes Trader 1’s
expected value at $\\{\omega_{3},\omega_{4}\\}$ Trader 2’s at
$\\{\omega_{2},\omega_{3}\\}$:

$a\frac{p}{p+m}+d\frac{m}{p+m}=b\frac{1-2m-p}{1-p-m}+d\frac{m}{1-p-m}.$

There is always a solution to this equation. For example, if $a+b\neq 0$, we
can normalise to $p+m=1/2$ and we have that

$p=\frac{b(1-2m)}{a+b}.$

Given this normalisation, the expected value of $X$ at all states and for all
traders is $u=2pa+2dm$. Fix any marginal cost $c$, which may be very close to
0. We can then choose $m$ very close to 0.5 (and corresponding $\mu$) such
that $u$ is very close to $d$, say $d-\epsilon$, assuming, without loss of
generality, that $a,b<d$, otherwise we have $d+\epsilon$ and the rest of the
proof proceeds accordingly. By repeating the consensus announcement of
$u=d-\epsilon$ and not acquiring any signals, both traders get 0 utility.
Suppose that one trader wants to buy a signal structure $Q$, which, in the
case of $X(\omega)=d$, will move his posterior expected value of $X$ from
$d-\epsilon$ to $d-\epsilon+\nu$, for some $0<\nu<\epsilon$. The upper bound
of his utility, net of the cost of the signal, is realised at $X(\omega)=d$
and it is $s(d-\epsilon+\nu,d)-s(d-\epsilon,d)$. This is positive because
$d-\epsilon+\nu$ is closer to $d$ than the previous announcement,
$1-\epsilon$. However, by decreasing $\epsilon$ appropriately it can be made
as small as needed, and therefore smaller than $cK(\mu,Q)$ for all $Q$. This
is a direct consequence of Assumption 1, which specifies that the cost gets
arbitrarily high for posteriors which assign probability that is arbitrarily
close to one for some state. For this result, we only need that the cost is
non-decreasing. Hence, no trader will buy any signal structure and $X$ is
$\kappa$ non-separable.

For the converse, suppose $X$ is $\kappa$ non-separable given some $\Pi$ and
for all $\kappa$. The second condition of Proposition 2 and the assumption in
Section 2.1 that $\underset{i\in I}{\bigcap}\Pi_{i}(\omega)=\\{\omega\\}$, for
all $\omega\in\Omega$, imply that there exist two states
$\omega_{a}\neq\omega_{b}$ such that $X(\omega_{a})=X(\omega_{b})=m$. Let $M$
be the set of states $\omega$ with $X(\omega)=m$. Given that $\Omega$ has at
least four states, we have the following cases.

Case 0. $M=\Omega$. Security $X$ is constant and therefore (trivially) always
separable, for all information structures $\Pi$. This is a contradiction
because we have assumed $X$ is $\kappa$ non-separable given some $\Pi$ and for
all $\kappa$.

Case 1. There is either a unique $\omega\notin M$, or there are exactly two
states $\omega,\omega^{\prime}\notin M$, such that
$X(\omega)>m>X(\omega^{\prime})$. From Proposition 3.2, $X$ is always
separable, for all information structures $\Pi$. As with Case 0, this is a
contradiction.

The only other remaining case is that There exist
$\omega,\omega^{\prime}\notin M$ with either $X(\omega),X(\omega^{\prime})<m$,
or $X(\omega),X(\omega^{\prime})>m$, which concludes the proof.

A Corollary is that if security $X$ pays differently across all states, then
it is $\kappa$ separable for some $\kappa$, given any information structure
$\Pi$. In other words, if information is sufficiently cheap to acquire, there
are easily describable securities where information aggregation is achieved
and this is robust to changes in who participates in the market and what is
their private information.

######  Corollary 1.

If $X(\omega)\neq X(\omega^{\prime})$ for all
$\omega,\omega^{\prime}\in\Omega$, then, given any $\Pi$, $X$ is $\kappa$
separable for some $\kappa$.

This Corollary echoes the following result within the context of perfectly
competitive markets, as pointed out in Laffont and Maskin (1990). As long as
there are “enough prices”, so that trade can be made contingent on
sufficiently many events, then the competitive equilibrium is generically
separating and the function relating information to prices is invertible
(Grossman (1976), Radner (1979), Allen (1981)). We could interpret Corollary 1
as a generalisation of this result, because our setting applies to all Nash
equilibria, not just competitive equilibria.

We conclude this section with a classification of all securities, with respect
to $\kappa$ separability. Note that the classification does not depend on the
information structure, just the payoff structure of the security.

Let $X$ be a security defined on $\Omega$ that has at least four states. If
there is $m$ such that $X(\omega^{\prime})=X(\omega^{\prime\prime})=m$ for at
least two states $\omega^{\prime},\omega^{\prime\prime}\in\Omega$, let $M$ be
the set of states $\omega$ with $X(\omega)=m$. Otherwise, set $M=\emptyset$.

Case 0. $M=\Omega$. Security $X$ is constant and therefore (trivially) always
separable, for all information structures $\Pi$.

Case 1. There is either a unique $\omega\notin M$, so that the security is
A-D, or there are exactly two states $\omega,\omega^{\prime}\notin M$, such
that $X(\omega)>m>X(\omega^{\prime})$. From Proposition 3.2, $X$ is always
separable, for all information structures $\Pi$.

Case 2. There exist $\omega,\omega^{\prime}\notin M$ with either
$X(\omega),X(\omega^{\prime})<m$, or $X(\omega),X(\omega^{\prime})>m$. From
Theorem 3, $X$ is $\kappa$ non-separable for all $\kappa$, for some
information structure $\Pi$.

Case 3. $M=\emptyset$, hence $X(\omega)\neq X(\omega^{\prime})$, for all
$\omega,\omega^{\prime}\in\Omega$. From Corollary 1, $X$ is $\kappa$ separable
for some $\kappa$, for all information structures $\Pi$.

It is interesting to note that, within our framework that has continuous
values and finitely many states, all cases except Case 3 are of measure zero.
Hence, if we were to randomly pick a security, generically we would pick one
with unique values which aggregates information in all equilibria and given
any information structure, as long as the cost of information is sufficiently
low.

###  4.4 Strategic traders

We now show that $\kappa$ separable securities characterize information
aggregation in strategic settings where there cost of information acquisition
is $\kappa$. Consider game
$\Gamma^{S}(\Omega,I,\Pi,X,h,y_{0},\mu,\underline{y},\overline{y},s,\beta)$,
where $I$ is the set of $n$ players, $s$ is a strictly proper scoring rule,
$y_{0}$ is the market maker’s initial announcement at time $t_{0}$, $\mu$ is
the common prior, $h$ is the cost of statistical experiments,
$[\underline{y},\overline{y}]$ is the set of possible announcements, and
$\beta$ is the common discount rate. Let
$(y_{1},\ldots,y_{k})\in[\underline{y},\overline{y}]^{k}$ be a history of
announcements and $(\tau_{1},\ldots,y_{k})\in T^{k}$ be a history of signals
up to time $t_{k}$. Denote by $T^{k}_{i}$ the collection of all histories
about $i$’s signals up to time $t_{k}$.

At time $t_{k}$, Trader $i$ with belief $\mu$ can choose to acquire
information in the form of a statistical experiment ${\cal R}\in{\cal E}$,
with cost $h(\mu,R)$. Each experiment ${\cal R}$ and belief $\mu$ uniquely
induce a Bayesian plausible belief $Q$ over posteriors, which costs
$cK(\mu,Q)=h(\mu,R)$. We can therefore analyse the game in terms of an
information cost structure $\kappa=(K,c)$, which is generated by $h$.

After receiving a signal $\tau$ of experiment ${\cal R}$ and updating to a
posterior $\gamma$ in the support of $Q$, Trader $i$ makes an announcement
$y_{k}$. Let $[\underline{y},\overline{y}]^{T}$ be the set of all functions
from signals to announcements in $[\underline{y},\overline{y}]$. His mixed
strategy at time $t_{k}$ is a measurable function

$\sigma_{i,k}:\Pi_{i}\times
T^{k-1}_{i}\times[\underline{y},\overline{y}]^{k-1}\times[0,1]\longrightarrow{\cal
E}\times[\underline{y},\overline{y}]^{T}.$

It specifies a statistical experiment and an announcement for each signal that
is drawn from that experiment, given the element of his partition, the history
of his past signals and everyone’s announcements up to time $t_{k}$, and the
realization of random variable $\iota_{k}\in[0,1]$, which is drawn from the
uniform distribution. One such draw takes place at each time $t_{k}$ and the
draws are independent of each other and of the true state $\omega$.

