# Random Attention Span ††thanks: I would like to thank Victor Aguiar, Maria
Goltsman, Nail Kashaev, Roy Allen, Elliot Lipnowski and John Quah for useful
comments and suggestions. I am grateful to Victor Aguiar, Nail Kashaev, Maria
Jose Boccardi and Jeongbin Kim for providing their experimental dataset for
the empirical estimation chapter in this paper.

Dazhuo Wei

(This version: April, 2024 / First Version: April, 2022)

###### Abstract

In this paper, I introduce a random attention span model (RAS) which uses
stopping time to identify decision-makers’ behavior under limited attention.
Unlike many limited attention models, the RAS identifies preferences using
time variation without any need for menu variation. In addition, the RAS
allows the consideration set to be correlated with the preference. I also use
the revealed preference theory that provides testable implications for
observable choice probabilities. Then, I test the model and estimate the
preference distribution using data from M-Turk experiments on choice behaviors
that involve lotteries; there is general alignment with the distribution
results from logit attention model.

JEL classification numbers: C50, C51, C52, C91.
Keywords: random utility, random consideration sets.

## 1\. Introduction

The stochastic revealed preference theory, also known as the random utility
model, was introduced by McFadden and Richter, (1990). Its aim is to explain
how decision-makers behave when choosing alternatives by assuming that they
maximize their utility. A major criticism is that the representation assumes
individuals consider all alternatives.111For example, if only two options are
available, apples and bananas, and people choose apples more often; then the
RUM model with full attention would conclude that apples are preferred over
bananas. However, if the model’s full consideration assumption is violated,
this conclusion may not be accurate as people may not have seen the bananas at
all.222Aguiar et al., (2023) rejects full consideration with the experimental
data consisting of lotteries; Goeree, (2008) shows that limited information
about a product contributes to differences in purchasing outcomes; van Nierop
et al., (2010) shows that there is considerable variation in choice shares and
considerations across laundry detergent brands; Honka et al., (2017) shows
that advertising makes consumers aware of more options, search more, and find
better alternatives. This has motivated the analysis of models that do not
assume full consideration. Most of these models use observed menu variation to
distinguish between preference and limited attention effects on choices.
Examples include models like Cattaneo et al., (2020) that requires menu
variation and relies on monotonicity on the size of menus, and models like
Brady and Rehbeck, (2016), Aguiar, (2017) and Manzini and Mariotti, (2014)
which use menu variation and rely on functional-form restrictions on the
attention rules. These models have produced exciting results, but obtaining
such menu variation in real-world field data can be challenging.333Aguiar et
al., (2023) points out that nobody has a clean menu variation setup for field
data. For one, it is costly for firms to change menus too frequently.
Additionally, some major products always stay on the menu, so the preference
ranking between these products cannot be obtained due to the lack of
variation.

In this paper, I propose a new model called random attention span model (RAS)
that can help identify preferences of individuals without any need for menu
variation. Additionally, I provide an empirical approach to estimating the
preference distribution and to testing the model given a data set of
stochastic choices and stopping times. The RAS model can cover a wide range of
nonparametric attention rules within a fixed and observed menu. This model has
a richer data set that comprises the information on the time it takes
individuals to make decisions that enables me to analyze the consideration
rules in a nonparametric way and draw implications on the identified
preferences for stochastic choices without observing individuals’ exact
attention.

The availability of stopping time data sets is a key motivation of RAS. In
contrast to the lack of menu variation, most online shopping platforms record
stopping time along with individuals’ choices. Therefore, a different approach
using time variation instead of menu variation would be convenient for
investigating preferences under limited attention.

The observation of stopping time is widely used in the neuroscience and
psychology literature (Ratcliff and McKoon,, 2008, Milosavljevic et al.,,
2010, Krajbich et al.,, 2010) and has also been introduced to the economics
literature by Reutskaja et al., (2011). Reutskaja et al., (2011) designs an
experiment to show that limited attention is true among decision-makers by
using an eye-tracking device to identify which alternatives each decision-
maker sees. The study finds that people are very good at choosing the best
alternative among the subset that the decision-maker really pays attention to.
This finding supports the limited attention models that indicate decision-
makers may have limited attention and are utility maximizing in the limited
attention set.

The RAS theory is based on the assumption that individuals’ preferences remain
stable over time and that their attention span follows a set of rules called
"time monotonicity." This set means that as individuals take more time to make
decisions, the probability that their attention is concentrated entirely on
any subset of the full menu weakly decreases. Essentially, the theory posits
that individuals tend to explore more new items as they spend more time on the
decision, without forgetting the items they have already seen. For example, if
someone is shopping online and starts by looking at item 1, their
consideration set is limited to that single item. As time passes, they are
likely to see item 2 and other options, but they will not forget about item 1.

The time monotonicity assumption is valid in sample models with time
introduced. In a model of search and satisfaction where the satisfaction
thresholds depend on time, a decision-maker will make a choice when: 1. this
choice offers an above-threshold utility; 2. this choice is the first item
they have ever seen which is above the satisfaction threshold. In this case,
if individuals who stop later are more patient and have higher thresholds, the
probability that their attention remains in a small range of considerations is
smaller. Therefore, as decision-makers who stop later are more patient, the
principle of time monotonicity is upheld. In a model of stochastic choice and
limited considerations where the probability of any certain item being
considered increases over time, the probability of considering anything beyond
any given set is non-decreasing over time.444Please see 2.4 for details.

In this paper, I derive testable implications through linear inequalities in a
homogeneous preference setting. I argue that the more time people spend making
a decision, the more likely they are to choose the better option. This is
because, as they spend more time, they are more likely to see the preferred
items. As a result, decision-makers tend to choose the most preferred items
with more time, leaving the probability of selecting the less preferred items
to be less likely.

When dealing with heterogeneous preferences, the paper shows that the RAS
representation is equivalent to having an equality link between attention
rules, a preference distribution, and a stochastic choice data set. This
equality means that researchers can estimate the preference distribution by
minimizing the distance between the data and the model by using a simulation
on the attention rule matrix that follows time monotonicity.

In this paper, I tested RAS with experimental data designed and collected by
Aguiar et al., (2023) using testing method in Kitamura and Stoye, (2018). The
test does not reject the null hypothesis that the choice data set is generated
by RAS.

I estimate the preference distribution in the experimental data. For easier
model comparison, the utility is limited to constant relative risk aversion
(CRRA). I compare my estimation with the benchmark model which constrains
attention in the logistic function (CRRA-LA).555A logit attention is as
follows: Consider a decision-maker who assigns a positive weight for each non-
empty subset of X. Psychologically,$w_{A}$ is a strength associated with the
subset $A$. The probability of considering $A$ in S can be written as
$\mu(A\mid S)=\frac{w_{A}}{\sum_{A^{\prime}\subset S}w_{A^{\prime}}}.$ 666A
CRRA-LA setup restricts the attention rule to the logit attention and its
utility is the CRRA utility.. The RAS which only uses stopping time variation
yields comparable estimation results to CRRA-LA which relies on data on menu
variation. However, the RAS identifies a significantly higher percentage of
decision-makers who prefer a more complex lottery.

The estimation results show that the extra data on stopping time is important
in restricting the attention rules and identifying preferences for items which
are much less likely to be considered. The RAS uses data with time variation
when decision-makers are facing the same menu, while the CRRA-LA (Aguiar et
al.,, 2023, Brady and Rehbeck,, 2016) estimation uses data when decision-
makers are facing all kinds of menus but without considering the stopping time
factor.

My approach differs from other works that have used stopping-time
observations. These works have focused more on the link between choice
accuracy and the time decision-makers spent, conditional on a specific payoff
function. In comparison, the RAS uses stopping time observations to derive the
preference distribution under limited attention. The RAS also differs from the
rational inattention model with time (Hebert and Woodford,, 2023) and
sequential search models (McCall,, 1965, Weitzman,, 1979, Aguiar et al.,,
2016) in the assumption of having formed the consideration set. In sequential
search models, one fundamental assumption is that people know the payoff
distribution. In the rational inattention model, there is some updating
process on the beliefs of what the payoff distribution is. However, in the
random attention model, decision-makers do not need to know any information
about the payoff distribution beyond their consideration set. Therefore, the
attention is "random," and the formation of the consideration set may not have
an optimal time to stop.

