Learning in a Small/Big World∗
Benson Tsz Kin Leung†
April 2, 2024

Abstract
Complexity and limited ability affect how we learn and make decisions under uncertainty. Using finite automata to model belief formation, this paper studies the characteristics of optimal learning behavior in small and big worlds, where the complexity ofthe environment is low and high, respectively, relative to the cognitive ability of the decision maker. Optimal behavior is well approximated by the Bayesian benchmark in verysmall worlds but is more different as the worlds get bigger. In addition, in big worlds, theoptimal learning behavior could exhibit a wide range of well-documented non-Bayesianlearning behavior, including heuristics, correlation neglect, persistent over-confidence,
inattentive learning, and other behaviors of model simplification or misspecification.
These results establish a clear and testable relationship among the prominence of nonBayesian learning behavior, complexity, and cognitive ability.
Keywords: Learning, Bounded Memory, Bayesian, Complexity, Cognitive Ability
JEL codes: D83, D91∗

I am grateful to Tilman Börgers, Jacques Crémer, Matthew Elliott, Renato Gomes, Philippe Jehiel,
Hamid Sabourian, Mikhael Safronov, Larry Samuelson, Tak-Yuen Wong, anonymous referees, and the audiences at various seminars and conferences for their insightful discussions and comments.
†
Hong Kong Baptist University. Email: btkleung@hkbu.edu.hk, ltkbenson@gmail.com.

1

1

Introduction

Savage (1972) argues that Bayesian decision theory applies only to small world problems
but not big world problems, where the latter refers to scenarios where it is difficult to form
prior beliefs on states and signal structures or construct the state space. This paper offers
an alternative distinction between “small” and “big” worlds based on the complexity of theinference problem relative to the cognitive ability of individuals. “Small” worlds refers toinference problems that are simple relative to the cognitive ability of individuals. Therefore,by comparing learning behaviors in the “small” and “big” worlds, this paper sheds light on
the heterogeneity of learning behaviors across decision problems and individuals. Analyzinga theoretical model with finite automata, I show that a wide range of ignorance behavior, including the use of heuristics (Kahneman et al. (1982)), correlation neglect (Enke andZimmermann (2019)), persistent over-confidence (Hoffman and Burks (2017)), inattentive
learning (Graeber (2019)), arises as optimal learning behaviors in face of complexity in big
worlds, but not in small worlds. Thus, these “biased” learning behaviors are more prominent
in more complex problems and among individuals with lower cognitive ability.
More specifically, I consider a decision maker (DM) who tries to learn the true state of
the world from a finite state space, where the number of possible states N measures thecomplexity of the inference problem. In each period t = 1, · · · , ∞, the DM guesses the true
state of the world and gets a higher utility if he is correct. After making a guess, he receives
a signal and updates his belief. To model limited cognitive ability, I assume that the DM’s
belief is confined to an M sized automaton that captures limited cognitive ability, as in theseminal work of Hellman and Cover (1970) and Wilson (2014).1 In period 0, he starts in
one of the M memory states, makes his guess given his memory state based on a decision
rule, receives a signal, transits to another based on a transition rule, and finally enters thenext period. In contrast to the Bayesian model, the DM has a coarser idea of the likelihoodof different states of the world, and the coarseness decreases in M . Thus M measures theNis small, otherwise thecognitive ability of the DM. I define small worlds as cases where M
decision problem is a big world: Whether a problem is a small or big world depends on therelative complexity of the world with respect to the individual’s cognitive ability.2
To shed light on how complexity affects learning, I compare the characteristics of the
optimal updating mechanisms that maximize the asymptotic utility of the DM in small andbig worlds. The results are summarized in Table 1. First, I analyze how the individual’s
decisions differ from the Bayesian benchmark in small and big worlds. This sheds light onunder what circumstances the Bayesian model serves as a good approximation of decisionmaking under uncertainty. I show that in small worlds, asymptotic behavior is close to1

See also Compte and Postlewaite (2012), Monte and Said (2014), Basu and Chatterjee (2015), Chauvin(2019) Chatterjee and Sabourian (2020) in the economic literature that model belief updating and the aversionof complexity with finite automata. See also Oprea (2020) and Banovetz and Ryan (2020) for experimental
evidence.
2
Note that if the individual tracks his belief not with a finite automaton but with a real number statistic,
the cardinality of the belief statistics is much larger than N , and the model collapses to a Bayesian model.

2Small Worlds:
low complexity relativeN
to cognitive ability M

Big Worlds:
high complexity relativeN
to cognitive ability M

Is behavior close to Bayesian?

Yes

No

Could ignorance in learning be “optimal”?

No

Yes

Could disagreement be persistent?

No

Yes

Table 1: Differences in learning behaviors in small/big worlds
Bayesian. In particular, the DM almost always makes the same guess as a Bayesian individualNN
as M→ 0. In contrast, when the world is bigger, i.e., when M
increases, the DM makes more
mistakes and his behavior becomes more different from Bayesian.
The second, and main, result of this paper shows that in big worlds, as the DM facesa trade-off in allocating his scarce cognitive resources, i.e., the M memory states, it could
be optimal to ignore some states and focus learning on a subset of states. In contrast, such“ignorant” behavior is never optimal in small worlds. This shows a relationship between
complexity and ignorance in learning, which nests different well-documented learning biases,
including heuristics, correlation neglect, persistent over-confidence, inattentive learning, andother model simplification and misspecification behaviors.
To see how “ignorance” captures different biases, consider the example of persistent overconfidence (Hoffman and Burks (2017), Heidhues et al. (2018)). Suppose that the state of
the world comprises the DM’s and his teammate’s ability, where both could be high or low,
and the DM observes team performances as signals. In the current setting, persistent overconfidence occurs when the DM never guesses the states where his ability is low, behaves as
if he always believes he has high ability, and only updates his belief about his teammate’s
ability, even after observing a sequence of bad team performance. Similarly, for correlationneglect (Enke and Zimmermann (2019)), consider that the state of the world comprises astrong or weak stock market and positive, no, or negative correlation among data. Correlationneglect is captured in the current setting as the individual ignores the possible correlation
among data and only updates his belief about the stock market as if he always believes there
is no correlation.
This paper shows that such ignorance behavior is optimal only when relative complexity islarge (big worlds), especially when the ignored state is a priori unlikely, or when information
supporting that state is weak. For example, if the DM is confident about his ability, hefocuses on learning his teammate’s ability; if it is difficult to distinguish state with andwithout correlation, the DM ignores the possibility that data are correlated. Importantly,
I also show that even if the states and information structures are symmetric, ignorance is
optimal in environments where it is difficult to learn, e.g., when signals are noisy or when
the state space is large. It echoes that complexity, or difficulty in learning, drives ignorance,even when there are no a priori reasons to ignore any specific states of the world.
3

Last, I analyze whether disagreements are persistent in small and big worlds. As asymptotic behaviors are close to the Bayesian model in small worlds, intuitively individuals would
always eventually agree with each other and make the same guesses. In contrast, in big
worlds, because individuals with different prior beliefs and/or cognitive ability adopt different optimal learning mechanisms and could ignore different states, they could disagreewith each other with probability 1 even after receiving the same infinite sequence of public
information. For example, after observing a large sequence of bad team performance, two
individuals with persistent over-confidence would disagree on the assessment of their abilities: they ignore the states where their ability is low and attribute the bad performanceto the other person. Moreover, I show a novel driving force of disagreement: even when
two individuals have the same prior beliefs and observe the same infinite sequence of public
signals with no uncertainties in signal structures, they could eventually disagree with eachother when they have different levels of cognitive ability M .
This paper is organized as follows. In the next section, I briefly discuss how this paper
relates to the literature. Section 3 presents the model. I analyze the optimal learning behavior
in small and big worlds in Sections 4. In Section 5, I conclude by presenting a discussion of
the results. The proofs and omitted results are presented in the Appendix, and the extensions
are presented in the online Appendix.3

2

Literature

In this section, I discuss the existing literature and the contribution of this paper.
First, this paper is obviously related to the literature using finite automata to model
learning with aversion to complexity (Hellman and Cover (1970), Börgers and Morales (2004),
Compte and Postlewaite (2012), Wilson (2014), Chatterjee and Hu (2021), etc.).4 While theliterature discusses the impact of complexity on learning, they do not analyze how different
levels of complexity and cognitive ability affect learning and the prominence of learning biases.
Moreover, the studies above focus on binary state space because of technical difficulties.
While this paper does not “fully” characterize the optimal automaton, the results shed lighton its characteristics when N > 2. In particular, the results about ignorance suggest thatfor larger state space, unlike when N = 2, ignorance is a crucial feature even when the states
are a priori the same.
Sims (2003) and Matějka and McKay (2015) study the implication of rational inattention
and show that it explains sticky prices in the market and micro-founds the multinomial
logit choice model, Steiner and Stewart (2016) shows that an optimal response to noises inperceiving the details of lotteries leads to probability weighting in prospect theory (Kahneman
and Tversky (1979)), and Jehiel and Steiner (2020) and Leung (2020) show that a capacity
3

