# Sequential Naive Learning111Itai acknowledges support from the Ministry of
Science and Technology (grant 19400214). Yakov acknowledges support from the
Israel Science Foundation (grant 2021296). Manuel acknowledges the support
from the Spanish Ministry of Science, Innovation, and Universities (ref.
ECO2015-63711-P) andthe Department of the Economy and Knowledge of the
Generalitat de Catalunya (ref. 2017 SGR 1244).

Itai Arieli, Yakov Babichenko, Manuel Mueller-Frank Faculty of Industrial
Engineering and Management, Technion: The Israel Institute of Technology,
<EMAIL_ADDRESS>of Industrial Engineering and Management,
Technion: The Israel Institute of Technology<EMAIL_ADDRESS>Business School, University of Navarra<EMAIL_ADDRESS>
###### Abstract

We analyze boundedly rational updating from aggregate statistics in a model
with binary actions and binary states. Agents each take an irreversible action
in sequence after observing the unordered set of previous actions. Each agent
first forms her prior based on the aggregate statistic, then incorporates her
signal with the prior based on Bayes rule, and finally applies a decision rule
that assigns a (mixed) action to each belief. If priors are formed according
to a discretized DeGroot rule, then actions converge to the state (in
probability), i.e., _asymptotic learning_ , in any informative information
structure if and only if the decision rule satisfies probability matching.
This result generalizes to unspecified information settings where information
structures differ across agents and agents know only the information structure
generating their own signal. Also, the main result extends to the case of $n$
states and $n$ actions.

##  1 Introduction

Many decisions under uncertainty require individuals to consider both their
own private information and the earlier decisions of other individuals who
faced the same decision problem. The inference based on the latter is called
observational learning. Important examples of observational learning are
social learning, where the earlier choices of friends or social contacts are
observed, and aggregate-statistic based learning, where the available
information consists of average reviews, sales statistics, and so on. The
importance of social learning for “real-world” decision making has been
established for the purchase of consumer goods (see Feick and Price [21]),
agricultural practice (see Munshi [41] and Conley and Udry [13]), and
microfinance (see Banerjee, Chandrasekhar, Duflo and Jackson [6]), among
others. Similarly, there is ample empirical evidence for aggregate statistics
impacting consumer choices; examples include book best-seller lists (Bonabeau
[11]), software download counts (Hanson and Putler [27]), online reviews
(Huang and Chen [30]), and opening weekend movie ticket sales (Moretti [37]).

In the literature, two predominant approaches to the study of observational
learning have emerged: Bayesian and boundedly rational. In the Bayesian
approach agents make fully rational inferences on the private information of
all agents based on the observed decisions. Despite being valuable as a
benchmark, the Bayesian approaches faces severe limitations. First, Hazla,
Jadbabaie, Mossel and Rahimian [29] have shown Bayesian updating in social
learning settings to be highly computationally intense. To clarify the
complexity, note that in a social learning model the information each agent
bases her action upon has two components: the past actions she observes and a
private signal. The complexity arises in the inference from the observed past
actions to the underlying information. This task is complicated by itself but
even more so for aggregate statistic-based learning since the aggregate
statistic provides no direct information about the observations each action
was based upon. Second, Bayesian inference crucially relies upon knowing
either the state-dependent distributions that generate the signals of each
agent, or a distribution over the state-dependent distributions. Thus Bayesian
updating is limited by agents’ computational sophistication and more
fundamentally by their knowledge of the environment

Since there is empirical validation for aggregate statistics impacting real-
world decisions, which due to the limitations described above are unlikely to
be based on Bayesian updating222For experimental evidence on the failure of
Bayesian updating in social learning environments see Kübler and Weizsäcker
[32], Eyster, Rabin, and Weizsäcker [20], March and Ziegelmeyer [35], and Enke
and Zimmermann [18]., a formal study of this case is warranted. While the
literature on Bayesian and boundedly rational social learning is extensive, to
the best of our knowledge there is no thorough analysis of boundedly rational
observational learning based on aggregate statistics. This paper can be seen
as a first step toward developing the theoretical foundations and implications
of aggregate statistics-based boundedly rational observational learning.

The core research question we address in this paper is the following: Do there
exist aggregate statistics-based boundedly rational heuristics that enable
long-run actions to optimally incorporate all privately held information, i.e.
asymptotic learning? We address this question in a setting with conditionally
independent and identically distributed signals as well as in an unspecified
information structure setting where agents do not know how the signals of
others are generated.

We borrow the standard herding model with binary states and binary actions
employed in the social learning literature.333See, for example, Bikhchandani,
Hirschleifer, and Welch [8], Smith and Sorensen [42], Mossel, Sly and Tamuz
[38], and Bohren [9]. The binary herding model is both well suited to our
analysis and objective since in many cases the observed aggregate statistics
are binary. For example, adoption statistics and sales statistics are binary
in the case of duopolies. However, in order to make our analysis more complete
we generalize our main result to a general finite set of states and actions.

The precise model we analyze can be described as follows. A countable infinite
set of agents take an irreversible binary action sequentially. Each agent
receives a payoff of $1$ if her action matches the realized state and a payoff
of $0$ otherwise. Prior to taking her action, each agent observes the
unordered set of previous actions and a private signal that is drawn
independently across agents conditional on the realized state. Each agent
takes an action based on an updating heuristic in three steps. First, a _prior
formation heuristic_ assigns a prior belief over the state space to the
observed aggregate statistic. Second, a _posterior formation heuristic_
aggregates the prior with the private signal into a posterior. Third, a
_decision rule heuristic_ maps the posterior of an agent to a mixed
strategy.444In most of the existing literature there is no distinction between
the prior formation, signal aggregation, and decision rule heuristics. In the
DeGroot model the average of observed beliefs is the new belief. In quasi-
Bayesian updating the selected action maximizes the expected utility
maximizing action given the quasi-Bayesian belief. Arieli, Babichenko, and
Mueller-Frank [2] is an exception that allows for general decision rules,
albeit in the repeated interaction setting. Each triple of a prior formation,
a posterior formation, and a decision rule heuristic defines an updating
heuristic. In this paper we focus on the following subclass of updating
heuristics:

##### Prior formation

We analyze the asymptotic learning properties of the two most common prior
formation functions studied in the literature in different settings: DeGroot
prior formation and quasi-Bayesian prior formation.

Under a DeGroot updating heuristic [16] continuous beliefs (or actions) are
updated by averaging over the observed set of beliefs (or actions).555The
literature on DeGroot updating is too vast to cite. A few notable examples
are, DeMarzo, Vayanos, and Zwiebel [17] and Golub and Jackson [24, 25]. We
adjust DeGroot updating to the binary action setting as follows: the DeGroot
prior belief of, say, state $\omega=1$ is equal to the proportion of observed
actions that equal $1$.666For the discretized version of DeGroot see
Chandrasekhar, Larreguy and Xandri [12], Leshem and Scaglione [33], and
Arieli, Babichenko and Mueller-Frank [2].  Thus DeGroot prior formation is
invariant in the properties of the information structure.

Under quasi-Bayesian prior formation, which was first introduced by Eyster and
Rabin [19]777Eyster and Rabin [19] refer to this as _best response trailing
naive inference._ each observed action is treated as if it were a Bayesian
best response to only the private signal of the agent taking the
action.888Further studies of this heuristic include Mueller-Frank and Neri
[40], Levy and Razin [34], Dasaratha and He [15], and Arieli, Babichenko, and
Shlomov [3]. For experimental findings supporting the quasi-Bayesian updating
heuristic see Eyster, Rabin, and Weizsacker [20], Mueller-Frank and Neri [40]
and Dasaratha and He [14]. Since agents observe only the unordered set of past
actions, the identities of the agents taking a given action are not observed.
This limits quasi-Bayesian updating to the conditionally i.i.d. signal
case.999Unless agents have a distribution over the order with which actions
are taken.

##### Posterior formation

Here we focus on a single heuristic: the Bayesian one. That is, the signal is
aggregated in a Bayesian manner for the given prior. Such a modeling choice is
common in the literature on boundedly rational learning; see, e.g., Eyster and
Rabin[19], Guarino and Jehiel [26], Jadbabaie, Molavi, Sandroni and Tahbaz-
Salehi[31], and Dasaratha and He [15].

##### Decision rule

The decision rule assigns a mixed strategy to every posterior belief, as
opposed to a deterministic strategy as is the norm in the literature. We
impose no restrictions on the decision rule beyond some mild natural
assumptions of monotonicity and semi-continuity.

We are interested in the information aggregation properties of the long-run
actions when agents follow a common updating heuristic. We say that such a
heuristic satisfies _asymptotic learning_ if the probability of agents
selecting the optimal action converges to one along the sequence of agents.

Our main result, Theorem 1, shows that if the prior formation function is
DeGroot, then asymptotic learning holds for _every_ (conditionally i.i.d.)
information structure if and only if the decision rule equals probability
matching. Surprisingly, Theorem 1 finds that in the binary action setting
combining the two most common prior and posterior formation heuristics, a
probability matching decision rule achieves asymptotic learning in any
information structure, and is the only decision rule to do so. This differs
sharply from the asymptotic learning properties of Bayesian learning. Smith
and Sorensen [43] show that asymptotic learning fails in the model considered
here if private signals are bounded.101010Private signals are bounded if the
support of the posterior belief conditional on the signal does not contain 0
or 1. Instead, the boundedly rational heuristic combining DeGroot prior
formation with probability matching achieves asymptotic learning for bounded
_and_ unbounded signal structures. Also, Theorem 1 generalizes to the general
case of $n$ states and $n$ actions, another distinction from the Bayesian
learning case.

Next we consider the case of unspecified information structures. Signals are
conditionally independent but not identically distributed across agents, and
each agent knows only the information structure generating her own signal. For
unspecified information structures, Theorem 2 shows that DeGroot prior
formation paired with probability matching achieves asymptotic learning, if
the signal distribution of each agent satisfies a weak informativeness
property. Our results can be seen as a normative foundation for the DeGroot
heuristic and the probability matching decision rule. Jointly they achieve
learning in environments where Bayesian learning fails to do so, and even in
environments where Bayesian learning is not applicable due to insufficient
information about the information structure.

