OPTIMAL INQUIRY
Tai-Wei Hu and Andriy Zapechelnyuk
Abstract. A decision maker acquires and processes information about an uncertain state of nature by an inquiry, which is a procedure prescribing a sequence ofquestions to be asked before a decision is reached. The decision maker bears a cognitive cost proportional to the length of inquiry. We characterize optimal inquiries
and uncover two behavioural implications: attention span reduction (the decision
maker favours shorter inquiries as cognitive cost rises by focusing on a subset ofdecisions and assigns them different priorities) and confirmation bias (the decision
maker seeks evidence through inquiry to confirm her prior guess of which decisions
are optimal).
JEL Classification: D81, D83.
Keywords: Bounded rationality; information theory; cognitive cost; rational inattention; attention span; confirmation bias; consideration set

Date: June 20, 2024
Tai-Wei Hu: School of Economics, University of Bristol, 12 Priory Road, Bristol, BS8 1TU, UK.
E-mail: taiwei.hu@bristol.ac.uk
Andriy Zapechelnyuk : School of Economics, University of Edinburgh, 31 Buccleuch Place, Edinburgh, EH8 9JT, UK. E-mail: azapech@gmail.com.
The authors are grateful to Larry Samuelson and Jakub Steiner for helpful comments. For the purpose of open access, the authors have applied a Creative Commons Attribution (CC BY) licence to
any Author Accepted Manuscript version arising from this submission.
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HU AND ZAPECHELNYUK

1. Introduction
Inquiry is one of the most frequent and important modes of information processingin our daily life. Examples are abundant. A doctor visit usually consists of a seriesof questions from reception to actual consultation of the patient’s conditions. A
crime investigation typically consists of a series of questions and processing theiranswers. Inquiry about product characteristics and payment schemes is an important
aspect of shopping experiences. In all these examples, information to be gathered canbe potentially overwhelming, whereas cognitive resources available to process it arelimited and precious. In this paper, we propose a theory of optimal inquiry to process
information that takes costly cognition seriously, with novel behavioral implications
on attention span and confirmation bias.
We formalize an inquiry as the decision maker’s strategy of asking questions about
the relevant state of nature. It starts with an initial question and a contingent planthat decides which question to ask depending on the answers to the previous ones. As
in the standard Bayesian paradigm, the answers to the inquiry hence determine theinformation set that guides the decision maker’s final decision. Unlike the standard
framework, however, our framework explicitly postulates a cognitive cost associatedwith the length of the inquiry.
Our framework provides an explicit and intuitive procedure for information processing. It has the same backbone motivation as what gave rise to the rational-inattention
literature (surveyed in Maćkowiak et al., 2023). The main departure of our approachfrom this literature is that we focus on the dynamic process of inquiry with an endogenous choice of the optimal procedure. This allows us to obtain behavioral implications
that are of dynamic nature, such as an endogenous preference for a shorter attention
span and a prioritization of certain salient decisions before considering others.
Moreover, our cost of inquiry is directly associated with the cognitive activity ofthe decision maker, namely, the acts of asking questions and processing their answers,
and thus it is independent of the decision maker’s beliefs. For example, the costof performing a blood-sugar test and processing its result (in terms of physical or
cognitive resources) is independent of the patient’s medical history. In contrast, inthe standard rational inattention model, the cost is a function of the decision maker’s

OPTIMAL INQUIRY

3prior beliefs, which can be unrealistic is certain applications and has a conceptual
problem if applied to game situations (Denti et al., 2022).
Our main result is a characterization of optimal inquiry. To obtain this result, weovercome a technical challenge: our set of possible inquiries is an infinite set, withsome of its elements (the inquiry tree) discrete and others (the information partition)
continuous. This is a nonstandard problem that does not admit the standard firstorder approach. Instead, we reformulate the task of finding an optimal inquiry as asimple finite optimization problem. Our characterization relies on the following two
principles for optimality of an inquiry.
First, we show that an optimal inquiry is dynamically consistent in the following
sense. Consider a decision maker who processes information according to an inquiry,
and suppose that she has asked a few questions but not yet ready for a final decision. At this point, she could stop and reconsider her inquiry strategy, taking all theinformation she already acquired so far as given. Dynamic consistency requires it be
optimal to stick to the original plan. We prove this property for any optimal inquiry
in our framework.
Second, we use two well-known results from the information theory—the Kraft
inequality (Kraft, 1949) and the Huffman coding (Huffman, 1952)—to characterize
the set of payoff-relevant outcomes that can be implemented by an optimal inquiry.
Any such outcome consists of two parts—form and content. The form includes aconsideration set, which is a subset of feasible decisions that can be implemented inthat outcome, and a length profile, which specifies how many questions are asked toreach each decision in the consideration set. The content consists of an information
partition, which describes the posterior information about the state upon reachingeach decision in the consideration set. We show that the form determines the content:
given the optimal consideration set and the length profile, the boundaries of the
optimal information partition are determined by simple indifference conditions.
However, the content is also informative about the form: given the information
partition, the optimal length profile is determined by the Huffman coding. This also
implies a negative correlation between the ex ante likelihood of choosing a decisionand the inquiry length that leads to that decision. That is, more likely decisions areprioritized and considered before other decisions. To determine the optimal form,

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HU AND ZAPECHELNYUK

optimal inquiry trades off the accuracy of information processed, expressed as thefinesse of the resulting information partition, against the cognitive resources needed toachieve it, measured by the number of questions to be asked. We draw two behaviouralimplications from this trade-off.
First we consider implications related to the form of optimal inquiry, and define
attention span as the expected number of questions the DM asks before reaching adecision. We show that the DM optimally reduces her attention span as the cognitive
cost rises. This is achieved either by dropping some decision out of the consideration
set, or by prioritising some decisions over others, or both. At the extreme, when thecost is very low, all feasible options are considered, and it takes as many questions
as needed to distinguish them all. On the other hand, when the cost is very high, no
information is processed, and the decision is chosen according to the prior belief.
Second, we consider implications to the content of the optimal inquiry. We showthat optimal inquiry always exhibits confirmation bias, in the following sense. The
decision maker optimally seeks information to confirm her prevalent hypothesis ofwhich decisions are optimal. This formalizes the informal definition of confirmation
bias in psychology such as Nickerson (1998): “It refers usually to unwitting selectivity
in the acquisition and use of evidence.” We uncover an economic mechanism for theconfirmation bias to occur optimally. Because of the cognitive cost, the decision
maker is willing to make suboptimal choices that are associated with fewer questions.
At the same time, ex ante more likely choices are optimally prioritized with fewer
questions to confirm them. These two forces together lead to the confirmation bias
endogenously.
Finally, we use our model to study two phenomena, misdiagnosis in primary healthcare and wrongful convictions in criminal investigation, where the literature has argued that cognitive factors are important for biases in the inquiries with dire consequences. Through the lens of our model, we show that the pressure to end inquiry
early can lead to a biased process. In the case of health care, we show features such as“premature diagnosis” and “search satisficing” can be explained by our confirmation
bias. In the case of criminal justice, we show that a “tunnel vision” that leads tohigher rate of wrongful conviction can be linked to higher cognitive cost.

OPTIMAL INQUIRY

5

Related Literature. This paper makes a conceptual and methodological contribution
to three strands of literature.
The first strand includes papers that formulate and study decision making with cognitive limitations. A prominent approach in this literature is the rational inattention
approach, initiated by Sims (2003). It treats limited cognition as costly informationacquisition. The cost of acquiring information is postulated as an ex-ante cost function, typically modelled as entropy reduction relative to the prior belief, as in Matějka
and McKay (2015) and Jung et al. (2019). Morris and Strack (2019) introduce analternative ex-ante cost function motivated by the classic sequential sampling problem of Wald (1945). Bloedel and Zhong (2021) provide general conditions for ex-antecost functions to arise from dynamic models of information acquisition, and Pomatto
et al. (2023) characterize ex-ante cost functions that satisfy several economically interpretable axioms.
Unlike this literature, we focus on a concrete but intuitive dynamic model where
the cost of information is directly associated with asking questions. The dynamic
nature of the process and the sequencing of questions matters and has behavioral
implications. Moreover, our dynamic consistency result shows the decision-maker’s
commitment to an information acquisition strategy—which is a typical assumption
in the above literature—has no power in our setting.
Cognitive limitations of a decision maker have also been modeled without reducing
them to an ex-ante cost function. Wilson (2014), following the approach of Cover andThomas (2006), formulates the decision-making process as a finite automaton. The
main result in Wilson (2014) is a dynamic-consistency type of result called multi-selfconsistency. The cognitive constraint is modelled via an exogenously given numberof memory states that capture the decision-maker’s memory capacity. In contrast,
we prove the dynamic consistency in the conventional sense and endogenize the sizeof the optimal inquiry via a cognitive cost. Cremer et al. (2007) propose a modelof organizational language using codes, with the main trade-off between the use ofbroader codes, which are easier to process, and the precision of such codes. While
our model shares a similar trade-off, our model of inquiry is dynamic in nature with
implications on the timing of information processing.

6

HU AND ZAPECHELNYUKThe second strand of literature includes papers that studies behavioural biases withcognitive frictions. These papers range from axiomatic to constrained optimization
approaches. For consideration set, the former include Masatlioglu et al. (2012) andManzini and Mariotti (2014) and the latter includes Caplin et al. (2019). While ourapproach is closer to the latter, we connect the two approaches by showing that our
optimal inquiry satisfies certain desirable axioms, such as dynamic consistency andthe attention-filter property of Masatlioglu et al. (2012).
The third strand rationalizes confirmation bias. The wisdom from the literatureis that frictions in information processing tend to cause the decision-maker to favour
signals that confirm the prior belief. Wilson’s (2014) model generates this form ofconfirmation bias based on limited memory. However, in her model the decision-makerdoes not seek evidence but passively processes it. In contrast, our decision-maker
actively seeks evidence to confirm her more likely options. Jehiel and Steiner (2020)
obtain confirmation bias in a model where the decision maker chooses whether or
not to continue to receive more signals, but can only remember the last one received.
Confirmation bias here means that the agent is more likely to stop when seeing asignal in favour of the prior.
In the rational-inattention literature, Steiner et al. (2017) obtain a “status quobias” in a dynamic rational-inattention model where the decision-maker tend to stick
to prior decisions. Nimark and Sundaresan (2019) also obtain a “confirmation effect,”meaning that the decision-maker adopts signal structures in favour of the prior belief.
All these papers argue that certain implications from the proposed models can beinterpreted as confirmation bias, and emphasize the importance of the prior belief. In
contrast, we define confirmation bias formally as the decision-maker seeking evidence
to confirm ex ante most likely guesses about which decision is optimal, a definition
that is based not on priors but on observable choices.

2. The model
2.1. Primitives. A decision-maker (DM) needs to process information about an uncertain state of nature before taking an action. The DM’s utility u(a, x) depends on

OPTIMAL INQUIRY

7

her action, a ∈ A, and an uncertain state, x ∈ X.1 The set of actions A is finite andcontains at least two actions. The set of states X is a convex subset of RL , L ∈ N.
State x is distributed according to a probability distribution G that is absolutely
continuous and has full support on X. We will use notation P[·] and E[·] to denotethe probability and expectation under G, respectively.
We say that action a dominates another action a0 if u(a, x) ≥ u(a0 , x) for all x ∈ X
and strictly so for some x ∈ X. Throughout the paper, we assume:(A1 ) For all a ∈ A, u(a, x) is continuous in x, and E[u(a, x)] is finite.
(A2 ) For all a, a0 ∈ A, a does not dominate a0 .
(A3 ) For all a0 , a00 ∈ A and any constant c ∈ R, the set {x ∈ X : u(a0 , x) − u(a00 , x) =c} has empty interior.
Assumption (A1 ) is needed for the DM’s optimization problem to be well defined.
Assumption (A2 ) is introduced to simplify exposition and it precludes existence ofdominated actions. Assumption (A3 ) is a generalization of the condition of “thin”
indifference curves between each pair of actions. It means that the utility curves ofany two actions are almost never parallel to each other. Many usual utility functions
satisfy this assumption. For example, (A3 ) is satisfied for the following two classes ofutility functions.
(U1 ) The Lancaster model of product characteristics: X ⊂ RL and, for each a ∈ A,
there is (αa , βa ) ∈ R × RL such that u(a, x) = βa · x + αa .
(U2 ) A tracking problem: A ⊂ RL and X ⊂ RL , and u(a, x) is the negative distance
between a and x, that is, u(a, x) = −||a − x||p + αa , where || · ||p is the Lp -norm
on RL and αa ∈ R for each a ∈ A.
There are two special cases of (U1 ) that we will use for illustrations. The firstcase has L = 1 and hence the utilities depend only on a one-dimensional state. The
second case has A = {a1 , ..., aL } and u(x, al ) = xl for l = 1, ..., L, where the values
xl are distributed independently. This is the case where the DM chooses between L
independently valued options.
1Variable x can be interpreted as a profile of observables or signals with quantitative information

about the true underlying state of nature (which may be ultimately unobservable) that the DM can
ask questions about.