The online Appendix could be found on https://sites.google.com/site/ltkbenson/research.
Also see Chatterjee and Sabourian (2020) for a review, and Oprea (2020) and Banovetz and Ryan (2020)
for empirical evidence.
44

constraint on the number of signals that individuals could update their belief with drives
confirmation bias and other biases in belief formation. This paper also contributes to a
growing literature that explains behavioral anomalies as optimal/efficient strategies in lightof limited cognitive ability. Steiner and Stewart (2016) shows that an optimal response tonoises in perceiving the details of lotteries leads to probability weighting in prospect theory,
and Jehiel and Steiner (2020) and Leung (2020) show that a capacity constraint on thenumber of signals that individuals could update their belief with drives confirmation bias. In
contrast, this paper explains a larger class of biases under the same framework and illustrates
a relationship between their prominence and the level of complexity.
Similar to Section 4.2 in this paper, Caplin et al. (2019) present conditions where theDM would ignore some actions in a rational inattention setting. Unlike this paper, they
only present conditions depending on the prior belief and the utility matrix, but not thelevel of complexity nor the informativeness of signals. Moreover, in contrast to Caplin et al.
(2019), in this paper, I show that it could be optimal to ignore some states even in symmetricenvironments where prior belief is uniform and the utility matrix is symmetric.
Last, this paper’s results on asymptotic disagreement contribute to the large literaturethat explains the phenomenon. In the existing literature, asymptotic disagreement is driven
by differences in signal distributions across states or differences in learning mechanisms(Morris (1994), Mailath and Samuelson (2020), Gilboa, Samuelson and Schmeidler (2020)),
the lack of identification or uncertainty in signal distributions (Acemoglu, Chernozhukov
and Yildiz (2016)), confirmation bias (Rabin and Schrag (1999)), or model misspecification(Freedman (1963, 1965), Berk (1966)). Differently, this paper looks into the connection
between limited ability and disagreement, and shows when asymptotic disagreement could
arise and when it will not occur, depending on the relative complexity of the inference problem. Moreover, I show a novel machanism that disagreement could arise solely because ofdifferences in cognitive abilities.

3

Model

I consider a world with N possible true states, i.e., ω ∈ Ω = {1, 2, · · · , N }, and a decisionmaker (DM), wherein each period t = 1, · · · , ∞, the DM tries to guess what the true state is.
Formally, in each period t, the DM takes an action at ∈ A = Ω and gets utility u(a, ω) ∈ R
where a = ω is the unique maximizer of u(a, ω).5 The DM does not observe his utility
after taking action; thus, u(a, ω) is best interpreted as an intrinsic utility of being correct.
Otherwise, the problem becomes trivial as the DM will experiment and learn perfectly thePNω
true state after observing the utility. The prior belief of the DM is (pω )Nω=1 where
ω=1 p = 1
and pω > 0 for all ω ∈ Ω.
5This assumption rules out “safe” actions that are not maximizers in any state but yield good payoffs in
multiple states. In the next section, I discuss how complexity affects the incentive of choosing these “safe”
actions.

5

In each period after taking an action, the DM receives a signal st ∈ S that is independently
drawn across different periods from a continuous distribution with p.d.f. f ω in state ω.6,7 I
assume that no signal perfectly rules out any states of the world: there exists ς > 0 such thatf ω (s)> ς for all ω, ω 0 ∈ Ω.
0s∈S f ω (s)
inf

(1)

Without loss of generality, no pairs of signal structures are the same, i.e., there are no0ω and ω 0 6= ω such that f ω (s) = f ω (s) for (almost) all s ∈ S. This implies that states are
identifiable, and a Bayesian learns the true state perfectly as t → ∞. In contrast, I focus on
the bounded memory setting I now describe.
The DM is subject to a memory constraint such that he can only update his belief using
an M memory states automaton. In each period, his belief is represented by a memory
state mt ∈ {1, 2, · · · , M }. An updating mechanism specifies an initial state m1 ∈ 4M , atransition function between the M memory states given a signal s ∈ S, which is denoted as
T : M × S → 4M , and a decision rule d : M → A.8 The set of memory states where the
DM chooses action ω is denoted as M ω . In each period t, as illustrated in Figure 1, the DM
starts with some memory state mt , take action at ∼ d(mt ), receives a signal st , and transit tomemory state mt+1 ∼ T (mt , st ).9 This paper analyzes the asymptotic learning of the DM,
i.e., the DM aims to choose an updating mechanism at period 0 that maximizes his expectedlong run per-period utility:10"
lim Em1 ,T ,d

T →∞

#
T
1X
u(at , ω) .
T t=1

Given state ω ∈ Ω, the sequence mt , together with some specified initial memory state
m1 , forms a Markov chain. Denote µωm as the long-run proportion of time that the DM isin memory state m when the true state of the world is ω, and Qω as the matrix of transiRtion probabilities, i.e., Qω = [ s Pr{T (m, s) = m0 }f ω (s) ds]mm0 . By the Birkhoff–Khinchintheorem, the distribution µω = (µω1 , µω2 , · · · , µωM )T solves the following system of equations:µω = (µω )T Qω ,(2)

By the Brouwer fixed-point theorem, a solution always exists. Moreover, when there are
multiple solutions, it implies that the Markov Chain is reducible, and the long-run distribution
6

The order, i.e., whether the DM receives a signal before or after taking an action in each period, does
not affect the result.
7
To ease exposition, I assume that signals follow a continuous distribution, but the results hold with moregeneral probability measures.
8
Note that without loss of generality, I restrict attention to deterministic decision rules unless statedotherwise.
9
Switching between multiple M memory state automatons requires more than M memory states, and thecurrent setting allows switching between smaller automatons, as illustrated in Online Appendix A.
10
An alternative is to maximize the discounted sum of utility as in Wilson (2014). As shown in Online
Appendix B, the results in this paper hold qualitatively when the discount factor is close to 1.

6

Starts at memory state mt .

Takes action at ∼ d(mt ).

Receives signal st .

Transits to memory state
mt+1 ∼ T (mt , st ).

Figure 1: Timeline at period t given an updating mechanism (T , d).
is uniquely pinned down by the initial memory state.
The asymptotic utility, or the long-run per-period utility, of an updating mechanism(m1 , T , d) is equal to:
U (m1 , T , d) =

N
X

"
p

M
X

ω

ω=1

!#
u (d(m), ω) µωm

(3)

m=1

and the asymptotic utility loss is equal to:L(m1 , T , d) =NX[pω u(ω, ω)] − U (m1 , T , d).

(4)

ω=1

The DM maximizes the asymptotic utility or, equivalently, minimizes the asymptotic utility
loss. I mostly refer the optimal design of the updating mechanism as the minimization of L.
In general, with similar arguments in Hellman and Cover (1970), an optimal mechanism may
not exist. Therefore, the rest of the paper focuses on -optimal updating mechanisms thatare defined as follows. DefineL∗M ≡ inf L(m1 , T , d).
m1 ,T ,dAn updating mechanism (m1 , T , d) is -optimal if and only if L(m1 , T , d) ≤ L∗M + .
PωThroughout the paper, I focus on the more interesting case where L∗M < mina N
ω=1 p [u(ω, ω) − u(a, ω)],such that learning strictly improves utility.
In the next section, I compare the characteristics of the -optimal updating mechanisms
in small and big worlds. Roughly speaking, N represents how complicated the world is, andM represents the cognitive resources/ability of the DM. This gives a natural definition ofsmall and big worlds based on relative (or perceived) complexity: an inference problem is aN
is small, and is a big world otherwise. Throughout the paper, unlesssmall world when the M
stated otherwise, I focus on the more interesting case that M ≥ N .

7

4

Results

4.1

Is behavior close to Bayesian?