Finally, Theorem 3 shows that asymptotic learning fails under quasi-Bayesian
prior formation, for every decision rule, in every (conditionally i.i.d.)
information structure. Under quasi-Bayesian prior formation the prior
formation is tailored to the information structure. By contrast DeGroot prior
formation is invariant in the information structure. Nevertheless, DeGroot
prior formation can achieve asymptotic learning in any information structure
while quasi-Bayesian prior formation fails to achieve asymptotic learning in
_every_ information structure.

The DeGroot belief-updating heuristic is arguably the most prominent in the
literature and it is compelling due to its simplicity. Our results highlight
that even in coarse action environments it enables long-run agents to
approximately learn the realized state from an unordered set of binary actions
(that are conditionally dependent) without any knowledge of the signal
structures underlying them.111111If all agents included in the group apply the
DeGroot probability matching heuristic. Thus DeGroot prior formation is an
attractive proposition for agents, particularly in light of Theorem 3.
Similarly, our results favor the use of probability matching as a decision
rule in an observational learning environment due to its positive learning
externality.

The rest of the paper is organized as follows. Section 2 discusses the
relation to the literature. Section 3 introduces the model. Section 4 presents
our asymptotic learning results for DeGroot and quasi-Bayesian prior
formation. Section 6 establishes the asymptotic learning result for signals
that are not identically distributed conditional on the state and discusses
further extensions. Section 7 concludes with a discussion and interpretation
of our results and some thoughts on future work. All proofs are relegated to
the appendix.

##  2 Related Literature

We note first that previous work on boundedly rational heuristics that has
provided positive information aggregation results considered rich action
environments where actions are equal to beliefs or allow a one-to-one
inference on the signal.121212See Golub and Jackson [24], Molavi, Tahbaz-
Salehi and Jadbabaie [36], Jadbabaie, Molavi, Sandroni, and Tahbaz-Salehi
[31], and Dasaratha and He [15]. In the binary action case the mapping from
actions to beliefs or signals is considerably coarser than in the rich action
case. Thus we analyze the binary action case to understand information
aggregation properties of boundedly rational heuristics in environments least
favorable for it.

The most related papers to ours are Eyster and Rabin [19] and Dasaratha and He
[15] who study the sequential setting. Both analyze the asymptotic learning
properties of the quasi-Bayesian belief updating heuristic paired with the
(Bayesian) deterministic decision rule that maximizes expected utility given
the belief. Eyster and Rabin [19] pioneered this modeling approach and show
that asymptotic learning fails as herds on the incorrect action occur with
positive probability. Dasaratha and He [15] analyze a rich action setting with
Gaussian signals. They provide a characterization of the observational network
structure that achieves asymptotic learning. Our paper generalizes both the
information structure and the class of updating heuristics considered in these
related papers. We complement their result by showing that in the binary
action setting no decision rule can achieve asymptotic learning under quasi-
Bayesian prior formation.

A series of recent papers analyzes the implications of Bayesian agents having
mis-specified beliefs about the environment in a social learning environment.
Bohren [9] considers the sequential social learning model with binary actions
where a proportion $p$ of agents observe the history of actions while others
do not. She shows that if agents assume that the proportion is $p^{\prime}\neq
p$, asymptotic learning might fail. Bohren and Hauser [10] extend this
approach in providing a general framework of misspecified beliefs and how they
impact social learning outcomes. Frick, Iijima, and Iishi [22] provide
conditions under which learning outcomes are not robust to minimal
misspecifications.131313See also Frick, Iijima, and Iishi [23]. Our Theorem 2
extends their approach by considering a unspecified information environment,
i.e., where agents have no knowledge of the information structure, but
nevertheless achieve asymptotic learning through the DeGroot probability
matching heuristic.

Finally, our asymptotic learning result extends the scope of previous naive
learning results from the rich action environment of Golub and Jackson [24]
and Jadbabaie, Molavi, Sandroni, and Tahbaz-Salehi [31] to a binary action
environment.141414Besides the cardinality of the action space, there are other
considerable differences as both papers mentioned consider a model of repeated
interaction. In this we coincide with Arieli, Babichenko, and Mueller-Frank
[2] who analyze a generalized version of the DeGroot probability matching
heuristic in the repeated interaction setting. In contrast to Arieli,
Babichenko, and Mueller-Frank [2] we provide a characterization of the
decision rules that achieve asymptotic learning under DeGroot prior formation
in the sequential herding model and show this decision rule to be probability
matching. Further, we differ in considering the unspecified information
environment and the result provided there.

##  3 The Model

Our setting is inspired by the standard herding model (see Banerjee [7],
Bikhchandani, Hirshleifer, and Welch [8], and Smith and Sorensen [42]). The
model is described by the tuple $(N,\Omega,A,\mu,(F,S))$, where $N$ is a
countably infinite set of agents who decide sequentially on a binary action
$a\in A=\\{0,1\\}$. The state of the world $\omega\in\Omega=\\{0,1\\}$ is
drawn at $t=0$ according to the prior $\mu$. The realized state is not
observed by the agents.

Thereafter, at every time $t\geq 1$ agent $t$ takes an action $a_{t}$. As in
the standard herding model, the utility of an agent is $1$ if her action
equals the state and it is $0$ otherwise. Unlike the standard model, we assume
that each agent observes “only” the unordered set of actions that preceded
her, as opposed to also the order in which they were taken. That is, the
aggregate statistic observed by agent $t$ can be represented as a pair of
numbers $(m,k)$ such that $m+k=t-1$ and where $m$ (respectively $k$)
corresponds to the number of $0$ (respectively $1$) actions that are observed
by agent $t$. We next describe in detail our assumptions on the information
structure and the process according to which actions are taken.

#### The Information Structure

The _information structure_ of our model is described by the tuple
$(\mu,F_{0},F_{1},S)$. We denote the common prior by $\mu$, where
$\mu=\Pr\left[\omega=1\right]$, $S$ is a measurable space of signals, and
$F_{\omega}\in\Delta(S)$ is a state-dependent distribution over the signals
given the state $\omega=0,1$. We throughout assume that signals are
informative, i.e., $F_{0}\neq F_{1}$.

In addition to observing the unordered history of actions $(m,k)$, each agent
$t$ receives a private signal $s_{t}\in S$ that is drawn independently from
the signal of all other agents according to $F_{\omega}\in\Delta(S).$ Let
$\mathbf{P}_{F,\mu}\in\Delta(\Omega\times S)$ denote the measure that is
generated by $\mu$ and $F$.

#### Belief Updating and Decision Rule

We now introduce a general approach to the agents’ deliberations that are
finalized in their action choice. The class of heuristics we analyze decouples
the belief formation process explicitly from the action choice while sharing
properties with Bayesian updating. Concretely, we decompose the procedure
according to which an action is chosen into three steps: 1) prior formation
given the history, 2) posterior formation incorporating the private signal,
and 3) the action choice given the posterior belief. We now describe the three
steps in detail.

Prior formation: The (boundedly rational) prior formation given the unordered
history occurs according to a _prior formation function_. Recall that the
history observed by agent $t$ can be represented as a pair of numbers $(m,k)$
such that $m+k=t-1$ and where $m$ (respectively $k$) corresponds to the number
of $0$ (respectively $1$) actions that are observed by agent $t$. A prior
formation function is a mapping $b:\mathbb{N}\times\mathbb{N}\rightarrow[0,1]$
that assigns a prior probability $b(m,k)$ to state $\omega=1$ given the number
of observed choices of $0$ and of $1$.

Posterior formation: Each agent incorporates her private signal with the prior
to form her posterior belief. We assume that the agent knows the signal-
generating distributions $(F_{0},F_{1})$ and applies Bayes’ rule to form her
posterior belief given her signal, treating the prior as a true Bayesian
prior. For every signal $s\in S$, let
$p(s)=\mathbf{P}_{F,\frac{1}{2}}(\omega=1|s)$ be the conditional probability
of state $\omega=1$, given the signal $s$ and the uniform prior
$\mu=\frac{1}{2}$. Thus, for every $s\in S$ and a given prior $b(k,m)=b$, the
agent’s posterior probability of state 1 equals:

$\displaystyle\frac{p(s)b}{p(s)b+(1-p(s))(1-b)}.$ (1)

Decision rule: Each agent selects her action given her posterior belief. We
assume that the action choice is determined by a _decision rule_ , i.e., an
increasing function $\sigma:[0,1]\rightarrow[0,1]$ that assigns a mixed
strategy to each belief. More precisely, for a posterior $y$ the term
$\sigma(y)$ represents the probability that the agent plays action $a=1$, and
$1-\sigma(y)$ is the probability that she plays action $a=0$. We restrict
attention to decision rules $\sigma$ that are continuous at $0$ and $1$ and
piecewise continuous elsewhere in the following sense: either it holds at any
$x\in(0,1)$ that $\lim_{y\rightarrow x^{+}}\sigma(y)=\sigma(x)$, or it holds
that at any point $x$ $\lim_{y\rightarrow x^{-}}\sigma(y)=\sigma(x)$. That is,
$\sigma$ is either right-continuous at any point or left-continuous at any
point. One decision rule that will play a prominent role in our results is
_probability matching_ where $\sigma(x)=x.$ That is, action $a=1$ is played
with a probability that coincides with the posterior probability of state
$\omega=1$.

Let us briefly discuss our approach in the context of the existing literature.
The use of a boundedly rational heuristic that assigns a prior belief to the
observed history in combination with the incorporation of the private signal
via Bayes’ rule is quite common in the literature: for example, Eyster and
Rabin [19], Guarino and Jehiel [26], and Dasaratha and He [15] employ this
approach in the strict sequential irreversible action setting, and Jadbabie et
al. [31] and Molavi, Tahbaz-Salehi and Jadbabaie [36] in the repeated
interaction setting. The main deviation from the boundedly rational literature
lies in our formulation of the decision rule so that it allows for
randomization. Existing papers overwhelmingly assume, either implicitly or
explicitly, that agents best respond to their (boundedly rational) beliefs,
resulting in deterministic decision rules.151515The one exception is the
companion paper Arieli, Babichenko, and Mueller-Frank [2], which analyzes the
repeated interaction setting.