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HU AND ZAPECHELNYUK

2.2. Inquiries. When confronted with a state x, the DM does not observe x directly.
Instead, she relies on a series of questions to process information about x. Formally,
we consider an inquiry as a series of true/false questions formulated as propositions.
A proposition is a statement about x in the form “x ∈ Y ” that can be true or false.
Denote the collections of Borel subsets of X by B(X), and let us identify a proposition
with a set Y ∈ B(X). We say that the proposition is true if x ∈ Y and it is false if
x 6∈ Y .
An inquiry Q = hN, T, σ, χ, di is a finite binary tree. Non-terminal nodes of thetree are associated with propositions, and terminal nodes are associated with actions.
Specifically:• a finite set N of nodes contains a root no and a nonempty set T of terminal
nodes (note that the tree may consist of a single terminal node, i.e., N = T ={no });• each non-terminal node n ∈ N − T is followed by exactly two edges labelledtrue and false;• successor function σ for the tree assigns to each non-terminal node n ∈ N − T
and each edge e = {true, f alse} a child σ(n, e) ∈ N of node n following edge e;• proposition mapping χ assigns to each non-terminal node n ∈ N − T a proposition χ(n) ∈ B(X);• decision rule d assigns to each terminal node t ∈ T an action dt ∈ A.
We denote the set of all possible inquiries given the set of states X by QX .
Given a state of nature x ∈ X, an inquiry Q = hN, T, σ, χ, di begins with theproposition χ(no ) at the root of the inquiry tree, and it ends whenever a terminal
node is reached. It proceeds by following the inquiry tree. At a non-terminal noden ∈ N − T , the inquiry asks whether it is true that x ∈ χ(n). If true, then theinquiry proceeds to the node σ(n, true); otherwise, the inquiry proceeds to the nodeσ(n, f alse). When a terminal node t ∈ T is reached, the DM takes action dt .
2.3. Information. The inquiry transforms a quantitative statement, say, “x ≥ r”,
into a qualitative one, say, “yes” or “no”, eventually leading to a qualitative recommendation of which action to choose. The underlying assumption is that the DM
cannot directly digest quantitative information. Knowing that his blood sugar levelis 6 mmol/L means little to a medical lay person, but knowing that it is below theOPTIMAL INQUIRY

9level that would be labelled as “normal” is very useful as it suggests a decision of notgoing to the GP. Indeed, our theory is aimed at the optimal thresholds for what itmeans by “normal” (do nothing), “concerning” (see the doctor soon), or “emergency”(call an ambulance).
Formally, the inquiry categorizes states of nature into subsets through a series ofquestions. When arriving any (terminal or non-terminal) node n ∈ N , the DM’s
information about the state is summarized by a subset of states, denoted by In (Q).
That is, given the answers to the questions in the previous nodes, the DM can inferthat the true state belongs to In (Q), recursively defined as follows. Clearly, at theroot, all states are possible, and hence Ino (Q) = X. Given a non-terminal noden ∈ N − T , let ntrue and nf alse be the successors of n after “true” and “false” answers
to the proposition χ(n), respectively. Then we define
Intrue (Q) = In (Q) ∩ χ(n) and Inf alse (Q) = In (Q) ∩ X − χ(n) .

(1)

Now, for each x ∈ X, the DM will reach some terminal node t at the end of theinquiry. Thus, the set It (Q) consists of all states under which terminal node t isreached, and we call it a category of states induced by Q. Note that the collection of
categories {It (Q) : t ∈ T } forms a partition of X. It is the information partition at
the end of the inquiry.
As zero probability events do not matter for payoffs, we adopt and use throughout
the paper a measure based notion of partition that disregards sets of measure zerounder G. Specifically:
Definition 2.1. A collection of disjoined sets {X1 , X2 , ..., XK } is a partition of X if
P
P(Xk ) > 0 for each k, and k P(Xk ) = P(X) = 1.
Note that a partition according to the above definition does not have to be exhaustive; it is sufficient for the partition to cover a measure-one set. We adopt thisdefinition to avoid discussions about measure-zero sets that have no rendering on theDM’s expected payoffs.
2.4. Payoffs. We assume that asking questions is costly. Let the DM’s cognitive costof any single question be λ > 0. Given an inquiry Q, let `t (Q) be the length of thepath from no to t in the tree, that is, `t (Q) is the number of questions asked to reach

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HU AND ZAPECHELNYUK

terminal node t. Then, the ex-post cost of inquiry at terminal node t is equal toλ`t (Q).
We can now formulate the DM’s optimization problem. Given an inquiry Q and astate x, if the inquiry reaches the terminal node t for the given x, the DM’s ex-post
payoff net of the cognitive cost isu(dt , x) − λ`t (Q).
Because each terminal node t ∈ T is reached whenever the state x is in It (Q), theDM’s ex ante expected utility from inquiry Q isXZ
W (Q; λ) =u(dt , x) − λ`t (Q) G(dx).
t∈T

(2)

x∈It (Q)We are interested in the optimal inquiry that solves
max W (Q; λ).

Q∈QX

(3)

The maximization problem (3) resembles the problem studied in the rational inattention literature (e.g., Matějka and McKay 2015, Jung et al. 2019, and Caplin et al.
2019). But this resemblance is more in formality than in substance. While frominformation theory the average length of investigation defined here is closely related
to entropy and the rational-inattention approach is motivated by measuring cognitive
cost as number of questions, the standard approach measures the cost of informationin terms of entropy reduction relative to the prior belief. In contrast, in our model
the primitive cost does not depend on the prior—it is simply the asking (and theimplied act of processing the answer) itself is costly. Moreover, in contrast to theusual setup in which the model is silent about the corresponding procedure that the
DM uses to arrive at her decision, in our model there is an explicit connection between the solution to (3) and the procedure used. In particular, we may say that therealized process is simpler for a decision if fewer questions are needed to arrive atthat decision, that is, ` is smaller.
2.5. Example. We illustrate our setting and, later, the results by the following example. The example is based on the case study in Croskerry et al. (2013), which
illustrates how cognitive factors affect misdiagnosis in healthcare. A detailed description and the implications from our theory to misdiagnosis will be given are Section
6.1. This example is based on the following stylized situation of a doctor visit. A

OPTIMAL INQUIRY

11θ
u(a1 , x)u(a3 , x)u(a2 , x)

0

x
0

1−θ

θ

1

Figure 1. The doctor’s utility from actions aR , aL , and aI .
patient comes to a family doctor about a common symptom but may in fact have arare condition that requires further investigations to avoid serious health implications.
The severity of the issue is summarized by a state x ∈ [0, 1]. The doctor has threepossible actions: to send the patient home to rest (labeled as action a1 ), to prescribe
the usual medication for the common symptom (labeled as action a2 ), or to refer thepatient for further investigation (labeled as action a3 ). Depending on the severity of
the issue, x, the doctor’s gross payoffs from these actions are given by the quadratic
loss relative to the respective ideal states 0, 1/2, and 1:
U (a1 , x) = −x2 ,U (a2 , x) = ( 14 − θ) − ( 12 − x)2 ,U (a3 , x) = −(1 − x)2 ,where θ ∈ (0, 1/2) is a parameter capturing the importance of the extreme actions a1
and a3 relative to the middle action a2 . In other words, a1 would be ideal for rather
healthy patients, i.e., patients with small x’s, a2 for x’s around the middle, and a3
for severe conditions. For convenience, fix a default action, say, a2 , and consider theutility u(x, a) from each action a ∈ {a1 , a2 , a3 } as compared to the default action,u(x, a) = U (x, a) − U (x, a2 ). Thus,u(a1 , x) = θ − x,u(a2 , x) = 0,u(a3 , x) = θ − (1 − x),as shown in Figure 1.
The doctor is initially uninformed about x. Note that the doctor does not need to
discover x precisely, she only needs to find out enough to choose a treatment. To learn
about x, the doctor asks several yes/no questions according to an inquiry that starts
with an initial question, specifies follow-up questions depending on earlier answers,

12

HU AND ZAPECHELNYUKIs x > θ/2?
true
Is x > 1 − θ?

Is x < θ?
false

true
a1

Is x > 1/2?
true

false

a3

a1

A.

false

Is x > 1 − θ?
true
a3

true

false

a3

Is x < θ?

false
a2

B.

true
a1

a1

false
a2

C.

Figure 2. Examples of inquiries

and prescribes actions. Each question is formulated as a proposition “x ∈ Y ” withY ⊂ [0, 1], which can be true or false. Examples of inquiries are shown in Figure 2.
A cost λ of a question is interpreted as the opportunity cost of time and cognitiveeffort spent on a patient that could have been spent to diagnose and treat other
patients. Indeed, in Croskerry et al. (2013) this cognitive cost is regarded as animportant factor that affects the doctor’s investigation and the resulting decision.
If the doctor reaches a decision a after asking ` questions, the resulting payoff isthus u(a, x) − λ`a . For example, in the inquiry B (Figure 2), the cost is λ if a1 isreached, and it is 2λ if either a2 or a3 are reached. The doctor would like to choose
an inquiry that maximizes her expected utility net of the cost of inquiry, given his
prior knowledge, modelled as a prior distribution over x.
As a benchmark, suppose that there is no cost of asking questions, λ = 0. Then,
as apparent from Figure 1, it is optimal to choose a1 when x is below θ, to choose
a2 when x is between θ and 1 − θ, and to choose a3 when x is above 1 − θ. Inquires
B and C (Figure 2) both achieve this outcome. However, when questions are costly,
λ > 0, the two inquiries differ significantly in terms of the cognitive cost: when action
a1 is taken, it takes only one question in inquiry B but it may take three in inquiry
C; when action a2 is taken, it takes two questions in B but three in C. Moreover,once we take the cognitive cost into account, the doctor may find it optimal to trade
off some accuracy of information about x to reduce the cost of inquiry; in otherwords, the optimal information partition would be endogenously determined by theOPTIMAL INQUIRY

13

cognitive cost. As we shall see later, because of these considerations inquiries B and
C in Figure 2 are neither equivalent to each other nor optimal.
3. Optimal Inquiries
We first establish two principles of optimality of inquiries. We show that optimalinquiry is dynamically consistent. We also show that an inquiry can be summarized
by its payoff-relevant outcome, and we then characterize the outcomes of optimalinquiries. Based on these principles, we will express the task of finding an optimalinquiry as a simple finite optimization problem, and analyze the properties of its
solution.
3.1. Dynamic Consistency. We show that it makes no difference whether the DMcommits to an optimal inquiry ex ante or she is free to update her strategy at any
interim stage, and hence, the inquiry is not only ex ante, but also sequentially optimal.
We use the following notion of dynamic consistency. Let Q = hN, T, σ, χ, di ∈ QX
be an inquiry. Consider a node n ∈ N . At that node, the DM infers that the state isin In (Q). Observe that every possible play after reaching n is itself an inquiry, whose
initial set of states is In (Q). Let us refer to it as a sub-inquiry at node n ∈ N . The
set of all possible sub-inquiries at n given information In (Q) is QIn (Q) . Denote by Qn
the specific sub-inquiry at n that prescribes to play according to the original inquiry
Q.
Suppose that the DM initially follows inquiry Q but, upon reaching node n, shereevaluates her strategy: whether to follow the original plan Qn or to deviate to another sub-inquiry Q̂ ∈ QIn (Q) . Let Wn (Q̂; λ) be the DM’s expected payoff conditionalon reaching node n if she chooses sub-inquiry Q̂ upon arrival to n. We say that theoriginal inquiry Q is dynamically consistent if no deviation is beneficial at any node.
Definition 3.1. An inquiry Q = hN, T, σ, χ, di is dynamically consistent if, for eachnode n ∈ N ,Wn (Qn ; λ) =max Wn (Q̂; λ).