The full state is $\phi=(\omega,\iota_{1},\iota_{2},\ldots)$ and describes the
initial uncertainty and the randomizations of the players. Let
$X(\phi)=X(\omega)$ be the true value of $X$ at
$\phi=(\omega,\iota_{1},\iota_{2},\ldots)$ and let
$\Phi=\Omega\times[0,1]^{\mathbb{N}}$ be the full state space. Denote by
$\sigma_{i}$ the collection $\sigma_{i,k}$ of $i$’s strategies, at all times
$t_{k}$ where it is his turn to make an announcement. Let
$\sigma=(\sigma_{1},\ldots,\sigma_{n})$ be a profile of strategies. Given a
strategy $\sigma$ and state $\phi$, let $y_{i+nk}(\sigma,\phi)$ be the
announcement of Trader $i$, $\mu_{i+nk}(\sigma,\phi)$ his belief, and
$cK(\mu_{i+nk}(\sigma,\phi),Q_{i+nk}(\sigma,\phi))$ his cost of information
acquisition in period $t_{i+nk}$ which is generated by the experiment he has
chosen and his current beliefs.

######  Definition 6.

A strategy profile $\sigma$ is a Nash equilibrium if, for every Trader $i$ and
every alternative strategy
$\sigma^{\prime}=(\sigma_{-i},\sigma^{\prime}_{i})$, we have

$E_{\mu}\Bigg{[}\sum_{k=0}^{\infty}\beta^{i+nk}\bigg{(}s\Big{(}y_{i+nk}(\sigma,\phi),X(\phi)\Big{)}-s\Big{(}y_{i+nk-1}(\sigma,\phi),X(\phi)\Big{)}-cK(\mu_{i+nk}(\sigma,\phi),Q_{i+nk}(\sigma,\phi))\bigg{)}\Bigg{]}\geq$
$E_{\mu}\Bigg{[}\sum_{k=0}^{\infty}\beta^{i+nk}\bigg{(}s\Big{(}y_{i+nk}(\sigma^{\prime},\phi),X(\phi)\Big{)}-s\Big{(}y_{i+nk-1}(\sigma^{\prime},\phi),X(\phi)\Big{)}-cK(\mu_{i+nk}(\sigma^{\prime},\phi),Q_{i+nk}(\sigma^{\prime},\phi))\bigg{)}\Bigg{]},$

where the expectation is taken with respect to the common prior $\mu$.

We now show that $\kappa$ separable securities characterize information
aggregation when the cost structure is $\kappa=(K,c)$.202020Note that
Assumption 1 is not needed for this result.

theorem Fix information structure $\Pi$ and cost of experiments $h$, which
generates cost structure $\kappa$. Then:

  * •

If security $X$ is $\kappa$ separable under $\Pi$, then in any Nash
equilibrium of game
$\Gamma^{S}(\Omega,I,\Pi,X,h,y_{0},\mu,\underline{y},\overline{y},s,\beta)$,
information gets aggregated.

  * •

If security $X$ is $\kappa$ non-separable under $\Pi$, then there exists prior
$\mu$ such that for all $s$, $y_{0}$, $\underline{y}$, $\overline{y}$, and
$\beta$, there exists a Perfect Bayesian equilibrium of the corresponding game
$\Gamma^{S}$ in which information does not get aggregated.

We follow the proof of Theorem 1 in Ostrovsky (2012), which proceeds in four
steps. In the first step, we show that there is a uniform lower bound on the
expected profits that at least one trader can make by improving the forecast.

Let $r$ be a distribution over $\Omega$ and $z$ be the previous announcement.
Following Ostrovsky (2012), we define the instant opportunity of Trader $i$ as
the highest expected payoff that he can achieve by changing the forecast from
$z$, if the state is drawn according to $r$. The difference from Ostrovsky
(2012) is that we allow the trader to acquire information before making an
announcement. Let $Q_{\omega}$ be the optimal experiment that Trader $i$
acquires at $\Pi_{i}(\omega)$, given his beliefs $r_{\Pi_{i}(\omega)}$. Trader
$i$’s instant opportunity is

$\underset{\omega\in\Omega}{\sum}r(\omega)\left(\underset{\gamma\in
Supp(Q_{\omega})}{\sum}Q_{\omega}(\gamma)\underset{\omega^{\prime}\in\Omega}{\sum}\gamma(\omega^{\prime})\Big{[}s(E_{\gamma}[X],X(\omega^{\prime}))-s(z,X(\omega^{\prime}))\Big{]}-cK(r_{\Pi_{i}(\omega)},Q_{\omega})\right).$

Let $\Delta$ be the set of probability distributions $r\in\Delta(\Omega)$ for
which there is uncertainty about security $X$, so that there are states $a,b$
with $r(a)>0$, $r(b)>0$ and $X(a)\neq X(b)$.

######  Lemma 1.

If security $X$ is $\kappa$ separable, then for every $r\in\Delta$, there
exist $\chi>0$ and Trader $i$ such that, for every $z\in{\mathbb{R}}$, the
instant opportunity of Trader $i$ given $r$ and $z$ is greater than $\chi$.

###### Proof.

Fix $r\in\Delta$. There are two cases. First, $X$ is separable with respect to
$r$. Then, the proof of Lemma 1 in Ostrovsky (2012) applies and we have the
result. Second, $X$ is non-separable at $r$ and value $v$, where
$v=E_{r}[X|\Pi_{i}(\omega)]$ for all $\omega\in Supp(r)$ and $i\in I$. Because
$X$ is $\kappa$ separable, there is information acquisition. This implies that
for some $\omega\in Supp(r)$, Trader $i$ receives a strictly positive payoff
by acquiring information and changing the previous announcement
$v=E_{r}[X|\Pi_{i}(\omega)]$. Note that the new announcement is not
deterministic but depends on the optimal $Q$, so he announces $E_{\gamma}[X]$
with probability $Q(\gamma)$. From continuity, there is a small enough
$\epsilon>0$, so that for all previous announcements
$z\in[v-\epsilon,v+\epsilon]$, Trader $i$ receives a strictly positive payoff
of at least $\chi_{1}>0$ by acquiring information and changing the
announcement. If $z\notin[v-\epsilon,v+\epsilon]$, then Trader $i$ receives a
strictly positive payoff of at least $\chi_{2}>0$, by not acquiring any
information and announcing the myopically best
$E_{r}[X|\Pi_{i}(\omega)]=v$.212121This is true because a proper scoring rule
is ‘order-sensitive’ so that the further away the previous announcement is
from the true expected value $E_{r}[X|\Pi_{i}(\omega)]=v$, the higher the
payoff is for Trader $i$. The lowest payoff from the myopically best
announcement is 0 and it is achieved when it is equal to the previous
announcement (see p. 2618 in Ostrovsky (2012)). By setting
$\chi_{3}=\min\\{\chi_{1},\chi_{2}\\}$, we have that at $\Pi_{i}(\omega)$,
Trader $i$ receives a strictly positive payoff of at least $\chi_{3}$, for all
previous announcements $z$. For any $\omega^{\prime}\in
Supp(r)\setminus\Pi_{i}(\omega)$, Trader $i$ can repeat $v$ and receive 0.
Hence, the instant opportunity of $i$ given $r$, which is the ex-ante
expectation over all $\omega\in Supp(r)$, is at least
$\chi=r(\Pi_{i}(\omega))\chi_{3}>0$ for all previous announcements.

∎

Steps 2-4 are identical to the proof of Ostrovsky (2012). The reason is that
we only need to use the uniform lower bound that we have established in Step
1. Note that, once we fix an equilibrium strategy, the only difference from
Ostrovsky (2012) is that players receive (up to) an extra signal in each
period where they make an announcement.

The proof of the second part of the Theorem is similar to that of Ostrovsky
(2012). Suppose that $X$ is $\kappa$ non-separable. Then, there exist $\mu$
and $v$ at which there is no information acquisition and $X$ is non-separable
at $\mu$. In the corresponding game $\Gamma^{S}$ where the initial
announcement is $v$, no trader will find it profitable to acquire any
information and their best response is to repeat $v$, in every period and
after every history, hence it is a Bayesian Nash equilibrium.

###  4.5 The value of the market

If the cost of information drops significantly, do we even need markets to
aggregate information through prices? Each trader could buy the necessary
signals and then trade. In this section, we argue that this intuition is not
correct, because markets become even more important in an environment with
information acquisition. The reason is that markets can aggregate information
before it becomes economically viable for each trader to acquire the required
information on their own.