We henceforth assume that the process starts at time $t=3$ and that the
history at this time is $(m,k)=(1,1)$. This assumption that both actions are
present where the first agent takes her action is needed in order to avoid
extreme priors of $0$ or $1$ in the DeGroot prior formation; both of these
extreme priors cause the entire process to be stabilized on a single action
($0$ or $1$).

#### DeGroot and quasi-Bayesian prior formation

As defined, our three-step approach captures a wide range of heuristics. In
particular, our assumptions capture three commonly applied prior formation
functions that have been extensively analyzed in the literature: Bayesian,
DeGroot, and quasi-Bayesian updating (also called naive inference). We discuss
each in turn to provide further insight into every step of our
heuristic.161616We include Bayesian prior formation in the discussion as a
benchmark and to illustrate the properties of the other two prior formation
heuristics.

So far we have only assumed that each agent knows the signal-generating
functions $(F_{0},F_{1})$. To capture Bayesian and quasi-Bayesian updating,
let us additionally assume that agents commonly know the information structure
$(\mu,F_{0},F_{1},S)$. Under Bayesian updating, the decision rule of each
agent assigns the action $1$ to every belief (weakly) above $\frac{1}{2}$ and
the action $0$ to every belief (strictly) below $\frac{1}{2}$. A _Bayesian
prior formation_ function then assigns a prior belief to each unordered set of
actions via Bayes’ rule, assuming that actions are chosen based on the
Bayesian decision rule and that the state and signals are generated according
to $(\mu,F_{0},F_{1})$.

A _quasi-Bayesian prior_ formation function treats each observed action as if
it were a best response to only the private signal of the agent taking the
action. In other words, each agent is treated as a Bayesian agent who observes
no history of actions and bases her decision only on the prior $\mu$ and her
signal $s$. The quasi-Bayesian decision rule coincides with the Bayesian one
in assigning the action $1$ to every belief (weakly) above $\frac{1}{2}$ and
the action $0$ to every belief (strictly) below $\frac{1}{2}$. We shall call
this decision rule the _Bayesian decision rule_ going forward. Let $e$ denote
the quasi-Bayesian prior formation function. Assuming a common prior
$\mu=\frac{1}{2}$, for $t\geq 3$ we have

$\displaystyle e(k,m)=\frac{q^{k}r^{m}}{q^{k}r^{m}+(1-q)^{k}(1-r)^{m}},$ (2)

where $q=\mathbf{P}_{F,\frac{1}{2}}(\omega=1|p(s)\geq\frac{1}{2})$ and
$r=\mathbf{P}_{F,\frac{1}{2}}(\omega=1|p(s)<\frac{1}{2})$. Thus $q$ represents
the probability a Bayesian agent assigns to state $\omega=1$ conditional on
observing $a=1$ for an agent who took her action in isolation based solely on
her private signal. Similarly, $r$ represents the probability a Bayesian agent
assigns to state $\omega=1$ conditional on observing the action $a=0$ taken by
an isolated agent. Hence $e(k,m)$ coincides with the posterior of a Bayesian
agent who observes $k$ actions $1$ and $m$ actions $0$ that were taken based
on the private signal only.

Finally, a _DeGroot prior formation_ function sets the prior equal to the
proportion of $1$ actions in the observed unordered set. Let $d$ denote the
DeGroot prior formation function. For an observed set $(m,k)$ we have
$d(k,m)=\frac{k}{k+m}$. This approach is inspired by DeGroot [16], where
continuous actions are updated by taking the average of observed actions. For
binary actions, the DeGroot prior formation function has been analyzed in the
repeated interaction model by Chandrasekhar et al. [12] and Arieli,
Babichenko, and Mueller-Frank [2]. The former paper combines the DeGroot prior
formation function with the Bayesian decision rule, and the latter paper
allows for mixed decision rules as in this paper.

##  4 Learning Dynamics under DeGroot Prior Formation

We first focus on the case where the information structure generating the
signals of each agent is commonly understood by all agents.171717Following the
convention in the literature we assume that signals are i.i.d. across agents
conditional on the state. Thus the use of boundedly rational heuristics is
justified only by the complexity of inference rather than additionally by
insufficient knowledge of agents for the application of Bayesian updating. We
turn to our main question, which concerns the information aggregation
properties of the class of heuristics we introduced above. In particular, we
will analyze two different notions of learning. Let
$\mathbf{P}_{F,\mu,b,\sigma}\in\Delta(\Omega\times A^{\infty})$ be the
probability measure over the product of the state space and the action process
that is generated by $\mu$, $F$, and an updating heuristic $(b,\sigma)$.

######  Definition 1.

For a given information structure $(\mu,F_{0},F_{1},S)$ we say that an
updating heuristic $(b,\sigma)$ satisfies _asymptotic learning_ if
$\mathbf{P}_{F,\mu,b,\sigma}(a_{t}=\omega)$ approaches one as $t$ goes to
infinity.

The second notion of learning, _adequate learning_ , is stronger and defined
as follows.

######  Definition 2.

An updating heuristic $(b,\sigma)$ satisfies _adequate learning_ if for every
information structure $(\mu,F_{0},F_{1},S)$ where $F_{0}$ and $F_{1}$ are
absolutely continuous with respect to each other, it holds that
$\mathbf{P}_{F,\mu,b,\sigma}(a_{t}=\omega)$ approaches one as $t$ goes to
infinity.

Asymptotic learning is the common information aggregation notion in the
literature: actions are required to converge in probability to the true state,
for a given information structure. By contrast, adequate learning requires
actions to converge to the truth for _every_ informative information
structure.181818An information structure is called informative if $F_{0}$ and
$F_{1}$ are absolutely continuous with respect to each other. Applying these
two notions, we now study the information aggregation properties of the
DeGroot and the quasi-Bayesian prior formation functions. We do so by imposing
the respective prior formation function but allow for any decision rule.

###  4.1 Adequate Learning for Specified Information Structures

We now present the first main result of the paper, which provides a
characterization of the decision rules that achieve adequate learning under
DeGroot prior formation.

######  Theorem 1.

Let $d$ denote the DeGroot prior formation function. The updating heuristic
$(d,\sigma)$ satisfies adequate learning if and only if $\sigma(x)=x.$

There are several features of this result that are worthwhile to discuss.
First, a very simple updating heuristic enables actions to converge to the
true state. In contrast, Bayesian prior formation would be exceedingly
complex. Each agent would be required to consider the unordered set each
predecessor might have observed, her Bayesian inference based on the signal
and the set of signals that in combination would lead to her action. In
contrast, DeGroot virtual prior formation simply averages the actions in the
unordered set. Second, the updating heuristic combining DeGroot averaging with
probability matching achieves asymptotic learning for _every_ informative
information structure. Instead, in the setting we consider, adequate learning
would fail for Bayesian agents. Any information structure that features
bounded signals191919Signals are bounded if the support of the Bayesian
posterior belief conditional on the signal contains neither 0 nor 1. fails to
achieve asymptotic learning. Third, probability matching is necessary and
sufficient for adequate learning under DeGroot prior formation. Thus, Theorem
1 gives a normative foundation for both Degroot averaging and probability
matching in an observational learning environment. We discuss this in more
detail in Section 7.

A puzzling feature of Theorem 1 is that probability matching not only enables
adequate learning but is in fact the _only_ strategy to do so. Clearly,
adequate learning is a strong condition since it requires asymptotic learning
to hold for _every_ information structure. A natural follow-up question would
weaken this requirement. For example, given a subclass of information
structures $\mathcal{F}$, one may ask what decision rules enable asymptotic
learning for this subclass. Our proof sheds some light on this question, but
we leave a full analysis for future research.

#### Proof Outline of Theorem 1

The proof of Theorem 1 relies on the analysis of generalized Polya urn models
due to Hill, Lane, and Sudderth [28]. In these Polya urn models an initial
composition of blue and red balls is given in an urn. In discrete time, at
every stage $t$, one new ball is added to the urn without replacement. This
additional ball is blue with probability $f(x_{t})$ and red with probability
$1-f(x_{t})$, where $x_{t}$ is the proportion of blue balls in the urn at the
beginning of state $t$ and $f:[0,1]\rightarrow[0,1]$ is a measurable _urn
function_. Note that conditional on state $\omega$, our action process is a
generalized urn process, where the blue and red balls correspond to the $1$
and $0$ actions respectively, and where the proportion $x_{t}$ corresponds to
the fraction of $1$ actions at the beginning of period $t$.

For sufficiency, consider the process of actions conditional on state
$\omega=1$. Let probability matching be the decision rule. Sufficiency then
follows from two auxiliary results. We first show that the urn function $f$
corresponding to probability matching, which further depends on the
information structure, has the property that given a virtual prior of $x$, the
probability of action $a=1$ is strictly larger than $x$ for all $x\in(0,1)$. A
direct corollary to the results of Hill, Lane, and Sudderth [28] establishes
that (conditional on state $\omega=1$) the proportion of $1$ actions, and thus
the virtual prior $x_{t}$, almost surely converges to $1$ as $t$ goes to
infinity. Thus, in particular, $\mathbf{P}_{F,\mu}(a_{t}=1|\omega=1)$
approaches one.

For necessity of probability matching for adequate learning, we consider a
decision rule $\sigma$ such that $\sigma(x)>x$ for some value $x\in[0,1]$ (the
case $\sigma(x)<x$ may be treated symmetrically). We establish the result by
providing one information structure for which asymptotic learning fails. Let
$\tilde{F}$ be a binary information structure, i.e., with two signals, which
is sufficiently uninformative. Consider the actions process conditional on
state $\omega=0$. We show that the urn function $\tilde{f}$ corresponding to
$\tilde{\sigma}$ and $\tilde{F}$ satisfies $\tilde{f}(x)>x$ for some $x$. By
standard coupling arguments this implies that the long-run fraction of $0$
actions is bounded away from $1$ with positive probability. This in turn
implies that $\mathbf{P}_{F,\mu}(a_{t}=0|\omega=0)$ does not converge to one
and thus asymptotic learning fails.