(4)

Q̂∈QIn (Q)Note that dynamic consistency implies that the DM behaves in a sequentially optimal way at each terminal node as well. Specifically, the DM chooses an action thatmaximizes her expected payoff given the information at that node. That is, if Q is

14

HU AND ZAPECHELNYUKdynamically consistent, then, for each terminal node t ∈ T , the action dt must be asolution of
Z
max
a∈A

u(a, x)G(dx|It (Q)).

(5)

x∈It (Q)We have the following theorem.
Theorem 3.1. Every optimal inquiry is dynamically consistent.
The theorem is proved by the typical argument for dynamic consistency. At any
node n, the proof shows the following: if there is a sub-inquiry that is superior tothe original one, then one can modify the original inquiry by plugging in the superior
sub-inquiry after node n and obtain a strictly higher ex ante payoff. Crucial to thisargument, however, is the fact that at any node n, the cognitive cost paid for thequestions asked to arrive at n is sunk because of the additive-cost structure.2
Theorem 3.1 demonstrates the procedural rationality of the optimal inquiry, aproperty that cannot be discussed without the explicit formulation of the decisionmaking process. Moreover, as we shall see later, although the optimal inquiry features
certain “biases” from the perspective of a model without the cognitive cost, thesebiases are not driven by inconsistent behavior between different stages of the decision
process, they are an inevitable part of the optimal response to the cognitive cost.
3.2. Outcomes. Here we show that it suffices to describe an optimal inquiry by its
payoff-relevant outcome. The outcome consists of two parts: the form and the content. The form consists of a consideration set—which is a subset of decisions that canbe implemented in that outcome—and a length profile—which specifies how manyquestions are asked to reach each decision in the consideration set. The content
consists of a collection of categories that forms an information partition, which describes the posterior information about the state upon reaching each decision in theconsideration set.
We begin by observing that if an inquiry is optimal, then every node must be
reached with positive probability. Indeed, if there was a node n that is only reachedwith probability zero, then, in some predecessor node n0 , the proposition χ(n0 ) or its
2For example, in the model where the ex-ante cost of inquiry is proportional to the longest pathin the tree, at any interim node n the cost is not sunk, and, thus, dynamic consistency may not
hold.

OPTIMAL INQUIRY

15complement would have had measure zero, so the associated costly question would
have been redundant.
Lemma 3.1. If an inquiry Q = hN, T, σ, χ, di is optimal, then every node n ∈ N isreached with positive probability.
Next, we observe that an optimal inquiry cannot induce the same action in two or
more terminal nodes. Indeed, if it was the case, there would have been no need to
distinguish between these terminal nodes, so the number of costly questions in theinquiry could be reduced. For example, in inquiry C (Figure 2), action a1 is chosen
after a single question when x ∈ [0, θ/2] and after three questions when x ∈ (θ/2, θ).
Let us merge these conditions into a single proposition, x < θ. Asking whether x < θ
first, and choosing a1 if true, and otherwise asking the remaining question, whetherx > 1 − θ, leads us to inquiry B. Inquiry B chooses each action on the same subsetof states as inquiry C, but asks fewer questions. This observation leads us to thefollowing property of optimal inquiry.
Lemma 3.2. If an inquiry Q = hN, T, σ, χ, di is optimal, then dt 6= dt0 for all pairsof distinct terminal nodes t, t0 ∈ T .
An immediate implication of Lemma 3.2 is that each terminal node corresponds toa unique action in A. In what follows, we will identify terminal nodes with actions
they induce. Specifically, let D(Q) be the set of actions induced in inquiry Q. We willrefer to D(Q) as the consideration set, and to actions in D(Q) as decisions. The set
D(Q) can be a proper subset of A, with the interpretation that the DM will process
information in a way that would lead her only to consider a strict subset of all feasible
actions. For each decision a ∈ D(Q), let `a (Q) denote the length of inquiry leading
to the terminal node where a is chosen, and let Ia (Q) denote the information set
or the category induced by Q in that terminal node. Let `(Q) = {`a (Q)}a∈A andI(Q) = {Ia (Q)}a∈A . We will refer to the triple (D(Q), `(Q), I(Q)) as the outcome
profile induced by Q, and hence (D(Q), `(Q)) describes the form of the inquiry Q andI(Q) describes the content. Note that the form of an inquiry is discrete in nature
while the content is continuous.
Note that actions in A − D(Q) are never chosen; for every action a ∈ A − D(Q) we
use the notation `a (Q) = ∅ and Ia (Q) = ∅. Since under the inquiry Q, a decision a

16

HU AND ZAPECHELNYUKwith a shorter inquiry length grasps the DM’s attention first before another a0 witha longer length, we may say that the DM prioritizes a decision a over another a0 if`a (Q) < `a0 (Q).
For illustration, consider inquiries in Figure 2:
Inquiry A’s outcome: D = {a1 , a3 },` = (1, ∅, 1), I = {[0, 1/2], ∅, (1/2, 1]}.

Inquiry B’s outcome: D = {a1 , a2 , a3 }, ` = (1, 2, 2), I = {[0, θ), [θ, 1 − θ], (1 − θ, 1]}.
Inquiry C has the same action a1 in two different terminal nodes, and thus cannot
be represented this way. Moreover, by Lemma 3.2, inquiry C cannot be optimal. In
fact, this example also shows that the subtlety needed for the proof of Lemma 3.2.
Indeed, the two terminal nodes for which the same action, in this case a1 , is taken,
occurs at different branches of the inquiry tree. Thus, to show that the inquiry C
is suboptimal, we construct another inquiry (namely, inquiry B) which leads to thesame information partition but with the categories for the two terminal nodes withaction a1 combined into one single category, and with all the inquiry shorter than thecorresponding ones in inquiry C.
This construction is based on a more general principle that leads to the following
characterization of inquiry outcomes. Let D ⊆ A, let ` = (`a )a∈A ∈ (N ∪ {∅})|A|
be a length profile, and let I be a partition of X. Denote by Z the set of such
triples (D, `, I). We say that an outcome profile (D, `, I) ∈ Z is implementable if
there exists an inquiry Q ∈ QX that induces this outcome profile, that is, (D, `, I) =(D(Q), `(Q), I(Q)). The following lemma characterizes implementable outcomes.
Lemma 3.3. An outcome profile (D, `, I) ∈ Z is implementable if and only ifX2−`d = 1.

(6)

d∈D

Equality (6) follows from the Kraft inequality in information theory that characterizes the path lengths of binary trees. Here we have equality instead of inequality because in our inquiry trees each non-terminal node has precisely two outgoingbranches.
Lemma 3.3 implies that the set of feasible outcomes only depends on the formof an inquiry, that is, the consideration set D and the length profile `, but it does
not depend on the content, that is, the information partition. In other words, theOPTIMAL INQUIRY

17

lemma shows that for any given form (D, `) that satisfies (6) and any given content I
with |D| non-empty categories, we can construct an inquiry with the corresponding
outcome. However, as we shall see later, not every information partition is optimal,
and that the form and the content are interdependent.
To illustrate this lemma, let us return to our example in Section 2.5. There are only
three actions in the example. Thus, only three length profiles for D = {a1 , a2 , a3 }satisfy equality (6), namely, ` = (1, 2, 2) (see inquiry B in Figure 2), ` = (2, 1, 2), and` = (2, 2, 1). If we increase the number of actions to four, so D = {a1 , a2 , a3 , a4 },then there will be 13 length profiles that satisfy (6), namely, the uniform profile,(2, 2, 2, 2), and 12 distinct permutations of the extreme profile (1, 2, 3, 3). This setof length profiles increases exponentially with the size of D. However, later on wedevelop optimal conditions with which we can identify a smaller candidate set.
Lemmas 3.1–3.2 lead us to a key observation. An outcome (D, `, I) captures all weneed to know to evaluate the DM’s expected payoff of an inquiry that leads to thatoutcome. Indeed, suppose that two different inquiries Q and Q0 implement the sameoutcome (D, `, I). Then, by (2), we have0

W (Q; λ) = W (Q ; λ) =

XZ
d∈D




u(d, x) − λ`d G(dx).

(7)

x∈Id

Moreover, Lemma 3.3 implies that for any outcome (D, `, I) that satisfies (6) there
exists an inquiry with that outcome.
Thus, without loss of generality, an inquiry can be equivalently represented by its
outcome (D, `, I). An outcome of an optimal inquiry will be called optimal outcome.

3.3. Optimal Inquiries. We have shown an inquiry can be summarized by its outcome (D, `, I). Moreover, Lemma 3.3 shows that the information partition I does
not affect whether or not an outcome profile if implementable. This characterization
allows us to solve the optimal inquiry problem in two stages. We first fix an arbitrary
form (D, `) that satisfies (6), and solve for the optimal content I = I ∗ (D, `). Then,
we maximize over all possible forms of (D, `)’s.
In the first stage, taking (D, `) as given, we find an information partition I ∗ (D, `)that maximizes the DM’s expected utility. Specifically, let I ∗ (D, `) = {Id∗ (D, `)}d∈A ,18

HU AND ZAPECHELNYUK

θ−λu(a1 , x) − λu(a3 , x) − 2λu(a2 , x) − 2λ

−2λ

x
0

θ+λ

1−θ

1

Figure 3. Determination of I ∗ from ` = (1, 2, 2)
where x ∈ X : u(d, x) − λ`d > maxa∈D−{d} u(a, x) − λ`a , d ∈ D,
Id∗ (D, `) =(8)∅,d ∈ A − D.
That is, for each decision d that can be chosen, so d ∈ D, Id∗ (D, `) is the set of states
where action d is the unique best-response action among all actions in D when the
DM takes into account the cost of inquiry associated with each action. Note thatI ∗ (D, `) is a partition of X according to Definition 2.1, because, by assumption (A3 ),
S
the set X − d∈D Id∗ (D, `) has measure zero.
We have the following lemma.
Lemma 3.4. If (D, `, I) is an optimal outcome, then I is identical to I ∗ (D, `) up toa measure zero set.
The key observation to understand this lemma is that I ∗ (D, `) is the optimal information partition given the form, (D, `), as the DM chooses the unique best-response
action for each state x ∈ X, except for a measure zero of states. Indeed, given (D, `),the “effective” utility for the DM by choosing action d is now u(d, x) − λ`d , when
taking the cognitive cost into account.
Lemma 3.4 is essential to solving the optimal inquiry. As mentioned earlier, theoutcomes of an inquiry include both a continuous element I and discrete element(D, `). Lemma 3.4 shows that of the optimal content I is determined by the form(D, `) through I ∗ (D, `). It also generates candidate optimal inquiries effectively. For

OPTIMAL INQUIRY

19

example, it immediately implies that Inquiry B in Figure 2 is suboptimal. Indeed,
Inquiry B has the partition
Ia1 = [0, θ), Ia2 = [θ, 1 − θ], Ia3 = (1 − θ, 1],
as shown in Figure 1. But, given the length profile is ` = (1, 2, 2) with the associatedcost λ of each question, the DM can do strictly better by using the partition
Ia∗1 = (0, θ + λ), Ia∗2 = (θ + λ, 1 − θ), Ia∗3 = (1 − θ, 1),(9)

as shown in Figure 3. Indeed, on the interval (θ, θ + λ), the DM optimally chooses
decision a1 , even though a2 would have been a better decision absent cognitive cost.
This is because a1 needs one less question to ask and, thus, saves λ, while u(x, a2 ) −u(x, a1 ) < λ for any state x in that interval.
As a consequence of Lemma 3.4, the maximization problem (3) can now be reduced
to the choice of the form, (D, `). Let F ∗ be the set of all forms (D, `) with D ⊆ A and` satisfying (6). The DM chooses a form (D, `) ∈ F ∗ , and the outcome is determinedby (D, `, I ∗ (D, `)). By Lemmas 3.2–3.4, we obtain the following characterization of
optimal inquiries.
Theorem 3.2. An inquiry Q is a solution of (3) if and only if the pair (D(Q), `(Q))
is a solution ofmax ∗(D,`)∈F

XZ
d∈D




u(d, x) − λ`d G(dx).