To make this point, we compare the prediction accuracy of the market with that
of a poll, where traders simultaneously make only one announcement and we
compute the average.222222Note that there are many ways of improving the
accuracy of a poll by aggregating announcements differently (Baron et al.,
2014). In our framework, markets will always be more accurate than polls
because more information is disseminated through multiple rounds of
announcements, and the value of information is positive. Several papers have
examined the two settings in experiments and real-life settings, and the
results are mixed. Snowberg et al. (2013) argue that prediction markets are
better. Berg et al. (2008) show that the Iowa Electronic Markets were more
accurate than 964 polls in predicting the outcomes of five presidential
elections between 1988 and 2004. Cowgill and Zitzewitz (2015) show that
internal prediction markets in Google and Ford were more accurate than the
predictions of professional forecasters. On the other hand, Atanasov et al.
(2017); Dana et al. (2019) argue that while prediction markets are more
accurate than the simple mean of forecasts from polls, the latter outperform
prediction markets when forecasts are aggregated with transformation
algorithms or made in teams. Camerer et al. (2016) show that markets are
equally accurate with a survey in predicting the replicability of economic
experiments. Recall from Section 4.4 that the full state
$\phi=(\omega,\iota_{1},\iota_{2},\ldots)$ describes the initial uncertainty
$\omega\in\Omega$ and the randomizations of the players, as well as the signal
realisations.

######  Definition 7.

For each $\phi\in\Phi$ and common prior $\mu$, the prediction of the poll is
the average of the myopic predictions, $y_{i}$, where each Trader $i$
optimally obtains a random posterior $Q$ and then they all announce
simultaneously in the first period of the corresponding game $\Gamma^{M}$:

$p^{p}(\phi,\mu)=\frac{\underset{i=1}{\overset{n}{\sum}}y_{i}}{n}.$

The prediction $p^{m}(\phi,\mu)$ of the market is the last price of
$\Gamma^{S}$.

We define the accuracy of the market given $\phi$ and $\mu$ as
$A^{m}(\phi,\mu)=1-|p^{m}(\phi,\mu)-X(\phi)|$, and similarly for the poll,
$A^{p}(\phi,\mu)=1-|p^{p}(\phi,\mu)-X(\phi)|$. The highest possible accuracy
is 1, when $p^{m}(\phi,\mu)=X(\phi)$. The expected accuracy of the market
given $\mu$ is defined as $A^{m}(\mu)=E_{\mu}A^{m}(\phi,\mu)$, whereas for the
poll it is $A^{p}(\mu)=E_{\mu}A^{p}(\phi,\mu)$.

Recall that a non-degenerate prior $\mu$ given $X$ does not assign probability
1 to a unique value of security $X$. If the security is $\kappa$ separable for
some $\kappa$, then the information gets aggregated for some positive marginal
cost of information, which is not true for polls. We, therefore, have the
following remark.

######  Remark 4.

Under Assumption 1, fix an information structure $\Pi\in{\cal P}$ and a cost
of information acquisition $\kappa=(K,c)$. If security $X$ is $\kappa$
separable, then for any non-degenerate $\mu$ given $X$ and for all
$0<c^{\prime}\leq c$, information gets aggregated by a market with cost
$\kappa^{\prime}=(K,c^{\prime})$, so that $A^{m}(\phi,\mu)=1$ for all $\phi\in
Supp(\mu)$, but it is not aggregated by the poll, so that $A^{p}(\phi,\mu)<1$.

###### Proof.

Remark 3 shows that $X$ is $\kappa^{\prime}$ separable, hence the first part
follows directly from Theorem 6, which shows that information gets aggregated.
Assumption 1 implies that individual traders would never acquire full
information in the first round. Because $\mu$ is non-degenerate given $X$, we
have that $p^{p}\neq X(\phi)$. ∎

We now show that $\kappa$ separability is equivalent to the market being
strictly more accurate than the poll, for all non-degenerate priors given $X$.

proposition Suppose Assumption 1 and cost of information $\kappa$. Security
$X$ is $\kappa$ separable given $\Pi\in{\cal P}$ if and only if
$A^{m}(\mu)>A^{p}(\mu)$ for all non-degenerate priors $\mu$ given $X$.

If $X$ is $\kappa$ separable, then from Theorem 6 we have that information
aggregates at all states and therefore $A^{m}(\mu)=1$. Assumption 1 implies
that no trader would acquire full information, hence $p^{p}(\phi,\mu)\neq
X(\phi)$ and $A^{p}(\mu)<1$.

If $X$ is $\kappa$ non-separable, then we can find non-degenerate $\mu$ given
$X$ for which there is no information acquisition and $X$ is non-separable at
$\mu$. This means that everyone agrees on the announcement at all states and
the game ends in the first round. The announcement is the same for everyone,
so the poll gives the same prediction as the market and
$A^{m}(\mu)=A^{p}(\mu)$.

To interpret this result, suppose we define the value of the market (with
security $X$) to be
$V(X)=\underset{\mu\in\Delta_{0}}{\min}A^{m}(\mu)-A^{p}(\mu)$, the minimum
improvement in accuracy given all non-degenerate priors given $X$. Then,
Proposition 4 implies that $X$ is $\kappa$ separable if and only if $V(X)>0$.

We conclude with the following example, where we show how the market compares
in prediction accuracy with the poll, if we fix a prior $\mu$ for which the
security is non-separable and we vary the marginal cost of information $c$.
Suppose that the security is $X=(0,1,2,3)$ and the common prior is
$\mu=(\frac{1}{8},\frac{3}{8},\frac{3}{8},\frac{1}{8})$ on state space
$\Omega=\\{\omega_{1},\omega_{2},\omega_{3},\omega_{4}\\}$. Trader 1’s
partition is $\\{\\{\omega_{1},\omega_{3}\\},\\{\omega_{2},\omega_{4}\\}\\}$
and Trader 2’s is
$\\{\\{\omega_{1},\omega_{4}\\},\\{\omega_{2},\omega_{3}\\}\\}$. Security $X$
is non-separable $\mu$ because the expected value of $X$ for both traders is
$v=\frac{3}{2}$ at all states, yet there is uncertainty about the value of the
security. Traders can acquire information, where $K$ is the Shannon cost of
information. We assume that traders are myopic and proper scoring rules are
used in both settings, hence each announcement is the expected value of the
security given the acquired information.

Figure 1 shows the expected accuracy of markets and polls for prior $\mu$.
When the marginal cost of information is high (c>4.5), no trader acquires any
information and the accuracy of the market is equal to that of the poll. As
the marginal cost decreases below $4.5$, Trader 2 starts acquiring information
if the state is either 1 or 4. This implies that his announcement differs
across partitions, revealing whether event $\\{\omega_{1},\omega_{4}\\}$ or
$\\{\omega_{2},\omega_{3}\\}$ is true. Trader 1 combines this with his private
information and learns which state is true, thus announcing $v=X(\omega)$.
Therefore, a small change in the marginal cost of information allows the
market to aggregate information, with a prediction accuracy of 1. In contrast,
the poll’s accuracy improves gradually as information gets cheaper. This means
that, as the cost of information decreases, the prediction accuracy of the
market suddenly jumps to 1, whereas the accuracy of the poll gradually
increases. Equivalently, the value $V(X)$ of the market is 0 for $c>4.5$, at
$c=4.5$ it becomes positive and it decreases as $c<4.5$ decreases, converging
to 0 as $c\rightarrow 0$.

Figure 1: Prediction accuracy ($A(\mu)=E_{\mu}A(\omega,\mu)$) for markets and
polls.

We conclude by observing that, in a strategic setting, markets can incentivize
traders to acquire and utilize information more efficiently when the cost is
close to the threshold of 4.5, because a small information acquisition could
enable the aggregation of all available information, whereas in polls it can
only marginally improve individual predictions.

##  5 The speed of information aggregation

Another interesting question about information acquisition is whether it makes
the process of information aggregation conclude faster. If this were the case,
the entire society would benefit by learning the intrinsic value of $X$, and
the corresponding event, faster.

Unfortunately, there are no good news. Even when traders are myopic, we show
that information acquisition can make the process of information aggregation
both slower and faster, depending on the parameters. We then show that this is
true for all securities except for the Arrow-Debreu (A-D) security, when the
environment is non-strategic. In particular, with A-D securities the
information aggregation can never be faster (or slower) as compared to the no
information acquisition case. If the process cannot get faster or slower, the
competition of traders to acquire more information and appropriate more of the
surplus produces neither negative nor positive externalities in society.

We begin with an example, which shows that information acquisition may make
information aggregation both faster and slower.

######  Example 3.

The state space is $\Omega=\\{\omega_{1},\omega_{2},\omega_{3},\omega_{4}\\}$,
the security is $X=(0,2,1,1)$, and there are two myopic traders with common
prior $\mu=(\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4})$. Trader 1’s
partition is $\\{\\{\omega_{1},\omega_{2}\\},\\{\omega_{3},\omega_{4}\\}\\}$
and Trader 2’s is
$\\{\\{\omega_{1},\omega_{4}\\},\\{\omega_{2},\omega_{3}\\}\\}$.