###  4.2 Learning Dynamics in the Unspecified Information Model

We next relax the assumption of conditionally i.i.d. signals, and analyze
adequate learning for conditionally independent but not identically
distributed signals. One critical assumption common to Bayesian observational
learning models is that all agents commonly understand how the signals of all
agents are generated. Thus they share a common specified model of the
information structure. A recently emerging literature considers the case where
Bayesian agents share a common yet incorrect model of the information
structure.202020See the discussion in the section on related literature. We go
one step further and consider the case of an unspecified model of information.
Agents know how their own signals are generated but have _no_ knowledge of the
signal generating process of others, including that of no distribution over
possible information structures. This prevents the use of both Bayesian and
quasi-Bayesian prior formation. The model is as follows. The state of the
world $\omega$ is drawn according to prior $\mu\in(0,1)$, and each agent $t$
receives a private signal that is drawn by an information structure $F^{t}$
independently across agents. Note that since agents observe only the aggregate
statistic of past actions, even knowing the joint information structure
$\\{F^{t}\\}_{t\in\mathbb{N}}$ would not allow them to distinguish more
informative from less informative actions, adding further complexity to the
Bayesian-updating task. We impose one mild condition on the set of information
structures ${F^{t}}_{t\in\mathbb{N}}$. We let $P^{t}\in\Delta([0,1])$ be the
probability measure that is generated by $F_{t}$ conditional on the prior
$\mu=\frac{1}{2}.$ Formally, recall that
$p(s)=\mathbf{P}_{F,\frac{1}{2}}(\omega=1|s)$ is the conditional probability
of $\omega=1$ given that the prior is $\mu=\frac{1}{2}$ and the realized
signal is $s$. For every Borel measurable subset $B\subseteq[0,1]$ we let

$P_{t}(B)=\mathbf{P}_{F,\frac{1}{2}}(p(s)\in B).$

Let $v_{t}$ be the variance of $P_{t}$. We say that the sequence
$\\{F_{t}\\}_{t}$ is _uniformly informative_ if $\inf_{t}v_{t}>0.$

We provide the following extension of Theorem 1.

######  Theorem 2.

Let $d$ denote the DeGroot prior formation function. If $\sigma(x)=x$ then
$(d,\sigma)$ satisfies adequate learning within the class of uniformly
informative information structures.

Theorem 2 shows that the DeGroot probability matching heuristic achieves
perfect information aggregation even in settings where agents have no
knowledge of the order in which past actions were chosen, no knowledge of the
information structure of others, and where information structures across
agents indeed differ. It is worthwhile to discuss this result’s distinguishing
features from existing results. Molavi, Tahbaz-Salehi, and Jadbabaie [36] and
Jadbabaie, Molavi, Sandroni, and Tahbaz-Salehi [31] analyze a setting where
agents receive private signals in each period. They show that a version of
DeGroot prior formation paired with a Bayesian decision rule can lead to
perfect information aggregation when signals are not identically distributed
across agents. However, they do so in a setting where agents communicate their
belief, and thus allow for much finer inference on the underlying signal of
the agent than in our setting of binary actions. Recent work by Frick, Iijima,
and Iishi [22] has highlighted the fragility of the asymptotic learning
outcomes under Bayesian learning to minor errors in the agents’ common
understanding of the environment.212121See also Acemoglu, Ozdaglar, and
ParandehGheibi [1] and Mueller-Frank [39] for the fragility of DeGroot
updating in the rich action setting. By contrast, Theorem 2 provides an
asymptotic learning result for an updating heuristic that is robust within an
open set of (conditionally independent) information structures.

We make two remarks on the uniform informativeness condition. First, one can
show that without it Theorem 2 can fail. Otherwise one could construct a
sequence of information structures where the informativeness of the signals is
vanishing along the sequence and asymptotic learning fails. The failure can
occur even if the condition $\lim\sup_{t}v_{t}>0$ is satisfied. Second, here
we chose to express informativeness in terms of the variance of private
beliefs. Other notions of informativeness will lead to the same result.

#### Proof Outline of Theorem 2

The proof is based on the following line of argument. Since the joint
information structure is uniformly informative, we can find an (informative)
information structure $F$ that is dominated (in the Blackwell-ordering sense)
by all information structures $F_{t}$. We then show that, conditional on state
$\omega=1$ and a proportion $x_{t}$, the probability of agent $t$ choosing
action $a_{t}=1$ under the information structure $F_{t}$ is larger than the
probability of her choosing the same action under the information structure
$F$. Theorem 2 then follows from Theorem 1.

##  5 Learning under Quasi-Bayesian Prior Formation

We next turn to quasi-Bayesian prior formation. Eyster and Rabin [19] consider
the sequential social learning model and show that Quasi-Bayesian prior and
Bayesian posterior formation paired with a Bayesian decision rule leads to
failure of asymptotic learning (in a setting with bounded signals). As we show
next, failure of asymptotic learning extends to every decision rule, in every
information structure. Recall that under quasi-Bayesian updating, every
observed action is treated as if it were based only on the private signal. To
assure that every observed unordered set of actions is consistent with its
interpretation, we assume that the prior belief is equal to half222222Under
the uniform prior a Bayesian agent, who takes his action based on his signal
alone, selects either action with positive probability, for any pair of state-
dependent signal-generating measures $(F_{0},F_{1})$., $\mu=\frac{1}{2}$. Here
we further assume that $F_{0}$ and $F_{1}$ are mutually absolutely continuous
with respect to each other. In making this assumption we avoid cases where
some signals fully reveal the realized state.

Recall that the quasi-Bayesian prior formation function $e(k,m)$ is given by

$\displaystyle e(k,m)=\frac{q^{k}r^{m}}{q^{k}r^{m}+(1-q)^{k}(1-r)^{m}}$ (3)

where $q=\mathbf{P}_{F,\frac{1}{2}}(\omega=1|p(s)\geq\frac{1}{2})$ and
$r=\mathbf{P}_{F,\frac{1}{2}}(\omega=1|p(s)<\frac{1}{2})$.

The following result provides a sharply negative result regarding the
information aggregation properties of Quasi-Bayesian prior formation.

######  Theorem 3.

Let $e$ denote the Quasi-Bayesian virtual prior formation function. The
updating heuristic $(e,\sigma)$ fails to achieve asymptotic learning for every
decision rule $\sigma$, in every informative information structure
$(\mu=\frac{1}{2},F,S)$.

This result raises the question as to what causes the distinction between the
sharp negative result of Theorem 3 and the positive result of Theorem 1. The
answer, we believe, relies on the fact that under the quasi-Bayesian heuristic
the prior of agent $t$ is determined by the difference between the number of
occurrences of actions $1$ and $0$. Thus the prior can be extreme and close to
zero (or one) even when the proportion of actions equal to $1$ (or $0$) is
close to $\frac{1}{2}$. This entails that, compared with the DeGroot prior,
the quasi-Bayesian prior is more sensitive and goes to the extremes much
faster. In particular, incorrect herds are harder to overturn under quasi-
Bayesian prior formation.

#### Proof Outline of Theorem 3

We next provide a proof outline for Theorem 3. Fix an information structure
$F$ and, by way of contradiction, assume that asymptotic learning holds for
some decision rule $\sigma$. We first show that conditional on state $0$, the
probability of selecting action $1$ goes to one as the observed fraction of
$1$ actions goes to one. We use this fact to couple the quasi-Bayesian action
process, conditional on state $\omega=0$ with a random walk over the integers.
The idea is that the coupled random walk moves to the right by one integer if
$a_{t}=1$ and otherwise moves to the left.

We then show, using the classic _gambler’s ruin probability formula_ , that if
the starting time $t_{0}$ of the random walk corresponds to a history where
the advantage of actions $1$ over $0$ is sufficiently large, then the random
walk converges to infinity without ever crossing state $0$ with positive
probability. Using our identification, this implies that $a_{t}$ converges to
$1$ with positive probability conditional on state $\omega=0$, in
contradiction to the fact that asymptotic learning holds.

##  6 Extensions

#### General Finite State Space

Following the norm in the social learning literature, we have focused up to
now on the binary model. A natural question to ask is whether Theorem 1
carries forward to the case of a general finite state space
$\Omega=\\{\omega_{1},\ldots,\omega_{n}\\}$. We address this now.

Let the information structure be given by
$(\mu,(F_{\omega})_{\omega\in\Omega},S)$, where $\mu\in\Delta(\Omega)$ is the
prior probability measure, $S$ is a measurable space, for every
$\omega\in\Omega$ we have $F_{\omega}\in\Delta(S)$, and $F_{\omega}\neq
F_{\omega^{\prime}}$ for $\omega\neq\omega^{\prime}$. The action set is
$A=\\{a_{1},\ldots,a_{n}\\}$, where for each $1\leq k\leq n$ action $a_{k}$
yields a utility of $1$ in state $\omega_{k}$, and a utility of $0$ in any
other state. As before, a decision rule is a mapping
$\sigma:\Delta(\Omega)\rightarrow\Delta(A).$ We can replace the right- or
left-continuity requirement with the requirement that for every
$p\in\Delta(\Omega)$ it holds that $\lim_{q\rightarrow
p}\sigma_{k}(q)=\lim_{q\rightarrow p}\sigma_{k}(q)$ for vectors $q$ such that
$q_{k}<p_{k}$ (or vice versa).

Agent $t$ now forms a prior probability $p^{t}\in\Delta(\Omega)$ according to
the proportion choice of actions in the unordered set. She then incorporates
her private signal in accordance with Bayes’ rule. We claim that Theorem 1
extends to this setting. To see why, let the realized state be
$\omega_{k}\in\Omega$ and consider the following reduction to the binary case:
at time $t$ we add a “blue ball” to the urn only if agent $t$ plays action
$a_{t}$, and otherwise we add a “red ball”. By Theorem 1, this reduction
demonstrates that when the decision rule $\sigma$ equals probability matching,
then the proportion of $a_{k}$ actions approaches one as time goes to
infinity. The converse direction also follows immediately from Theorem 1.