(10)

x∈Id∗ (D,`)

Because F ∗ is a finite set, and the expected utility is bounded for each d ∈ D byassumption (A1 ), we establish the existence of optimal inquiry.
Corollary 3.1. An optimal inquiry exists.
Another straightforward implication of Theorem 3.2 is that the optimal inquiry
satisfies a minimal rationality property: the independence of irrelevant alternatives
principle. This property is also known in the literature as “attention filter” (Masatlioglu et al., 2012). It is defined as follows. Suppose that DM’s consideration set D
is a strict subset of A. Then, the attention filter property requires that, for any asmaller action set A0 ⊂ A that contains D, the optimal consideration set is still D.
This property holds under optimal inquiry.

20

HU AND ZAPECHELNYUK

separate {{Id2 , Id4 }, Id1 } from Id3

separate {Id2 , Id4 } from Id1

separate Id2 from Id4

Id2

Id1

Id4

separate {Id2 , Id4 } from {Id1 , Id3 }Id3

separate Id2 from Id4

Id1(a) ` = (2, 3, 1, 3)

Id4

separate Id1 from Id3

Id1

Id3

(b) ` = (2, 2, 2, 2)

Figure 4. Huffman coding with |D| = 4 and P(Id3 ) > P(Id1 ) >P(Id2 ) > P(Id4 )Corollary 3.2. If (D, `, I) is an optimal outcome for action set A, then it is also an
optimal outcome for each action set A0 such that D ⊆ A0 ( A.
Although F ∗ is a finite set, as mentioned earlier, it can be a relatively large set.
However, there is an additional optimality condition that helps determine the optimal
length profile more efficiently, and it also helps characterize its behaviour. Indeed,
while Lemma 3.4 characterizes the optimal categories for a given length profile, one
can also look for optimality conditions for the length profile for given categories.
Specifically, given a partition {Id }d∈D , the optimal ` must minimize the average lengthwith respect to the distribution such that `d occurring with probability P(Id ) andsubject to the constraint (6). This is a well-known problem in information theory, andthe solution is described by the algorithm called Huffman coding. Here we show how
the algorithm works for |D| = 4. The generalization to arbitrary D is straightforward.
We refer to, e.g., Thomas and Cover (2006), Section 5.6, for formal details.
Consider D = {d1 , d2 , d3 , d4 } with the following probability ranking of the decisions:P(Id3 ) > P(Id1 ) > P(Id2 ) > P(Id4 ).

(11)Specifically, letP(Id1 ) = 0.25, P(Id2 ) = 0.2, P(Id3 ) = 0.4, P(Id4 ) = 0.15.
In stage t = 0, let us define p0d = P(Id ) for each d ∈ D, and order the decisions
according to their probabilities: p0d3 = 0.4 > p0d1 = 0.25 > p0d2 = 0.2 > p0d4 = 0.15.

OPTIMAL INQUIRY

21

We then merge the last two, {d2 , d4 }, and treat the pair as a single decision whose
probability is p1{d2 ,d4 } = 0.2 + 0.15 = 0.35. The other decisions and their probabilities
stay the same: p1d1 = p0d1 and p1d3 = p0d3 . In stage t = 1, we reorder the decisions ofstage t = 0 according to their probabilities: p1{d3 } = 0.4 > p1{d2 ,d4 } = 0.35 > p1{d1 } =0.25. We then merge the last two, {{d2 , d4 }, {d1 }}, and treat the set as a single
decision with probability p2{{d2 ,d4 },{d1 }} = 0.35 + 0.25 = 0.6. In stage t = 2, again, wereorder the decisions of stage t = 1: p2{{d2 ,d4 },{d1 }} = 0.6 > p2{d3 } = 0.4. We then merge
the remaining decisions to obtain {{{d2 , d4 }, {d1 }}, {d3 }}. Finally, we construct theinquiry tree by unraveling the nested set {{{d2 , d4 }, {d1 }}, {d3 }} from the top layer
down, as shown in Figure 4(a). The length profile for this tree is ` = (2, 3, 1, 3).
If we consider the same initial probability ranking of the decisions (11) but different
probabilities,
P(Id1 ) = 0.25, P(Id2 ) = 0.2, P(Id3 ) = 0.37, P(Id4 ) = 0.18,
then the Huffman coding procedure yields a different inquiry tree, with length profile` = (2, 2, 2, 2), as shown in Figure 4(b). In fact, as follows from the next proposition,` = (2, 3, 1, 3) and ` = (2, 2, 2, 2) are the only length profiles that can be obtained
given the probability ranking (11).
We now show that for any candidate consideration set D, the optimal length profile` is determined by the partition I. Moreover, decisions that take longer to reach areless likely to be chosen.
Proposition 3.1. If (D, `, I) is an optimal outcome, then:(a) ` is obtained from the Huffman coding w.r.t. the distribution {P(Id )}d∈D ;(b) for all d, d0 ∈ D, if P(Id ) > P(Id0 ), then `d ≤ `d0 .
Proposition 3.1 (b) highlights a negative correlation between the ex ante probabilityof a decision and the inquiry length to reach that decision. It allows us to simplify thesolution of problem (10), but reducing the set of candidate outcomes (D, `, I ∗ (D, `)).
Indeed, as illustrated by the above example with D = {d1 , d2 , d3 , d4 } and probability
ranking of decisions (11), out of 13 feasible length profiles that satisfy (6), only twoare consistent with (11), namely, ` = (2, 3, 1, 3) and ` = (2, 2, 2, 2).
Proposition 3.1(a) is particularly useful to determine optimal ` for λ small. Indeed,for λ sufficiently small, the optimal length profile is determined by the information

22

HU AND ZAPECHELNYUK

Ia∗10x̄01 x̄1

Ia∗2

0.35
0.35 + λx̄02 = x̄2

Ia∗3

0.65

1

(a) Optimal information partitions

D = {a1 , a2 , a3 }D = {a1 , a3 } D = {a2 }` = (1, 2, 2)` = (1, ∅, 1)` = (∅, 0, ∅)

λ
0

0.036

0.1

(b) Optimal (D, `) as λ increases

Figure 5. Optimal inquiry for θ = 0.35 as λ increases
partition I 0 = {Ia0 }a∈A that is optimal under standard Bayesian analysis with zero
cost, λ = 0. Specifically, when the DM learns the state x for free, she simply chooses
the best action for each x ∈ X. That is, for each a ∈ A,
Ia0 =



0



x ∈ X : u(x, d) > 0 max u(x, a ) .
a ∈A−{a}As λ increases, the information partition is continuously adjusted according to I ∗ (D, `)given by (8). However, as long as λ is small enough, the optimal consideration set remains D = A, and the optimal length profile remains the same as the one determinedby the Huffman coding for λ = 0.
For illustration, we return to the example in Section 2.5 with A = {a1 , a2 , a3 } andutility functions given by Figure 1, and the parameter θ = 0.35. When λ = 0, we
have D = A and P(Ia01 ) = P(Ia03 ) = 0.35 and P(Ia03 ) = 0.3. The optimal information
partition for λ = 0 is shown in Figure 5(a), where x̄01 and x̄02 denote the thresholds
between the partition elements. Using the Huffman coding, we obtain length profiles(1, 2, 2) and, by symmetry, (2, 2, 1). Let us fix ` = (2, 2, 1).
When λ ∈ (0, 0.036), the same (D, `) remain optimal, but optimal information
partition I ∗ (D, `) is adjusted to take into account the cost, namely, it is given by(9). The optimal information partition for such λ is shown in Figure 5(a), where x̄1
and x̄2 denote the thresholds between the partition elements. It can be seen thatthe category Ia∗1 where a1 is chosen expands as λ increases. This expansion reflects
the general principle to resolve the key trade-off in our setting. On the one hand,
the DM likes the categories to match with the benchmark partition I 0 to achieve
higher utilities. On the other hand, the DM likes to have shorter inquiry lengths in
expected terms. As λ increases, the latter becomes more important, and the DM
optimally adjusts her categories to shorten the average inquiry length. Indeed, as Ia∗1

OPTIMAL INQUIRY

Ia∗1
0

x̄01

x̄1

Ia∗2 x̄02 = x̄2 Ia∗3

0.335
0.335 + λ

0.665(a) λ < 0.01 and ` = (1, 2, 2)

Ia∗1
1

0

23

x̄1

x̄01

Ia∗2

0.335
0.335 − λ

x̄02

x̄2

Ia∗3

0.665
0.665 + λ1(b) λ > 0.01 and ` = (2, 1, 2)

Figure 6. Optimal information partitions for D = {a1 , a2 , a3 } and θ = 0.335.
is associated with length 1 and others with length 2, expanding Ia∗1 leads to shorter
expected inquiry length.
For λ higher we have a similar effect but at the extensive margin by changing
the consideration set. That is, for λ ∈ (0.036, 0.1), the optimal D becomes {a1 , a3 },and a2 is no longer considered. As a result, at λ = 0.036, the category Ia∗1 expands
discontinuously, from (0, 0.35 + λ) to (0, 0.5), with a discrete jump of average inquiry
length to a lower level, because of the change of the optimal consideration set.
Lastly, for λ > 0.1, it is optimal for the DM not to ask any questions, and simply
choose the ex-ante optimal decision a2 . Figure 5(b) shows how the optimal pair (D, `)changes as λ increases.
Curiously, there can also be changes to optimal information partition due to changes
in the optimal length profile alone, while the consideration set remains the same. To
illustrate this, consider the same example, but now with θ = 0.335. In this case, for
λ < 0.043, optimal D = A. As before, for λ very small, an optimal length profileis (1, 2, 2), and the optimal information partition is shown in Figure 6(a). However,in this case for λ > 0.01, the optimal length profile changes to (2, 1, 2), and the
optimal information partition becomes as shown in Figure 6(b). This curious switch
illustrates the aforementioned principle that the DM likes to shorten average inquiry
as the cognitive cost rises. Indeed, as λ increases, the expected inquiry length of thelength profile (2, 1, 2) with the associated category I ∗ decreases faster than that of(1, 2, 2), because with (2, 1, 2) the expansion of Ia∗2 happens on both sides while with
(1, 2, 2) on one side. As a result, the increasing importance of shorter inquiry length
prompts the DM to switch to (2, 1, 2) for λ’s sufficiently high.
These two examples show that, as the cognitive cost rises, the DM respond byeither expand the category with the shortest inquiry length (as in the left panel of
Figure 5), or by dropping some decisions out of consideration altogether (as in the

24

HU AND ZAPECHELNYUKleft panel of Figure 5 for higher λ’s). Note that, consistent with Proposition 3.1, in
the former case the expanded category is also the one with higher probability, andhence this expansion leads to lower average inquiry length, so as the latter case. In
the next section we discuss this effect more generally.
4. Attention Span
Our model of inquiry can be interpreted as an attention strategy, whereby the DM
focuses on various decisions during her inquiry process. With this interpretation, anatural question is then how cognitive cost affects the DM’s attention span, defined as
how long she would concentrate on the task of gathering information before making
a decision. Formally, we measure attention span in our framework as the expectedinquiry length given by¯`(D,`, I) =X`d P(Id ).