The drifting diffusion model (DDM) is also popular for using stopping time.
While both the RAS and the DDM use stopping time, the two models do so from
quite different angles. There are fundamental differences in the underlying
assumptions of the RAS and the DDM. In the DDM, only two alternatives are
present and one alternative is chosen if the net evidence is above a
particular threshold. In contrast, in the RAS, multiple alternatives are
present; and the decision-maker will choose the most preferred option if it is
in their consideration set. The combination of noise and a threshold in the
DDM leads to the possibility of mistakes if the threshold is very low or if
there is an unusually high level of noise. However, in the RAS, decision-
makers are assumed to make the best choice among the alternatives that they
actually consider. In the RAS, the assumption of time monotonicity is
exogenous to decision-makers’ attention and is conditional on time, although
the stopping time at the decision is endogenous. In general, in random
attention models, researchers rule out the possibility of mistakes, and the
choices decision-makers make are based only on their preferences and the
consideration set at the time they make the decision. This difference leads
the DDM and the RAS to have different focuses of study. The DDM has a focus on
choice accuracy, while the RAS has a focus on the preference distribution.

The RAS is close to the random attention model (RAM) (Cattaneo et al.,, 2020)
in terms of the nonparametric assumption on attention rules. The RAM uses a
nonparametric restriction named monotonicity in the size of the attention-
forming process. It states that the probability of considering a given set
cannot increase if the choice set grows. The RAM provides a general framework
to test different models of stochastic consideration when preferences are
homogeneous. Therefore, their framework is suitable for data on individuals’
choices but not so much when the population has a distribution of different
preferences.

Aguiar and Kashaev, (2022) proposes a model that involves heterogeneous
preferences and random attention. Their approach is very general, which allows
for a correlation between attention rules and preference rankings. However,
both Aguiar and Kashaev, (2022) and Cattaneo et al., (2020) rely on the use of
menu variation. The RAS instead accommodates a nonparametric restriction on
attention while proposing a different monotonicity feature (namely time
monotonicity). This paper complements Aguiar and Kashaev, (2022) and Cattaneo
et al., (2020) in the sense that it generalizes the use of variation in
identifying preferences.

The rest of the paper is structured as follows: In Section 2, I introduce the
model setup. In Section 3, I present the random attention span model (RAS) in
a homogeneous preference setting. In section 4, I introduce the heterogeneous
preference version of RAS. Section 5 has a discussion on the estimation and
testing methods. In Section 6, I present the empirical estimation results. In
Section 7, I provide the concluding remarks and a discussion.

## 2\. Model

In the world of RAS, decision-makers begin by facing an observed fixed menu.
Nature then picks a stopping time, consideration set, and preference for each
decision-maker. Decision-makers then choose the best alternative according to
their preferences within their consideration set.

Menu $S$Consideration set $A$Stopping time $t$Preference $\succ$Choice $x$
Figure 1: Decision-makers begin by facing an observed fixed menu. Nature then
picks a stopping time, consideration set, and preference for each decision-
maker. Decision-makers then choose the best alternative according to their
preferences within their consideration set.

A finite set $S$ is a universal set of all mutually exclusive alternatives
that decision-makers face, also known as the menu. An element of $S$ is
denoted as $a$, and its cardinality is $|S|=K$. Let $\mathcal{X}$ denote the
set of all non-empty subsets of $S$. In this paper, the menu $S$ is fixed and
observed by researchers. Therefore, there is no menu variation being used for
revealed preference implications.

The stopping time observation $t$ has a support from $[0,T^{*}]$ where $T^{*}$
is the maximum amount of time one can spend. The stopping time here represents
the duration of time for the decision-makers to make their choice. Let $\succ$
denote the set containing all possible strict preference rankings to rule out
ties.

For a scenario where a decision-maker is browsing through Amazon to buy a TV,
the menu $S$ displays all the TV listings on Amazon. Item $a$ refers to a
specific TV model listed on Amazon. A stopping time $t$ is the period from the
start of browsing the TV section on Amazon to the point where the purchase is
completed.

######  Definition 1.

(Stochastic Choice Data Set). A stochastic choice dataset is a map
$\pi:S\times T\rightarrow[0,1]$ such that for all $t\in T$,

$\begin{split}&\pi(a\mid t)\geq 0,\forall a\in S,\\\ &\sum_{a\in S}\pi(a\mid
t)=1\\\ \end{split}$

This expression, $\pi(a\mid t)$, denotes the probability of selecting option
$a$ in the menu $S$ at the time at which the decision is made, $t$. This
formulation accommodates both deterministic and stochastic decision rules. If
$\pi(a\mid t)$ equals either one or zero, then the decision is deterministic.
For instance, in an online shopping scenario, $\pi(a\mid t=5min)$ is the
probability of choosing TV $a$ when customers are shopping on Amazon and
making their purchases within exactly five minutes.

It is important to understand that decision-makers may not pay attention to
all possible options. To determine the likelihood of agents considering
different options, it is critical to define the probabilities of potential
consideration sets. For any given menu $S$, every non-empty subset of options
may be a potential consideration set, and each set may have a certain
probability of being considered. The frequency of each consideration set
should fall between zero and one, and the total frequency should add up to
one. Formally:

######  Definition 2.

(Attention Rule). An attention rule is a map $\mu:\mathcal{X}\times
T\times\succ\rightarrow[0,1]$ such that for all $A\in\mathcal{X}$, $t\in T$,
$\succ_{i}\in\succ$

$\begin{split}&\mu(A\mid t,\succ_{i})\geq 0,\\\ &\sum_{A\subseteq S}\mu(A\mid
t,\succ_{i})=1\\\ \end{split}$

The notation $\mu(A\mid t,\succ_{i})$ represents the probability of focusing
explicitly on $A\subseteq S$ with stopping time $t$ when presented with the
menu $S$. However, focusing exclusively on $A$ means that the decision-maker
is seeing the exact set A, not just some of its subsets. For instance,
$\mu(\\{a,b\\}\mid t,\succ_{i})$ represents the probability that the decision-
maker is seeing both $a$ and $b$, and not $a$ only or $b$ only. In addition,
the possibility of individuals considering nothing or considering anything
beyond the menu is ruled out.

Furthermore, by using a conditional attention rule, the RAS is different from
Aguiar et al., (2023) in that it does not limit the correlation between
attention and a preference. This is especially significant as the preference
often determines the consideration sets in various model settings. Conversely,
Aguiar et al., (2023) enforces the fact that the random rule for the
consideration set and the heterogeneous preferences are independent.

###  2.1. Random Attention Span (RAS)

In this subsection, I introduce the Random Attention Span model with stopping
time. Decision-makers of type $i$ have a strict preference $\succ_{i}$ for X,
where $d_{\succ}$ is the number of possible strict preferences. The preference
ranking is an asymmetric, transitive, and complete binary relation. With
probability $p(\succ_{i})\in[0,1]$, decision-makers are endowed with
preferences $\succ_{i}$ drawn from the set of all linear orders (strict
preferences) on $S$, $\succ$. The probability of all possible preferences sum
up to one: $\sum_{\succ_{i}\in\succ}p(\succ_{i})=1$. Decision-makers always
pick the best alternative within their consideration set.

######  Definition 3.

(Random Attention Span Representation). A stochastic choice data set $\pi$ has
a random attention span representation if there is a preference ordering set
$\succ$ for X, with the distribution of preference
$p=[p(\succ_{1}),p(\succ_{2}),\cdots,p(\succ_{d_{\succ}})]$, and an attention
rule $\mu$ such that:

$\pi(a\mid t)=\sum_{\succ_{i}\in\succ}p(\succ_{i})\sum_{A\subseteq S}1(a\text{
is }-\succ_{i}\text{ best in }A)\cdot\mu(A\mid t,\succ_{i})$

for all $a\in S$.

In this case, $\pi$ is represented by $(\mu,p)$. I also define $\pi$ as a
random attention span (RAS). This RAS representation is very general and can
be used to derive the implications of the revealed preference under time
monotonicity.

###  2.2. Preference Stability

The RAS representation is so general that it cannot identify preferences
without any further restrictions. The unrestricted distribution of preferences
is permissive enough to explain any behavior. The use of random-attention-like
models is based on establishing a connection between the data set of
stochastic choices and the unobserved attention probabilities and with certain
limitations on the attention rules and preference distributions. In the RAS,
the attention rules are subject to two fundamental restrictions: time
monotonicity and preference stability.

In the RAS, the preference distribution is assumed to be stable over time:

######  Assumption 1.

(Preference Stability). There exists a $p^{*}\in\Delta(\succ)$ such that
$p^{*}(\succ_{i}\mid t)=p^{*}(\succ_{i})$ for all $t$, $\succ_{i}\in\succ$.

This stability limitation leads to the marginal distribution of preferences of
the general population being equivalent to the true distribution of
heterogeneous preferences in the population. The true distribution of
preferences is the distribution of choices that controls behavior in a
counterfactual situation where there are no limited considerations.