#### General Updating Heuristics

Without imposing any restrictions on the prior formation function and on the
posterior formation function one may ask: which triplets $(b,a,\sigma)$
satisfy adequate learning? Here $b:\mathbb{N}^{2}\to[0,1]$ is a general prior
formation function and $a:[0,1]\times S\to[0,1]$ is a general function that
generates a posterior belief based on the prior and the signal $s\in S$.
Unfortunately, we are not yet able to fully answer this question and we leave
it open for future research. Nevertheless, a few observations about this more
general question are worth mentioning.

Consider the case where the prior formation function
$b:\mathbb{N}\times\mathbb{N}\rightarrow[0,1]$ is _size invariant_. That is,
there exists an increasing continuous function $g:[0,1]\rightarrow[0,1]$ such
that for every $(k,m)$,

$b(k,m)=g(\frac{k}{k+m})=g(d(k,m)).$

Under size invariant prior formation the prior is determined by a general
function of the proportion of $1$ actions, not necessarily the identity
function.

One particular case of the question above, would assume size invariant prior
formation and Bayesian posterior formation. The objective then would be to
characterize the class $(g,\sigma)$ of size invariant prior formation
functions and decision rules for whom adequate learning holds. Let $x$ denote
the proportion of $1$ actions and $y$ the posterior belief. Consider the case
where for every $x$ and $y$ we have

$g(x)=\frac{xc}{cx+(1-c)(1-x)}\text{ and
}\sigma(y)=\frac{(1-c)y}{(1-c)y+c(1-y)}.$

We call such a pair of functions a _complementary pair_. In words, the
function $g$ maps a proportion $x$ to a prior $g(x)$ by treating $x$ as a
Bayesian prior and updating it via Bayes rule with an imaginary signal $s$
such that $p(s)=c.$ The strategy $\sigma$ on the other hand takes the
posterior $y$ and “corrects” the bias imposed on the proportion $x$ by
updating $y$ via Bayes’ rule with the imaginary countersignal $s^{\prime}$
such that $p(s^{\prime})=1-c$. The following is a simple corollary of the
proof of Theorem 1.

######  Corollary 1.

If $(g,\sigma)$ is a _complementary pair_ , then adequate learning holds.

To see this, note that such a complementary pair $(g,\sigma)$ is
_observationally equivalent_ to the case where $g=id$ and $\sigma=id$ (as
considered in Theorem 1) in the following sense: for every proportion $x$,
state $\omega$, and information structure $F$, the probability that an agent
who uses $(g,\sigma)$ will choose action $1$ is the same as $(id,id)$. Thus,
conditional on any state $\omega$ the action process that is induced by
$(g,\sigma)$ and by $(id,id)$ have exactly the same distribution. The
corollary readily follows from this fact. One can further show that only
complementary pairs are observationally equivalent232323This claim is by no
means straightforward. to $(id,id)$. Based on the corollary, one may
conjecture that observational equivalence with DeGroot prior formation paired
with probability matching is a necessary condition for adequate learning
within the class of size-invariant prior formation functions.

##  7 Concluding Remarks

Our results provide a normative foundation for the DeGroot probability-
matching heuristic in a observational learning environment with binary
actions. We now briefly discuss several aspects of our contribution and
mention some open questions for future work. First, for models of repeated
interaction on social networks DeGroot updating has been shown to achieve
learning in a setting where the action space is rich; see Golub and Jackson
[24]. We show that in the sequential model where observations are limited to
an aggregate statistic, DeGroot prior formation can achieve asymptotic
learning even for binary actions in _every_ informative information structure,
but only if the decision rule satisfies probability matching. For some
information structures where probability matching achieves asymptotic
learning, pairing the DeGroot prior formation with the (deterministic)
Bayesian decision rule induces long-run actions that match the state only with
probability half. Thus probability matching can dramatically improve long-run
outcomes relative to the Bayesian decision rule. Despite probability matching
violating expected utility maximization, there is a vast experimental
literature documenting it in laboratory settings. The reasons behind the
widespread use of probability matching have long puzzled researchers. In the
words of Kenneth Arrow [5]:

“We have here an experimental situation which is essentially of an economic
nature in the sense of seeking to achieve a maximum of expected reward, and
yet the individual does not in fact, at any point, even in a limit, reach the
optimal behavior. I suggest that this result points out strongly the
importance of learning theory, not only in the greater understanding of the
dynamics of economic behavior, but even in suggesting that equilibria may be
different from those that we have predicted in our usual theory.”

Our results point to one possible evolutionary justification for probability
matching: it can achieve optimal asymptotic outcomes in observational learning
environments where expected utility maximization fails. More precisely,
probability matching induces a positive learning externality. The existence of
what is seen as a violation of rationality might instead point to an intrinsic
utility that weights outcomes of others. Our characterization leaves no room
for any mixed learning heuristic other than _exact_ probability matching.
However, we believe that for a fixed information structure, under the
appropriate notion of proximity, any decision rule that is close to
probability matching also achieves asymptotic learning.

Second, we derive the positive results in a model where agents observe only
the unordered set of past actions. Note that the exact same results would hold
if instead they were to observe the ordered set of past actions as is the norm
in the sequential social learning literature.

Finally, our analysis leaves room for further work. We established probability
matching as necessary and sufficient for adequate learning if prior formation
is according to DeGroot. A more universal approach would instead allow for
general prior formation and aim at characterizing the properties of the pair
of the prior formation function and decision rule that achieve adequate
learning. The size-invariant prior formation function introduced above
provides one avenue of approach to this problem. We leave this for future
work.

##  8 Proofs

We start with the proof of Theorem 1. The proof of the theorem relies on [28]
which is a classic paper on _generalized urn processes_. A generalized urn
process $\\{x_{t}\\}_{t\in\mathbb{N}}$ comprises an initial urn composition of
$m>0$ blue balls and $k>0$ red balls and a measurable function
$f:[0,1]\rightarrow[0,1]$. At any point in time $t\geq 1$, if the blue balls’
proportion is $x_{t}$, then a new blue ball is added with probability
$f(x_{t})$ and a new red ball is added with probability $1-f(x_{t})$. We first
note that by Theorem 2.1 in [28] the process $\\{x_{t}\\}_{t\geq 0}$ converges
almost surely to a limit. The following two auxiliary lemmas follow directly
from the main results in [28].

######  Lemma 1.

Consider an urn process and assume that $f$ is a continuous function. If
$f(x)>x$ ($f(x)<x$) for every $x\in(0,1)$, then
$\lim_{t\rightarrow\infty}x_{t}=1$ ($\lim_{t\rightarrow\infty}x_{t}=0$).

######  Lemma 2.

Consider an urn processes and assume that $f$ is a continuous function. If
there exists $x^{\prime}\in(0,1)$ such that $f(x)>x$ in a left neighborhood of
$x^{\prime}$ and $f(x)<x$ in a right neighborhood of $x^{\prime}$, then there
exist an open interval $I$ where $x^{\prime}\in I$ and a constant $n$ such
that for $x_{0}\in I$ and $m+k\geq n$ it holds that
$\lim_{t\rightarrow\infty}x_{t}=x^{\prime}$ with positive probability.

Moreover we make use of the obvious fact that if $g:[0,1]\rightarrow[0,1]$ is
a piecewise constant such that $f\geq g$ and $\\{y_{t}\\}_{t}$ is the
corresponding process with the same initial composition as $x_{t}$, then we
can couple the two processes together such that $x_{t}\geq y_{t}$ for every
$t$.

We start with a Lemma that is based on Lemma 2.

######  Lemma 3.

Consider an urn process and assume that the function $f$ is picewise
continuous as defined above. If $f(\tilde{x})>\tilde{x}$ (or
$f(\tilde{x})<\tilde{x}$) for some $\tilde{x}\in(0,1)$, then there exist an
interval $J\subseteq(0,1)$ and some $n>0$ so that for every initial condidions
$(m,k)$ with $x_{0}\in J$ and $m+k>n$ it holds with positive probability that
$\lim_{t\rightarrow\infty}x_{t}\neq 0$ ($\lim_{t\rightarrow\infty}x_{t}\neq
1$).

###### Proof of Lemma 3.

It follows from the facts that $f$ is either left- or right-continuous that
there exists a closed interval $I=[a,b]\subset(0,1)$ such that $x\in I$ and
$f(x)>x+\delta$ for every $x\in I$. Therefore, we can find a continuous
function $g:[0,1]\rightarrow[0,1]$ and a point $x^{\prime}\in I$ such that
$f(y)\geq g(y)$ for every $y\in[0,1]$, and $g(y)>x$ in a left neighborhood of
$x^{\prime}$ and $g(y)<y$ on a right neighborhood of $x^{\prime}$. It follows
from Lemma 2 that there exist an interval $J$ of $x^{\prime}$ and a constant
$n$ such that for every initial condition $(m,k)$ such that $x_{0}\in J$ the
process $\\{y_{t}\\}_{t}$ converges to $x^{\prime}$ with positive probability.
Since for the corresponding process $\\{x_{t}\\}$ we have $f\geq g$, from the
above comment we can couple the processes together such that $x_{t}\geq y_{t}$
for every $t$. Therefore if $m+k>n$ and $x_{0}\in J$, then
$\lim_{t\rightarrow\infty}x_{t}\geq x^{\prime}>0$ with positive probability. ∎

###### Proof of Theorem 1.

We note that if the prior is $\mu$, then it follows directly from Bayes rule
that the conditional probability of $\omega=1$ given a signal $s\in S$ is

$p_{\mu}(s)=\mathbf{P}_{F,\mu}(\omega=1|s)=\frac{p(s)\mu}{p(s)\mu+(1-p(s))(1-\mu)}.$

It follows from standard martingale considerations that

$E_{F,\mu}[p_{\mu}(s)]=\mu.$

Consider a prior $\mu$ and let $G^{\mu}$ be the posterior belief distribution.
That is, for every measurable $A\subseteq[0,1],$