(12)

d∈D

Importantly for our purpose, it captures whether there is a lot of probability weight
on a few decisions with short inquiry length, or whether this weight is more spread¯out among many decisions. A smaller `(D,`, I) means a shorter attention span.
In the extreme, the DM has no attention span at all when she chooses a single¯
decision without asking any questions, in which case we have `(D,`, I) = 0. The
opposite extreme occurs when the lengths are all equal and the probabilities spreadout. Given an outcome (D, `, I), we say that the length profile ` is uniform if it
assigns the same length to all decisions in D, so `d = `d0 for all d, d0 ∈ D. In otherwords, the inquiry outcome is uniform if the same number of questions is asked for allstates of nature. Note that this can only happen if |D| = 2k for some k ∈ {0, 1, ...}.
Since under |D| = 2k it is always feasible to set all lengths equal to k, k is also an
upper bound for the expected inquiry length for an optimal inquiry. In other words,¯
in any optimal inquiry with |D| = 2k , we have `(D,`, I) ≤ k, and this upper boundis achieved if and only if the length profile is uniform.
Our key result in this section is that, under optimal inquiry, a higher cognitive cost
shortens the DM’s attention span. Specifically, we show that the expected inquiry
length decreases with λ, and strictly so when the optimal inquiry is non-uniform.
Since the expected length may not be differentiable w.r.t. λ, we use the following
notion of strict monotonicity. A function is strictly increasing (decreasing) locally

OPTIMAL INQUIRY

25

at λ if there exists an interval [λ1 , λ2 ] with λ1 < λ2 and λ ∈ [λ1 , λ2 ] such that thefunction is strictly increasing (decreasing) on [λ1 , λ2 ].
Theorem 4.1. Given λ, let (Dλ , `λ , I λ ) be an optimal outcome. The average inquiry¯ λ , `λ , I λ ) is decreasing in λ. Moreover, it is strictly decreasing locally at λlength `(D
whenever `λ is not uniform.
Theorem 4.1 shows that higher cognitive cost always shortens the attention span,and strictly so as long as the optimal inquiry length is not uniform. The intuition for
Theorem 4.1 is based on the following trade-off that the optimal inquiry resolves. On
the one hand, to achieve a high (expected) utility from actions, it needs to minimize
the mismatch between its category for the action and the set of states for which theaction is ex post optimal; on the other hand, it needs to minimize the expected lengthof inquiry. As the cognitive cost increases, the latter motive becomes more important,and optimal inquiry shifts probabilities toward categories with shorter inquiries at theexpense of more mismatches.
This preference for shorter inquiries generates a “bias” if we compare the information partition thus generated to the ones that would be used by a Bayesian DM underzero cognitive cost. We call this effect confirmation bias. The bias is endogenously
determined by the cost and the utility function. As shown in Figure 6, in case of
λ < 0.01 the bias favors action a1 by expanding the set of states where a1 is chosen(at the detriment of a2 ), but for λ > 0.01 it favors action a2 (at the detriment of botha1 and a3 ).
As mentioned earlier, this bias is generated by the motive to decrease the expectedinquiry length, and this can be achieved by adjusting the inquiry either through theform or through the content. The content affects the intensive margin, and the DM
can increase the probability of choosing decisions with shorter inquiry length. The
form affects the extensive margin, and the DM can simply drop certain actions fromthe consideration set and in this way the overall inquiry length may be reduced. The
former factor is addressed in detail in the next section. In the next subsection, we
analyze the latter, namely how optimal consideration set is determined.
4.1. Optimal Consideration Sets. In our setting, a consideration set can be regarded as a set of actions that the DM deems “cognitively viable”, and any action

26

HU AND ZAPECHELNYUK

outside this set is simply ignored in the decision-making process, even though it might
be ex post optimal. One important factor that determines which actions are viable
is the underlying preferences. Specifically, it is useful to distinguish two actions only
if they produce sufficiently different payoffs in different states of nature. In contrast,
if two actions are similar, it will not be worthwhile to differentiate them. Formally,let δ(a0 , a00 ) measure how close actions a0 and a00 are in the payoff space:δ(a0 , a00 ) = sup |u(a0 , x) − u(a00 , x)|.
x∈X

The following result shows that a consideration set will optimally drop one of the twosimilar actions.
Proposition 4.1. If actions a and a0 are such that δ(a0 , a00 ) < λ, then at most oneof them will be in the optimal consideration set.
Next, we show that when the cognitive cost is small enough, then all actions are
optimally considered, and when the cognitive cost is large enough, then the DM asks
no questions and chooses the same action in all states of nature.
Proposition 4.2. There exist two thresholds λ2 > λ1 > 0 such that for all λ < λ1 ,the optimal consideration set is Dλ = A; and for all λ > λ2 , the optimal consideration¯ λ ) = 0.
set is a singleton, |Dλ | = 1, and `(Z
It is tempting to generalize Proposition 4.2 by conjecturing that, as λ increases, theoptimal consideration set monotonically shrinks in the set inclusion order. However,this is not true in general. To illustrate this, consider the example in Section 2.5 with
A = {a1 , a2 , a3 }, parameter θ ∈ (0, 1/2) that captures the preference for extreme
actions a1 and a3 relative to middle action a2 , and utilities given by Figure 1.
Figure 7 shows how the optimal consideration set depends on the cost λ and thepreference parameter θ. We point out three features of the optimal inquiry that may
be of interest. First, the optimal consideration sets can be disjoint: for a fixed λthat is relatively high (say, for λ = 0.1), as θ increases, the optimal D changes from{a2 } to {a1 , a3 }. In words, preferences can affect the “cognitive viability” faced by
the DM, which can change in a discontinuous way. Second, the optimal consideration
sets can change by multiple actions at a time: for a fixed θ that is relatively low (say,for θ = 0.2), as λ increases, the optimal D changes from {a1 , a2 , a3 } to {a2 }. Thus,

OPTIMAL INQUIRY

27

λD = {a2 }0.0625D = {a1 , a3 }D = {a1 , a2 , a3 }
θ
0

0.33

0.5

Figure 7. Optimal consideration sets
for two DM’s with exactly the same preference, a small difference of cognitive cost
can make one DM to consider all actions while making the other to consider only
one. Third, an option can be phased out and then phased back in. For a given θthat is close to 1/3, say, for θ = 0.335, the optimal consideration set changes from
{a1 , a2 , a3 } to {a1 , a3 } to {a2 }, so action a2 is dropped, but then reintroduced.
4.2. Example with Independent Values. In the above example where the stateis of one-dimensional, there is no natural sense of how to rank the actions underassumption (A2 ). However, there is a natural ranking in environments where thevalues of the actions are independently distributed. Here we study how this ranking
affects the optimal consideration set.
Consider the model where X = RL and A = {a1 , ..., aL }, withu(al , x) = xl for all l = 1, ..., L.

(13)

Assume that the values x1 , ..., xL are independently distributed. Specifically, each xl
has a distribution Gl , and G(x) = ΠLl=1 Gl (xl ). We have the following result.
Proposition 4.3. Suppose that G1 F OSD G2 F OSD · · · F OSD GL . Then there
exists K ∈ {1, ..., L} such that D = {a1 , ..., aK } is the optimal consideration set.
According to Proposition 4.3, when the actions are ranked by the first-order stochastic dominance, an action can be in the optimal consideration set only if all thehigher-ranked actions are in there. Moreover, if an action al is not in the consideration set, the DM will ask no questions about its value (xl ), nor about values of all

28

HU AND ZAPECHELNYUKλD = {a1 }0.167
D = {a1 , a2 }
0.14
0.126

D = {a1 , a2 , a3 }D = {a1 , a2 , a3 , a4 }` = (2, 2, 2, 2)

0

0.033

` = (1, 2, 3, 3)

0.07

0.14 0.15

τFigure 8. Optimal consideration set and length profile

the lower-ranked actions. That is, the DM will only spend cognitive resources on thedimensions that she deems most likely to be optimal from the ex ante perspective.
To illustrate Proposition 4.3, consider the following example. Let L = 4 andA = {a1 , a2 , a3 , a4 }, and the utility is given by (13). Let τ ∈ [0, 0.15] be a parameter.
Suppose that the values x1 and x2 are each uniformly distributed on [0, 1], while x3
and x4 are each uniformly distributed on [−τ, 1 − τ ]. Clearly, x1 and x2 first-orderstochastically dominate x3 and x4 .
Figure 8 shows the optimal consideration sets and length profiles (up to the symmetry between a1 and a2 and between a3 and a4 ) for different values of τ and λ. Notethat when |D| = 1 and |D| = 2 are optimal (the white and the red areas, respectively), the expected values do not vary with τ , and hence the boundary between thewhite and the red areas is a horizontal line. The area where |D| = 3 is optimal (thegreen area) appears only if τ < 0.07, and the corresponding optimal length profile is(1, 2, 2). In all those areas, as λ increases, the adjustment comes from the extensive
margin, where the number of actions considered decreases from 3 to 2, to 1. This may
be regarded as focused consideration: the DM only asks questions about the first twoor three values and decides based on this evidence, but ignores any potential evidence
from other dimensions (e.g., x4 ).

OPTIMAL INQUIRY29

In contrast, in the area where |D| = 4 is optimal (the blue area), there is a shift
in the length profile. For small τ (to the left of the dotted line), the optimal length
profile is the uniform one, (2, 2, 2, 2), as the probabilities of each action being the
optimal one are not too different from one another. However, for large τ (to the rightof the dotted line), the optimal length profile will switch from the uniform one to theextreme one, (1, 2, 3, 3).
Thus, although the optimal inquiry does not adjust the consideration set D as
λ increases, it does change the inquiry strategy discontinuously: for higher λ’s, theaction a1 has the shortest length, and that set of states where a1 is chosen will expand
with λ. The latter may be interpreted as a confirmation bias in terms of the content
of the inquiry: among those actions the DM is willing to consider, she is happy to
expand the evidence to admit a certain action as acceptable (in this case, a1 ), which
would be her most likely action ex ante. In the next section we show that this is aprevalent feature of the optimal inquiry.
5. Confirmation Bias
We have seen from the previous section that, under optimal inquiry, the DM often
prioritizes some actions by asking questions that lead to these decisions first, andturning to other actions only if the initial answers are negative. Moreover, some actions may not be considered at all. This may be interpreted as a form of confirmation
bias in the extensive margin, as the DM searches for evidence to support higherpriority actions, and does not attempt to find evidence in support of actions outside
the chosen consideration set. This is mainly related to the form of the inquiry.
In this section, we turn to confirmation bias in terms of the content, taking the formof the inquiry as given. We will show that, given the form, (D, `), the DM optimallyexpands the categories associated with the more likely actions, relative to the zerocognitive-cost benchmark. This can be interpreted as the DM searching for evidence
to confirm the desirability of the actions in D that are most likely to be optimal.
To define confirmation bias, let us consider the zero-cost case as a benchmark, andcompare the set of states under which the most likely actions are taken under the
optimal inquiry with and without the cognitive cost. To do so, we first rank theactions according to their likelihood under the optimal inquiry. For a fix λ > 0, let(D, `, I) be an optimal outcome under cost λ, and let K = |D|. We order the actions

30

HU AND ZAPECHELNYUKin D according to how likely they are chosen under optimal inquiry, so D = {dk }K
k=1 ,such that
P(Id1 ) ≥ P(Id2 ) ≥ ... ≥ P(IdK ),(14)with a tie-breaking rule P(Idk ) = P(Idk+1 ) =⇒ `dk ≤ `dk+1 .
For each k = 1, ..., K −1, let Dk = {d1 , ..., dk } be the subset of k most likely actions in
D. Let Ekλ be the event that some action in Dk is preferred to all actions in D − Dk ,when the cost λ is taken into account:λEk = x ∈ X : 0max u(dk0 , x) − λ`dk0 >k =1,...,kmax

m=k+1,...,Ku(dm , x) − λ`dm

,

(15)

∗
d∈Dk Id (D, `) except, possibly, on ameasure zero set. In words, conditional on event Ekλ , the optimal inquiry almost
surely leads to an action in Dk . Similarly, let Ek0 be the event of choosing one of theIt can be easily seen that Ekλ coincides withS

actions from Dk in the zero-cost benchmark.
Definition 5.1. An inquiry Q with outcome (D, `, I) has confirmation bias if forevery order {dk }K
k=1 that satisfies (14),
Ek0 ⊆ Ekλ for all k = 1, 2, ..., K − 1.

(16)

It has strict confirmation bias if (16) holds, and there exists k ∈ {1, ..., K − 1} suchthat
Ekλ − Ek0 has a non-empty interior.