In the current setup, a Bayesian individual (almost) perfectly learns the true state of theworld asymptotically and achieve asymptotic utility loss close to 0. Thus, we could interpretL∗M as the distance between the DM’s behavior and that of a Bayesian individual. The
Proposition presents the formal results and the paragraph after offers interpretations. Forease of exposition, I denote uω ≡ (uω (a, ω))N
a=1 .
Proposition 1. We have the following results regarding L∗M :(i) L∗M strictly decreases in M ;(ii) For each N and (uω , pω , f ω )N
ω=1 , there exists some constant r < 1 and K > 0 such that
M −1
L∗M < Krb N c (also implies limM →∞ L∗M = 0);(iii) For each N and (f ω )N
ω=1 , there exists a sequence of updating mechanism (m1 , TM , dM )such that limM →∞ L(m1 , TM , dM ) = 0 for all (uω , pω )N
ω=1 .
The behavioral implications of Proposition 1 are as follows: (i) shows that the DM’s
behavior gets closer to that of a Bayesian individual as M increases, or equivalently asNNdecreases, i.e., as the world gets smaller.11 (ii) shows that as M
converges to 0, theM
DM’s asymptotic decisions are well-approximated by Bayesian, and shows a higher relativecomplexity of the world decreases the convergence rate. (iii) demonstrates the robustness ofNis close to 0, as no knowledge of prior or theperfect learning in very small worlds where M
12utility matrix is needed.
Before I describe the the proof, it is important to first define confirmatory signals for eachstate ω, especially because the model goes beyond the binary state setting. When N = 2, theconfirmatory signals for state 1 are S1 = {s ∈ S : f 1 (s) > f 2 (s)}. The set of confirmatory
signals for both states defined in this way is non-empty, and because of the binary nature,
it is also more likely that the DM receives confirmatory signals for the correct state thanRR
for the wrong state, i.e., s∈S1 f 1 (s) ds > s∈S2 f 1 (s) ds. However, these observations do not
generalize to N > 2. For example, consider the following signal structure with N = 3 andS = {s1 , s2 , s3 }:f 1 (s1 ) = 0.5, f 1 (s2 ) = 0.4, f 1 (s3 ) = 0.1;
f 2 (s1 ) = 0.4, f 2 (s2 ) = 0.2, f 2 (s3 ) = 0.4;
f 3 (s1 ) = 0.1, f 3 (s2 ) = 0.4, f 3 (s3 ) = 0.5.
N
In the main text, N is fixed, and thus a world is small enough, i.e., M
is small enough, if and only if
M is large enough. In online Appendix C, I analyze the -optimal updating mechanism of a sequence ofinference problems where both N, M → ∞, and show that the behavioral implications depend on the limit
NNof M
instead of M . In particular, the results of small worlds hold qualitatively when the limit of M
is 0 andN
the results of big worlds hold qualitatively when the limit of M is strictly greater than 0.
12
I thank an anonymous referee for pointing this out.
11

8

0

If I define signals supporting state ω as Sω = {s ∈ S : ω = arg maxω0 f ω (s)}, as analogue
to N = 2, there won’t be any signals that support state 2. One could also show that no
deterministic definitions of confirmatory signals will guarantee that the DM receives signals
supporting the correct state more likely than signals supporting a wrong state in every stateof the world. Instead, I stochastically label signals as confirmatory signals for each state. A
signal s is labeled as a confirmatory signal for state ω with probability proportional to f ω (s),with appropriate normalization such that each signal is labeled as a confirmatory signal forone of the state with probability less than 1, and is labeled as a confirmatory signal for nostate with complementary probability. In this example, s2 is labeled as a confirmatory signal
for state 1 with probability 0.4, a confirmatory signal for state 2 with probability 0.2 and aconfirmatory signal for state 3 with probability 0.4. s1 and s3 are labeled analogously. Thus,in state 2, the DM receives a signal supporting state 2 with probability 0.42 +0.22 +0.42 = 0.36,
a signal supporting state 1 with probability 0.5 ∗ 0.4 + 0.4 ∗ 0.2 + 0.1 ∗ 0.4 = 0.32, and a signal
supporting 3 with probability 0.1 ∗ 0.4 + 0.4 ∗ 0.2 + 0.5 ∗ 0.4 = 0.32. Similarly, in state 1, theDM receives a confirmatory signal for state 1 more likely than a signal supporting state 2(or 3), and in state 3, the DM receives a confirmatory signal for state 3 more likely than asignal supporting state 1 (or 2). In all ω, the DM receives a signal supporting the correctstate more likely than a signal supporting a wrong state.
Now using the definition of confirmatory signals, I briefly describe the proof of (i) of
Proposition 1. Consider an -optimal mechanism (m1 , TM , dM ) at memory size M in which
M 1 , M 2 6= ∅ and M 1 ∪ M 2 = M , i.e., the DM either chooses action 1 or action 2, andsuppose the memory size increases to M + 1. The following construction strictly improves
the asymptotic utility of (m1 , TM , dM ). First, keep m1 and d unchanged (for memory state
1, · · · , M ). Second, add the following transition to (m1 , TM , dM ): the DM transits to M + 1
with some probability δ1 if he was at a memory state in M 1 and received a signal supportingstate 1, and transits out of M + 1 to, randomly, one of the memory states in M 1 with someprobability δ2 if he received a signal supporting state 2. Last, the DM chooses action 1 inmemory state M + 1. The proof involves choosing the appropriate δ1 and δ2 such that the
DM chooses action 1 in state 1 with the same probability as before. As it is less likely thatthe DM will transit to memory state M + 1 in state 2, he chooses action 1 less likely in state 2
and hence strictly improves his asymptotic utility.
Now to prove (ii) and (iii) of Proposition 1, I construct a simple updating mechanism,
illustrated in Figure 2. The mechanism tracks only the DM’s favorable action and the corresponding confidence level over time. At any period t, the DM believes one of the N
actions or no action is favorable, while his confidence level of his favorable action, if he has
one, is an integer between 1 and b MN−1 c. The memory states could thus be represented bymt ∈ {0}∪{1, · · · , N }×{1, · · · , b MN−1 c} where memory state 0 stands for no favorable action.
The decision rule is such that he takes the favorable action if he has one, and takes action 1
if he does not have a favorable action.13 The transition rule is described as follows.
13

The decision rule when the DM does not have a favorable action does not affect the proof and result.

9

action 1

action 2

action N − 1

action N

..
.

..
.

confidence
level

···
..
.

..
.

..
.

with lengthb MN−1 c···0

Figure 2: A simple updating mechanism that achieves perfect learning in small worlds for allω N
N , (pω )Nω=1 and (f )ω=1 .
First, the DM starts with no favorable action.14 If he receives a confirmatory signal for astate ω, he changes his favorable action to action ω with a confidence level 1; if he receives
signals that is not confirmatory for any states, he stays in the same memory state 0 in which
he has no favorable action. Second, suppose at some period t the DM’s favorable action isaction ω with confidence level k. If he receives a confirmatory signal for state ω, he revises
his confidence level upwards to k + 1 if k is not already at the maximum b MN−1 c, and stays inthe same memory state if k is at the maximum. Third, if he receives a confirmatory signal
for state ω 0 6= ω, he revises his confidence level downwards to k − 1 with probability 1δ < 1
if k ≥ 2, transits to the memory state 0 with no favorable action with probability 1δ < 1 if
k = 1, and stays in the same memory state with probability 1 − 1δ . Lastly, if he receives
signals that are not supporting any states, he stays in his current memory state with his
favorable action and confidence level unchanged. This simple updating mechanism could
thus be interpreted as an algorithm that tracks the confidence level of only one state/action
at a time, with underreaction to belief-challenging signals (captured by 1δ < 1).
The proof involves choosing a sufficiently large δ such that it is more likely for the DM
to adjust his confidence level upwards than to adjust it downwards. This ensures enough“exploitation” that the DM doesn’t switch between actions too often. Crucially, when M
Nincreases, the maximum number of confirmatory signals the DM can “store” increases, andthe more likely the DM will be at the correct branch choosing the correct action. WhenM
→ ∞, the DM almost surely learns perfectly the true state as t → ∞.
N
Here I point out several noteworthy implications of the analysis. First, as argued above,
unlike Cover (1969) and Wilson (2014) with N = 2, when N > 2, it is in general necessary
14

The starting memory state has no impact on the long-run distribution over the memory states and does
not affect the asymptotic payoff.

10to define confirmatory signals for each state stochastically. Thus, a stochastic transition
matrix is generally necessary. It implies that upon receiving the same signal, the individual
will sometimes regard it as supporting one state, and sometimes regard it as supporting a
different state. It resembles empirical evidence that documents heterogeneous interpretationsof same pieces of information, especially when the piece of information is imprecise (Gaines
et al. (2007)).
Second, as shown in Proposition 1(iii), the simple updating mechanism described in thissection approximates perfect learning for all N and (uω , pω , f ω )N
ω=1 in small worlds when M
ω N
is large. Therefore, no knowledge of prior belief (pω )Nω=1 and utility matrices (u )ω=1 arerequired. It is in particularly consistent to the result that behavior is close to Bayesian insmall worlds as the simple updating mechanism is parsimonious and easy to implement.
Third, perfect learning with the simple updating mechanism in very small worlds is robust
to “implementation errors”, as shown in the online Appendix E. Roughly speaking, I assume
that the DM mistakenly transits to a neighboring memory state with some probability γ
in each period regardless of the signal realization s. Such local mistakes could be inducedby mistakes in the perception of signals or imperfect tracking (local fluctuation) of memory
states. Online Appendix E shows that Proposition 1(iii) hold for all γ ∈ [0, 1), further
strengthening the result of perfect learning in very small words.

4.2

Is ignorance optimal?