$G_{\mu}(A)=\mathbf{P}_{F,\mu}(p_{\mu}(s)\in A).$

Similarly, for every state $\omega$ we let

$G_{\mu}^{\omega}(A)=\mathbf{P}_{F,\mu}(p_{\mu}(s)\in A|\omega)$

be the posterior distribution conditional on state $\omega$. Let
$\underline{\beta}_{\mu},\overline{\beta}_{\mu}\in[0,1]$ be the lower point
and the upper point of the support of $p(s)$. Since
$\frac{dG_{\mu}^{1}}{dG_{\mu}}(x)=\frac{x}{\mu}$ and
$\frac{dG_{\mu}^{0}}{dG_{\mu}}(x)=\frac{1-x}{1-\mu}$ at every point in the
support of $p_{\mu}(s)$ (see, e.g., Lemma 1 in [4]), it follows that
$G_{1}([0,x])<G_{0}([0,x])$ for every
$x\in\underline{(}\beta_{\mu},\overline{\beta}_{\mu})$. Thus $G^{\mu}_{1}$
first-order stochastically dominates $G^{\mu}_{0}$. We further note that, by
definition, $E_{F,\mu}[p_{\mu}(s)]=\int_{[0,1]}xdG_{\mu}(x)$, and
$E_{F,\mu}[p_{\mu}(s)|\omega]=\int_{[0,1]}xdG^{\omega}_{\mu}(x)$. By the law
of iterated expectation,

$\displaystyle\mu=E_{F,\mu}[p_{\mu}(s)]=\mu
E_{F,\mu}[p_{\mu}(s)|\omega=1]+(1-\mu)E_{F,\mu}[p_{\mu}(s)|\omega=0].$

Since $E_{F,\mu}[p_{\mu}(s)|\omega=1]>E_{F,\mu}[p_{\mu}(s)|\omega=0]$ we must
have that

$E_{F,\mu}[p_{\mu}(s)|\omega=1]>\mu>E_{F,\mu}[p_{\mu}(s)|\omega=0].$

We note that for the strategy $\sigma$ the learning process conditional on any
state $\omega\in\Omega$ is an urn process. In particular, in the case
$\sigma(x)=x$, for every proportion of $\frac{m}{m+k}=x_{t}$ agents who play
action $1$, agent $t$ plays action $a_{t}=1$ with probability
$f(x_{t})=E_{F,x_{t}}[p_{x_{t}}(s)|\omega]$, which is a continuous function of
$x_{t}$. Thus, in particular, conditional on state $\omega=1$, the probability
that agent $t$ plays action $1$ is $E_{F,x_{t}}[p_{x_{t}}(s)|\omega=1]>x_{t}$.
We further note that $E_{F,x_{t}}[p_{x_{t}}(s)|\omega=1]$ is a continuous
function of $x_{t}$ since $p_{x_{t}}(s)$ is a continuous function of $x_{t}$
for any signal $s$. Thus it follows from the auxiliary Lemma 1 that the
proportion $x_{t}$ converges to $1$ conditional on state $\omega=1$.
Similarly, since $E_{F,x_{t}}[p_{x_{t}}(s)|\omega=0]<x_{t}$ the process
$x_{t}$ converges to $0$ conditional on state $\omega=0$. Note that
$f(x)=E_{F,\mu}[p_{x}(s)|\omega]$ approaches $0$ as $x$ approaches $0$ and
$f(x)=E_{F,\mu}[p_{x}(s)|\omega]$ approaches $1$ as $x$ approaches $1$ for
every state $\omega$. Therefore since $\lim_{t\rightarrow\infty}x_{t}=\omega$
with probability one, we have that the probability of $a_{t}=\omega$
approaches one as $t$ goes to infinity. Hence asymptotic learning holds.

We next show the converse direction. We need to show that if $\sigma\neq id$,
then for any prior $\mu\in(0,1)$ there exists a state $\omega$ such that
conditional on state $\omega$ it holds that
$\lim_{t\rightarrow\infty}x_{t}\neq\omega$ with positive probability.

Since $\sigma\neq id$ it holds that $\sigma(\tilde{x})\neq\tilde{x}$ for some
$\tilde{x}\in(0,1)$. Therefore, either $\sigma(\tilde{x})>\tilde{x}$ or
$\sigma(\tilde{x})<\tilde{x}$. Without loss of generality consider the first
case. Since $\sigma$ is either left- or right-continuous at $\tilde{x}$, we
can find an interval $I\subset(0,1)$ and a constant $\eta$ such that
$\sigma(x)>x+\delta$ for every $x\in I$.

We recall that conditional on state $\omega$ action $1$’s proportion
$\\{x_{t}\\}_{t\in\mathbb{N}}$ is a generalized urn process where the
probability of playing action $1$ is
$f(x)=E_{F,\mu}[\sigma(p_{x}(s))|\omega=0]$. To see this, note that since for
every $x$ the function $p_{x}(s)$ is continuous and increasing in $x$, it
follows that if $\sigma$ is right-continuous at any point so is $f$ (and
similarly with respect to left continuity).

We claim that we can find an information structure $F$ and a point $x\in(0,1)$
such that $E_{F,\mu}[\sigma(p_{x}(s))|\omega=0]>x$.

To see this, consider a binary information structure $F$ with
$S=\\{s_{0},s_{1}\\}$ and $p(s_{\omega}|\omega)=\frac{1}{2}+\epsilon$ for
every $\omega$. It follows from Bayes’ rule that
$p_{x}(s_{1})=\frac{(\frac{1}{2}+\epsilon)x}{(\frac{1}{2}+\epsilon)x+(\frac{1}{2}-\epsilon)(1-x)}$
and
$p_{x}(s_{0})=\frac{(\frac{1}{2}-\epsilon)x}{(\frac{1}{2}-\epsilon)x+(\frac{1}{2}+\epsilon)(1-x)}$.
Moreover, conditional on state $\omega=0$, the variable $p_{x}$ equals
$p_{x}(s_{1})$ with probability $\frac{1}{2}-\epsilon$ and $p_{x}(s_{0})$ with
probability $\frac{1}{2}+\epsilon$.

We note that $\lim_{\epsilon\rightarrow 0}p_{x}(s_{\omega})=x$. Therefore we
can find $\eta>0$ and $x\in I$ such that $\sigma(p_{x}(s_{\omega}))>x$ for
$\omega=0,1$. This implies in particular that
$f(x)=E_{F,x}[\sigma(p_{x}(s))|\omega=0]>x.$

It therefore follows from Lemma 3 that there exist an open interval
$J\subset(0,1)$ and $n$ such that if $m+k>n$ and $x_{0}\in J$, then
$\\{x_{t}\\}$ does not converge to $0$ with positive probability.

First we may assume that, conditional on state $\omega=1$, it holds that
$\lim_{t\rightarrow\infty}x_{t}=1$ with positive probability. Otherwise
$(d,\sigma)$ does not satisfy asymptotic learning. Therefore, conditional on
state $\omega=0$ for every $n^{\prime}$ and $\epsilon>0$ there exists a time
$t$ such that $x_{t}\geq 1-\epsilon$ and $t\geq n^{\prime}$. This follows from
the fact that any finite sequence of actions that has a positive probability
conditional on state $\omega=1$. This holds true since $F_{0}$ and $F_{1}$ are
absolutely continuous with respect to each other.

We note that for small enough $\epsilon$ the value $n^{\prime}$ must be
arbitrarily large. Hence, for small enough $\epsilon$, if $x_{t}$ reaches a
point $x_{t}\geq 1-\epsilon$, then if $x_{t}$ crosses the interval $J$ from
right to left we must have a time $t^{\prime}$ such that $x_{t}\in J$ and
$t>n$. In such a case we have that $\\{x_{t}\\}$ does not converge to $0$ with
positive probability. Thus either the process crosses $x_{t}\geq 1-\epsilon$
and after that does not cross $J$ with positive probability or $x_{t}\in J$
and $t>n$. In any case $\\{x_{t}\\}$ does not converge to $0$ with positive
probability. Hence asymptotic learning fails.

Without loss of generality we consider the first case. We claim that we can
find an information structure $F$ and an interval $J$ such that
$E_{F,\mu}[\sigma(p_{\mu}(s))|\omega=0]>\mu+\eta$ for every $\mu\in J$ and
some $\eta>0$.

To see this, consider a binary information structure $F$ with
$S=\\{s_{0},s_{1}\\}$ and $p(s_{\omega}|\omega)=\frac{1}{2}+\epsilon$ for
every $\omega$. It follows from Bayes’ rule that
$p_{x}(s_{1})=\frac{(\frac{1}{2}+\epsilon)x}{(\frac{1}{2}+\epsilon)x+(\frac{1}{2}-\epsilon)(1-x)}$
and
$p_{x}(s_{0})=\frac{(\frac{1}{2}-\epsilon)x}{(\frac{1}{2}-\epsilon)x+(\frac{1}{2}+\epsilon)(1-x)}$.
Moreover, conditional on state $\omega=0$, the variable $p_{x}$ equals
$p_{x}(s_{1})$ with probability $\frac{1}{2}-\epsilon$ and $p_{x}(s_{0})$ with
probability $\frac{1}{2}+\epsilon$.

We note that $\lim_{\epsilon\rightarrow 0}p_{x}(s_{\omega})=x$. Therefore we
can find $\eta>0$ and $x\in I$ such that $\sigma(p_{x}(s_{\omega}))>x$ for
$\omega=0,1$. This implies in particular that
$f(x)=E_{F,x}[\sigma(p_{x}(s))|\omega=0]>x.$

It now follows from Lemma 3 that there exist an open interval $J\subset(0,1)$
and $n$ such that if $m+k>n$ and $x_{0}\in J$, then $\\{x_{t}\\}$ does not
converge to $0$ with positive probability.

First we may assume that, conditional on state $\omega=1$, it holds that
$\lim_{t\rightarrow\infty}x_{t}=1$ with positive probability. Otherwise
$(d,\sigma)$ does not satisfy asymptotic learning. Therefore, conditional on
state $\omega=0$ for every $n^{\prime}$ and $\epsilon>0$ there exists a time
$t$ such that $x_{t}\geq 1-\epsilon$ and $t\geq n^{\prime}$. This follows from
the fact that any finite sequence of actions that has a positive probability
conditional on state $\omega=1$ also has a positive probability conditional on
state $\omega=0$. This holds true since $F_{0}$ and $F_{1}$ are absolutely
continuous with respect to each other.