###  2.3. Example for RAS Representation

The RAS representation accommodates Manzini and Mariotti, (2014) while adding
attention development with time.

######  Example 1.

Manzini and Mariotti, (2014) with Heterogeneous Preference.

Suppose for a menu $S$, each element $a$ has an probability
$\gamma_{t,\succ_{i}}(a)$ that $a$ is in the consideration set conditional on
stopping time $t$ and preference ranking $\succ_{i}$. Let $\succ$ be the set
of all possible preference orderings and $\succ_{i}\in\succ$. Let
$p_{t}(\succ_{i})$ be the probability of $\succ_{i}$ conditional on time t.
Since the marginal distribution of preferences is assumed to be stable over
time: $p_{t}(\succ_{i})=p(\succ_{i})$ for all $t$ and $\succ_{i}\in\succ$. The
RAS representation is:

$\begin{split}\pi(a\mid
t)=&\sum_{\succ_{i}\subseteq\succ}p(\succ_{i})\sum_{A\subseteq S}1(a\text{ is
}-\succ\text{ best in }A)\cdot\mu(A\mid t,\succ)\\\
=&\sum_{\succ_{i}\subseteq\succ}[p(\succ_{i})\gamma_{t,\succ_{i}}(a)\prod_{b\in
S:b\succ a}(1-\gamma_{t,\succ_{i}}(b))]\end{split}$

Moreover, in the RAS world, $\gamma_{t,\succ_{i}}(a)$ can depend on the
preference ranking, and the correlation between it and the stopping time is
expressed by the $\gamma_{t,\succ_{i}}(a)$.

In this paper, I explore how people pay attention to different options when
making decisions. Imagine there is a menu of all possible choices. At any
given time, each option has a chance of catching the decision-maker’s
attention. When an option does catch their attention, it gets added to the
decision-maker’s consideration set that is like a mental pocket for options
they are considering. As time passes, the collection of options in the
consideration set grows. This is the basis for the time monotonicity
assumption.

######  Definition 4.

(Accumulated Attention)

The accumulated attention at A is the probability of paying attention to any
subset of A and to A itself:

$\alpha(A\mid t,\succ_{i})=\sum_{B\subseteq A}^{A}\mu(B\mid t,\succ_{i})$

Logically, the accumulated attention is the probability that one’s
consideration set is within A or A’s subsets and includes nothing apart from
the universe of set A.

######  Assumption 2.

(Monotonicity in Time).

For any $t<t^{\prime}$, $A\subset S$,

$\alpha(A\mid t,\succ_{i})\geqslant\alpha(A\mid t^{\prime},\succ_{i})$

and

$\alpha(S\mid t,\succ_{i})=1$

According to Assumption 2, as the stopping time increases, the accumulated
attention at any consideration set A should not increase. In other words, the
probability of considering items beyond consideration set A weakly increases
for all possible consideration sets. This means that decision-makers are more
likely to consider a larger set of alternatives over time. The reason for this
consideration is that as time passes, decision-makers tend to move away from
smaller consideration sets. The intuition behind this time monotonicity is
that the decision-makers are exploring more options as they spend more time.

It is important to understand that with the time monotonicity, for any item
$a$ and a given preference, the probability of $a$ being considered
conditional on preference is weakly increasing over time. However, the weakly
increase in the marginal probability of each item being considered does not
guarantee time monotonicity.

######  Example 2.

Consider a menu that consists of three items, namely $a$, $b$, and $c$, and
two time periods.

In period 1, $\mu(\\{a\\}\mid t=1)=0.5$, $\mu(\\{b,c\\}\mid t=1)=0.5$;

while in period 2, $\mu(\\{a,b\\}\mid t=2)=0.5$, $\mu(\\{c\\}\mid t=2)=0.5$.

In this example, the probabilities for each item being considered are all
weakly increasing, yet the time monotonicity assumption is not satisfied. This
is because in period 2, the probability that both b and c are considered
together decreases. One reason is that people who considered both b and c in
period 1 forget item b in period 2. The time monotonicity assumption rules out
this forgetting behavior.

###  2.4. Examples for Time Monotonicity

Many traditional frameworks satisfy time monotonicity. Here are three examples
demonstrating why it is a reasonable assumption to make.

######  Example 3.

[Top N(Weitzman,, 1979).]

Consider a menu that consists of three items, namely $a$, $b$, and $c$, and
three time periods. The search order is fixed. In period 1, only item $a$ is
visible to the decision-maker; while in period 2, both $a$ and $b$ are
visible; and in period $3$, all three items are visible. Thus, the accumulated
attention for all non-empty subsets of the menu $S$ cannot increase. This
result can be extended to a menu with $n$ items.

######  Example 4.

[Manzini and Mariotti, (2014) Extension.]

Suppose for a menu $S$ and discrete stopping time $t$ that at each time period
$t$, conditional on preference $\succ_{i}$, each element $a$ has an
probability $\gamma_{t,\succ_{i}}(a)$ that $a$ is in the consideration set
conditional on stopping time $t$ and preference ranking $\succ_{i}$.777For
example, $\gamma_{t=1,a\succ b}(a)$ represents the probability that item $a$
is being considered at time period 1 for people who prefer $a$ to $b$ If I
assume people do not forget what they already considered previously, then
$\gamma_{t,\succ_{i}}(a)$ naturally increases as time moves on (the
probability that $a$ is considered increases as t increases for all $a\in S$).

Further, the accumulated attention for a certain set $A$ is the probability
that one is not considering anything beyond set $A$. Then the accumulated
attention can be written as:

$\alpha(A\mid t,\succ_{i})=\prod_{b\notin A}(1-\gamma_{t,\succ_{i}}(b))$

Since $\gamma_{t,\succ_{i}}(b)\leq\gamma_{t^{\prime},\succ_{i}}(b)$ for all
$t<t^{\prime}$, $b\in S$. Then

$\prod_{b\notin A}(1-\gamma_{t,\succ_{i}}(b))\geq\prod_{b\notin
A}(1-\gamma_{t^{\prime},\succ_{i}}(b)),\forall t<t^{\prime}$

Therefore,

$\alpha(A\mid t,\succ_{i})\geqslant\alpha(A\mid t^{\prime},\succ_{i}),\forall
t<t^{\prime}$

######  Example 5.

[Search Model with Satisficing (Fixed Utility)(Aguiar et al.,, 2016).]

A search model with satisficing has the following procedure. First, decision-
makers will search for items using a random search index in which items having
higher search indexes being considered first. Each decision-maker has their
own threshold, and once they find the first item that is above this threshold,
they will stop searching and choose that item.

For $S=\\{y_{1},\dots,y_{n}\\}$, let $s=(s_{y})_{y\in S}\in\mathbb{R}^{n}$ be
a random search index with a continuous c.d.f. $F_{(s)}$ and
$\mathbf{u}=\left(\mathbf{u}_{y}\right)_{y\in S}\in\mathbb{R}^{n}$ be a fixed
utility. The search index represents the order in which an item will be
considered; the larger an index an item has, the more up-front this item will
be considered. So, for any $y_{s}$ and $y_{z}$, $y_{s}$ is searched before
$y_{z}$ if and only if $s_{y_{s}}\geq s_{y_{z}}$. The distribution of search
indexes is independent from the preference orderings.

For example, if the fixed utility has the following order:
$u_{y_{1}}>u_{y_{2}}>u_{y_{3}}$, then it means that $y_{1}\succ y_{2}\succ
y_{3}$. And if the search index is $s_{y_{2}}>s_{y_{1}}>s_{y_{3}}$, then
$y_{1}$ is searched after $y_{2}$ but before $y_{3}$.

$\tau(t)\in\mathbb{R}$ is a random variable where its distribution depends on
time $t$. $\tau(t)$ denotes a threshold conditional on time $t$. For any
$t^{\prime}>t$, $\tau(t^{\prime})$ is assumed to have a first-order stochastic
dominance over $\tau(t)$ in which the people who stay longer are drawn from a
distribution where they have higher thresholds.

For any consideration set A with cardinality $|A|=N$, denote
$y_{{a_{1}}^{A}}\succ y_{{a_{2}}^{A}}\succ...\succ y_{{a_{n}}^{A}}$, where
${a_{i}}^{A}$ represents the ith highest utility element in the consideration
set A.