In this subsection, I present the main results on whether ignorance behavior is optimal insmall and big worlds. Ignorance is formally defined as follows: an updating mechanism
ignores state ω if the DM almost never chooses action ω no matter what the true state is,
i.e.,
#
"P
T
1
a
=ω
t
t=1
= 0 for all ω 0 .
lim Eω0 ,m1 T ,dT →∞T
An updating mechanism is ignorant if it ignores some state. As argued in the introduction, ignorance nests a large set of behavioral biases that depart from the Bayesian model. Note thatPω
given the assumption that information strictly improves utility, i.e., L∗M < mina N
ω= p [u(ω, ω)−u(a, ω)], when N = 2, no ignorant updating mechanism is -optimal when  is sufficiently
small.I present the formal results on ignorance in the following Proposition when N > 2, andoffer interpretations in the next paragraph.
Proposition 2. Regarding ignorance behavior:
000(i) For all N > 2 and M , there exists some (uω , pω , f ω )N¯ > 0 such that allω 0 =1 and ¯-optimal updating mechanism ignores state ω for some ω;000(ii) In contrast, take N > 2 and (uω , pω , f ω )N
ω 0 =1 , when M is big enough, all -optimal
update mechanisms are non-ignorant for some small enough  .

11First, (i) shows that generally, there exist decision environments such that ignorance is
optimal. As M
is finite, the DM cannot allocate infinite cognitive resources to every state ofN
the world. The DM is bound to make mistakes and faces trades-off between the probabilityof mistakes in different states of the world. (i) implies that it could be optimal for the DM
to ignore some states altogether to improve learning in other states. On the other hand,(ii) shows that the set of decision environments where ignorance is optimal vanishes as M
grows large, or equivalently as the world gets smaller. As M increases, the DM can allocate
many more memory states to each action. Trade-off between learning in different states isNconverges to 0, the DM learns almost perfectly for all
less important. In particular, when M
states of the world, and has no incentive to ignore any of the states. Proposition 2 also implies
Nis small, the DM will not choose actions that are “safe” but are sub-optimal inthat when M
every state of the world.
Elaborating on Proposition 2, the following Corollary presents three conditions as examples on when such ignorance happens.
Corollary 1. When N > 2, u(ω 0 , ω) = 0 for all ω and ω 0 6= ω, there exists some thresholdξp , ξu > 0 and F > 1 such that if(i) pω < ξp , or(ii) min 00u(ω,ω)< ξu , oru(ω 00 ,ω 00 )
ω 6=ω

ω

ω0

(s)
(iii) sups ffω0(s)× sups ff ω (s)≤ F for some ω 0 6= ω,(s)all ¯-optimal mechanisms are ignorant for some ¯ > 0.
Intuitively, (i) shows that when the prior probability of a state is low, the DM would
rather ignore that state and allocate cognitive resources to learn other states of the world.
For example, an individual who is confident about his ability would not update his belief in
his but only his teammates’ ability, exhibiting persistent overconfidence. Similarly, (ii) showsthat when a state is relatively unimportant that the utility of guessing correctly that stateis relatively small, the DM would ignore that state to focus on learning others. Lastly, (iii)
shows that when it is difficult to distinguish states ω and ω 0 , the DM ignores one of the twostates. That is, the DM ignores states that are difficult to identify. To see the intuition,0
imagine two states ω and ω 0 where f ω (s) ≈ f ω (s) for all s, and that state ω is a priori lessfavorable than state ω 0 . The DM has to receive a large number of signals that support ωagainst ω 0 such that he prefers to take action ω instead of ω 0 . To use many memory states
to record signals supporting ω and still be (almost) unsure about choosing ω over ω 0 is aninefficient use of memory states. Instead, saving those numerous cognitive resources to learn
other states of the world, for example, by recording signal supporting ω 0 against some ω 00
and vice versa, he can improve his utility in the other states (ω and ω 00 ) and will be better
off. Condition (iii) is particularly applicable in correlation neglect: when it is difficult todistinguish positive or no correlation in the data, and if positive correlation is slightly less12

likely, the DM ignores the possibility of positive correlation and focuses on learning otherstates of the world.

Optimal ignorance in symmetric environments
Proposition 2 and Corollary 1 shows that ignorance is optimal in asymmetric environments
when Mis small, but ignorance is also optimal in some symmetric environments. I considerN
the following setup where states and actions are ex-ante identical: There are N possibleω ω
signal realizations, i.e., S = {s1 , · · · , sN }, and pω , u(ω, ω), u(ω 0 , ω), FFω (s(sω0)) > 1 are invariant
across all ω and ω 0 6= ω. Note that for computational simplicity, I consider a discrete signal
structure in this Subsection.
The result is shown in Figure 5, where N = M ∈ {4, 6, 8}. The y-axis is the asymptotic
utility of a symmetric updating mechanism that ignores half of the states, illustrated in
Figure 4, minus that of the best non-ignorant updating mechanism, illustrated in Figure 3.
Thus, if the y-axis is above 0, all -optimal mechanisms must be ignorant for small enoughω ω
. The x-axis is the informativeness of the signal structure, i.e., FFω (s(sω0)) where ω 0 6= ω. The
analysis and the code is presented in the Appendix B. Figure 5 shows that ignorance is
optimal when the informativeness is smaller than some threshold, and the threshold is bigger
when N is bigger. In other words, ignorant is optimal when it is difficult to learn, whichstrengthens the result of Proposition 2.15 The result also contrasts with the setting when
N = 2, in which the optimal updating mechanism is symmetric and non-ignorant in such
symmetric environments.
The intuition of the result is as follows. To consider all states, the DM allocates one
memory state to each action. It is thus easy for the DM to alternate between different
actions and unavoidably make mistakes. Put differently, the updating mechanism is “noisy”.
This is especially true when N is large as one memory state constitutes only a small part of theautomaton or when signals are very noisy. If, in contrast, the DM ignores half of the actions,
he allocates two memory states to each of the actions that he considers and he switches
between actions less frequently. This improves his decision making among the smaller set ofωω 0
is small, the improvement outweighsstates that he considers. When N is large or when l
the loss he incurs among the states that he ignores, because the loss is small to begin with,i.e., the asymptotic utility of a “noisy” updating mechanism that considers all states is small.

4.3

Is disagreement persistent?

Lastly, I turn to the question of whether disagreement could persist asymptotically in small
and big worlds. Consider two individuals A and B who have different utility matrices (uωA )N
ω=1
ω N
ω N
and (uωB )N,
and/ordifferent
prior
beliefs
(p
)
and
(p
)
,
and/ordifferent
abilities
ω=1
A ω=1
B ω=1
15

The magnitude of the thresholds also suggests that ignorance is not an extreme event. For example,
when N = 3, ignorance is better when the likelihood ratio of the signals is smaller than (around) 3.

13

δ23 × S3

δ41 × S1
δ21 × S1
1

2
δ12 × S2

δ13 × S3

δ31 × S1

δ24 × S4

δ42 × S2

δ43 × S3
3

4
δ34 × S4

δ14 × S4

δ32 × S2

Figure 3: The optimal non-ignorant updating mechanism that considers all states, with
N = M = 4. The number in the node denotes the action that the DM takes when he isthis memory state. Moreover, in memory state ω, and upon receiving a signal that supports
state ω 0 6= ω, the DM transits to memory state ω 0 with probability δωω0 < 1 and stays in his
current memory state otherwise.

δ × S1S1
1

2
δ × S2

δ × S1
3

δ × S2

4
S2

Figure 4: An example of an updating mechanism that ignore two states, with N = M = 4.
The DM takes action 1 in memory states 1 and 2, and takes action 2 in memory states 3 and4. In memory state 3, if the DM receives a signal supporting state 2, he transits to memory
state 4; if the DM receives a signal supporting state 1, he transits to memory state 2 withprobability δ and stays in his current memory state otherwise. In memory state 4, if theDM receives a signal state 1, he transits to memory state 3 with probability δ and stays in
his current memory state otherwise. The transition function in memory state 1 and 2 aredefined accordingly. δ is chosen to be close to 0 to maximize the asymptotic utility.

14

Figure 5: The y-axis is the (estimated) asymptotic utility of the ignorant mechanism illustrated in Figure 4 minus that of the non-ignorant mechanism illustrated in Figure 3, whileωω 0
the x-axis is l . The ignorant mechanism outperforms the non-ignorant mechanism whenωω 0
lis smaller than some threshold. Moreover, the threshold is larger as N increases from 4
to 6, and from 6 to 8.
16
ω N
Their updating mechanisms,
of information acquisition captured by (fAω )N
ω=1 and (fB )ω=1 .
(mA1 , FA , dA ) and (mB1 , FB , dB ), induce a (random) sequence of actions over time. To define
disagreement, I sample one action from each individual A and B and define the “disagreement
between (mA1 , FA , dA ) and (mB1 , FB , dB ) in state ω” as the probability that the two sampledactions are different.17 For a given , the disagreement between individual A and B in state ωis the supremum of disagreement between all pairs of -optimal updating mechanisms in state
ω. Lastly, the two individuals almost always disagree (resp. agree) with each other for a given if their disagreement equals 1 (resp. 0) in all states.