We note that for small enough $\epsilon$ the value $n^{\prime}$ must be
arbitrarily large. Hence, for small enough $\epsilon$, if $x_{t}$ reaches a
point $x_{t}\geq 1-\epsilon$, then if $x_{t}$ crosses the interval $J$ from
right to left we must have a time $t^{\prime}$ such that $x_{t}\in J$ and
$t>n$. In such a case we have that $\\{x_{t}\\}$ does not converge to $0$ with
positive probability. Thus, either the process crosses $x_{t}\geq 1-\epsilon$
and after that does not cross $J$ with positive probability or $x_{t}\in J$
and $t>n$. In any case $\\{x_{t}\\}$ does not converge to $0$ with positive
probability. Hence, since the proportion $x_{t}$ does not converge to zero,
this implies that
$\limsup_{t\rightarrow\infty}\mathbf{P}_{F,b,\sigma}(a_{t}=1|\omega=0)>0$ and
asymptotic learning fails.

∎

We next turn to the proof of Theorem 2.

###### Proof of Theorem 2.

Recall that the variance of $P_{t}$ is $v_{t}$ and $\inf_{t}v_{t}=v>0$. Hence
there exists an information structure $F$ that is Blackwell-dominated $F_{t}$
for every $t$. This can be shown by constructing a binary information
structure $F$ whose corresponding belief distribution function $P$ is a _mean
preserving contraction_ of $P_{t}$ for every $t$. (That is, $P_{t}$ is a mean-
preserving spread of $P$.) Such a binary information structure exists since
$v_{t}\geq v$ for every $t$.

Consider the learning process both under $P_{t}$ and under $P$ conditional on
state $\omega=1$. It follows from $\frac{dP_{t}^{1}}{dP_{t}}(y)=\frac{y}{\mu}$
(again see, e.g., Lemma 1 in [4]) that conditional on $\omega=1$ and the
observed proportion $x_{t}=x$, the probability of choosing action $1$ equals

$f_{t}(x)=\int\frac{2xy^{2}}{xy+(1-x)(1-y)}\mathrm{d}P_{t}(y).$

And, similarly, for $P$ this probability equals
$f(x)=\int\frac{2y^{2}}{xy+(1-x)(1-y)}\mathrm{d}P(x).$ Since
$\frac{2y^{2}}{xy+(1-x)(1-y)}$ is convex in $y$ and since $P_{t}$ is a mean-
preserving spread of $P$ it follows that $f_{t}(x)\geq f(x)$ for every $x$. It
follows from Theorem 1 that under $P$ the proportion of actions $1$ converges
to $1$ as time goes to infinity. Therefore the coupling argument mentioned
after the auxiliary lemmas implies that the proportion of actions $1$
converges to $1$ also with respect to the generalized processes and hence the
probability $a_{t}=1$ converges to one conditional on state $\omega=1$. A
similar derivation holds for $\omega=0$ with respect to $a_{t}=0$. Therefore
asymptotic learning holds.

∎

###### Proof of Theorem 3.

Assume by way of contradiction that $(e,\sigma)$ satisfies asymptotic learning
for some information structure $F$. We first claim that $\lim_{x\rightarrow
1}\sigma(x)=\sigma(1)=1$ (and similarly $\lim_{x\rightarrow
0}\sigma(x)=\sigma(0)=0$). To see this, consider the case where the realized
state is $\omega=1$. Since asymptotic learning holds, we must have that
$\mathbf{P}(a_{t}=1|\omega=1)$ converges to one. Therefore, as time goes to
infinity, the proportion of actions $1$ agent $t$ observes converges to one
(in probability). Hence, if we let $m_{t}$ and $k_{t}$ be the number of $1$’s
and $0$’s in the realized observation set of $t$, it follows that
$e(k_{t},m_{t})$ approaches one. This implies that the posterior probability
$\frac{p(s_{t})e(k_{t},m_{t})}{p(s)e(k_{t},m_{t})+(1-p(s_{t}))(1-e(k_{t},m_{t}))}$
of agent $t$ also converges to one in probability. Since agent $t$ chooses
action $1$ with probability
$\int_{S}\sigma\Big{(}\frac{p(s)e(k_{t},m_{t})}{p(s)e(k_{t},m_{t})+(1-p(s))(1-e(k_{t},m_{t}))}\Big{)}dF_{1}(s)=\mathbf{P}(a_{t}=1|\omega=1,e(k_{t},m_{t}))$,
which approaches one with $t$, we must have that $\lim_{x\rightarrow
1}\sigma(x)=\sigma(1)=1$.

We note that since $F_{0}$ and $F_{1}$ are mutually absolutely continuous with
respect to each other, and since $\lim_{x\rightarrow 1}\sigma(x)=1$ we must
have that $\mathbf{P}(a_{t}=1|e(k_{t},m_{t}),\omega=0)=\\\
\int_{S}\sigma\Big{(}\frac{p(s)e(k_{t},m_{t})}{p(s)e(k_{t},m_{t})+(1-p(s))(1-e(k_{t},m_{t}))}\Big{)}dF_{0}(s)$
approaches one as $e(k_{t},m_{t})$ goes to one. This implies that for every
value $c\in(0,1)$ there exists $b\in(0,1)$ such that if $e(k_{t},m_{t})>b$,
then $\mathbf{P}(a_{t}=1|e(k_{t},m_{t}),\omega=0)>c$.

To conclude the proof, we couple our process of actions conditional on the
realized state $\omega=0$ with a biased random walk such that if the random
walk converges to infinity before ever crossing the origin, then the public
belief of the learning process $e(k_{t},m_{t})$ converges to $1$. This will
establish the desired contradiction. First consider a biased simple random
walk with a success probability of $\frac{3}{4}$ starting at $1$. Thus at
every stage the random walk position increases by $1$ with probability
$\frac{3}{4}$ and it decreases by $1$ with probability $\frac{1}{4}$. It
follows using a simple recursive argument that the probability that the random
walk converges to infinity before ever reaching $0$ is $\frac{2}{3}.$

To apply the random walk argument to our action process, let
$l=\Big{\lceil}\frac{\ln(\frac{r}{1-r})}{\ln(\frac{q}{1-q})}\Big{\rceil}$.
Note that $l$ is a positive integer since $r<\frac{1}{2}$ and $l>\frac{1}{2}$.
Choose $b$ such that $\mathbf{P}(a_{t}=1|e(k_{t},m_{t})\geq
b,\omega=0)>(3/4)^{\frac{1}{l}}:=v.$ Take a time $t^{\prime}$ such that
$e(k_{t^{\prime}},m_{t^{\prime}})>b$. As long as
$e(k_{t^{\prime}},m_{t^{\prime}})>b$, the probability that $a_{t}=1$ will be
played for $l$ consecutive periods is, by assumption, at least
$[(\frac{3}{4})^{\frac{1}{l}}]^{l}=\frac{3}{4}$. Moreover, the probability of
choosing $a_{t}=0$ is less than $\frac{1}{4}$.

We now define a biased random walk $\\{Z_{n}\\}_{n\geq 0}$ and a corresponding
increasing sequence of times $\\{t(n)\\}_{n\geq 0}$ as follows. We let
$t(0)=t^{\prime}$ and let $Z_{0}=1$. If starting at time $t^{\prime}$ the
action $a_{t}=1$ was played for $l$ consecutive periods, i.e., $a_{t}=1$ for
$t=t^{\prime},\ldots,t^{\prime}+l-1$, we let $Z_{1}=2$ and
$t(1)=t^{\prime}+l$. Otherwise, we let $Z_{1}=0$ and set $t(1)=\tilde{t}+1$,
where $\tilde{t}\in\\{t^{\prime},\ldots,t^{\prime}+l-1\\}$ is the first time
$t$ such that $a_{t}=0$. Inductively, given that $Z_{k}$ is an integer and
$t(k)$ is defined, we set $Z_{k+1}=Z_{k}+1$, and $t(k+1)=t(k)+l$ if $a_{t}=1$
was played in every time period $t=t(k),\ldots,t(k)+l-1$. Otherwise we let
$Z_{k+1}=Z_{k}-1$, and $t(k+1)$ is defined by $\tilde{t}+1$ where
$\tilde{t}\in\\{t(k),\ldots,t(k)+l-1\\}$ is the first time such that
$a_{t}=0$.