In general, for all $A\not=S$ and $|A|=N$, the accumulated attention can be
constructed as:

$\alpha(A\mid t)=\sum^{A}_{B\subseteq A}\mu(B\mid
t)=\sum_{i=1}^{N}Pr(u_{{a_{i}}^{A}}\geq\tau(t))\times P_{{a_{i}}^{A}}$

where $u_{{a_{i}}^{A}}$ is the utility index of the $i$th highest utility item
in consideration set $A$ and $P_{{a_{i}}^{A}}$ is a non-negative number.

And when $A=S$,

$\alpha(S\mid t)=\sum^{A}_{B\subseteq A}\mu(B\mid t)=1.$

For example, if there are only three items: $y_{1}$, $y_{2}$, and $y_{3}$; the
fixed utility has the following ordering: $u_{y_{1}}>u_{y_{2}}>u_{y_{3}}$,
$A=\\{y_{1},y_{2}\\}$. So, I have

$\begin{split}\alpha(A\mid t)=&\mu(\\{y_{1},y_{2}\\}\mid
t)+\mu(\\{y_{1}\\}\mid t)+\mu(\\{y_{2}\\}\mid t)\\\
=&[Pr(u_{y_{1}}\geq\tau(t))-Pr(u_{y_{2}}\geq\tau(t))]\times Pr(s_{y_{2}}\geq
s_{y_{1}}\geq s_{y_{3}})\\\ &+Pr(u_{y_{1}}\geq\tau(t))\times Pr(s_{y_{1}}\geq
s_{y_{2}},s_{y_{1}}\geq s_{y_{3}})+Pr(s_{y_{2}}\geq\tau(t))\times
Pr(s_{y_{2}}\geq s_{y_{1}},s_{y_{2}}\geq s_{y_{3}})\\\
=&Pr(u_{y_{1}}\geq\tau(t))\times[Pr(s_{y_{2}}\geq s_{y_{1}}\geq
s_{y_{3}})+Pr(s_{y_{1}}\geq s_{y_{2}},s_{y_{1}}\geq s_{y_{3}})]\\\
&+Pr(u_{y_{2}}\geq\tau(t))\times[Pr(s_{y_{2}}\geq s_{y_{1}},s_{y_{2}}\geq
s_{y_{3}})-Pr(s_{y_{2}}\geq s_{y_{1}}\geq s_{y_{3}})]\end{split}$

This equation can be rewritten as:

$\alpha(A\mid t)=Pr(u_{y_{1}}\geq\tau(t))\times
P_{a_{1}^{A}}+Pr(u_{y_{2}}\geq\tau(t))\times P_{a_{2}^{A}}$

where $P_{a_{1}^{A}}=[Pr(s_{y_{2}}\geq s_{y_{1}}\geq
s_{y_{3}})+Pr(s_{y_{1}}\geq s_{y_{2}},s_{y_{1}}\geq s_{y_{3}})]\geq 0$ and
$P_{a_{2}^{A}}=[Pr(s_{y_{2}}\geq s_{y_{1}},s_{y_{2}}\geq
s_{y_{3}})-Pr(s_{y_{2}}\geq s_{y_{1}}\geq s_{y_{3}})]\geq 0$.

If $\tau(t^{\prime})$ has a first-order stochastic dominance over $\tau(t)$
for all $t^{\prime}>t$, then $Pr(u_{y_{i}}\geq\tau(t))\geq
Pr(u_{y_{i}}\geq\tau(t^{\prime}))$ for all items in the consideration set A.
Then, the accumulated attention probability is non-increasing in t for all
strict subsets of the full menu and equals one if $A=S$. Further the
monotonicity assumption in time is the following:

For any $t<t^{\prime}$, $A\subset S$,

$\sum_{B\subseteq A}^{A}\mu(B\mid t)\geqslant\sum_{B\subseteq
A}^{A}\mu\left(B\mid t^{\prime}\right)$

and

$\sum_{B\subseteq S}^{S}\mu(B\mid t)=\sum_{B\subseteq S}^{S}\mu\left(B\mid
t^{\prime}\right)$

This is an example where stopping time is endogenous and people who stop at
different times have different distributions of satisfactory thresholds. The
search model with satisfying behavior satisfies monotonicity in time as long
as the conditional threshold $\tau(t^{\prime})$ has a first-order stochastic
dominance over $\tau(t)$ for all $t^{\prime}>t$.

The intuition of this first-order stochastic dominance is that as stopping
time goes up, the distribution of the satisficing threshold moves to the
right, meaning that people who stop later are more patient and have higher
thresholds. Therefore, as decision-makers who stop later are more patient, the
principle of time monotonicity is upheld in Aguiar et al., (2016).

######  Example 6.

[Attention Diffusion.]

This example shows an attention-forming process which mimics Ratcliff and
McKoon, (2008). However, unlike the DDM in which decision-makers collect
information for decisions, decision-makers in this example collect the
saliency scores for items to be in their consideration set. And time
monotonicity is satisfied if the saliency threshold is non-increasing over
time in this example.

The diffusion attention example is as follows: For any item $a_{i}\in S$, a
decision-maker collects its saliency score over time until it reaches a
certain saliency threshold $\tau_{a_{i}}(t)\geq 0$. If the score is above
$\tau_{a_{i}}(t)$ at time $t$, then item $a_{i}$ is in the consideration set
at time $t$.

Evidence accumulation in favor of $a_{i}$ is represented by the Brownian
motion with drift $V_{a}(t)\equiv v_{a_{i}}t+B_{a_{i},t}$, where
$B_{a_{i},t}\sim N(0,t\sigma^{2})$.

Therefore at time $t$, the probability that item $a_{i}$ is in the
consideration set $A$ is:

$\begin{split}\gamma_{t,\succ}(a_{i})=&Pr(v_{a_{i}}t+B_{a_{i},t}\geq\tau_{a_{i}}(t))\\\
=&Pr(B_{a_{i},t}\geq\tau_{a_{i}}(t)-v_{a_{i}}t)\\\
=&1-\Phi(\frac{\tau_{a_{i}}(t)-v_{a_{i}}t}{\sqrt{t}\sigma})\\\ \end{split}$

Time monotonicity is satisfied if the saliency threshold does not increase
over time.

The derivative of the probability with respect to time is:

$\begin{split}\frac{\partial\gamma_{t,\succ}(a_{i})}{\partial
t}=&-\frac{1}{\sigma}\phi(\frac{\tau_{a_{i}}(t)-v_{a_{i}}t}{\sqrt{t}\sigma})\times(\frac{\tau^{\prime}_{a_{i}}(t)}{\sqrt{t}}-\frac{\tau_{a_{i}}(t)}{2\sqrt{t^{3}}}-\frac{v_{a_{i}}}{2\sqrt{t}})\\\
\end{split}$

If $\tau^{\prime}_{a_{i}}(t)\leq 0$, then
$\frac{\partial\gamma_{t,\succ}(a_{i})}{\partial t}\geq 0$. So, the
probability that item $a_{i}$ is in the consideration set $A$ increases over
time.

Note that the accumulated attention to a certain set $A$ is the probability
that one is not considering anything beyond set $A$. Then the accumulated
attention can be written as:

$\alpha(A\mid t,\succ)=\prod_{b\notin A}(1-\gamma_{t,\succ}(b))$

Since $\gamma_{t,\succ}(b)\leq\gamma_{t^{\prime},\succ}(b)$ for all
$t<t^{\prime}$, $b\in S$. Then

$\prod_{b\notin A}(1-\gamma_{t,\succ}(b))\geq\prod_{b\notin
A}(1-\gamma_{t^{\prime},\succ}(b)),\forall t<t^{\prime}$

Therefore,

$\alpha(A\mid t,\succ)\geqslant\alpha(A\mid t^{\prime},\succ),\forall
t<t^{\prime}$

## 3\. Revealed Preference Implications in a Homogeneous Preference
Environment

In this section, I provide the implications of the revealed preferences solely
with stopping time variation when the population share a homogeneous
preference. These implications apply for the occasions when firms do not
change the menu of goods that they sell but manage to record how long
customers take to make choices.

I develop a theorem of the implications of the revealed preferences and a
corresponding algorithm to reject the "wrong" preference orderings. To begin
with, I first build a lower contour set $A_{y}$ includes all items that are
less preferred than y,

$A_{y}=\\{x\in S\mid y\succ_{i}x\\}$

######  Proposition 1.

Suppose that the time monotonicity is satisfied. With a lower contour set of
y, where $A_{y}$ includes all items that are less preferred than y, the RAS
representation with preference $\succ_{i}$ can be characterized by the
following condition:

$\sum_{i_{x}\in A_{y}}\pi\left(i_{x}\mid
t^{\prime},\succ_{i}\right)\leqslant\sum_{i_{x}\in A_{y}}\pi(i_{x}\mid
t,\succ_{i})\qquad\forall y\in S,t<t^{\prime},$

One implication of Proposition 1 is that the most preferred item is more
likely to be picked over time, while the least preferred item is less likely.