Corollary 2. Regarding disagreement:ωωω N(i) For all N > 2, MA and MB , there exists some (uωA , pωA , fAω )N
ω=1 and (uB , pB , fB )ω=1 suchthat individual A and B almost always disagree as  → 0.
ωω
ω N(ii) In contrast, take (uωA , pωA , fAω )N
ω=1 and (uB , pB , fB )ω=1 , there exists some K1 , K2 > 0and
1 such that the disagreement
between individual A and B is bounded above by r<
min{MA ,MB }−1
b
c
N
K1 r
+ K2  in all states.
16

For example, individual A could receive noisier signals than individual B, i.e., fAω = γ + (1 − γ)fBω forsome γ ∈ (0, 1); or individual A could have different learning advantages in identifying some states better but
0
0
00
000
other states worse than individual B, i.e., sups fAω (s)/fAω (s) > sups fBω (s)/fBω (s) but sups fAω (s)/fAω (s) <00000sups fBω (s)/fBω (s) for some ω, ω 0 , ω 00 , ω 000 .
17
For example, if both individuals alternate between action 1 and 2, their disagreement is 21 . An alternativeB
definition is to measure the proportion of time t where aA
t 6= at : If both individuals alternate between action 1
and 2, their disagreement is 0 if they both start with the same action and is 1 otherwise. Corollary 2 (i)holds with this definition, while the limit result of (ii) holds, i.e., disagreement approaches 0 as M goes toinfinite.

15

(i) shows that in general there exists decision environments such that disagreement almostN
always happens and is persistent. However, as shown in (ii), when the world gets smaller ( Mgoes to 0), such set of decision environments with disagreement vanishes: different individuals with different prior beliefs and/or information acquisition abilities who adopt (almost)
optimal updating mechanisms are bound to agree with each other.
The intuition of the result is as follows: as different individuals with different prior beliefs
and utilities would adopt different updating mechanisms, they could focus their learning on
different subsets of states of the world when they have limited memory. In particular, consider
an example with N = 4, if individual A ignores state 1 and 2 and individual B ignores state 3
and 4, they will never choose the same action and thus disagree with certainty.18 However, asNgoes to 0, decisions gets closer to Bayesian, and the two individuals almost always chooseM
the same actions.
The following result further shows that disagreement could be driven solely by differencesin cognitive ability. For simplicity, I allow M to be smaller than N in the following result.
ω Nωω
Corollary 3. There exists MA 6= MB and (uωA , pωA , fAω )N
ω=1 = (uB , pB , fB )ω=1 such that individual A and B almost always disagree as  → 0.

I prove the Corollary with an example where MA = 1 < MB = 2, N = 3, and u(ω, ω) =1 > u(ω, ω 0 ) = 0 for all ω and ω 0 6= ω. Importantly, state 1 is a priori more likely than state 2
and 3, while it is easier to distinguish state 2 and 3 than to distinguish state 1 and state 2,
or state 1 and state 3.19 In this example, individual A always chooses action 1 as he does nothave sufficient cognitive resources to learn. On the other hand, if p1 is not too large comparedto p2 and p3 , individual B ignores state 1. He takes advantages of the informative information
structure to distinguish state 2 and 3, such that he can be confident that he doesn’t takeaction 2 in state 3 and vice versa. Thus, as long as state 1 is not a priori too likely, ignoring
it would yields a higher utility.20 To sum up, as individual A has a lower cognitive ability,
his learning and actions would depend a lot on the prior belief, while individual B has theability to take advantage of the information structure.21 As a result, they adopt different
updating mechanisms which leads to disagreement.
18
Note that the result of disagreement continues to hold in a framework where individuals observe eachother’s actions. More specifically, one could re-define the signal structures in the current setting to incorporate
the information conveyed by the actions taken by the two individuals.
23
f 1 (s)f 2 (s)(s)
19
sups ff 3 (s)Mathematically, sups ff 2 (s)(s) > sups f 2 (s) sups f 1 (s) .
20
One may argue that after seeing individual B choosing action 2 or 3, individual A should change his
action. However, this is not possible as he has only one unit of memory capacity M = 1 and thus has toeffectively commit to one action. In particular, one can generalize this framework to which the two individuals
also see each others’ actions as signals and Proposition 3 would still hold.
21
Note that although this example imposes strong assumptions in particular on the size of bounded memoryof individual A, it generates a strong form of disagreement in which the two individuals disagree asymptotically with certainty. Similar intuition implies that even when the assumption is relaxed, the difference in M
would lead to asymptotic disagreement at least probabilistically.

16

5

Discussion and Conclusion

Heterogeneity of learning and heuristics under different environments or acrossdifferent individuals This paper explains a wide range of behavioral anomalies with thesame mechanism, i.e., efficient allocation of limited cognitive resources in light of complexity.
Importantly, the comparison of small and big worlds illustrates a link between the degree of(relative) complexity of the inference problem and the aforementioned non-Bayesian learning behaviors, which is supported by the experimental results in Enke and Zimmermann(2019) and Graeber (2019). Enke and Zimmermann (2019) shows that correlation neglect
negatively correlates with the cognitive ability of subjects and “an extreme reduction in theenvironment’s complexity eliminates the bias”, while Graeber (2019) shows that a reductionin the complexity of the problem by removing a decipher stage of signals reduces inattentive
learning behavior.
On the other hand, Enke and Zimmermann (2019) and Graeber (2019) also show that
simply reminding subjects about the neglected variables reduces inattentive learning and improves inference. This “reminder effect” can be reconciled in the current setup via an effect ofa change in the state space. Consider the behavior of inattentive inference in Graeber (2019).
The author shows that when subjects are asked to guess the realization of a variable A, theyoften ignore the effect of another variable B on the signal distribution. Applying to the settingin this paper, consider that before being reminded about the ignored variable, the state spaceis supp(A)×supp(B)×{B affects the signal distribution, B does not affect the signal distribution},in which subjects might ignore the states that say “B affects the signal distribution”. Afterbeing reminded about the effect of B, the set of states of the world is effectively reduced
to supp(A) × supp(B) × {B affects the signal distribution}, the complexity decreases, andsubjects adopt another learning mechanism that does not involve ignorant learning.
Future research directions The mechanism mentioned in the previous paragraph brings
forth an open question that is not answered in this paper. In reality, individuals face different(sets of) inference problems and are likely endowed with different learning mechanisms fordifferent sets of states of the world. Like in the example mentioned in the previous paragraph, upon receiving new information, individuals could revise the state space and transit
from one learning mechanism to another. This is also related to the question of how individuals construct the state space given an inference problem. Arguably, there are infinitely
many variables that might affect the signal distributions, and their realizations could be incorporated in the set of possible states. Roughly speaking, the result of ignorance seems tosuggest that individuals may only include the most “important” or “a priori probable” states,
while the “reminder effect” suggests that the construct of the state space also depends on theinformation received by the individual. Moving forward, I believe that the question of how
individuals construct their perceived state space and the corresponding prior belief deserves
more in-depth and careful analysis as it is fundamental to individuals’ learning behavior.

17

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20

A

ProofsA.1

Proof of Proposition 1

I first define “confirmatory signals” for each state. A signal s supports state ω with probabilityPωGω (s) and supports no state with probability 1 − N
ω=1 G (s), whereω

G (s) ∝ qR

f ω (s)(A.1)(f ω (s))2 dsPωωwith normalization such that N
ω=1 G (s) ≤ 1. With some abuse of notations, I use F
to denote the probability of receiving a signal supporting state ω 0 , or equivalently “a signal
R000Gω ”, in state ω, i.e., F ω (Gω ) ≡ f ω (s)Gω (s) ds. Importantly, Equation A.1 implies it ismore likely that the DM receives a signal supporting a correct state than a signal supportinga wrong state:
sZ
R ω
R ω
0
2
(f (s)) ds
f (s)f ω (s)ds
0
ω
ω
ω
2
F (G ) = qR=
(f (s)) ds > qR
= F ω (Gω ),(f ω (s))2 ds(f ω0 (s))2 ds
implied by the Cauchy–Schwarz inequality.
Proof of Proposition 1 (i). Consider a -optimal mechanism with memory size M denoted as(m1 , T , d). I now construct a mechanism with memory size M + 1 denoted as (m01 , T 0 , d0 )that delivers a strictly higher asymptotic utility than (m1 , T , d). The result follows when goes 0. More specifically, pick an m̃ ∈ M 1 and without loss suppose arg mina u(a, N ) = 1.
−10
0 0With some constants (δi )N
i=0 that I will describe later, (m1 , T , d ) follows:
m01 = m1 .
d(m) = d0 (m) for all m = 1, · · · , M .
d(M + 1) = 1.
Pr [T 0 (m, ·) = m0 ] ∝ Pr [T (m, ·) = m0 ] for all m ∈/ {m̃, M + 1} and all m0 6= m, M + 1.
δ Pr [T (m̃, s) = m] if m 6= M + 1.;00Pr [T (m̃, s) = m] ∝ P
 N δ Gi (s) if m = M + 1.
i=1 i
c if m = m̃;Pr [T (M + 1, s) = m] ∝
0 if m 6= m̃, M + 1.
P
for some constant c and with appropriate normalization such that m0 6=m Pr [T 0 (m, s) = m0 ] ≤1 for all m.22 (δi )Ni=1 is chosen such that they satisfy the following system of linear equations
for some chosen ∆ > 1:
Note that the long-run distribution is invariant if the transition matrix Pr [T 0 (m, s) = m0 ] is scaled up
by a common factor.
22

21




 R

R 1
R
R
G (s)f 1 (s) ds − ∆ GN (s)f 1 (s) ds
G2 (s)f 1 (s) ds − ∆ GN (s)f 1 (s) ds
R
R
 R 1

 R 2



G (s)f 2 (s) ds − ∆ GN (s)f 2 (s) ds G (s)f 2 (s) ds − ∆ GN (s)f 2 (s) ds 
 + ···+
δ1 
..
..