(17)

In words, the DM has confirmation bias when affected by the cognitive cost if, foreach k = 1, ..., K − 1, she confirms to k most likely actions: she chooses one of theseactions on a larger set of states, as compared to what she would have done without
cognitive cost. The difference Ekλ − Ek0 is the set of states where an error relative
to the zero-cost benchmark occurs. In any state that belongs to Ekλ − Ek0 , the DM
is biased, as she chooses an action that is suboptimal from the perspective of theBayesian DM who knows the state.
This definition formalizes the notion of confirmation bias usually adopted in psychology. According to Nickerson (1998), “it refers usually to unwitting selectivity
in the acquisition and use of evidence,” which he believes is also the definition usedby general psychologists. Our definition has the advantage of formally defining both

OPTIMAL INQUIRY

31

confirming sets and biases: the actions the DM confirms to are the most likely onesin her optimal strategy, and the errors are defined against the zero-cost benchmark.
Theorem 5.1. Every optimal inquiry has confirmation bias. Moreover, an optimalinquiry has strict confirmation bias if and only if its length profile is not uniform.
An immediate implication of Theorem 5.1 is that the probability distribution overactions over D = {dk }K
k=1 induced by the optimal inquiry first order stochasticallydominates that induced by the zero-cost benchmark:P(Ekλ ) ≥ P(Ek0 ), for each k = 1, ..., K − 1,(18)

and the inequality is strict for some k if the confirmation bias is strict.
Theorem 5.1 shows that the optimal inquiry always features confirmation bias, andit has strict confirmation bias when its length profile is not uniform. From Proposition
3.1(b) we know that the most likely actions are associated with shorter inquiry lengths.
At the same time, given the optimal length profile, the optimal information partition
given by (8) has the feature that decisions associated with shorter inquiry lengths
will be chosen on a larger set of states relative to the no-cost benchmark. These two
factors reinforce each other and give rise to the confirmation bias in our setting.
We have define confirmation bias as a comparison against the benchmark casewithout the cognitive cost. Now we show that, this bias grows as the cognitive cost
increases, in the sense that likelihood of choosing one of k ex ante most likely actions
under optimal inquiry only increases as λ goes up.
Definition 5.2. Confirmation bias is locally increasing at λ if P(Ekλ ) is locally increasing at λ for each k = 1, ..., K − 1. Moreover, confirmation bias is locally strictlyincreasing in λ if P(Ekλ ) is locally strictly increasing in λ for some k.
We have the following result.
Proposition 5.1. The confirmation bias is locally increasing in λ. Moreover, it isstrictly increasing locally at λ if and only if the optimal length profile at λ is not
uniform.
Proposition 5.1 shows that as the cognitive cost rises, the DM would optimally
make more “errors” and is biased more toward the most likely actions. This result is32

HU AND ZAPECHELNYUK

D = {a1 , a2 , a3 , a4 }` = (2, 2, 2, 2)` = (1, 2, 3, 3)
0

0.516

D = {a1 }λ

0.543

Figure 9. Optimal consideration set and length profile under i.i.d.
exponential distribution
closely related to Theorem 4.1, which states that the DM has more “focus” on most
likely actions. Naturally, as the cognitive cost rises, the DM prefers to shorten theinquiry length, and Proposition 5.1 shows that the way to optimally achieve that isby decreasing the accuracy of her categories in favour of most likely actions, which
are also actions associated with the shortest inquiry lengths.
This endogenous emergence of confirmation bias in our model gives novel implications to behaviour as a result of cognitive limitations. Here we give an example
that illustrates this novelty and the difference from the literature on rational inattention. Matějka and McKay (2015) show that symmetric actions will be treatedsymmetrically in the rational inattention model. In contrast, in our model the DM
may optimally treat symmetric actions asymmetrically to save the cognitive cost.
Consider the model of independent values, as in Section 4.2, with A = {a1 , a2 , a3 , a4 },X = R4+ , and each xl has exponential distribution Gl (xl ) = 1 − e−xl . Figure 9 depicts
the optimal consideration set and the optimal length profile as a function of λ, up
to the symmetry between the actions. As apparent from the figure, once λ increases
above 0.516 so it becomes too costly to differentiate all actions and treat them equally,
the DM prioritizes an arbitrary action (in this case, a1 ) and increases the set of states
where this action is chosen, eventually choosing this action alone as λ increases above
0.543. This implies endogenous bias: the DM is biased towards a1 because it is ex
ante more likely under optimal inquiry, but it is ex ante more likely because the DM
is biased towards it, even though this choice may be suboptimal ex post.
6. Case studies
We offer two case studies that illustrate the potential applications of our model.
There are two important social institutions where information processing primarily
takes the form of explicit inquiries: doctor visits and criminal investigations. In
recent years, research has indicated that misdiagnosis and wrongful convictions, both

OPTIMAL INQUIRY

33x ∈ [x̄1 , x̄2 ]?
true

false
x < x̄1 ?

laxative
true
rest“rest”
false

further investigation

0“extensive
investigation”“laxative”

x̄01 = θx̄1 = θ − λx̄02 = 1 − θx̄2 = θ + λ1

Figure 10. Optimal doctor inquiry tree (left) and optimal categories (right)of which impact the life of the people affected significantly, are in fact closely linked
to cognitive factors in the inquiries.
6.1. Medical Misdiagnosis. Singh et al. (2017) argue that “...diagnosis in primarycare (i.e., first-contact, accessible, continued, comprehensive and coordinated care)
represents a high-risk area for errors. PCPs typically face high patient volumes andmake decisions amid uncertainty.” They claim that the amount of errors is significant:“...a recent study estimated that about 5% of US adult patients experience diagnostic
errors (defined as missed opportunities to make a correct or timely diagnosis) [...]
every year.” They point out that diagnostic reasoning is an important factor: “Several
experimental studies have highlighted reasoning biases, in relation to both hypothesis
generation and information interpretation in PCP’s.” Croskerry et al. (2013) discuss
the mode of diagnostic reasoning (which they call “type 2”), which is the more typicalone and is subject to “biases”: “Our systematic errors are termed biases and there
are many of them–biases over a hundred cognitive and approximately one dozen orso affective biases (ways in which our feelings influence our judgment).”
Croskerry et al. (2013) also describe a case study, in which a patient complained
about constipation but actually was suffering from Cauda Equina Syndrome. Thissituation may be examined using our one-dimensional example with three actions:send the patient home to rest, prescribe a laxative, or refer for an extensive investigation. The value of each action depends on the true condition (the state x) of
the patient. Suppose that the utility function is as in Figure 1 (interpreting a1 as“rest”, a2 as “laxative”, and a3 as “extensive investigation”), the state x is uniformly
distributed, and θ < 1/3, so that “laxative” is the most likely correct action ex ante.
For a moderate value of λ, the optimal inquiry would assign the category associatedwith prescribing laxative with inquiry length of one. That is, the doctor should

34

HU AND ZAPECHELNYUKprioritize to enquire about whether or not it is best just to prescribe laxative. We
depict the optimal inquiry tree and the optimal categories in Figure 10, where x̄1
and x̄2 are thresholds between the categories under optimal inquiry, and x̄01 and x̄02
are thresholds under zero-cost benchmark. Theorem 5.1 predicts that the doctor
would prescribe laxative on a larger set of states (the interval [x̄1 , x̄2 ]), as comparedto the benchmark where the state of the patient can be discovered at zero cost (theinterval [x̄01 , x̄02 ]). Thus, in the intervals of states (x̄1 , x̄01 ) and (x̄02 , x̄2 )), the doctor
makes an error. This is consistent with the argument in Croskerry et al. (2013) thatthe error is due to the following: “The principle biases for the physician who saw
him in the clinic were framing, search satisficing and premature diagnostic closure.”In our model, framing and search satisficing can be explained by the confirmation
bias we identified. Indeed, as in our model, e.g., for x ∈ (x̄02 , x̄2 ), the doctor could
have continued the investigation and potentially reached the conclusion that a more
extensive investigation is needed, but stopped prematurely and prescribed a laxative.
An advantage of our model is that we can define “bias” rigorously. In this example,
when the doctor prescribes “laxative” instead of “extensive investigation” for x ∈(x̄B
2 , x̄2 ), we may call it an “error” and claim that the process is “biased”. However,
it is the process that is biased but not the decision per se. Moreover, relative to othermodels of imperfect information processing, our model is able to make prediction
about the inquiry process, in this case, about the “premature diagnosis,” and linkthat to the behavioural biases endogenously.
6.2. Wrongful Conviction. Gould and Leo (2010) review the literature on the extent and factors leading into wrongful convictions and believe that it is the process
and factors affecting the process that are important. In their words, “...it is better tounderstand the sources of wrongful convictions not so much as dichotomous causes– a witness correctly or incorrectly identified the defendant and the identification
directly led the jury to convict – but as contributing factors in a path analysis thatmight have been broken at some point before conviction.” Among the leading factors the article identifies, we are interested in “tunnel vision”, which is described in
Gould and Leo (2010) as “the more law enforcement practitioners become convincedof a conclusion – in this case, a suspect’s guilt – the less likely they are to consider
alternative scenarios that conflict with this conclusion.”

OPTIMAL INQUIRY“charge B”

0“no charge”

θBx̄01 = 1+θ
= x̄1
B

1−λ
x̄2 = 1+θA

35“charge A”
1x̄02 = 1+θ
A

1

Figure 11. Optimal categories for the policy inquiry
We illustrate this tunnel vision with the following example. Suppose that there
are two suspects, A and B, and one of them is surely guilty. Given all the possible
observables the police can investigate, suppose that state x ∈ [0, 1] represents theposterior belief that A is guilty. There are three actions: charge A, B, or neither,denoted by aA , aB , and a∅ , respectively. Suppose that the police obtains utility θA
if they charge A when A is guilty, θB if charge B when B is guilty, −1 if a wrong
suspect (either A or B) is charged, and 0 if neither. Thus,
u(aA , x) = θA x + (−1)(1 − x), u(aB , x) = θB (1 − x) + (−1)x, u(a∅ , x) = 0.
Assume that x is distributed according to the uniform distribution, and that 1 >θA > θB ≥ 0.52, so that ex ante the most likely suspect is A, and no action isdominated. We depict the optimal categories in Figure 11, where x̄1 and x̄2 arethresholds between the categories under optimal inquiry, and x̄01 and x̄02 are thresholds
under zero-cost benchmark. As indicated in the figure, a positive λ leads to an
expansion of the category for charging A, which was the primary suspect. In otherwords, the confirmation bias leads the police to lower the threshold of evidence needed
to charge suspect A relative to the benchmark case without such cost. Specifically,on the interval of states (x̄2 , x̄02 ), the police makes an error by charging A when theyshould have let them go. Moreover, this interval expands with λ. This may be an
explanation of the tunnel vision: the police under pressure to end the investigation
optimally focuses on the prime suspect and is willing to charge the prime suspecteven with relatively weak evidence.
Furthermore, if the cognitive cost were to rise even higher, the police would havefound it optimal to drop the no-charge option out of the consideration set altogether.
Thus, if we define “type-I” error as the situation where the police does not charge
anyone, and “type-II” error as the situation where police charges the wrong suspect,then the optimal inquiry is always biased toward the type-II error. That is, it always

36

HU AND ZAPECHELNYUK

has higher type-II error than the no-cost benchmark. This may be an explanation of
the tunnel vision: the police under pressure to end the investigation optimally focuses
on fewer options that what they would have considered with no such pressure.
Appendix A. Proofs
A.1. Proof of Theorem 3.1. To prove Theorem 3.1, we use the results presented
in Sections 3.2 and 3.3.
Let Q∗ = hN, T, σ, χ, di be an optimal inquiry, and let Z ∗ = (D∗ , `∗ , I ∗ ) be theoutcome implemented by Q∗ . By Lemma 3.2, D∗ = T . By Lemma 3.4, for eachd ∈ D∗ , category Id∗ is given by (8). To simplify notation, let Xn = In (Q∗ ) for eachn ∈ N . By Lemma 3.1,
P(Xn ) > 0 for all n ∈ N .