Suppose that the true state is $\omega_{1}$. Initially, traders cannot acquire
any information. In the first period, Trader 1 announces 1. This reveals no
information to Trader 2 because his private information is
$\\{\omega_{1},\omega_{4}\\}$, and the same announcement would be made by
Trader 1 in both states. In period 2, Trader 2 announces 0.5. This reveals to
Trader 1 that the true state is $\omega_{1}$, hence, in period 3 he announces
0 and the game ends.

Suppose now that traders are able to acquire the same signal as in the
previous example. When the true state is $\omega_{1}$, Trader 1 acquires
signal $z$, updates his beliefs to $(\frac{2}{3},\frac{1}{3},0,0)$ and
announces $\frac{2}{3}$. The announcement reveals to Trader 2 that the state
is $\omega_{1}$, because at $\omega_{4}$ Trader 1 would have announced 1. As a
result, Trader 2 announces 0 in period 2 and the game ends, hence information
aggregation happens faster with information acquisition.

We now show that the converse is also true, so that information acquisition
may delay information aggregation. Suppose that the common prior is
$\mu=(\frac{3}{16},\frac{6}{16},\frac{7}{32},\frac{7}{32})$. At $\omega_{1}$,
if there is no information acquisition, Trader 1 announces $\frac{4}{3}$. The
announcement reveals to Trader 2 that the state is $\omega_{1}$, hence Trader
2 announces 0 and the game ends in the second round. If there is information
acquisition, Trader 1 acquires the signal $z$ and updates his beliefs to
$(\frac{1}{2},\frac{1}{2},0,0)$, and, therefore, he announces 1. This reveals
no information to Trader 2, and he announces 0.5. This reveals to Trader 1
that the true state is $\omega_{1}$, hence in the third round he announces 0
and the game ends.

Interestingly, if the security is the Arrow-Debreu (A-D), then information
acquisition has no impact on the speed of information aggregation. To
understand why, consider an A-D security that pays 1 at $\omega$ and 0
otherwise. At any period, there are two cases. First, the trader knows that
either $\omega$ is true, or that it is not true, accordingly announcing 1 or
0. Second, he is unsure about whether $\omega$ is true or not. Irrespective of
whether he buys a signal structure, he will announce some value $0<v<1$.
Crucially, other traders already know that his partition cell includes
$\omega$, hence his announcement does not reveal any public information and
the speed of information aggregation is unaffected.

We now provide the formal results. Proposition 2 shows that for any security
that is not A-D, we can find information structures for which $X$ is
separable, and depending on the prior the process can be faster or slower.
Proposition 2 shows that for an A-D security, allowing for information
acquisition neither speeds up nor delays the information aggregation process.

Recall from Section 4.4 that the full state
$\phi=(\omega,\iota_{1},\iota_{2},\ldots)$ describes the initial uncertainty
$\omega$ and the randomizations of the players. In this section, traders have
a pure strategy of an announcement and the choice of the random posterior,
hence $\iota_{n}$ denotes the realisation of the posterior in period $t_{n}$.

Let $t^{*}_{I}(\phi)$ be the period where information aggregation is achieved,
given a separable security and state $\phi$, in an environment where traders
can get information before they make an announcement. Similarly, let
$t^{*}_{NI}(\phi)$ be the period where information aggregation is achieved, in
an environment where there is no information acquisition.

For Proposition 2, we make the following mild assumption.

######  Assumption 2.

Given a state space $\Omega$ with two states and for any proper scoring rule
$s$ and non-constant security $X$, there is posterior separable cost function
$K$ such that, for all $c$, the solution to (4) is unique.

The assumption is true for a quadratic scoring rule and entropy cost function,
as shown in Appendix 11 in Ilinov et al. (2024). More generally, consider cost
$K$ and the payoff that is generated by making the myopic best announcement,
both as functions of the posterior $\gamma\in[\mu,1]$, where $\mu$ is the
prior on one of the two states. Both these functions are strictly convex and a
sufficient condition for a unique solution is that they do not have the same
first-order condition for more than one $\gamma$. As we have complete freedom
in choosing $K$, it can be that we achieve this for any $s$.232323See Tsakas
(2020) for an analysis of this decision problem.

proposition Suppose Assumption 2 and that non-constant security $X$ is not
A-D. Then, there is information structure $\Pi\in{\cal P}$ for which $X$ is
separable, such that for some information cost $\kappa=(K,c)$ and
$\phi\in\Phi$,

  * (i)

$t^{*}_{NI}(\phi)<t^{*}_{I}(\phi)$ for some common prior $\mu$,

  * (ii)

$t^{*}_{NI}(\phi)>t^{*}_{I}(\phi)$ for some common prior $\mu^{\prime}$.

Consider non-constant security $X$ which is not A-D. Then, $X$ maps to at
least three values, $a<b<d$, and we denote the respective payoff-relevant
states as $a,b$, and $d$. Note that each payoff-relevant state
$a,b,d\in\Omega$ is associated with several full states $\phi\in\Phi$, which
resolve any uncertainty about which posteriors a trader receives at each
period. To ease the notation, we omit mentioning $\phi$ when it is
straightforward how statements about $\Omega$ are translated to statements
about $\Phi$.

Let the partition of Trader 1 be $\Pi_{1}=\\{\\{a,d\\},\\{b\\}\\}$ for these
three states, whereas for any other state, we have
$\Pi_{1}(\omega)=\\{\omega\\}$. For Trader 2 it is
$\Pi_{2}=\\{\\{a,b\\},\\{d\\}\\}$ and for any other state we have
$\Pi_{2}(\omega)=\\{\omega\\}$. Hence, the information structure is in ${\cal
P}$. For Trader 1, let $p_{1}$ be the probability of state $a$ conditional on
$\\{a,d\\}$. For Trader 2, let $p_{2}$ be the probability of state $a$
conditional on $\\{a,d\\}$.

We first show that $X$ is separable with respect to $\Pi$, so that
$t^{*}_{I},\>t^{*}_{NI}$ are well-defined for all priors. If $\Omega$ has more
than three states, then for all $\omega^{\prime}\in\Omega\setminus\\{a,b,d\\}$
and $i\in I$ we have $\Pi_{i}(\omega^{\prime})=\\{\omega^{\prime}\\}$ and
$E_{\mu}[X|\Pi_{i}(\omega^{\prime})]=X(\omega^{\prime})$. Therefore,
separability only depends on the three states, $a,b$, and $d$. From
Proposition 3.2, security $X$ is separable with respect to the restriction of
$\Pi$ on $\\{a,b,d\\}$. But then, $X$ is separable with respect to $\Pi$.

Second, we show that (i) is true at state $a$, given $\Pi$ and some
$\kappa=(K,c)$. Take a full support prior $\mu$ such that
$E_{\mu}[X|\Pi_{1}(a)]=v\neq b$. This is possible because $X(a)<X(b)<X(d)$. At
state $a$, Trader 1 announces $v$, so Trader 2 learns that $b$ is not true and
announces a. We therefore have $t^{*}_{NI}(a)=2$.

Because $X(a)<X(b)<X(d)$, there exists prior $\mu^{*}$ with full support on
states $\\{a,b,d\\}$ and posterior probability $p_{1}^{*}$ on $a$, such that
$E_{\mu^{*}}[X|\Pi_{1}(\omega)]=v=b$ for all $\omega\in Supp(\mu^{*})$.

Let $K$ be a posterior separable cost function which satisfies Assumption 2,
so that the optimal random posterior is unique. Let $g(c)$ be the function
that maps each cost $c$ to the set of optimal random posteriors. Applying the
Maximum Theorem, $g$ is an upper semicontinuous correspondence. From
Assumption 2, $g$ is a function and therefore continuous. Let $g_{1}(c)$ be
the restriction of $g(c)$ on one of the two posteriors that comprise the
optimal random posterior, and in particular the one that takes values in
$[\mu,1]$.242424From Bayesian plausibility, the other posterior will take
values in $[0,\mu]$. For sufficiently high $c$, $g_{1}(c)$ will choose a
posterior very close to the prior $\mu$, whereas for sufficiently low $c$,
$g(c)$ will choose a posterior very close to 1. From the Intermediate Value
Theorem, there exists $c$ such that the optimal solution is $p^{*}_{1}$.

This implies that Trader 1 optimally acquires information given $c$ and
updates his beliefs from $p_{1}$ to $p^{*}_{1}$ at some $\phi$, announcing
$b$. Trader 2 does not gain any new information from the announcement, because
the same announcement would be made at state $b$. The game proceeds to the
next period, so $t^{*}_{I}(\phi)>2.$ However, without information acquisition,
$p_{1}$ is such that $E_{p}[X|\Pi_{i}(\omega)]\neq b$ for all $i=1,2$ and
$\omega\in Supp(p)$. Therefore, without information acquisition, after Trader
1 announces his prediction, Trader 2 will know the state and
$t^{*}_{NI}(\phi)\leq 2$.