 + δ2 
.
.




R 1
R
R
R
G (s)f N −1 (s) ds − ∆ GN (s)f N −1 (s) ds
G2 (s)f N −1 (s) ds − ∆ GN (s)f N −1 (s) ds
 R
 R


R
R
GN −1 (s)f 1 (s) ds − ∆ GN (s)f 1 (s) ds
GN (s)f 1 (s) ds − ∆ GN (s)f 1 (s) ds
RR R N −1

 R N



G
(s)f 2 (s) ds − ∆ GN (s)f 2 (s) ds G (s)f 2 (s) ds − ∆ GN (s)f 2 (s) ds 



=0
δN −1 ..
..
 + δN 

.
.




R N −1R
R
R
G(s)f N −1 (s) ds − ∆ GN (s)f N −1 (s) ds
GN (s)f N −1 (s) ds − ∆ GN (s)f N −1 (s) ds(A.2)

Such (δi )N
i=1 exists as there are more variables than number of Equations. Now, denote0 ω M +1(µm )m=1 as the long-run distribution of (m01 , T 0 , d0 ) and (µωm )M
m=1 as the long-run distributionof (m1 , T , d). Equation (A.2) ensures thatPN R i00µMω +1 µMN+1
G (s)f ω (s) ds
i=1
/
=
=∆>1
R
P
0
0
N
µm̃ω
µm̃N
Gi (s)f N (s) ds
i=1
for all ω = 1, · · · , N − 1. Next, set δ0 = 1 +

PN

i=1

R

Gi (s)f N (s) ds∆, we have for ω = 1, · · · , N − 1,c0µmω = µωm for all m 6= m̃, M + 1,10µm̃ω = µωm̃δ!
PN R i
N
G
(s)f
(s)
ds
0
0
0
∆ = µωm̃ .
µm̃ω + µMω +1 = µm̃ω 1 + i=1c
One the other hand, in state N ,
1

0

µmω =

1−

PN

R

PN

R

i=1

Gi (s)f N (s) ds(∆ − 1)µωm̃
c

10µm̃ω =001−0µm̃ω + µMω +1 = µm̃ωµωm for all m 6= m̃, M + 1,1
( µωm̃ )Gi (s)f N (s) ds
i=1(∆ − 1)µωm̃ δc!
PN R i
N
G
(s)f
(s)
ds
i=1

1+

c

.

Pick ∆ > 1 (but not too big), the DM chooses action 1 with a lower probability in state N ,1
PN R iand other actions with a higher probability (by a factor of). As
G (s)f N (s) ds
1−

i=1

c

(∆−1)µωm̃

arg mina u(a, N ) = 1, the result follows.
Proof of Proposition 1 (ii) and (iii). I formally describe the simple updating mechanism described in the main text. Denote λ = b(M − 1)/N c, relabel the memory states as 0, 11, · · · , 1λ,21, · · · , 2λ, · · · , N λ and denote the unused memory states as N λ + 1, N λ + 2, · · · , M − 1.

22

The decision rule is as follows:d(0) = 1;
d(ωk) = ω for all ω = 1, . . . , N and k = 1, · · · , λ;d(m) = 1 for all m > N λ.
The transition function between the memory states is defined below for some big enoughδ > 1 such that0
δF ω (Gω )P> 1 for all ω, ω 0 ∈ Ω.
00ωω
ω 00 6=ω 0 F (G )Suppose the DM receives some signal s, he follows the following transition rule:
T (0, s) =ω1with probability Gω (s) for all ω = 1, · · · , N ;Pωwith probability 1 − N
ω=1 G (s).

0




ω2


T (ω1, s) = 0



ω1
T (ωλ, s) =with probability Gω (s);Pjwith probability j∈Ω\{ω} G δ(s) ;Pjwith probability 1 − j∈Ω\{ω} G δ(s) − Gω (s).

ω(λ − 1)ωλGj (s);
δPjwith probability 1 − j∈Ω\{ω} G δ(s) .

with probabilityPj∈Ω\{ω}while for k = 2, 3, · · · , λ − 1,ω(k + 1)T (ωk, s) = ω(k − 1)



ωk

with probability Gω (s);Pjwith probability j∈Ω\{ω} G δ(s) ;Pjwith probability 1 − j∈Ω\{ω} G δ(s) − Gω (s).

Finally, for m > N λ, T (m, s) = m for all s. By restricting the initial memory state toone of 0, 11, 12, · · · , 1λ, 21, 22, · · · , 2λ, · · · , N λ, the DM will never transit to memory statesm > N λ.
Note that this updating mechanism does not depend on pω nor uω . Now I compute thelong-run distribution µω and the utility loss L(m1 , T , d). Fix a state ω, at the two extreme
memory states in branch ω 0 , i.e., memory states ω 0 λ and ω 0 (λ − 1),
1 X ω ω00
F (G )δ ω00 6=ω0
"
#−1
ω
ω0
δF
(G
)
µωω0 (λ−1) = µωω0 λ Pωω 00
ω 00 6=ω 0 F (G )0µωω0 (λ−1) F ω (Gω ) = µωω0 λ

23

for all ω 0 . Next, at memory state ω 0 (λ − 1),
"
#
X
X
1
1
00
00
0
F ω (Gω ) + µωω0 (λ−2) F ω (Gω0 ) = µωω0 (λ−1)
F ω (Gω ) + F ω (Gω )µωω0 λδ ω00 6=ω0
δ ω00 6=ω0
"
#−1
ω
ω0
δF
(G
)
µωω0 (λ−2) = µωω0 (λ−1) Pωω 00
ω 00 6=ω 0 F (G )Repeating the same procedures implies that for all k = 1, · · · , λ and ω 0 = 1, · · · , N"µωω0 k = µωω0 λ

0δF ω (Gω )Pωω 00
ω 00 6=ω 0 F (G )#−(λ−k)and"µωω0 k = µω0AsPNω 0 =10

δF ω (Gω )Pωω 00
ω 00 6=ω 0 F (G )

#k
(A.4)

ω0

ω

Pλ

(A.3)

0

δF (S )
ωωωωP
, we haveωω 00 ) by r
k=1 µω 0 k + µ0 = 1, and denote00
0 F (S
ω 6=ω

µωωλ

λ
X

[rωω ]−(λ−k) + µωωλ [rωω ]−λ + µωωλ [rωω ]−λ

rωω

0

ik

µωωλ = Pλk=1 [r1
ωω ]−(λ−k) + [r ωω ]−λ + [r ωω ]−λP

ω 0 6=ωPλk=1 [rand the probability of choosing actions in Ω \ ω in state ω is smaller thanwhich is as follows:[rωω ]−λ + [rωω ]−λPλk=1 [rωω −λ< [r ]=1

ω 0 6=ω k=1

k=1

⇒

λ h
XX

P

ω 0 6=ωωω ]−(λ−k) + [r ωω ]−λ + [rωω −λ

+ [r ]

λ h
XX

rωω

0

Pλ

k=1
P
ωω −λ

]

 ωω0 kr
Pλ

ω 0 6=ωk=1 [rωω 0 ]kPωm∈{ω1,···/,ωλ} µmωω 0 ]kik

ω 0 6=ω k=1#" 0 λrωω− [rωω ]−λrωω" 0 λ #X rωω0rωω−λ< [rωω ] +0rωω − 1rωω
ω 0 6=ω" ωω0 λ #
ωω 0
0
max
r
r
ω 6=ω
< [rωω ]−λ + (N − 1)max
0
ωωmaxω0 6=ω r − 1 ω0 6=ω rωω( ωω0 λ )0
maxω0 6=ω rωωr< (N − 1)+ 1 max [rωω ]−λ , max0ωωω 0 6=ω
maxω0 6=ω r − 1
rωω
0

rωω
= [rωω ]−λ +rωω0 − 1
ω 0 6=ω
X(A.5)