We call a sequence of realizations $\\{z_{n}\\}_{n\geq 0}$ of the random walk
_good_ if it does not reach zero. As mentioned above, the probability of a
good sequence is at least $\frac{2}{3}$. We claim that if the sequence is
good, then $e(k_{t},m_{t})>b$ for every $t\geq t^{\prime}$. Since $z_{n}=j>0$
for every $n$ it follows by definition that
$k_{t}-k_{t^{\prime}}\geq(m_{t}-m_{t^{\prime}})l$ for every time $t$. We note
that $e(k,m)=\frac{1}{1+(\frac{1-q}{q})^{k}(\frac{1-r}{r})^{m}}$. Therefore,
for every $t\geq t^{\prime}$ we can write

$\displaystyle
e(k_{t},m_{t})=\frac{1}{1+(\frac{1-q}{q})^{k_{t^{\prime}}}(\frac{1-r}{r})^{m_{t^{\prime}}}(\frac{1-q}{q})^{k_{t}-k_{t^{\prime}}}(\frac{1-r}{r})^{m_{t}-m_{t^{\prime}}}}$
$\displaystyle\geq\frac{1}{1+(\frac{1-q}{q})^{k_{t^{\prime}}}(\frac{1-r}{r})^{m_{t^{\prime}}}(\frac{1-q}{q})^{({m_{t}-m_{t^{\prime}}})l}(\frac{1-r}{r})^{m_{t}-m_{t^{\prime}}}},$
(4)

where inequality (4) follows since $\frac{(1-q)}{q}<1$ and
$k_{t}-k_{t^{\prime}}\geq(m_{t}-m_{t^{\prime}})l$. Thus replacing
$(\frac{1-q}{q})^{({k_{t}-k_{t^{\prime}}})}$ with
$(\frac{1-q}{q})^{({m_{t}-m_{t^{\prime}}})l}$ increases the denominator and
therefore decreases the expression. Taking log of
$(\frac{1-q}{q})^{({m_{t}-m_{t^{\prime}}})l}(\frac{1-r}{r})^{m_{t}-m_{t^{\prime}}}$
gives

$({m_{t}-m_{t^{\prime}}})l\ln(\frac{1-q}{q})+({m_{t}-m_{t^{\prime}}})\ln(\frac{1-r}{r}).$

Since
$l=\Big{\lceil}\frac{\ln(\frac{r}{1-r})}{\ln(\frac{q}{1-q})}\Big{\rceil}\geq\frac{\ln(\frac{r}{1-r})}{\ln(\frac{q}{1-q})}$
and since $\ln(\frac{r}{1-r})$ is negative we have that

$({m_{t}-m_{t^{\prime}}})l\ln(\frac{1-q}{q})+({m_{t}-m_{t^{\prime}}})\ln(\frac{1-r}{r})\leq
0.$

Therefore
$(\frac{1-q}{q})^{({m_{t}-m_{t^{\prime}}})l}(\frac{1-r}{r})^{m_{t}-m_{t^{\prime}}}\leq
1$. Integrating this inequality into (4) gives $e(k_{t},m_{t})\geq
e(k_{t^{\prime}},m_{t^{\prime}})$, as desired.

Thus, overall we have shown that if conditional on state $\omega=0$ the
history reaches a time period $t^{\prime}$ where
$e(k_{t^{\prime}},m_{t^{\prime}})>b,$ then $e(k_{t},m_{t})>b,$ for every
$t\geq t^{\prime}$ with probability at least $\frac{2}{3}$. Note that, by the
contradiction assumption, conditional on state $\omega=1$ the history reaches
a time period $t^{\prime}$ where $e(k_{t^{\prime}},m_{t^{\prime}})>b$ with
probability one. Since $F_{0}$ is absolutely continuous with respect to
$F_{1}$ and since existence of a time period $t^{\prime}$ where
$e(k_{t^{\prime}},m_{t^{\prime}})>b$ depends on finitely many signal
realizations, we conclude that conditional on state $\omega=0$ there exists a
positive probability of reaching a time period $t^{\prime}$ such that
$e(k_{t^{\prime}},m_{t^{\prime}})>b$. This is a contradiction since
$\lim_{t\rightarrow\infty}\mathbf{P}(a_{t}=0|\omega=0)=1$ only when
$e(k_{t},m_{t})$ converges to zero in probability, conditional on state
$\omega=0$.

∎

## References

  * [1] Daron Acemoglu, Asuman Ozdaglar, and Ali ParandehGheibi.  Spread of (mis) information in social networks.  Games and Economic Behavior, 70(2):194–227, 2010.
  * [2] Itai Arieli, Yakov Babichenko, and Manuel Mueller-Frank.  Naive learning through probability matching.  Available at SSRN 3338015, 2019.
  * [3] Itai Arieli, Yakov Babichenko, and Segev Shlomov.  Robust non-bayesian social learning.  In Proceedings of the 2019 ACM Conference on Economics and Computation, pages 549–550, 2019.
  * [4] Itai Arieli, Yakov Babichenko, and Rann Smorodinsky.  Identifiable information structures.  Games and Economic Behavior, 120:16–27, 2020.
  * [5] Kenneth J Arrow.  Utilities, attitudes, choices: A review note.  Econometrica: Journal of the Econometric Society, pages 1–23, 1958\.
  * [6] Abhijit Banerjee, Arun G Chandrasekhar, Esther Duflo, and Matthew O Jackson.  The diffusion of microfinance.  Science, 341(6144), 2013.
  * [7] Abhijit V Banerjee.  A simple model of herd behavior.  The quarterly journal of economics, 107(3):797–817, 1992.
  * [8] Sushil Bikhchandani, David Hirshleifer, and Ivo Welch.  A theory of fads, fashion, custom, and cultural change as informational cascades.  Journal of political Economy, 100(5):992–1026, 1992.
  * [9] J Aislinn Bohren.  Informational herding with model misspecification.  Journal of Economic Theory, 163:222–247, 2016.
  * [10] J Aislinn Bohren and Daniel N Hauser.  Social learning with model misspecification: A framework and a characterization.  2020\.
  * [11] Eric Bonabeau.  The perils of the imitation age.  Harvard business review, 82(6):45–54, 2004.
  * [12] Arun G Chandrasekhar, Horacio Larreguy, and Juan Pablo Xandri.  Testing models of social learning on networks: Evidence from two experiments.  Econometrica, 88(1):1–32, 2020.
  * [13] Timothy G Conley and Christopher R Udry.  Learning about a new technology: Pineapple in ghana.  American economic review, 100(1):35–69, 2010.
  * [14] Krishna Dasaratha and Kevin He.  An experiment on network density and sequential learning.  arXiv preprint arXiv:1909.02220, 2019.
  * [15] Krishna Dasaratha and Kevin He.  Network structure and naive sequential learning.  Theoretical Economics, 15(2):415–444, 2020.
  * [16] Morris H DeGroot.  Reaching a consensus.  Journal of the American Statistical Association, 69(345):118–121, 1974.
  * [17] Peter M DeMarzo, Dimitri Vayanos, and Jeffrey Zwiebel.  Persuasion bias, social influence, and unidimensional opinions.  The Quarterly journal of economics, 118(3):909–968, 2003.
  * [18] Benjamin Enke and Florian Zimmermann.  Correlation neglect in belief formation.  The Review of Economic Studies, 86(1):313–332, 2019.
  * [19] Erik Eyster and Matthew Rabin.  Naive herding in rich-information settings.  American economic journal: microeconomics, 2(4):221–43, 2010.
  * [20] Erik Eyster, Matthew Rabin, and Georg Weizsacker.  An experiment on social mislearning.  Available at SSRN 2704746, 2015.
  * [21] Lawrence F Feick and Linda L Price.  The market maven: A diffuser of marketplace information.  Journal of marketing, 51(1):83–97, 1987.
  * [22] Mira Frick, Ryota Iijima, and Yuhta Ishii.  Misinterpreting others and the fragility of social learning.  Econometrica, 88(6):2281–2328, 2020.
  * [23] Mira Frick, Ryota Iijima, and Yuhta Ishii.  Stability and robustness in misspecified learning models.  2020\.
  * [24] Benjamin Golub and Matthew O Jackson.  Naive learning in social networks and the wisdom of crowds.  American Economic Journal: Microeconomics, 2(1):112–49, 2010.
  * [25] Benjamin Golub and Matthew O Jackson.  How homophily affects the speed of learning and best-response dynamics.  The Quarterly Journal of Economics, 127(3):1287–1338, 2012.
  * [26] Antonio Guarino and Philippe Jehiel.  Social learning with coarse inference.  American Economic Journal: Microeconomics, 5(1):147–74, 2013.
  * [27] Ward A Hanson and Daniel S Putler.  Hits and misses: Herd behavior and online product popularity.  Marketing letters, 7(4):297–305, 1996.
  * [28] Bruce M Hill, David Lane, and William Sudderth.  A strong law for some generalized urn processes.  The Annals of Probability, pages 214–226, 1980.
  * [29] Jan Hkazla, Ali Jadbabaie, Elchanan Mossel, and M Amin Rahimian.  Reasoning in bayesian opinion exchange networks is pspace-hard.  In Conference on Learning Theory, pages 1614–1648. PMLR, 2019.
  * [30] Jen-Hung Huang and Yi-Fen Chen.  Herding in online product choice.  Psychology & Marketing, 23(5):413–428, 2006.
  * [31] Ali Jadbabaie, Pooya Molavi, Alvaro Sandroni, and Alireza Tahbaz-Salehi.  Non-bayesian social learning.  Games and Economic Behavior, 76(1):210–225, 2012.
  * [32] Dorothea Kübler and Georg Weizsäcker.  Limited depth of reasoning and failure of cascade formation in the laboratory.  The Review of Economic Studies, 71(2):425–441, 2004.
  * [33] Amir Leshem and Anna Scaglione.  The impact of random actions on opinion dynamics.  IEEE Transactions on Signal and Information Processing over Networks, 4(3):576–584, 2018.
  * [34] Gilat Levy and Ronny Razin.  Information diffusion in networks with the bayesian peer influence heuristic.  Games and Economic Behavior, 109:262–270, 2018.
  * [35] Christoph March and Anthony Ziegelmeyer.  Even experts follow large herds: Naıve herding in the laboratory.  Technical report, Mimeo, 2015.
  * [36] Pooya Molavi, Alireza Tahbaz-Salehi, and Ali Jadbabaie.  A theory of non-bayesian social learning.  Econometrica, 86(2):445–490, 2018.
  * [37] Enrico Moretti.  Social learning and peer effects in consumption: Evidence from movie sales.  The Review of Economic Studies, 78(1):356–393, 2011.
  * [38] Elchanan Mossel, Allan Sly, and Omer Tamuz.  Strategic learning and the topology of social networks.  Econometrica, 83(5):1755–1794, 2015.
  * [39] Manuel Mueller-Frank.  Manipulating opinions in social networks.  Available at SSRN 3080219, 2018.
  * [40] Manuel Mueller-Frank and Claudia Neri.  A general analysis of boundedly rational learning in social networks.  Forthcoming at Theoretical Economics, 2020.
  * [41] Kaivan Munshi.  Social learning in a heterogeneous population: technology diffusion in the indian green revolution.  Journal of development Economics, 73(1):185–213, 2004.
  * [42] Lones Smith and Peter Sørensen.  Pathological outcomes of observational learning.  Econometrica, 68(2):371–398, 2000.
  * [43] Lones Smith and Peter Sorensen.  Rational social learning with random sampling.  Available at SSRN 1138095, 2008.