Finally, we show that (ii) is true at state $a$ given the same $\Pi$ and some
$K$. Following the same argument as before, because $X(a)<X(b)<X(d)$, there
exists $p$ with full support on the first three states and resulting unique
$p_{1}$ such that $E_{p}[X|\Pi_{1}(\omega)]=v=b$ for all $\omega\in Supp(p)$.
Therefore, after Trader 1 announces his prediction, Trader 2 does not know the
state and $t^{*}_{NI}(\phi)>2$. With information acquisition, we can find
sufficiently low $c$ such that Trader 1 acquires information, updates his
beliefs to $p^{*}_{1}\neq p_{1}$, and announces
$E^{*}_{p^{*}}[X|\Pi_{i}(\omega)]\neq v$ at state $\phi$ which projects to
$a$. Therefore, Trader 2 realises that the state is not $b$ and
$t^{*}_{I}(\phi)=2.$

Finally, we show that the speed of aggregation is unchanged if the security is
A-D.

proposition If $X$ is A-D, then $t^{*}_{NI}(\omega)=t^{*}_{I}(\omega)$ for any
state $\omega\in\Omega$, cost $\kappa$, information structure $\Pi\in{\cal
P}$, and common prior $p$. Without loss of generality we assume that
$X(\omega^{*})=1$ and $X(\omega)=0$ for all other $\omega$. In every period
$t$, there are two cases. First, the trader announces 0 and the process ends.
In that case, the trader does not acquire any information, because he knows
that the price is 0. Second, the trader considers $\omega^{*}$ with
$X(\omega^{*})=1$ to be possible, acquires information, and forms a posterior.
As the posterior cannot reveal any state with certainty, his private
information $\Pi_{i}(\omega)\cap E$, where $E$ is the public event revealed by
previous announcements, stays the same, but his posterior might change. Hence,
no information is revealed to other traders, because he does not make a
different announcement based on which partition cell he is in. Because no
information is revealed to other traders conditional on the process
continuing, the common knowledge event that is created by each announcement is
the same with and without information acquisition, hence, the process ends in
the same number of periods. Note, however, that announcements may differ
across the two environments.

The property that speed is unaffected is related to the fact that the A-D
security is not very informative. A trader will buy a posterior only in the
partition cell where 1 is possible, as in all others he knows that the value
is 0. This action does not have the positive externality of revealing to other
traders what is his partition cell and therefore it does not provide any
public information about what the true state is. Note, however, that some
positive externality persists. The other traders can solve the announcer’s
problem and therefore know the optimal signal structure he has purchased. By
hearing the announcement, they also update their posteriors and they can
benefit as long as it is their turn to announce and there is still some
surplus to be obtained. For example, if Trader 1 buys a signal structure and
moves the price from 0.5 to 0.99 (when the correct price is 1), then all other
traders can only benefit by moving it from 0.99 to 1, hence the remaining
surplus is very small. Finally, even though the number of periods for full
aggregation remains the same, the price will get faster to the true value,
hence the market will still benefit by attaching faster a higher probability
to the true value of the security.

##  6 Concluding Remarks

One interesting question is what happens to information aggregation when the
security is non-separable but the prior is generic. By perturbing the common
prior slightly, a non-separable security could become separable, and therefore
information could aggregate. Ostrovsky (2012) shows that information gets
aggregates in all pure-strategy equilibria with a generic prior, even for non-
separable securities. However, in his setting, it is an open question what
happens with mixed-strategy equilibria. Intuitively, even if we perturb the
initial prior, and given that agents have infinitely many available
strategies, it is not clear whether they will converge to some belief at which
the security is separable.

The current paper provides a different perspective on the issue of genericity,
by endogenizing the perturbation of beliefs. If information is not aggregated
at time $t$, a trader could buy an additional signal, if the cost is not too
high, and change her belief. By examining what happens when the cost is
arbitrarily low, we effectively allow for arbitrarily small perturbations of
beliefs to be feasible for traders. A “generic” security pays differently
across states and Corollary 1 shows that if the cost is sufficiently low, such
a security is $\kappa$ separable, for all information structures. Therefore,
using Theorem 6 we can say that generically information gets aggregated, in
all mixed-strategy equilibria.

##  Appendix A Appendix

###  A.1 Cost of Information

Let $T$ be a Polish space of possible signals with Borel $\sigma$-algebra
${\cal T}$. Let $\Delta(T)$ be the set of Borel probabilities on $T$, with
generic element $\xi$. We endow $\Delta(T)$ with the weak* topology: a
sequence of probabilities $\\{\xi_{n}\\}$ converges to a probability $\xi$ if
for every bounded continuous function $f:T\rightarrow{\mathbb{R}}$, we have
$\int fd\xi_{n}\rightarrow\int fd\xi$. Following Blackwell (1951), we model
the acquisition of information using statistical experiments. A statistical
experiment is a function from states into probabilities on signals, ${\cal
R}:\Omega\rightarrow\Delta(T)$.

Following Bloedel and Zhong (2020), we say that an experiment ${\cal R}$ is
bounded if (i) the conditional signal distributions $\\{{\cal
R}_{\omega}\\}_{\omega\in\Omega}$ are mutually absolutely continuous, and (ii)
there exists a constant $B>0$ such that the Radon-Nikodym derivatives
$\frac{d{\cal R}_{\omega}}{d{\cal R}_{\omega^{\prime}}}\in\left[1/B,B\right]$
for all $\omega,\omega^{\prime}\in\Omega$. In other words, a bounded
experiment does not definitively rule out any state and, moreover, has
uniformly bounded likelihood ratios. Let ${\cal E}$ be the set of all
experiments and ${\cal E}_{b}\subseteq{\cal E}$ be the collection of all
bounded experiments. Note that ${\cal E}$ contains experiments that are not
bounded. We say that ${\cal R}^{\prime}:\Omega\rightarrow\Delta(T^{\prime})$
is a garbling of ${\cal R}:\Omega\rightarrow\Delta(T)$ given
$E\subseteq\Omega$ if there exists $\psi:T\rightarrow\Delta(T^{\prime})$ such
that ${\cal R}^{\prime}_{\omega^{\prime}}(t^{\prime})=\int_{t\in
T}\psi(t^{\prime}|t){\cal R}_{\omega}(t)dt$.

Given a prior belief $\mu\in\Delta(\Omega)$, a statistical experiment ${\cal
R}$ induces via Bayesian updating a probability distribution over posteriors,
or random posterior, $B(\mu,{\cal R})=Q\in\Delta(\Delta(\Omega))$. Let
$Q(\gamma)$ be the probability of posterior $\gamma\in\Delta(\Omega)$, ${\cal
Q}(\mu)=\\{B(\mu,{\cal R}):{\cal R}\in{\cal E}\\}$ be the set of all random
posteriors that can be generated by some experiment ${\cal R}\in{\cal E}$, and
${\cal Q}=\underset{\mu\in\Delta(\Omega)}{\bigcup}{\cal Q}(\mu)$. Note that
$Q\in{\cal Q}(\mu)$ if and only if $\underset{\Delta(\Omega)}{\int}\gamma
Q(d\gamma)=\mu$. We endow ${\cal Q}$ with the weak* topology, so that it is a
compact and separable topological space. The subsets of ${\cal Q}$ are endowed
with the appropriate relative topologies.

Given a full support prior $\mu$, experiment ${\cal R}$ is bounded if and only
if the induced random posterior $Q=B(\mu,{\cal R})$ satisfies
$Supp(Q)\subseteq\Delta_{\epsilon}=\\{q\in\Delta(\Omega):q(\omega)>\epsilon\textrm{
for all }\omega\in\Omega\\}$ for some $\epsilon>0$. Let ${\cal
D}_{\epsilon}=\\{Q\in{\cal Q}:Supp(Q)\subseteq\Delta_{\epsilon}\\}$. Then,
${\cal D}_{b}=\underset{\epsilon>0}{\bigcup}{\cal D}_{\epsilon}$ denotes the
set of random posteriors that are induced by some full support prior and
bounded experiment. Denote by $\delta_{\mu}$ the degenerate random posterior
that puts probability 1 on $\mu$.

######  Definition 8.

A cost on experiments is a map $h:\Delta(\Omega)\times{\cal
E}\rightarrow[0,\infty]$, where $h(\mu,\cdot)$ is Borel measurable for each
prior $\mu\in\Delta(\Omega)$ and has the following properties.