The first inequality of Equation (A.5) is implied by the fact that the denominator is strictly

24



ω

maxω0 6=ω rωω

greater than 1. Now, using Equation (A.5) and denote K = (N − 1) max 0ω 6=ω r0

ωω 0 −1+ 1 wecan compute the upper bound of the utility loss L(m1 , T , d)L(m1 , T , d) ≤

N
X




X

pω max{u(ω, ω) − u(a, ω)} 
a6=ω

ω=1

µωm 

m∈{ω1,···/,ωλ} ωω0 λ )r<
p max{u(ω, ω) − u(a, ω)}K max [r ] , max
0
a6=ω
ω 6=ω
rωω
ω=1
"
(
 ωω0 λ )#r< max max{u(ω, ω) − u(a, ω)}K ω max [rωω ]−λ , maxω
a6=ω
ω 0 6=ω
rωω
N
X

ωω −1

As [r ]

ωω −λ

ω

(A.6)

 ωω0 rrωωare strictly smaller than 1 for all ω, Proposition 1 (ii) follows.
 ωω0 Moreover, as [rωω ]− 1 and maxω0 6=ω rrωω are strictly smaller than 1 for all ω, the right-handside of Equation (A.5) converges to 0 as λ goes to ∞. This proves Proposition 1 (iii).

A.2

and maxω0 6=ω

(ω

Proof of Proposition 2 and Corollary 1

Proof of Proposition 2 (ii). First, an ignorant updating mechanism induces utility loss weakly
greater than minω∈Ω mina6=ω [pω (u(ω, ω)−u(a, ω))] > 0. On the other hand, as shown in Proposition 1, L∗M converges to 0 as M → ∞. The monotonicity of L∗M implies there exists somebig enough M̄ such that for M ≥ M̄ , L∗M ≤ L∗M̄ < minω∈Ω mina6=ω [pω (u(ω, ω) − u(a, ω))].
The result follows.
Proof of Proposition 2 (i) and Corollary 1. First note that Corollary 1 implies Proposition 2(i). I first prove Corollary 1(i). Consider an updating mechanism (m1 , T , d) such thatµω̃m > 0 for some ω̃ and m where d(m) = ω. Now I prove that there exists a different
updating mechanism that yields higher asymptotic utility if pω is small enough.
Consider state ω and ω 0 , recall that the long-run distribution in state ω and ω 0 are thesolution of the following fixed point equations:µω = (µω )T Qω ;
0

0

(A.7)

0

µω = (µω )T Qω .
0

Equation (A.7) shows that, given an updating mechanism (m1 , T , d), f ω and f ω , the longω0ω 00 Nω 00 00
run distribution µω , µω and the ratio ( µµωm0 )M)ω 6=ω,ω0 .
m=1 are invariant of N , (p )ω 00 =1 and (fmTherefore, we can apply Theorem 2 of Hellman and Cover (1970) where N = 2 such that:ω0maxm µµm
ω

m
0
µω
m
m µω
m

≤ (l

ωω 0 ω 0 ω (M −1)

l

)

.

(A.8)

min
where l

ωω 0

ω

ω0

0
. First, minm µµm= sups ffω0(s)ω > 0 for all ω 6= ω. Suppose to the contrary that for
(s)
m

25

ω0

ω0

0

µm
ω
some ω 0 , minm µµm
ω = 0, Equation (A.8) implies that maxm µω = 0 and µm = 0 for all m. Itmmωω
in turn implies that maxm µµωm0 is unbounded and minm µµωm0 = 0 and it contradictsmmωmaxm µµωm0
m
µω
m
m µω0
m

min

≤ (l

ω 0 ω ωω 0 (M −1)

l

)

.

ω0

0
ω
= u(ω, ω), we have
Now since minm µµm
ω > 0 for all ω , denote u
m

0

0

µω
µωm
ωω 0 ω 0 ω −(M −1)
minm m
≥
(l
l
)
max
m ω
µωm
µm
0
0
µω
µωm
2(M −1)minm m
≥
ς
max
m
µωm
µωm
0
µω
minm m≥ ς 2(M −1)
µωm
0
0
0
ω0 ω0
uω pω µωm2(M −1) u p≥ς
for all m.
uω pω µωmuω pωThe second inequality follows Equation (1) while the third inequality is implied by the factPP
0that m µωm = m µωm = 1. Now, denote u = maxω∈Ω mina6=ω [u(ω, ω) − u(a, ω)] and u =
ω0 ω0

1−pω

0
minω∈Ω mina6=ω [u(ω, ω)−u(a, ω)], as min maxω0 uuω ppω = uu Np−1
ω , there exists some ω such that
0

0

0

0

0

1−pω

uω pω µωmuω pωu2(M −1)≥ςminmax≥ ς 2(M −1) N −1
for all m ∈ M .
ω
ω
ω
ω
ω
0
ω
u p µm
u p
u pω

(A.9)

ω

ω

p
2(M −1) u 1, or equivalently, 1−p, Equation (A.9) implies that
When pω < ς 2(M −1) uu 1−p
ω < ς
N −1
u N −1000uω pω µωm > uω pω µωm , and the DM is better off choosing ω 0 than choosing ω for all m ∈ M .
ω0 ω0 ω0
ω pω µωm
Setting ¯ = u pP µ00mpω−uproves the first bullet point of Corollary 1. Similar argument
00 ω 00µωmproves Corollary 1(ii). Check!
I now prove Corollary 1(iii). Again, consider an updating mechanism T such that µω̃m > 0
for some ω̃ and m where d(m) = ω. Similar to the proof of the first part of Corollary 1,
0

0

µωm
µωm
ωω 0 ω 0 ω −(M −1)
≥
(l
l
)
max
m ω
µωm
µm
0
0
µω
µωm
−(M −1)≥minm m
F
max
m
µωm
µωm
0
µω
−(M −1)minm m
≥F
ω
µm
0
0
0
ω0 ω0
uω pω µωm−(M −1) u p≥Ffor all m.
uω pω µωmu ω pω
minm

ω0 ω0As u pω ω> u p , for F <

h

uω pω
uω0 pω0i M1−1(A.10)000, Equation (A.10) implies that uω pω µωm > uω pω µωm ,26

ω0 ω0

ω0

ω ω ω

p µm
, the result
i.e., the DM is better to choose action ω 0 instead of ω. Setting ¯ = u pP µ00mpω−u00 ω 00µmωfollows.

A.3

Proof of Corollary 2

Proof. First, (i) is directly implied by Corollary 1. If pωA is small enough for all ω ∈ NA ⊂ N
and pωB is small enough for all ω ∈ NB = N \ NA , individual A (almost) never picks action ω
for all ω ∈ NA and individual B (almost) never picks action ω for all ω ∈ N \ NA . Therefore,
they must disagree with each other.
To prove (ii), note that by Proposition 1, we know that if an individual adopts an M −1
optimal updating mechanism, his utility loss is bounded above by Krb N c +  for some
K > 0 and r < 1. Thus, for all ω, theprobability that the individual chooses a “wrong”
action ω 0 6= ω is strictly smaller than

M −1

Krb N c +minω pω. For all ω, the probability that bothindividuals chooses action ω and thus agree with each other is greater than




1 −

MA −1
c
N

b
K
r
A
A


minω pωA +   


1−

MB −1
c
N

b
K
r
B
B


minω pωB+  


min{MA ,MB }−1
1
1
c
N
+ )
+
>1 −
(max{KA , KB }(max{rA , rB })bωωminω pA minω pB
for all -optimal updating mechanisms of individual A and B. The result follows.

A.4

Proof of Corollary 3

Proof. I prove the Corollary using the following simple example. Consider a setting with
N = 3 and two individuals, A and B, who share the same prior belief and the same objective
signal structure:
1
+ 2ν
3
1
p2 = p 3 = − ν3(A.11)f 1 (s)f n (s) √sup n= sup 1
= 1 + τ for n = 2, 3
s f (s)s f (s)f 2 (s)f 3 (s) √sup 3= sup 2
= 1 + Ψ where Ψ > τ .
s f (s)s f (s)(A.12)

p1 =

1

+2ν

with 1 + τ ≥ 31 −ν .23 Moreover, assume that u(1, 1) = u(2, 2) = u(3, 3) = 1 and u(ω, ω 0 ) = 0
3
for all ω 6= ω 0 . The only difference the two individuals have is their levels of cognitive ability,
23

It ensures that if M ≥ 2, the DM never chooses action 1 with probability 1 and he can achieve a strictly
lower utility loss compared to the benchmark of no information.