According to Proposition 1, if the stochastic data set has RAS representation
and satisfies time monotonicity, the probability of selecting the least k
favorite items, based on a certain preference ordering, should not increase as
individuals spend more time. This reasoning helps to understand the
relationship between time and probability in the context of choosing favorite
items from a stochastic data set.

Unlike the RAM result (Cattaneo et al.,, 2020), Proposition 1 does not
directly reveal the preference between two items, but it provides an algorithm
to eliminate incorrect preference orders. The algorithm is as follows:

######  Definition 5.

(Rejection Method) A preference ordering $\succ=\\{a_{1}\succ a_{2}\succ
a_{3}\succ...\succ a_{n}\\}$ is rejected if there exist some i,
$t<t^{\prime}$, such that the stochastic choice dataset has:

$\sum_{a_{x}\in A_{i}}\pi\left(a_{x}\mid
t^{\prime},\succ\right)>\sum_{a_{x}\in A_{i}}\pi(a_{x}\mid t,\succ)$

where $A_{i}=\\{a_{x}:x\geq i\\}$ is the lower contour set containing $a_{i}$
and all the items less favorable than $a_{i}$.

The logic of rejection directly follows Proposition 1. In a homogeneous
preference setup, for a proposed preference ordering $\succ=\\{a_{1}\succ
a_{2}\succ a_{3}\succ...\succ a_{n}\\}$, if there exist a time point
$t<t^{\prime}$ and a number $i<n$, then the probability of picking the last i
favorable item (according to the proposed preference ordering) increases, then
the proposed preference ordering can not be the true one that allows the
stochastic dataset to admit RAS representation. Therefore the proposed
preference ordering is rejected.

######  Theorem 1.

(Survival Characterization) A set of all possible orders of preferences
$\succ$ (which survives from the rejection method with the data) is non-empty
if and only if the stochastic choice data set has RAS representation with a
homogeneous preference.

###### Proof.

(Theorem 1)

The proof consists of two parts:

(i) Prove that if the stochastic choice dataset has a RAS representation with
a homogeneous preference, then the set of possible preference ordering $\succ$
is non-empty.

If the stochastic choice data set has a RAS representation with a homogeneous
preference $\succ^{*}$; then according to Proposition 1, we know that this is
equivalent to the stochastic data set having for all y that $t<t^{\prime}$,

$\sum_{i_{x}\in A_{y}}\pi\left(i_{x}\mid
t^{\prime},\succ^{*}\right)\leqslant\sum_{i_{x}\in A_{y}}\pi(i_{x}\mid
t,\succ^{*})$

where $A_{y}=\\{i_{x}:i_{x}\succ^{*}y\\}$ contains all available items that
are ranked higher than item y according to $\succ^{*}$.

Then the preference $\succ^{*}$ survives the rejection algorithm. Therefore,
the set of all non-rejected possible orders of preferences is non-empty.

(ii) Prove that if the set of non-rejected possible preference orderings
$\succ$ is non-empty, then the stochastic choice dataset has a RAS
representation with a homogeneous preference.

If the set of non-rejected possible preference orderings $\succ$ is non-empty,
there exists a preference ordering $\succ^{**}$ such that for all y,
$t<t^{\prime}$, the stochastic dataset satisfies

$\sum_{i_{x}\in A_{y}}\pi\left(i_{x}\mid
t^{\prime},\succ^{**}\right)\leqslant\sum_{i_{x}\in A_{y}}\pi(i_{x}\mid
t,\succ^{**})$

where $A_{y}=\\{i_{x}:i_{x}\succ^{**}y\\}$ contains all available items that
are ranked higher than item y according to $\succ^{**}$.

According to Proposition 1, the stochastic dataset has a RAS representation
with preference ordering $\succ^{**}$.

∎

The survival characterization offers a set of all possible preference
orderings that allows the stochastic dataset to admit RAS representation if
the preference ordering is homogeneous.

The details of a survival algorithm are as follows:

For n items, suppose the preference ordering in the population is homogeneous:

1\. Each single item $i_{x}$ could be a potential item that is ranked highest.
Then define set $C^{1}_{x}$ as one with $n-1$ items that excludes the item
$i_{x}$. With the rejection algorithm 5, check if

$\sum_{i\in C^{1}_{x}}\pi\left(i\mid t^{\prime}\right)\leqslant\sum_{i\in
C^{1}_{x}}\pi(i\mid t)$

for all t and t’. If not, I reject that $i_{x}$ is ranked first.

2\. Based on the unrejected items $i_{x}$, check a second item $i_{x2}$ with
corresponding $n-2$ item set $C^{2}_{x2}$ and check if

$\sum_{i\in C^{2}_{x2}}\pi\left(i\mid t^{\prime}\right)\leqslant\sum_{i\in
C^{2}_{x2}}\pi(i\mid t)$

for all t and t’. If not, I reject that $i_{x}$ is ranked first and $i_{x2}$
is ranked second.

3\. Follow the procedure until all the items are checked.

This algorithm allows me to reject the unreasonable preference orderings. Here
is one example using this algorithm:

######  Example 7.

Suppose there are three items $a\succ b\succ c$. At time t, let only set
$\\{b,c\\}$ be considered. At time t’, one will only consider set
$\\{a,b,c\\}$. So, $\pi(a\mid t^{\prime})=1$ and $\pi(b\mid t)=1$. We observe
this choice probability yet do not know the preference orderings.

Now following the algorithm: First check when $i_{x}=a,b,c$

1.If $i_{x}=a$, $\pi(b\mid t^{\prime})+\pi(c\mid t^{\prime})=0<\pi(b\mid
t)+\pi(c\mid t)=1$, fails to reject that a is ranked first.

If $i_{x}=b$, $\pi(a\mid t^{\prime})+\pi(c\mid t^{\prime})=1>\pi(a\mid
t)+\pi(c\mid t)=0$, rejects that b is ranked first.

If $i_{x}=c$, $\pi(b\mid t^{\prime})+\pi(a\mid t^{\prime})=1=\pi(b\mid
t)+\pi(a\mid t)=1$ fails to reject that c is ranked first. However, since
$\pi(c\mid t)=0$ for all t, c cannot be ranked first. So, c is also rejected.

2\. Given a is ranked first, check $i_{x2}=b$ or $c$

If $i_{x2}=b$, $\pi(c\mid t^{\prime})=0=\pi(c\mid t)$, fails to reject that a
is ranked first and b is ranked second.

If $i_{x2}=c$, $\pi(b\mid t^{\prime})=0<\pi(b\mid t)=1$, fails to reject that
a is ranked first and c is ranked second. However, since $\pi(c\mid t)=0$ for
all t, c as the second preferred item is rejected.

So, we have $a\succ b\succ c$.

Further, this is a set identification of preference orderings. Theorem 1 does
not guarantee point identification. Further research is needed to guarantee
singleton survivor preference sets.

## 4\. Heterogeneous Preferences

However, in a heterogeneous environment, decision-makers’ attention can be
influenced by their preference type. For instance, in the search model with
satisficing, individuals will stop searching when the last item they see
offers a payoff above a certain threshold. This means that their attention
development will stop when they come across such items. While this feature is
reasonable, it makes it difficult for the classic limited attention models to
accurately identify the preference distribution.

This section investigates the RAS under heterogeneous preferences. The RAS
provides revealed preference implications and potential applications while
allowing the attention rule to be correlated with preference type.

The RAS representation can be stated as:

$U\cdot A\cdot P=\Pi$ (1)

Where $U$ is the matrix consisting of probabilities of different attention
sets being considered, conditional on time and preference. $A$ is the choice
rule matrix that transforms attention sets into choices given preferences. And
$P$ represents the preference distribution.

Denote n as the number of alternatives, $d_{t}$ as the number of clustered
time periods, $d_{\succ}$ as the number of all possible orderings, $d_{c}$ as
the number of all possible consideration sets, and denote that
$d_{u}=d_{\succ}\times d_{c}$.