  1. (i)

If $B(\mu,{\cal R})=B(\mu,{\cal R}^{\prime})$ and ${\cal R},{\cal
R}^{\prime}\in{\cal E}_{b}$, then $h(\mu,{\cal R})=h(\mu,{\cal R}^{\prime})$,

  2. (ii)

If $B(\mu,{\cal R})=\delta_{\mu}$ and ${\cal R}\in{\cal E}_{b}$, then
$h(\mu,{\cal R})=0$,

  3. (iii)

Let $\\{\mu_{j},{\cal R}_{j}\\}_{j\in{\mathbb{N}}}$ be a sequence of
experiment-prior pairs inducing random posteriors $Q_{j}=B(\mu_{j},{\cal
R}_{j})$. If $Q_{j}\rightarrow Q^{*}$ and there exists some $\delta>0$ such
that $\\{Q_{j}\\}_{j\in{\mathbb{N}}}\subseteq{\cal D}_{\delta}$, then
$h(\mu_{j},{\cal R}_{j})\rightarrow h(\mu^{*},{\cal R}^{*})$, where
$Q^{*}=B(\mu^{*},{\cal R}^{*})$,

  4. (iv)

If ${\cal R}^{\prime}$ is a garbling of ${\cal R}$ then $h(\mu,{\cal R})\geq
h(\mu,{\cal R}^{\prime})$.

Point $(i)$ specifies that if two bounded experiments generate the same random
posterior given $\mu$, then they have the same cost. Point $(ii)$ says that a
bounded experiment that is completely uninformative has a cost of zero. Point
$(iii)$ is a continuity condition that is weaker than weak* continuity, in
order to allow for important classes of unbounded cost functions.252525See
Bloedel and Zhong (2020) for the motivation behind this continuity condition.
Finally, point $(iv)$ specifies that more informative experiments are more
costly.

The cost on experiments $h$ generates a cost structure $\kappa=(K,c)$ on
random posteriors, where $c>0$ is the unit cost of information and $K:{\cal
F}\rightarrow[0,\infty]$ maps elements of ${\cal
F}=\\{(\mu,Q):\mu\in\Delta(\Omega),Q\in{\cal Q}(\mu)\\}$, the set of all
priors $\mu$ and all posterior distributions $Q$ that can be generated by some
experiment ${\cal R}\in{\cal E}$, to the extended real line.

We assume that the cost $cK(\mu,Q)$ of acquiring random posterior $Q$, given
belief $\mu$, is determined by optimally acquiring the relevant experiment
${\cal R}$, so that

$cK(\mu,Q)\equiv\min\\{h(\mu,{\cal R}):B(\mu,{\cal R})=Q\\}.$

Note that Point (i) in Definition 8 ensures that all bounded experiments
${\cal R}$ which induce $Q$ given $\mu$ have the same cost, hence in that case
$cK(\mu,Q)=h(\mu,{\cal R})$. If only unbounded experiments generate $Q$ given
$\mu$, we have $cK(\mu,Q)=\infty$. Moreover, the cost of learning nothing is
zero, so that $K(\mu,Q)=0$ whenever $Supp(Q)=\\{\mu\\}$. A cost structure
$\kappa=(K,c)$ consists of a cost function $K$ and a unit cost of information
$c>0$. Let ${\cal K}$ be the collection of cost structures $\kappa=(K,c)$ that
are generated from the experiments.

If all unbounded experiments have infinite cost, then generating a random
posterior with support on a posterior $\delta_{\omega}$, which assigns
probability 0 to a state $\omega\in Supp(\mu)$, is infinitely costly.

######  Corollary 2.

Suppose Assumption 1. Given any $\omega\in Supp(\mu)$ and sequences $\\{{\cal
R}_{j}\\}_{j\in{\mathbb{N}}}$ with $Q_{j}=B(\mu,{\cal R}_{j})$ and
$\\{\gamma_{j}\\}_{j\in{\mathbb{N}}}$ with $Q_{j}(\gamma_{j})>\epsilon$ for
all $j$ and some $\epsilon>0$, if $\gamma_{j}\rightarrow\delta_{\omega}$ then
$h(\mu,{\cal R}_{j})\rightarrow\infty$.

The Corollary implies that for any induced sequences
$\\{Q_{j}\\}_{j\in{\mathbb{N}}}$ and $\\{\gamma_{j}\\}_{j\in{\mathbb{N}}}$
with $Q_{j}(\gamma_{j})>\epsilon$ for all $j$ and some $\epsilon>0$, if
$\gamma_{j}\rightarrow\delta_{\omega}$ then $K(\mu,Q_{j})\rightarrow\infty$.

##  Appendix B Proofs

Proposition 2 provides two independent conditions which are necessary for a
security to be $\kappa$ non-separable for all $\kappa$. The first condition
specifies that, at some state $\omega$, all traders agree that some value
$m\neq X(\omega)$ is possible. The second is an extension of Theorem 7 in
Ostrovsky (2012) (see Proposition 1), which provides a characterization of
separable securities. It says that for every $v^{\prime}$ value of $X$ and
within all intervals around $v^{\prime}$, there exists value v for which we
cannot pick multipliers $\lambda_{i}(\pi)$, for all elements $\pi$ of
partitions $\Pi_{i}$ and for all traders, such that the sign of $X(\omega)-v$
is the same as the sign of $\underset{i\in
I}{\sum}\lambda_{i}(\Pi_{i}(\omega))$, for all $\omega$ with $X(\omega)\neq
v$.

######  Proposition 2.

Suppose $X$ is $\kappa$ non-separable given an information structure $\Pi$ and
for all $\kappa\in{\cal K}$. Then, the following independent conditions are
true.

  * •

There exists $\omega$ with $X(\omega)\neq m$ and $m\in\underset{i\in
I}{\bigcap}X(\Pi_{i}(\omega))$.

  * •

For some $v^{\prime}\in X(\Omega)$, for all $\epsilon>0$, there exists
$v\in(v^{\prime}-\epsilon,v^{\prime}+\epsilon)$ for which there are no
functions $\lambda_{i}:\Pi_{i}\rightarrow{\mathbb{R}}$ such that for all
states $\omega$ with $X(\omega)\neq v$,

$(X(\omega)-v)(\underset{i\in I}{\sum}\lambda_{i}(\Pi_{i}(\omega))>0,$ (5)

###### Proof.

For the first condition, we prove the contrapositive. Suppose that for all
$m\in X(\Omega)$, for all $\omega$ with $X(\omega)\neq m$ we have
$m\notin\underset{i\in I}{\bigcap}X(\Pi_{i}(\omega))$. We will show that $X$
is $\kappa$ separable for some $\kappa=(K,c)$.

For any $K$ and sufficiently low marginal cost $c$, Assumption 1 implies that
the only priors $\mu$ for which there is no information acquisition are the
ones where the resulting posterior assigns probability close to 1 to some
state $\omega^{\prime}\in\Pi_{i}(\omega)$, for all $i\in I$ and $\omega\in
Supp(\mu)$. Suppose that there exists such $\mu$, with
$E_{\mu}[X|\Pi_{i}(\omega)]=v$ for all $i=1,...,n$ and $\omega\in Supp(\mu)$,
where $v$ is arbitrarily close to some $m$ with $X(\omega)=m$.

For the security to be $\kappa$ non-separable, there is also should be
uncertainty about its value. This means that for some $\omega^{\prime}\in
Supp(\mu)$, $X(\omega^{\prime})\neq v$. From our initial hypothesis, there
exists trader $i$ such that $m\notin X(\Pi_{i}(\omega^{\prime}))$, so that he
considers $m$ to be impossible. Because he assigns probability close to 1 to
some state, from continuity it must be that
$E_{\mu}[X|\Pi_{i}(\omega^{\prime})]\neq v$. Therefore, $X$ is separable at
$\mu$. As this is true for all $\mu$ that assign probability close to 1 to
some state $\omega^{\prime}\in\Pi_{i}(\omega)$, we have that the the security
$X$ is $\kappa$ separable for some $\kappa$.

For the second condition, suppose $X$ is $\kappa$ non-separable for all
$\kappa$. Take $\kappa$ with very small marginal cost $c$, so that for any
prior $\mu$ for which there is no information acquisition, at each state
$\omega\in Supp(\mu)$, and for each $i\in I$, $\mu_{\Pi_{i}(\omega)}$ assigns
probability which is arbitrarily close to 1 to some state
$\omega^{\prime}\in\Pi_{i}(\omega)$. This means that
$E_{\mu_{\Pi_{i}(\omega)}}[X]\in(v^{\prime}-\epsilon,v^{\prime}+\epsilon)$,
for small enough $\epsilon\in{\mathbb{R}}$ and
$X(\omega^{\prime})=v^{\prime}$.