27

where MA = 1 and MB = 2.
First, as M = 1 for individual A, his action is constant in all periods for all signal
realizations and thus d(1) = 1. Individual A always take action 1. Now I characterize the optimal updating mechanism of individual B. With some abuse of notations, denote L∗2 (nn0 )as optimal utility loss where the DM chooses action n in memory state 1 and action n0 inmemory state 2. We have, obviously,
2
− 2ν,
3
2
L∗2 (22) = L∗2 (33) = + ν.
3L∗2 (11) =Moreover, as argued in the proof of Proposition 2 (i), suppose d(1) = n and d(2) = n0 6= n,
we have0µn1
1 − µn1µn1 µn2f n (s)f n (s)= n0 / n0 ≤ sup n0
sup n .
0/0
1 − µn2µn2µ1 µ2
s f (s) s f (s)The upper bound of L∗ (nn0 ) is given by the following minimization problem:000min0 1 − pn − pn + pn (1 − µn1 ) + +pn (1 − µn2 )

n
µn
1 ,µ2

0

1 − µn1f n (s)f n (s)
µn1
/
≤
sup
sup
,
given
0
0
0
n
n
1 − µn2µn2
s f (s) s f (s)Hellman and Cover (1970) shows that the upper bound is tight with the updating mechanism
n0 (s)
n0 0
where T (1, s0 ) = 2 if and only if ff n (s(s0 )) is close to sups ff n (s), and T (2, s0 ) = 1 if and only ifnf n (s0 )is close to sups ffn0(s). Therefore,
f n0 (s0 )(s)q
2 (1 + τ )1L∗2 (12) = L∗2 (13) = − ν +321L∗2 (23) = + 2ν +
3
1

1
+ 2ν
3



τ
√
1
−ν
1+Ψ−
3
Ψ

1
−ν
3



2
−ν
3

−

2
− 2ν
3


,


.

+2ν

where 1 + τ ≥ 31 −ν implies L∗2 (22) = L∗2 (33) > L∗2 (11) ≥ L∗2 (12) = L∗2 (13). In the following,3
I prove L∗2 (12) > L∗2 (23) if ν is small enough which implies that aB
t = 2 or 3 for all t in all
-optimal updating mechanism when  is smaller than L∗2 (12)−L∗2 (23). Now, L∗2 (12) > L∗2 (23)
if and only if

L∗2 (12) − L∗2 (23) = 3ν +2( 13 − ν)

√

1+Ψ

Ψ

−

2
− 2ν
3

Ψ

−

q
2 (1 + τ )( 13 + 2ν)( 31 − ν)τWhen ν = 0,
2
L∗2 (12) − L∗2 (23) =
3

√

1+Ψ
−
Ψ

28

√

1+τ
τ



2
−
3



1
1
−
Ψ τ


.

2

+3

+ν
< 0.
τ

√

As both 1+x
and x1 decreases in x, Ψ > τ implies that L∗2 (12) − L∗2 (23) < 0 when ν = 0.
x
The result follows by continuity.

B

An example of ignorance with uniform prior belief
and symmetric signal structures

In this section, I consider a case where N = M ≥ 4 and states of the world are a priori
uniformly distributed, i.e., pω = N1 for all ω = {1, · · · , N }. Moreover, u(ω, ω) = 1 >u(ω 0 , ω) = 0 for all ω and ω 0 6= ω. For simplicity, consider a class of “symmetric” discrete
signal structures where S = {s1 , · · · .sN } and sω is a signal that supports state ω. Morespecifically,0F ω (sω ) = I F ω (sω ) for all ω and ω 0 6= ω where I > 1.
Thus, under all states of the world, it is I times more likely to receive a signal that supports
the true state than a signal that supports one of the other states.
In such a symmetric environment, there seems to be no reason to ignore any of the states.
However, I will present an example that shows that it is beneficial to ignores some states
when N is large or I is small. First, consider a simple “symmetric” updating mechanismthat ignores no states, illustrated in Figure 3 with an example of N = 4. As the DM ignores
no state, he allocates one memory state to each action. Without loss of generality, assume
he takes action ω in memory state ω. When the DM is in memory state m = ω, upon0
receiving a signal sω , i.e., a signal that supports state ω 0 , he transits to memory state ω 0with some probability δmω ≤ 1, and stays in his current memory state otherwise. Formally,
the transition function is as follows:
T (m, sω ) = δmω × {ω} + (1 − δmω ) × {m} for all m and ω.
In the following I show that such updating mechanism is optimal among the class of all
non-ignorant mechanism.
Suppose for some non-ignorant mechanism that the DM chooses action ω in memory stateω, and the DM transits from memory state m 6= ω to memory state ω upon receiving signal
0sω with strictly positive probability where ω 0 6= ω. Similar to the proof of Proposition 1 (i),I construct an updating mechanism that strictly improves asymptotic utility. First, decrease0
the probability of transiting from memory state m to ω upon receiving signal sω by p. Second,
increase the probability of transiting from memory state m to ω upon receiving signal sω byp (re-normalize if necessary to ensure the probability of transiting from memory state mto ω upon receiving signal sω is not greater than 1). These two steps do not change the00
000
utility in all state ω 00 6= ω, ω 0 because F ω (sω ) = F ω (sω ), increase the utility in state ω but
potentially decrease the utility in state ω 0 . Last, scale up all the transitions out of memory29state ω (again, re-normalize if necessary) such that the utility in state ω is the same before
the first step of construction. As it is less likely to receive sω in state ω 0 than in state ω, thelast step “over-scale-up” the transition out of memory state ω in ω 0 , and thus increases theutility in state ω 0 such that it is higher before the construction. It also increases the utility
in state ω 00 6= ω, ω 0 such that it is higher before the construction.
Now consider an ignorant mechanism that follows the similar idea of the non-ignorant
updating mechanism illustrated in Figure 3, but ignores half of the states of the worlds. For
simplicity, assume that N is plural. The ignorant mechanism is illustrated in Figure 4, withan example of N = 4. By ignoring half of the states, the DM allocates two memory states toeach action that he does not ignore. Without loss of generality, assume that the DM takes
action ω in memory states 2ω − 1 and 2ω for ω ≤ N2 , and ignores all actions ω 0 > N2 . In the“more confident” memory state 2ω, upon receiving a signal supporting state ω 0 6= ω whereω 0 ≤ N2 , the DM transits to state 2ω − 1 with probability δ < 1 and stays in his current
memory state otherwise. In the “less confident” memory state 2ω − 1, upon receiving a signalthat supports state ω, he transits to the “more confident” memory state 2ω; upon receiving
a signal that supports state ω 0 6= ω where ω 0 ≤ N2 , the DM transits to state 2ω 0 − 1 withprobability δ < 1 and stays in his current memory state otherwise.
Formally, the transition function is as follows:
T (2ω, sω ) = 2ω2ω − 1 with probability δω0T (2ω, s ) =2ω with probability 1 − δ

for all ω ≤N,2

for all ω ≤N 0N, ω 6= ω and ω 0 ≤ ,2
2

T (2ω − 1, sω ) = 2ω2ω 0 − 1 with probability δ0T (2ω − 1, sω ) =2ω with probability 1 − δ

for all ω ≤N,2

for all ω ≤N
N 0, ω 6= ω and ω 0 ≤ ,2
2

for all ω >

N
.
2

0

T (m, sω ) = m

Suppose the true state is 1, In the stationary distribution, we must have
δµ12ω

X

0

F 1 (sω ) = µ12ω−1 F 1 (sω ) for all ω ≤ω 0 6=ω,ω 0 ≤ N
2

N
,
2
(B.1)

N
δµ12ω−1 F 1 (s ) = δµ12ω0 −1 F 1 (sω ) for all ω, ω 0 ≤and ω 6= ω 0 .
2
ω0From the first Equation of Equation (B.1), we know that when δ is close to 0, µ12ω−1 is close30

to 0 for all ω ≤ N2 . Moreover, we haveF 1 (s1 )δF 1 (s1 ) δ
1
P
×
µ2 =
×
δ ω0 6=1,ω0 ≤ N F 1 (sω0 ) δF 1 (sω )
2


I
N
= N
×I ×
− 2 + I × µ12ω
2
−1
2
=

P

F
ω 0 6=ω,ω 0 ≤ N
2F 1 (sω )

1

0

(sω )

× µ12ω

I 2 (N + 2I − 4) 1
µ2ω
N −2

for all ω 6= 1 and ω ≤ N2 . We have
X

µ12 +

µ12ω = 1

ω6=1,ω≤ N
2

µ12 +

N −2
I 2 (N + 2I − 4)N
− 1 µ12 = 12µ12 =2I 2 (N + 2I − 4).
2I 2 (N + 2I − 4) + (N − 2)2

By repeating the same computation for all ω ≤ N2 , the asymptotic utility equals:N

2I 2 (N + 2I − 4)1
I 2 (N + 2I − 4)×=.
2 (N + 2I − 4) + (N − 2)22 (N + 2I − 4) + (N − 2)2
2I
N
2I
ω=1

2
X

(B.2)

Then the R code on https://sites.google.com/site/ltkbenson/research produces Figure 5 in the main text.

31