For example, if the attention rule is as follows:

$U_{\succ_{i}}=\begin{bmatrix}\mu(a\mid t_{1},\succ_{i})&\mu(b\mid
t_{1},\succ_{i})&...&\mu(S\mid t_{1},\succ_{i})\\\ \mu(a\mid
t_{2},\succ_{i})&\mu(b\mid t_{2},\succ_{i})&...&\mu(S\mid t_{2},\succ_{i})\\\
...&...&...&...\\\ \mu(a\mid t_{d_{t}},\succ_{i})&\mu(b\mid
t_{d_{t}},\succ_{i})&...&\mu(S\mid t_{d_{t}},\succ_{i})\\\
\end{bmatrix}_{d_{t}\times d_{c}}$

and

$U=[U_{\succ_{1}}\quad U_{\succ_{2}}\quad...\quad
U_{\succ_{d_{\succ}}}]_{d_{t}\times d_{u}}$

Combining the linear equality restrictions for all menus, choices,
consideration sets, and preference orders, I can construct a matrix $A$ that
consists of zero and one that does not depend on $\pi$. This $A$ transforms
the attention rule conditional on the preference into a choice probability for
the different alternatives conditional on the preference. For example, if an
individual has a preference of $a\succ b\succ c$ and the agent’s consideration
set is $\\{b,c\\}$, then this agent will choose $b$. The corresponding element
in matrix A is one.

######  Example 8.

(Generating A Matrix)

Suppose there are two items $a$ and $b$ and two possible preferences
$a\succ_{a}b$ and $b\succ_{b}a$. The A matrix is constructed by blocks of
discrete choices conditional on the preference orderings:

$\footnotesize A=\begin{bmatrix}1(a\text{ is }-\succ_{a}\text{ best in
}\\{a\\})&0&1(b\text{ is }-\succ_{a}\text{ best in }\\{a\\})&0\\\ 1(a\text{ is
}-\succ_{a}\text{ best in }\\{b\\})&0&1(b\text{ is }-\succ_{a}\text{ best in
}\\{b\\})&0\\\ 1(a\text{ is }-\succ_{a}\text{ best in }\\{a,b\\})&0&1(b\text{
is }-\succ_{a}\text{ best in }\\{a,b\\})&0\\\ 0&1(a\text{ is }-\succ_{b}\text{
best in }\\{a\\})&0&1(b\text{ is }-\succ_{b}\text{ best in }\\{a\\})\\\
0&1(a\text{ is }-\succ_{b}\text{ best in }\\{b\\})&0&1(b\text{ is
}-\succ_{b}\text{ best in }\\{b\\})\\\ 0&1(a\text{ is }-\succ_{b}\text{ best
in }\\{a,b\\})&0&1(b\text{ is }-\succ_{b}\text{ best in
}\\{a,b\\})\end{bmatrix}$

In this example, the first two columns in matrix $A$ contain blocks
conditional on the preference orderings that indicate whether one will choose
item $a$ under each consideration set.

$A_{a}=\begin{bmatrix}1(a\text{ is }-\succ_{a}\text{ best in }\\{a\\})&0\\\
1(a\text{ is }-\succ_{a}\text{ best in }\\{b\\})&0\\\ 1(a\text{ is
}-\succ_{a}\text{ best in }\\{a,b\\})&0\\\ 0&1(a\text{ is }-\succ_{b}\text{
best in }\\{a\\})\\\ 0&1(a\text{ is }-\succ_{b}\text{ best in }\\{b\\})\\\
0&1(a\text{ is }-\succ_{b}\text{ best in }\\{a,b\\})\end{bmatrix}$

For instance, the block $\begin{bmatrix}1(a\text{ is }-\succ_{a}\text{ best in
}\\{a\\})\\\ 1(a\text{ is }-\succ_{a}\text{ best in }\\{b\\})\\\ 1(a\text{ is
}-\succ_{a}\text{ best in }\\{a,b\\})\\\ \end{bmatrix}$ represents whether
item $a$ will be chosen conditional on $a\succ_{a}b$ under different
consideration sets. Trivially, $a$ will not be chosen if the consideration set
is $\\{b\\}$, and will be chosen as long as $a$ is the best item (according to
the preference ordering) in the consideration set.

Therefore, in this example,

$A=\begin{bmatrix}1&0&0&0\\\ 0&0&1&0\\\ 1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&1\\\
0&0&0&1\end{bmatrix}$

With the rule that at each time point the decision-maker will pick the best
among their consideration set, the transformation matrix yields:

$U\times A=\Pi_{\succ}$

where the $\Pi_{\succ}$ denotes the probability of choices conditional on
preferences:

$\Pi_{\succ}=\begin{bmatrix}\pi(a\mid t_{1},\succ_{1})&\pi(a\mid
t_{1},\succ_{2})&...&\pi(a\mid t_{1},\succ_{d_{\succ}})&...&\pi(i_{n}\mid
t_{1},\succ_{d_{\succ}})\\\ \pi(a\mid t_{2},\succ_{1})&\pi(a\mid
t_{2},\succ_{2})&...&\pi(a\mid t_{2},\succ_{d_{\succ}})&...&\pi(i_{n}\mid
t_{2},\succ_{d_{\succ}})\\\ ...&...&...&...&...&...\\\ \pi(a\mid
t_{d_{t}},\succ_{1})&\pi(a\mid t_{d_{t}},\succ_{2})&...&\pi(a\mid
t_{d_{t}},\succ_{d_{\succ}})&...&\pi(i_{n}\mid t_{d_{t}},\succ_{d_{\succ}})\\\
\end{bmatrix}_{d_{t}\times(n\times d_{\succ})}$

######  Example 9.

The setup is the same as the previous example. There are two items $a$ and $b$
and two possible preferences $a\succ_{a}b$ and $b\succ_{b}a$. Therefore,

$A=\begin{bmatrix}1&0&0&0\\\ 0&0&1&0\\\ 1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&1\\\
0&0&0&1\end{bmatrix}$

Suppose there are two time periods: $t_{1}$ and $t_{2}$. The attention rule
is:

$U=\begin{bmatrix}\mu(\\{a\\}\mid t_{1},\succ_{a})&\mu(\\{b\\}\mid
t_{1},\succ_{a})&\mu(\\{a,b\\}\mid t_{1},\succ_{a})&...&\mu(\\{a,b\\}\mid
t_{1},\succ_{b})\\\ \mu(\\{a\\}\mid t_{2},\succ_{a})&\mu(\\{b\\}\mid
t_{2},\succ_{a})&\mu(\\{a,b\\}\mid t_{2},\succ_{a})&...&\mu(\\{a,b\\}\mid
t_{2},\succ_{b})\\\ \end{bmatrix}_{2\times 6}$

Then

$\Pi_{\succ}=U\times A=\begin{bmatrix}\pi(a\mid t_{1},\succ_{a})&\pi(a\mid
t_{1},\succ_{b})&\pi(b\mid t_{1},\succ_{a})&\pi(b\mid t_{1},\succ_{b})\\\
\pi(a\mid t_{2},\succ_{a})&\pi(a\mid t_{2},\succ_{b})&\pi(b\mid
t_{2},\succ_{a})&\pi(b\mid t_{2},\succ_{b})\\\ \end{bmatrix}_{2\times 2}$

where $\pi(a\mid t,\succ_{i})=\sum_{B\subseteq S}1(a\text{ is }-\succ\text{
best in }B)\cdot\mu(B\mid t,\succ)$

Let
$p=\begin{bmatrix}p(\succ_{1})&p(\succ_{2})&...&p(\succ_{d_{\succ}})\end{bmatrix}$
be the distribution over all possible preferences. Then $P$ is a block
diagonal of $p$ with $n$ (number of items in the menu) blocks.

$P=\begin{bmatrix}p^{T}&0&...&0\\\ 0&p^{T}&...&0\\\ ...&...&...&...\\\
0&0&...&p^{T}\\\ \end{bmatrix}_{(n\times d_{\succ)}\times n}$

where

$\Pi=\begin{bmatrix}\pi(a\mid t_{1})&\pi(b\mid t_{1})&...&\pi(i_{n}\mid
t_{1})\\\ \pi(a\mid t_{2})&\pi(b\mid t_{2})&...&\pi(i_{n}\mid t_{2})\\\
...&...&...&...\\\ \pi(a\mid t_{d_{t}})&\pi(b\mid t_{d_{t}})&...&\pi(i_{n}\mid
t_{d_{t}})\\\ \end{bmatrix}_{d_{t}\times n}$

###  4.1. Theorem with Heterogeneous Preference

To identify preferences, the characterization of the RAS is needed as follows:

######  Theorem 2.

The following are equivalent:

(i) The stochastic choice dataset admits RAS representation with time
monotonicity,

(ii) There exists

a) A matrix $U\in\mathbb{R}_{+}^{d_{t}\times d_{u}}$ such that for all
$i\in[1,d_{\succ}]$, $k\in[1,d_{t}]$, $j\in[1,d_{c}]$

$\sum_{j\in[1,d_{c}]}U_{k,j+(i-1)\times d_{c}}=1$

b) $P$ where the sum of the elements in $p^{\prime}$ is one.

c) $U$ follows time monotonicity inequality (2).

such that the stochastic dataset satisfies the equality:

$U\cdot A\cdot P=\Pi$

In order to find the preference distribution in the stochastic data set, I
need a bundle $(U,P)$ with (a) a matrix $U$ that satisfies time monotonicity;
(b) a bundle $(U,P)$ that minimizes the distance between $U\cdot A\cdot P$ and
the choice data set $\Pi$. This method does not offer point identification
since I have more parameters than the choice probabilities, yet I can get a
set of preference distributions that are not rejected by theorem 2 and could
potentially be the true distribution under the heterogeneous preference data.