(19)

Fix a node n ∈ N . Let Tn ⊂ T be the set of terminal nodes that can be reachedfrom n under Q∗ . Note that if n is terminal (that is, if n ∈ T ), then Tn = {n}.
Let `n (Q∗ ) be the length of the path from no to n. Let Q∗n be the sub-inquiry at n
induced by the optimal inquiry Q∗ . Conditional on reaching n, the DM’s expectedˆ ∈ QXn is given by
payoff from a sub-inquiry Q̂ = hN̂ , T̂ , σ̂, χ̂, di


Z
X
1 
Wn (Q̂; λ) =u(dˆt , x) − λ`t (Q̂) G(dx|Xn ),(20)
P(Xn )
x∈It (Q̂)
t∈T̂where {It (Q̂)}t∈T̂ is a partition of Xn induced by Q̂, and `t (Q̂) is the length of inquiry
beginning from node n and terminating at node t ∈ T̂ . Recall that Q∗n is the subinquiry at n that prescribes to follow the optimal inquiry Q∗ , so the DM’s expected
payoff from Q∗ conditional on reaching n is given by (20) with Q̂ = Q∗n .
Let us prove (4). Clearly, Wn (Q∗n ; λ) ≤ maxQ̂∈QXn Wn (Q̂; λ). Suppose by contradiction that this inequality is strict. That is, there is a deviation Q̂ ∈ QXn ) in noden such that Wn (Q∗n ; λ) < Wn (Q̂; λ), or equivalently, by (20),
X Z
t∈Tn

x∈It (Q∗n )u(dt , x) − λ`t (Q∗n )<G(dx|Xn )
X Z
t∈T̂

x∈It (Q̂)

u(dˆt , x) − λ`t (Q̂) G(dx|Xn ). (21)OPTIMAL INQUIRY37

Let T̃ = (T − Tn ) ∪ T̂ , and construct an outcome Z̃ = (I˜t , `˜t , d˜t )t∈T̃ as follows:(It (Q∗ ), `t (Q∗ ), dt ),for each t ∈ T − Tn ,(I˜t , `˜t , d˜t ) =(I (Q̂), ` (Q∗ ) + ` (Q̂), dˆ ), for each t ∈ T̂ .
t

n

t

t

By construction, Z̃ is an implementable outcome by an inquiry in QX . Namely,
inquiry Q̃ that implements Z̃ is obtained from Q∗ by replacing the branch that follows
node n with Q̂. Then, we haveW (Q̃; λ)−W (Q∗ ; λ) = P(Xn ) Wn (Q̂; λ) − Wn (Q∗n ; λ)


X Z
∗

= P(Xn )u(dt , x) − λ(`n (Q ) + `t (Q̂)) G(dx|Xn )
t∈T̂

−

X Z
t∈Tn

x∈It (Q∗n )x∈It (Q̂)#
u(d∗t , x) − λ`t (Q∗ ) G(dx|Xn ) > 0.

The first equality is by definition of W and that Q̃ and Q∗ differ only in the branch atnode n. The second equality is by definition of Wn and the fact that the total lengthof path from no to t under Q̃ is the sum of the length from no to n under Q∗ and thelength from n to t under Q̂. The inequality is by (19) and (20). Thus, we reached acontradiction to the assumption that Q∗ is optimal.



A.2. Proof of Lemma 3.1. Let Q be an optimal inquiry. By contradiction, let
n0 ∈ N be a node that is reached with probability zero, but all the predecessors arereached with positive probability. Let n be the immediate predecessor of n0 , and let
n00 be the second successor of n. Consider now a new inquiry Q̂ obtained by modifying
Q as follows. The question at node n and the entire branch following n0 are removed.
Instead, upon reaching node n, the inquiry Q̂ will follow the branch of Q startingfrom the node n00 . Clearly, every terminal node t ∈ T that is reached with positive
probability under Q is reached with the same probability under Q̂, and the DM’s
expected payoff conditional on reaching any such node is unchanged. But the lengthof inquiry for the terminal nodes in the branch that starts from n00 is shorter underQ̂. This contradicts the optimality of Q.



A.3. Proof of Lemma 3.2. To prove Lemma 3.2, we use the following three claims.

38

HU AND ZAPECHELNYUKClaim A.1. Let (N, T, σ) be a binary tree with a set of nodes N , a set of terminal
nodes T ⊂ N , and a successor function σ. For each t ∈ T , let `t be the length of thePpath from the root to t. Then t∈T 2−`t = 1.
Proof. This claim directly follows from Theorem 5.2.1. in Cover and Thomas (2006)
and its proof. As in that proof, one can convert an instantaneous code into a binary
so that the lengths of paths to the terminal nodes correspond exactly to the codeword
lengths. We have an equality here instead of inequality because in our inquiry tree
every non-terminal node branches down to two further nodes.

Claim A.2. Let K ≥ 1. If ` = (`1 , ..., `K+1 ) ∈ NK+1 satisfies

PK+1 −`
k = 1, then
k=1 2

there exists `0 = (`01 , ..., `0K ) ∈ NK such that `0k ≤ `k for all k = 1, ..., K, `0k0 < `k0 for
P
−`0k
= 1.
some k0 ∈ {1, ..., K}, and K
k=1 2
Proof. Without loss of generality assume that `1 ≤ · · · ≤ `K+1 . It follows that
`K = `K+1 ; for otherwise the terminal node corresponding to `K+1 must be the only
successor of its predecessor. Let `0k = `k for k = 1, ..., K − 1 and let `0K = `K − 1.
Thus,
K
X

−`0k

2

k=1

=

K−1
X

−`k

2

+2

−`0K

=

K−1
X

−`k

2

+2

−`K +1

k=1

k=1

=

K+1
X

2−`k = 1,

k=1

where the second last inequality follows from the fact that `K = `K+1 .



Claim A.3. Let I = {Ik }K
k=1 be a partition of X into K elements, let D = {d1 , ..., dK } ⊂A, and let ` = (`1 , ..., `K ) ∈ NK be a length profile such thatK
X

2−`k = 1.

(22)

k=1

Then, there exists an inquiry Q = hN, T, σ, χ, di with a set T = {t1 , ..., tK } of terminal
nodes such that
Itk (Q) = Ik and `tk (Q) = `k for all k = 1, ..., K.

(23)

Proof. By Theorem 5.2.1. in Cover and Thomas (2006) (with the argument as in the
proof of Claim A.1 that translate instantaneous codes into binary trees), (22) implies
that there exists a finite binary tree with a set of nodes N and a successor relation

OPTIMAL INQUIRY

39

over N , with K terminal nodes labeled t1 , ..., tK , such that, for each k = 1, ..., K, thelength of the path from the root to each terminal node tk is exactly `k .
We now construct an inquiry Q = hN, T, σ, χ, di that satisfies (23). Let N be
as above, and let T = {t1 , ..., tK }. For each nonterminal node n ∈ N − T , let us
associate two edges leading out of n with true and f alse, and define the map σ so that
σ(n, true) = ntrue if nntrue along the edge labelled true and σ(n, f alse) = nf alse

nf alse along the edge labelled f alse. Let decision rule d be given by the choice

if nof dk in terminal node tk for each k = 1, ..., K.
It remains to construct a proposition mapping χ that yields the partition I in theterminal nodes. First, we associate each node in N with a set, In (Q), as follows. For
each k = 1, ..., K, let Itk (Q) = Ik . Then, by backward induction, for each nonterminal
node n ∈ N −T , let In (Q) = Iσ(n,true) (Q)∪Iσ(n,f alse) (Q). This implies that Ino (Q) = X
at the root no , since {Ik }K
k=1 is a partition.
Finally, define a proposition map χ as follows. For each nonterminal node n ∈ N −T , let χ(n) = Iσ(n,true) (Q). By induction from the root of the tree, it is straightforward
to verify that χ satisfies (1), so, for each n ∈ N , In (Q) is indeed the information set
induced by Q at node n.

We now prove Lemma 3.2. Let Q = hN, T, σ, χ, di be an optimal inquiry. Suppose,by contradiction, that dt0 = dt00 for some t0 , t00 ∈ T with t0 6= t00 . Let K = |T | − 1, andlet us label the terminal nodes consecutively, T = {t1 , ..., tK , tK+1 }, such that tK = t0
and tK+1 = t00 .
Now we construct an alternative inquiry, Q0 = (N 0 , T 0 , σ 0 , χ0 , d0 ), with |T 0 | = K
terminal nodes that leads to a strictly higher expected value to the DM. Let
0= ItK (Q) ∪ ItK+1 (Q),Ik0 = Itk (Q) for each k = 1, ..., K − 1, and IK(24)and letd0k = dtk for each t = 1, ..., K.
P−`tk (Q)Now, by Claim A.1, we have K+1= 1. By Claim A.2, there exists `0 ∈ NK
k=1 2
such that`tk (Q) ≤ `0k for all k = 1, ..., K, `tk (Q) < `0k for some k ∈ {1, ..., K},(25)−`0k (Q)000= 1. By Claim A.3 applied to I 0 = {Ik0 }K
k=1 , ` = (`1 , ..., `K ), and
k=1 2
d0 = (d01 , ..., d0K ), there exists an inquiry Q0 = hN 0 , T 0 , σ 0 , χ0 , d0 i with T 0 = {t1 , ..., tK }

and

PK

40

HU AND ZAPECHELNYUKsuch that
It0k (Q0 ) = Ik0 and `0tk (Q0 ) = `0k for all k = 1, ..., K.

(26)

Thus, we obtain
0

W (Q ; λ) =

K Z
X
k=1

>

Itk (Q0 )

K+1
XZ
k=1

(u(d0k , x) − λ`0tk (Q0 ))G(dx) =

K Z
X
k=1

Ik0

(u(d0k , x) − λ`0k )G(dx)(u(dtk , x) − λ`tk (Q))G(dx) = W (Q; λ),Itk (Q)

where the first and last equalities are by (2), the second equality is by (26), and theinequality is by (24), (25), and that, by Lemma 3.1, all the terminal nodes in T arereached with positive probability under Q.



A.4. Proof of Lemma 3.3.
Necessity. Suppose that an outcome profile (D, `, I) is implementable by an inquiry
Q = hT, N, σ, χ, di. Let (D, `, I) = (T, `(Q), I(Q)). By Lemma 3.2, D ⊂ A, and, by
Claim A.1, (D, `) satisfies (6).
Sufficiency. Immediate by Claim A.3.



A.5. Proof of Lemma 3.4. Let (D, `) be given. For any partition I = {Id : d ∈ D},
let
W (I; D, `) =XZd∈D[u(d, x) − λ`d ]G(dx).

Id

Now, by (8), for any I and any d ∈ D, if x ∈ Id∗ (D, `) ∩ Id0 with d 6= d0 then[u(d, x) − λ`d ] > [u(d0 , x) − λ`d0 ].
Thus, since P(X − ∪d∈D Id∗ ) = 0 by (A3 ) and the fact that G has full support,
W (I ∗ ; D, `) − W (I; D, `)X Z{[u(d, x) − λ`d ] − [u(d0 , x) − λ`d0 ]} G(dx)=d,d0 ∈D−d∈D=

Id∗ ∩Id0

XZ[u(d, x) − λ`d ]G(dx)

Id ∩(X−∪d∈D Id∗ )X Z
d6=d0 ∈DId∗ ∩Id0{[u(d, x) − λ`d ] − [u(d0 , x) − λ`d0 ]} G(dx) ≥ 0,

OPTIMAL INQUIRY

41

and the inequality is strict if P(Id∗ ∩ Id0 ) > 0 for some d 6= d0 . This proves theresult.



A.6. Proof of Theorem 3.2. By Lemma 3.4, if (D, `, I) is the outcome of an optimalinquiry, then W (I; D, `) = W (I ∗ ; D, `). To be optimal, it then must solve (10).
A.7. Proof of Proposition 3.1. (a) If (D, `) solves (10), given the partition, thelength profile must deliver the lowest average length and hence must be given by
Huffman coding.
(b) Let Z = (D, `, I) be the outcome of an optimal inquiry. First we show that if`d < `d0 , then P(Id ) ≥ P(Id0 ). Suppose, by contradiction, that P(Id ) < P(Id0 ). Now,let `0d = `d0 and `0d0 = `d , and keep other outcomes unchanged. Note that the new
outcome still satisfies (6) and hence can be induced by some inquiry. But now[P(Id )`0d + P(Id0 )`0d0 ] − [P(Id )`d + P(Id0 )`d0 ] = [P(Id )`d0 + P(Id0 )`d ] − [P(Id )`d + P(Id0 )`d0 ]= −[P(Id0 ) − P(Id )](`d0 − `d ) < 0.
Thus, the new inquiry decreases the average length but keeps the utilities unchanged.
This is a profitable deviation and a contradiction to the optimality of the original
inquiry.