Because $X$ is $\kappa$ non-separable, there exists a prior $\mu$ with no
information acquisition and value
$v\in(v^{\prime}-\epsilon,v^{\prime}+\epsilon)$ for which $X$ is non-
separable. This implies that, for any
$\lambda_{i}:\Pi_{i}\rightarrow{\mathbb{R}}$,

$E_{\mu}\left[(X(\omega)-v)\underset{i\in
I}{\sum}\lambda_{i}(\Pi_{i}(\omega))\right]=$ $\underset{i\in
I}{\sum}E_{\mu}\left[(X(\omega)-v)\lambda_{i}(\Pi_{i}(\omega))\right]=$
$\underset{i\in I}{\sum}\underset{\omega\in
Supp(\mu)}{\sum}\mu(\Pi(\omega))\lambda_{i}(\Pi_{i}(\omega))E_{\mu_{\Pi_{i}(\omega)}}\left[(X(\omega)-v)\right]=0,$

where the last equality derives from the definition of non-separability. But
then we cannot have $(X(\omega)-v)\underset{i\in
I}{\sum}\lambda_{i}(\Pi_{i}(\omega))>0$ for all $\omega$, hence there are no
such $\lambda_{i}$ functions.

To show that the first two conditions are independent, consider first security
$(1,0,0,-1,2)$. Let the partition of Trader 1 be
$\\{\\{\omega_{1},\omega_{2}\\},\\{\omega_{3},\omega_{4}\\},\\{\omega_{5}\\}\\}$
and for Trader 2 be
$\\{\\{\omega_{1},\omega_{3}\\},\\{\omega_{2},\omega_{5}\\},\\{\omega_{4}\\}\\}$,
which satisfies the first condition. We show that it fails the second
condition. First, if $v\leq-1$, then condition (1) is satisfied by setting
$\lambda_{i}(\Pi_{i}(\omega))=1$ for all $i\in I$ and $\omega\in\Omega$.
Similarly, if $v\geq 2$, we set $\lambda_{i}(\Pi_{i}(\omega))=-1$ for all
$i\in I$ and $\omega\in\Omega$. Then, suppose there exist functions such that
$\lambda_{1}(\\{\omega_{1},\omega_{2}\\})=a$,
$\lambda_{1}(\\{\omega_{3},\omega_{4}\\})=b$,
$\lambda_{1}(\\{\omega_{5}\\})=c$,
$\lambda_{2}(\\{\omega_{1},\omega_{3}\\})=d$,
$\lambda_{2}(\\{\omega_{2},\omega_{5}\\})=e$ and
$\lambda_{2}(\\{\omega_{4}\\})=f$. Consider $0\geq v\geq-1$. We then have that
$X(\omega)-v<0$ for $\omega_{4}$, and $X(\omega)-v>0$ for
$\omega_{1},\omega_{2},\omega_{3},\omega_{5}$. Set $f=-2$ and $a=b=c=d=e=1$
and (5) is satisfied. Consider $2\geq v\geq 1$. We then have that
$X(\omega)-v>0$ for $\omega_{5}$, and $X(\omega)-v<0$ for
$\omega_{1},\omega_{2},\omega_{3},\omega_{4}$. Set $c=2$ and $a=b=c=d=e=-1$
and (5) is satisfied. Finally, consider $1\geq v\geq 0$. We then have that
$X(\omega)-v<0$ for $\omega_{2},\omega_{3},\omega_{4}$, and $X(\omega)-v>0$
for $\omega_{1},\omega_{5}$. Set $c=3$, $f=b=e=-2$ and $a=d=1$ and (5) is
satisfied. Therefore, the second condition is not satisfied.

Consider now security $(-2,-1,0,1,2)$. Suppose there are three traders. The
partition of Trader 1 is
$\\{\\{\omega_{2},\omega_{4}\\},\\{\omega_{1},\omega_{3},\omega_{5}\\}\\}$,
for Trader 2 it is
$\\{\\{\omega_{1},\omega_{5}\\},\\{\omega_{2},\omega_{3},\omega_{4}\\}\\}$ and
for Trader 3 it is
$\\{\\{\omega_{1},\omega_{4}\\},\\{\omega_{2},\omega_{3},\omega_{5}\\}\\}$.
Therefore, such security violates the first condition. We will show that it
satisfies the second. Let $v^{\prime}=0$ and suppose that $v\in(0,\epsilon)$,
for any $\epsilon>0$. We then have that $X(\omega)-v<0$ for
$\omega_{1},\omega_{2},\omega_{3}$, and $X(\omega)-v>0$ for
$\omega_{4},\omega_{5}$.

Suppose there exist functions such that
$\lambda_{1}(\\{\omega_{2},\omega_{4}\\})=a$,
$\lambda_{1}(\\{\omega_{1},\omega_{3},\omega_{5}\\})=b$,
$\lambda_{2}(\\{\omega_{2},\omega_{3},\omega_{4}\\})=c$,
$\lambda_{2}(\\{\omega_{1},\omega_{5}\\})=d$,
$\lambda_{3}(\\{\omega_{2},\omega_{3},\omega_{5}\\})=e$ and
$\lambda_{3}(\\{\omega_{1},\omega_{4}\\})=f$. For (5) to be satisfied, we have
the following inequalities.

$\begin{cases}b+d+f<0\\\ a+c+e<0\\\ b+c+e<0\\\ a+c+f>0\\\ b+d+e>0\end{cases}$
(6)

If we add the first with the fifth, we have $e>f$. If we add the second with
the fourth, we have $f>e$. This is a contradiction and such equations do not
exist, which means that the second condition is satisfied.