## 5\. Estimation and Testing Methods

###  5.1. Estimation Method

One way to estimate the preference distribution is to simultaneously search
for preference distribution $\hat{p}$ and attention rule $\hat{U}$ together to
minimize the distance between stochastic choice data set $\hat{\Pi}$ and
$U\cdot A\cdot P$. However, due to the large size of the restrictions on time
monotonicity, this approach is hard to implement.

Therefore, I use simulation. The idea is to simulate a large number of
attention rules that satisfy time monotonicity. I use the hit-and-run method
mentioned in Aguiar and Kashaev, (2021) to simulate the attention rules (See
appendix F for details). In each simulation $k$, the preference distribution
$\hat{P}_{k}$ can be easily found since $U^{sim}_{k}$ is fixed and time
monotonicity is already satisfied. Then, I can find the simulation that offers
the smallest minimized distance between $\hat{\Pi}$ and $U^{sim^{*}}\cdot
A\cdot\hat{P}^{sim^{*}}$. The $\hat{P}^{sim^{*}}$ is the estimation result for
the preference distribution.

With the RAS representation $U\cdot A\cdot P=\Pi$, the estimation method here
is to simulate the attention rules $U^{sim}_{k}$ (according to time
monotonicity) and minimize the distance between the estimation and the data by
finding the proper $\hat{P}_{k}$ given the simulated $U^{sim}_{k}$.

$\displaystyle\hat{P}_{k}=\operatorname{ArgMin}_{P}$
$\displaystyle\quad||U^{sim}_{k}\cdot A\cdot P-\hat{\Pi}||^{2}$ (2)

and the distance is

$\displaystyle d_{k}=\operatorname{Min}_{P}$
$\displaystyle\quad||U^{sim}_{k}\cdot A\cdot P-\hat{\Pi}||^{2}$ (3)

With K number of simulations,

$\displaystyle\hat{P}=\\{\hat{P}_{k}\mid d_{k}\geq d_{i},\quad\forall i\leq
K\\}$ (4)

With the choice dataset $\Pi$, one can draw some empirical implications about
the preference distribution $P$. In fact, if the conditional choice matrix
$U(S)\cdot A$ does not have multicollinearity and the maximum time period is
greater than $(n_{m}-1)!$ ($n_{m}$ represents the number of items in the
menu), then the distribution of heterogeneous preferences can be uniquely
identified.

###  5.2. Testing Method

Theorem 2 allows me to test whether a given stochastic choice rule $\Pi$ can
be generated by the RAS representation. Testing for RUM is a well-understood
problem and amounts to solving a quadratic optimization with cone constraints
(Kitamura and Stoye,, 2018). However, the steps for testing the RAS need to be
adjusted since the time monotonicity constraints in the attention rule grow
exponentially. Therefore, in order to test whether the data have a RAS
representation, I first simulate a large number N of attention rules
$U^{sim}_{i}$ ($i\in N$) that follows time monotonicity. For each
$U^{sim}_{i}$, I minimize the distance:

$\displaystyle d_{i}=\operatorname{Min}_{P}$
$\displaystyle\quad||U^{sim}_{i}\cdot A\cdot P-\Pi||^{2}$ (5)

and the approximated attention rule $\hat{U}$ is the simulated $U^{sim}_{i}$
that has the least distance: $\hat{U}=min\\{U^{sim}_{i}|i\in N\\}$. In
appendix G, I have shown that given a large enough number of simulations, the
true attention rule lies in the set of $U^{sim}_{i}$ and should be close
enough to $\hat{U}$. Therefore, I am testing whether the stochastic choice
dataset admits RAS with the specific attention rule $\hat{U}$. According to
Theorem 2, this is equivalent to testing the null hypothesis:

$\inf_{P\in\mathbb{R}_{+}^{d}}\left\|\Pi-\hat{U}AP\right\|=0$

This testing problem can be directly linked to the testing problem in Kitamura
and Stoye, (2018).

Further, the true choice probability $\Pi$ is not observed, but the realized
choice frequencies $\hat{\Pi}$ are. Given the estimator of the true choice
probability $\hat{\Pi}$, the attention rule estimator $\hat{U}$ is computed.
The test statistics is:

$\mathrm{T}_{n}=n\min_{\left[P-\tau_{n}\iota/d\right]\in\mathbb{R}_{+}^{d}}\left(\hat{\Pi}-\hat{U}AP\right)^{\prime}\hat{\Omega}^{-}\left(\hat{\Pi}-\hat{U}AP\right)$

where n is the sample size; $\hat{\Omega}^{-}$ is a generalized inverse of a
diagonal matrix $\hat{\Omega}$ such that the i-th diagonal element
$\hat{\Omega}_{i,i}$ is a consistent estimator of the asymptotic variance in
the i-th component of $\hat{\Pi}$; $\tau_{n}$ is a tuning parameter; and
$\iota$ is a vector of ones of dimension d. Let $\hat{\Pi}_{l}$,
$l=1,\ldots,L$, be bootstrap replications of $\hat{\Pi}$. The critical values
of $T_{n}$ are computed following the bootstrap procedure in Kitamura and
Stoye, (2018):

(i) Compute $\hat{\eta}_{\tau_{n}}=\hat{U}AP_{\tau_{n}}$, where $P_{\tau_{n}}$
solves

$n\min_{\left[P-\tau_{n}\iota/d\right]\in\mathbb{R}_{+}^{d}}\left(\hat{\Pi}-\hat{U}AP\right)^{\prime}\hat{\Omega}^{-}\left(\hat{\Pi}-\hat{U}AP\right)$

(ii) Compute

$\hat{\Pi}_{l}=\hat{\Pi}_{l}-\hat{\Pi}+\hat{\eta}_{\tau_{n}}$

for every $l=1,\ldots,L$ and $\Omega^{*}$

(iii) Compute the bootstrap test statistic

$\mathrm{T}^{*}_{n,l}=n\min_{\left[P-\tau_{n}\iota/d\right]\in\mathbb{R}_{+}^{d}}\left(\hat{\Pi}_{l}-\hat{U}AP\right)^{\prime}\hat{\Omega}^{*-}\left(\hat{\Pi}_{l}-\hat{U}AP\right)$

$l=1,\ldots,L$.

(iv) Use the empirical distribution of the bootstrap statistic to compute
critical values of $T_{n}$.

For a specific confidence level $\alpha\in(0,\frac{1}{2})$, the decision rule
for the test is to reject the null hypothesis of the RAS with a particular
$U^{sim}$ value if the test statistic $\mathrm{T}_{n}$ is greater than the
$(1-\alpha)$ quantile of the empirical distribution of the bootstrap statistic
that is denoted as $\hat{c}_{1-\alpha}$. It is important to note that if the
null hypothesis is incorrect and the asymptotic variance of $\hat{\Pi}$ is
bounded above and is not equal to zero, then the test statistic diverges
toward infinity as the sample size increases. Therefore, as the sample size
grows, I can reject the wrongly specified null hypothesis with a probability
that approaches one.

## 6\. Empirical Results

In this section, the RAS model is tested and applied to estimate the
preference distribution in the experimental data collected by Aguiar et al.,
(2023). When time variation is used with the RAS and the CRRA preference, the
result is consistent with the estimation obtained through the menu variation
with a logit-attention rule structure and the CRRA preference (Aguiar et al.,,
2023).