A.8. Proof of Theorem 4.1. Let λ1 < λ2 . For each j = 1, 2, Let Qλj be an optimalinquiry for j = 1, 2, and let Zλj = (Dj , `j , I j ) be the associated outcome. DenoteXZū(Zλj ) =u(d, x)G(dx), j = 1, 2.
d∈Dj

x∈Idj

¯ λ ). By the optimality of ZλBy (7) and (12), we have W (Qλj ; λj ) = ū(Zλj ) − λ`(Zjjgiven λj , for each j = 1, 2, we have¯ λ ) ≥ ū(Zλ ) − λ1 `(Z¯ λ ) and ū(Zλ ) − λ2 `(Z¯ λ ) ≥ ū(Zλ ) − λ2 `(Z¯ λ ).
ū(Zλ1 ) − λ1 `(Z
1
2
2
2
2
1
1
Combining these inequalities yields¯ λ ) − `(Z¯ λ ) ≤ ū(Zλ ) − ū(Zλ ) ≤ λ2 `(Z¯ λ ) − `(Z¯ λ) .
λ1 `(Z121212¯ λ ) ≥ `(Z¯ λ ) whenever λ1 < λ2 .
Thus, `(Z
1
2
Next, let (D, `) be a solution of problem (10) under λ > 0 such that ` is not
uniform. Because (D, `) is finite, there exists an interval [λ0 , λ00 ] that contains λ42

HU AND ZAPECHELNYUK(possibly, λ = λ0 or λ = λ00 ) such that (D, `) is a solution of problem (10) for eachλcost in [λ0 , λ00 ]. Consider arbitrary λ1 , λ2 ∈ [λ0 , λ00 ] with λ1 < λ2 . Let Ek j be given by(15). Observe that for each j = 1, 2 we haveλP(Ek j ) = Pk[!
Id∗k0 (D, `; λj )

=

k0 =1

k
X

P(Id∗k0 (D, `; λj ).

k0 =1Thus, by Lemma A.1 (see Section A.11 below), for each k = 1, ..., K − 1 we obtaink
X

P(Id∗k0 (D, `; λ2 )) ≥

k0 =1

K
X

P(Id∗k (D, `; λ1 )),

(27)

k=1

with strict inequality for some k. In other words, given (D, `), the probability distribution over actions in D = {dk }K
k=1 under λ2 first-order stochastically dominatesthat under λ1 . Thus, by (12), we obtain¯ λ )=
`(Z
2

|D|
X

`dk P(Id∗k (D, `; λ2 )) <

k=1

|D|
X

¯ λ ).
`dk P(Id∗k (D, `; λ1 )) = `(Z1k=1¯ λ ) is strictly increasing on [λ0 , λ00 ].
Thus, we have shown that `(Z



A.9. Proof of Proposition 4.1. Let (D, `, I) be the outcome of an optimal inquiry
Q. Suppose that δ(a0 , a00 ) < λ for some a0 , a00 ∈ A. Suppose by contradiction thata0 , a00 ∈ D. There are two cases.
Case 1. Suppose that `a0 6= `a00 . W.l.o.g., let `a0 < `a00 . By Lemma 3.4 andassumption (A3 ), a0 ∈ D implies that the set
Ia0 = {x ∈ X : u(a0 , x) > u(a, x) − λ(`a0 − `a ) for all a ∈ D − {a0 }}(28)

has nonempty interior. Therefore, because a00 ∈ D, we must haveu(a0 , x) > u(a00 , x) − λ(`a0 − `a00 ) ≥ u(a00 , x) + λ for each x ∈ Ia0 ,where the first inequality is by (28), and the second inequality is because `a0 < `a00
and both `a0 and `a00 are integers. This contradicts the assumption that δ(a0 , a00 ) < λ.
ˆ I)ˆCase 2. Suppose that `a0 = `a00 . Consider an inquiry Q̂ with the outcome (D̂, `,given by D̂ = D − {a00 }, `ˆa0 = `a0 − 1, `ˆa = `a for all a ∈ D − {a0 }, Iˆa0 = Ia0 ∪ Ia00 ,and Iˆa = Ia for all a ∈ D − {a0 }. In words, Q̂ is the same as Q except that Q̂ merges
actions a0 and a00 and removes the question that distinguishes these actions. Because`a0 = `a00 = `ˆa0 + 1,

(29)

OPTIMAL INQUIRY43

Pˆ
we obtain 2−`a0 + 2−`a00 = 2−`a0 . Since d∈D 2−`d = 1, we obtain that




X
X
X ˆ
ˆ
ˆ
2−`d  + 2−`a0 + 2−`a00 = 1.
2−`d  + 2−`a0 = 2−`d = d∈D̂d∈D−{a0 ,a00 }d∈D̂−{a0 }Thus, by Lemma 3.3, there exists an inquiry Q̂ constructed as above. As Q and Q̂
differ only for x ∈ Ia0 ∪ Ia00 , we obtainZ 
(u(a0 , x) − λ`ˆa0 ) − (u(a0 , x) − λ`a0 ) G(dx)W (Q̂; λ) − W (Q; λ) =
I 0

+

Z a 


(u(a0 , x) − λ`ˆa0 ) − (u(a00 , x) − λ`a00 ) G(dx)

I 00

Za
=

Z
λG(dx) +

Ia0



0

00



u(a , x) − u(a , x) + λ G(dx)x∈Ia00> 0,
where the first equality is by (7), the second equality is by (29), and the inequality
is because δ(a0 , a00 ) < λ and Ia0 ∪ Ia00 has nonempty interior. We thus obtain acontradiction to the optimality of Q.

A.10. Proof of Proposition 4.2. Let λ2 = supx∈X (maxa∈A u(a, x)−mina∈A u(a, x)).
Then, for all λ ≥ λ2 , the utility gain from distinguishing any actions is smaller thanthe cost, so the optimal consideration set D is a singleton.
Next, by Theorem 3.2 and assumptions (A1 ) and (A3 ), there exists a nonempty set
F0 ⊂ F ∗ of pairs (D, `) that are optimal for a small enough cost. Specifically, for each(D, `) ∈ F0 , there exists a small enough λ(D,`) > 0 such that (D, `) solves problem(10) for all λ ∈ [0, λ(D,`) ]. Let λ1 = min(D,`)∈F0 λ(D,`) . Observe that by (A2 ) and (A3 )and the assumption of full support on X, if λ = 0, then each action a ∈ A is optimal
on a positive-measure subset of X. Hence, the unique optimal consideration set for
λ = 0 is D = A. It follows that (D, `) ∈ F0 implies D = A. Thus, A is the uniqueoptimal consideration set for all λ ∈ [0, λ1 ].



A.11. Proof of Theorem 5.1. Before proving Theorem 5.1, we state a lemma.
Lemma A.1. Let (D, `) ∈ F ∗ , and let K = |D|. W.l.o.g, let actions in D be orderedaccording to their lengths of inquiry, so D = {dk }K
k=1 , such that`d1 ≤ `d2 ≤ ... ≤ `dK̄ .

(30)

44

HU AND ZAPECHELNYUKFor each λ1 , λ2 ∈ R+ with λ1 < λ2 ,Ekλ1 ⊆ Ekλ2 for all k = 1, 2, ..., K − 1.

(31)

Moreover, if ` is not uniform, then there exists k ∈ {1, ..., K − 1} such that the set
Ekλ2 − Ekλ1 has a non-empty interior.

(32)

Proof. First, we prove (31). Let k ∈ {1, ..., K − 1}. Suppose by contradiction that
there exists x ∈ Ekλ1 such that x 6∈ Ekλ2 . By (15), x ∈ Ekλ1 and x 6∈ Ekλ2 imply that
there exist k ∗ ≤ k < m∗ such thatu(dk∗ , x) − λ1 `dk∗ = 0max u(dk0 , x) − λ1 `dk0 > u(dm∗ , x) − λ1 `dm∗ ,(33)u(dk∗ , x) − λ2 `dk∗ ≤(34)

k =1,...,kmax

m=k+1,...,Ku(dm , x) − λ2 `dm = u(dm∗ , x) − λ2 `dm∗ .

Combining (33) and (34), we obtainλ2 (`dm∗ − `dk∗ ) ≤ u(dm∗ , x) − u(dk∗ , x) < λ1 (`dm∗ − `dk∗ ),which is impossible, since λ2 > λ1 ≥ 0, and `dm∗ ≥ `dk∗ by (30). We reached acontradiction.
Next, suppose that ` is not uniform. Then there exists k ∗ ∈ {1, ..., K − 1} suchthat
`d1 = ... = `dk∗ < `dk∗ +1 ≤ ... ≤ `dK .

(35)

We prove (32) for k = k ∗ . Definew̄ = max ∗ u(dk , x) and wλ (x) =k=1,...,k
max

m=k∗ +1,...,Ku(dm , x) − λ`dm

− (w̄ − λ`dk∗ ).

Observe thatwλ2 (x) < wλ1 (x) for all x ∈ X and all λ1 < λ2 .

(36)This is because for any given x ∈ X there exists m∗ > k ∗ such thatwλ2 (x) = u(dm∗ , x) − λ2 `dm∗ + λ2 `dk∗ − w̄ < u(dm∗ , x) − λ1 `dm∗ + λ1 `dk∗ − w̄ ≤ wλ1 (x),where the strict inequality is by λ1 < λ2 and `k∗ < `m∗ .
Next, by (15) and (35), we havex ∈ Ekλ∗ ⇐⇒ wλ (x) < 0.

(37)

OPTIMAL INQUIRY45

Fix λ1 < λ2 . By assumptions (A2 )–(A3 ), the sets Ekλ∗1 and X − Ekλ∗2 have nonempty
interiors. Lety ∈ Int(Ekλ∗1 ) and z ∈ Int(X − Ekλ∗2 ).
By (36) and (37), we havewλ1 (y) < 0 < wλ2 (z) < wλ1 (z).
Letx∗ = α∗ y + (1 − α∗ )z, where α∗ = sup {α ∈ [0, 1] : wλ1 (αy + (1 − α)z) ≤ 0} .
Since X is convex, and points y and z are in Int(X), x∗ is an interior point of X. Since
wλ1 (x) is continuous in x by assumption (A1 ), we have wλ1 (x∗ ) = 0. Moreover, by(36), there exists ε > 0 such that wλ2 (x∗ ) = ε > 0. Let Ox∗ be the open neighborhoodof x∗ given by
Ox∗ = {x ∈ X : |wλ1 (x) − wλ1 (x∗ )| < ε}.
By the continuity of wλ1 (x), Ox∗ is an open nonempty set. Recall that by assumption(A3 ), (X − Ekλ∗1 ) has nonempty interior. Since the set Ox∗ ∩ (X − Ekλ∗1 ) contains x∗ , it
has nonempty interior. Finally, since Ox∗ ∩ (X − Ekλ∗1 ) ⊂ Ekλ∗2 − Ekλ∗1 , we obtain (32)for k = k ∗ .

We now prove Theorem 5.1. Let λ > 0 and let (D, `, I) be an outcome of an optimalinquiry. Observe that D = {dk }K
k=1 satisfies (14) if and only if it satisfies (30). Then,the statement of the theorem is immediate by Definition 5.1 and Lemma A.1 with
λ1 = 0 and λ2 = λ.



A.12. Proof of Proposition 5.1. To prove the proposition, we apply Lemma A.1.
Let λ > 0 and let (D, `) be a solution of problem (10) under λ. Because (D, `) isfinite, there exists an interval [λ0 , λ00 ] that contains λ (possibly, λ = λ0 or λ = λ00 )such that (D, `) is a solution of problem (10) for each cost in [λ0 , λ00 ]. Consider
arbitrary λ1 , λ2 ∈ [λ0 , λ00 ] with λ1 < λ2 . By Lemma A.1, it is immediate that, for eachk = 1, ..., K − 1,P(Ekλ2 ) ≥ P(Ekλ1 ).

(38)Moreover, if ` is not uniform, inequality (38) is strict for some k. We thus obtain that
confirmation bias is locally increasing, and strictly so whenever ` is not uniform. 

46

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