∎

## References

  * Allen [1981] Beth Allen.  Generic existence of completely revealing equilibria for economies with uncertainty when prices convey information.  _Econometrica_ , pages 1173–1199, 1981.
  * Atakan and Ekmekci [2023] Alp E Atakan and Mehmet Ekmekci.  Information aggregation in auctions with costly information.  2023\.
  * Atanasov et al. [2017] Pavel Atanasov, Phillip Rescober, Eric Stone, Samuel A. Swift, Emile Servan-Schreiber, Philip Tetlock, Lyle Ungar, and Barbara Mellers.  Distilling the wisdom of crowds: Prediction markets vs. prediction polls.  _Management Science_ , 63(3):691–706, 2017.
  * Baron et al. [2014] Jonathan Baron, Barbara A Mellers, Philip E Tetlock, Eric Stone, and Lyle H Ungar.  Two reasons to make aggregated probability forecasts more extreme.  _Decision Analysis_ , 11(2):133–145, 2014.
  * Berg et al. [2008] Joyce E. Berg, Forrest D. Nelson, and Thomas A. Rietz.  Prediction market accuracy in the long run.  _International Journal of Forecasting_ , 24(2):285–300, 2008.
  * Blackwell [1951] David Blackwell.  Comparison of experiments.  In _Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability_ , pages 93–102. University of California Press, 1951\.
  * Bloedel and Zhong [2020] Alexander W Bloedel and Weijie Zhong.  The cost of optimally-acquired information.  _Unpublished Manuscript_ , 2020.
  * Camerer et al. [2016] Colin F Camerer, Anna Dreber, Eskil Forsell, Teck-Hua Ho, Jürgen Huber, Magnus Johannesson, Michael Kirchler, Johan Almenberg, Adam Altmejd, Taizan Chan, et al.  Evaluating replicability of laboratory experiments in economics.  _Science_ , 351(6280):1433–1436, 2016.
  * Caplin and Dean [2015] Andrew Caplin and Mark Dean.  Revealed preference, rational inattention, and costly information acquisition.  _American Economic Review_ , 105(7):2183–2203, 2015.
  * Caplin and Martin [2015] Andrew Caplin and Daniel Martin.  A testable theory of imperfect perception.  _The Economic Journal_ , 125(582):184–202, 2015\.
  * Caplin et al. [2019] Andrew Caplin, Mark Dean, and John Leahy.  Rational inattention, optimal consideration sets, and stochastic choice.  _The Review of Economic Studies_ , 86(3):1061–1094, 2019.
  * Caplin et al. [2022] Andrew Caplin, Mark Dean, and John Leahy.  Rationally inattentive behavior: Characterizing and generalizing shannon entropy.  _Journal of Political Economy_ , 130(6):1676–1715, 2022.
  * Chen et al. [2010] Yiling Chen, Stanko Dimitrov, Rahul Sami, Daniel M Reeves, David M. Pennock, Robin D. Hanson, Lance Fortnow, and Rica Gonen.  Gaming prediction markets: Equilibrium strategies with a market maker.  _Algorithmica_ , 58(4):930–69, 2010.
  * Chen et al. [2012] Yiling Chen, Mike Ruberry, and Jennifer Wortman Vaughan.  Designing informative securities.  Mimeo, 2012.
  * Corgnet et al. [2022] Brice Corgnet, Cary Deck, Mark DeSantis, Kyle Hampton, and Erik O Kimbrough.  When do security markets aggregate dispersed information?  _Management Science_ , 2022.
  * Cowgill and Zitzewitz [2015] Bo Cowgill and Eric Zitzewitz.  Corporate prediction markets: Evidence from Google, Ford, and Firm X.  _The Review of Economic Studies_ , 82(4):1309–41, 2015.
  * Dana et al. [2019] Jason Dana, Pavel Atanasov, Philip Tetlock, and Barbara Mellers.  Are markets more accurate than polls? the surprising informational value of “just asking”.  _Judgment and Decision Making_ , 14(2):135–147, 2019.
  * DeGroot [1974] Morris H DeGroot.  Reaching a consensus.  _Journal of the American Statistical association_ , 69(345):118–121, 1974.
  * DeMarzo and Skiadas [1998] Peter DeMarzo and Costis Skiadas.  Aggregation, determinacy, and informational efficiency for a class of economies with asymmetric information.  _Journal of Economic Theory_ , 80(1):123–52, 1998\.
  * DeMarzo and Skiadas [1999] Peter DeMarzo and Costis Skiadas.  On the uniqueness of fully informative rational expectations equilibria.  _Economic Theory_ , 13(1):1–24, 1999.
  * Denti [2022] Tommaso Denti.  Posterior separable cost of information.  _American Economic Review_ , 112(10):3215–59, 2022.
  * Denti et al. [2022] Tommaso Denti, Massimo Marinacci, and Aldo Rustichini.  Experimental cost of information.  _American Economic Review_ , 112(9):3106–3123, 2022.
  * Dimitrov and Sami [2008] Stanko Dimitrov and Rahul Sami.  Non-myopic strategies in prediction markets.  In _Proceedings of the 9th ACM Conference on Electronic Commerce_ , pages 200–9, 2008.
  * Farboodi et al. [2022] Maryam Farboodi, Adrien Matray, Laura Veldkamp, and Venky Venkateswaran.  Where has all the data gone?  _The Review of Financial Studies_ , 35(7):3101–3138, 2022.
  * Galanis [2011] Spyros Galanis.  Syntactic foundations for unawareness of theorems.  _Theory and Decision_ , 71(4):593–614, 2011.
  * Galanis [2013] Spyros Galanis.  Unawareness of theorems.  _Economic Theory_ , 52(1):41–73, 2013.
  * Galanis [2021] Spyros Galanis.  Dynamic consistency, valuable information and subjective beliefs.  _Economic Theory_ , 71:1467–1791, 2021.
  * Galanis and Kotronis [2021] Spyros Galanis and Stelios Kotronis.  Updating awareness and information aggregation.  _B.E. Journal of Theoretical Economics_ , 2021.
  * Galanis et al. [2024] Spyros Galanis, Christos A. Ioannou, and Stelios Kotronis.  Information aggregation under ambiguity: Theory and experimental evidence.  _Review of Economic Studies, forthcoming_ , 2024.
  * Geanakoplos and Polemarchakis [1982] John Geanakoplos and Heraklis Polemarchakis.  We can’t disagree forever.  _Journal of Economic Theory_ , 28:192–200, 1982.
  * Golub and Jackson [2010] Benjamin Golub and Matthew O Jackson.  Naive learning in social networks and the wisdom of crowds.  _American Economic Journal: Microeconomics_ , 2(1):112–149, 2010.
  * Grossman [1976] Sanford Grossman.  On the efficiency of competitive stock markets where trades have diverse information.  _The Journal of Finance_ , 31(2):573–85, 1976\.
  * Hanson [2003] Robin Hanson.  Combinatorial information market design.  _Information Systems Frontiers_ , 5(1):107–19, 2003.
  * Hanson [2007] Robin Hanson.  Logarithmic market scoring rules for modular combinatorial information aggregation.  _Journal of Prediction Markets_ , 1(1):3–15, 2007\.
  * Hayek [1945] Friedrich August Hayek.  The use of knowledge in society.  _The American Economic Review_ , pages 519–30, 1945.
  * Hébert and Woodford [2021] Benjamin Hébert and Michael Woodford.  Neighborhood-based information costs.  _American Economic Review_ , 111(10):3225–3255, 2021.
  * Hébert and Woodford [2023] Benjamin Hébert and Michael Woodford.  Rational inattention when decisions take time.  _Journal of Economic Theory_ , 208:105612, 2023.
  * Ilinov et al. [2024] Pavel Ilinov, Andrei Matveenko, Maxim Senkov, and Egor Starkov.  Optimally biased expertise.  2024\.
  * Kim et al. [2023] Alex Kim, Maximilian Muhn, and Valeri Nikolaev.  Bloated disclosures: Can chatgpt help investors process financial information?  _arXiv preprint arXiv:2306.10224_ , 2023.
  * Kraemer et al. [2006] Carlo Kraemer, Markus Nöth, and Martin Weber.  Information aggregation with costly information and random ordering: Experimental evidence.  _Journal of Economic Behavior & Organization_, 59(3):423–432, 2006.
  * Kyle [1985] Albert S. Kyle.  Continuous auctions and insider trading.  _Econometrica_ , 53(6):1315–35, 1985.
  * Laffont and Maskin [1990] Jean-Jacques Laffont and Eric S Maskin.  The efficient market hypothesis and insider trading on the stock market.  _Journal of Political Economy_ , 98(1):70–93, 1990.
  * Lim and Brooks [2011] Kian-Ping Lim and Robert Brooks.  The evolution of stock market efficiency over time: A survey of the empirical literature.  _Journal of Economic Surveys_ , 25(1):69–108, 2011.
  * Maćkowiak et al. [2023] Bartosz Maćkowiak, Filip Matějka, and Mirko Wiederholt.  Rational inattention: A review.  _Journal of Economic Literature_ , 61(1):226–273, 2023.
  * Matějka and McKay [2015] Filip Matějka and Alisdair McKay.  Rational inattention to discrete choices: A new foundation for the multinomial logit model.  _American Economic Review_ , 105(1):272–298, 2015\.
  * McKelvey and Page [1990] Richard D. McKelvey and Talbot Page.  Public and private information: An experimental study of information pooling.  _Econometrica_ , 58(6):1321–39, 1990.
  * Mele and Sangiorgi [2015] Antonio Mele and Francesco Sangiorgi.  Uncertainty, information acquisition and price swings in asset markets.  _The Review of Economic Studies_ , page rdv017, 2015.
  * Ostrovsky [2012] Michael Ostrovsky.  Information aggregation in dynamic markets with strategic traders.  _Econometrica_ , 80(6):2595–647, 2012.
  * Page and Siemroth [2017] Lionel Page and Christoph Siemroth.  An experimental analysis of information acquisition in prediction markets.  _Games and Economic Behavior_ , 101:354–378, 2017.
  * Page and Siemroth [2021] Lionel Page and Christoph Siemroth.  How much information is incorporated into financial asset prices? experimental evidence.  _The Review of Financial Studies_ , 34(9):4412–4449, 2021.
  * Pavan et al. [2022] Alessandro Pavan, Savitar Sundaresan, and Xavier Vives.  (in) efficiency in information acquisition and aggregation through prices.  2022\.
  * Pelster and Val [2023] Matthias Pelster and Joel Val.  Can chatgpt assist in picking stocks?  _Finance Research Letters_ , page 104786, 2023.
  * Radner [1979] Roy Radner.  Rational expectations equilibrium: Generic existence and the information revealed by prices.  _Econometrica_ , 47(3):655–78, 1979.
  * Schlegel et al. [2022] Jan Christoph Schlegel, Mateusz Kwaśnicki, and Akaki Mamageishvili.  Axioms for constant function market makers.  _Available at SSRN_ , 2022.
  * Sims [2003] Christopher A. Sims.  Implications of rational inattention.  _Journal of Monetary Economics_ , 50(3):665–690, April 2003.
  * Snowberg et al. [2013] Erik Snowberg, Justin Wolfers, and Eric Zitzewitz.  Prediction markets for economic forecasting.  In _Handbook of economic forecasting_ , volume 2, pages 657–687. Elsevier, 2013.
  * The Economist [2021] The Economist.  How spooks are turning to superforecasting in the cosmic bazaar.  https://www.economist.com/science-and-technology/2021/04/15/how-spooks-are-turning-to-superforecasting-in-the-cosmic-bazaar, 2021\.  Accessed: 04-17-2021.
  * Tsakas [2020] Elias Tsakas.  Robust scoring rules.  _Theoretical Economics_ , 15(3):955–987, 2020\.
  * Tsallis [1988] Constantino Tsallis.  Possible generalization of boltzmann-gibbs statistics.  _Journal of statistical physics_ , 52:479–487, 1988.
  * Wolfers and Zitzewitz [2004] Justin Wolfers and Eric Zitzewitz.  Prediction markets.  _Journal of Economic Perspectives_ , 18(2):107–26, 2004.