###  6.1. The Experimental Dataset

The experiment is conducted through Amazon MTurk for a large cross-section
with at most two disjoint choice sets per individual. The experimental design
produces a stochastic choice data set with variation in the full choice set.
The experiment contains six lottery alternatives with different expected
values and variances to induce (potential) preference heterogeneity. Table 1
shows the alternatives and implied preference orderings if the decision-makers
are expected utility maximizers with a CRRA Bernoulli utility function

$u(x)=\begin{cases}\frac{x^{1-\sigma}}{1-\sigma},&\sigma\neq 1\\\
\ln(x),&\sigma=1\end{cases}$

Table 1: Lotteries measured by tokens, expected values, and variance

Lottery | Expectation | Variance | Preference Rank $u(x)=\frac{x^{1-\sigma}}{1-\sigma}$ With $\sigma$
---|---|---|---
|  |  | -1 | 0 | 0.25 | 0.3 | 0.5 | 0.75 | 1
(1) $\frac{1}{2}50+\frac{1}{2}0$ | 25.000 | 625.00 | 1 | 1 | 2 | 5 | 5 | 6 | 6
$(2)$ $\frac{1}{2}30+\frac{1}{2}10$ | 20.000 | 100.00 | 5 | 5 | 5 | 2 | 1 | 1 | 1
$(3)$ $\frac{1}{4}50+\frac{1}{4}30+\frac{1}{4}10+\frac{1}{4}0$ | 22.500 | 368.75 | 3 | 3 | 4 | 4 | 3 | 4 | 4
$(4)$ $\frac{1}{4}50+\frac{1}{5}48+\frac{3}{20}14+\frac{2}{5}0$ | 24.125 | 511.73 | 2 | 2 | 1 | 3 | 4 | 5 | 5
(5) $\frac{1}{5}48+\frac{1}{4}30+\frac{3}{20}14+\frac{1}{4}10+\frac{5}{20}0$ | 21.625 | 251.11 | 4 | 4 | 3 | 1 | 2 | 3 | 3
(o) 12 with probability 1 | 12.000 | 0.00 | 6 | 6 | 6 | 6 | 6 | 2 | 2

I have restricted the decision-makers to follow CRRA utility function in my
empirical estimation. According to Aguiar et al., (2023), the CRRA-LA model is
tested with experimental data and the model is not rejected. Moreover,
although I have estimated the preference distribution under the CRRA, the
estimation does not require any restrictions on the utility’s functional form
and is fully nonparametric for utility. The use of the CRRA is only to compare
my estimation result with Aguiar et al., (2023).

The RAS uses an estimation approach that only requires time variation without
the need for menu variation. Therefore, I only consider the segment of the
data where individuals face the full menu that includes all possible
lotteries. This segment of the data consists of 171 observations that comprise
the lottery choices made by individuals and the time they took to make their
choices.

To obtain the choice probability conditional on time $\Pi(t\mid S)$, I cluster
the observations into six time periods. The first period is grouped by
observations spending zero seconds in making choices. The remaining five time
periods are clustered using the K-mean clustering method. Table 2 shows the
K-mean clustered choice data set. Each row represents the time period the
probability is conditional on, and each column represents the probability of
picking the according lottery during the time period. Interestingly, all
players who made a choice in period 1 (where they spend almost no time in
making choices) chose the outside option $l_{O}$. This choice indicates that
at the beginning of the experiment, individuals’ consideration set was limited
to the outside option.

Table 2: Kmean Clustered Choice Data

period | l1 | l2 | l3 | l4 | l5 | lO
---|---|---|---|---|---|---
1 | 0 | 0 | 0 | 0 | 0 | 1
2 | 0.142857 | 0.204082 | 0.142857 | 0.173469 | 0.112245 | 0.22449
3 | 0.153061 | 0.326531 | 0.204082 | 0.132653 | 0.102041 | 0.0816327
4 | 0.142857 | 0.459184 | 0.122449 | 0.0510204 | 0.112245 | 0.112245
5 | 0.208333 | 0.395833 | 0.114583 | 0.020833 | 0.20833 | 0.0520083
6 | 0.14 | 0.31 | 0.17 | 0.11 | 0.17 | 0.1

###  6.2. Estimation Results

The goal here is to find the simulated attention rule $U^{sim}$ and preference
distribution $P$ bundle that offers the least distance between the estimation
and the data. To reduce the computational burden, I assume the preference
follows the CRRA rule with $\sigma\in[-1,1]$. I also assume $l_{O}$ as the
outside option lottery (the robustness check in the appendix shows this
assumption is plausible). With this CRRA structural restriction and $l_{O}$ as
the outside option, there are six available preference orderings, and each
preference ordering corresponds to a range of risk aversion. Table 3 provides
a summary of the six preference orderings and their corresponding risk
aversion ranges.

Table 3: CRRA Preference Orderings and Corresponding Risk Aversion Preference Order | Implied $\sigma$
---|---
$l_{1}\succ l_{4}\succ l_{3}\succ l_{5}\succ l_{2}$ | $[-1,0.2287)$
$l_{4}\succ l_{1}\succ l_{5}\succ l_{3}\succ l_{2}$ | $(0.2287,0.2606)$
$l_{4}\succ l_{5}\succ l_{1}\succ l_{3}\succ l_{2}$ | $(0.2606,0.2728)$
$l_{5}\succ l_{4}\succ l_{2}\succ l_{3}\succ l_{1}$ | $(0.2728,0.2832)$
$l_{5}\succ l_{2}\succ l_{4}\succ l_{3}\succ l_{1}$ | $(0.2832,0.3001)$
$l_{2}\succ l_{5}\succ l_{3}\succ l_{4}\succ l_{1}$ | $(0.3001,1]$

Based on my results in Table 4, there is a general alignment with the previous
research (Table 5) in terms of preference distribution. This general alignment
of results shows that RAS can use time variation instead of menu variation and
offer a similar outcome. Importantly, RAS complements random attention models
when there is a lack of menu variation in data sets.

Table 4: Estimation Results Preference Order | $\hat{p}$ | Implied $\sigma$
---|---|---
$l_{1}\succ l_{4}\succ l_{3}\succ l_{5}\succ l_{2}$ | $0.30181$ | $[-1,0.2287)$
$l_{4}\succ l_{1}\succ l_{5}\succ l_{3}\succ l_{2}$ | $0.00000$ | $(0.2287,0.2606)$
$l_{4}\succ l_{5}\succ l_{1}\succ l_{3}\succ l_{2}$ | $0.00000$ | $(0.2606,0.2728)$
$l_{5}\succ l_{4}\succ l_{2}\succ l_{3}\succ l_{1}$ | $0.22979$ | $(0.2728,0.2832)$
$l_{5}\succ l_{2}\succ l_{4}\succ l_{3}\succ l_{1}$ | $0.00000$ | $(0.2832,0.3001)$
$l_{2}\succ l_{5}\succ l_{3}\succ l_{4}\succ l_{1}$ | $0.46839$ | $(0.3001,1]$
Table 5: Estimation Results from CRRA-LA Model with Menu Variation Preference Order | $\hat{p}$ | Implied $\sigma$
---|---|---
$l_{1}\succ l_{4}\succ l_{3}\succ l_{5}\succ l_{2}$ | $0.30500$ | $[-1,0.2287)$
$l_{4}\succ l_{1}\succ l_{5}\succ l_{3}\succ l_{2}$ | $0.04905$ | $(0.2287,0.2606)$
$l_{4}\succ l_{5}\succ l_{1}\succ l_{3}\succ l_{2}$ | $0.04905$ | $(0.2606,0.2728)$
$l_{5}\succ l_{4}\succ l_{2}\succ l_{3}\succ l_{1}$ | $0.04905$ | $(0.2728,0.2832)$
$l_{5}\succ l_{2}\succ l_{4}\succ l_{3}\succ l_{1}$ | $0.04905$ | $(0.2832,0.3001)$
$l_{2}\succ l_{5}\succ l_{3}\succ l_{4}\succ l_{1}$ | $0.49880$ | $(0.3001,1]$

However, the RAS identifies a higher percentage of decision-makers who prefer
lottery 5. The estimation in the CRRA-LA setup, where the full menu variation
is necessary, is shown in Table 5. Compared to the CRRA-LA setup, the RAS
estimates in Table 4 show a similar percentage of decision-makers with
preference 1888Preference 1: $l_{1}\succ l_{4}\succ l_{3}\succ l_{5}\succ
l_{2}$ ($30.2\%$ versus $30.5\%$ for preference 1). On the other hand, it
shows a slightly lower percentage of decision-makers with preference
6999Preference 6: $l_{2}\succ l_{5}\succ l_{3}\succ l_{4}\succ l_{1}$
($46.8\%$ versus $49.9\%$ for lottery 2). Therefore, the estimations for
preference 1 and preference 6 are stable in both the RAS and CRRA-LA models.

Further, the RAS identifies a significantly higher percentage of individuals
who prefer Lottery 5 the most. Thus, Lottery 5 has the highest number of
operations, and one possible reason for this different result is that people
tend to avoid considering items that look more complicated, yet they might
prefer these items in terms of payoffs and variations. The RAS successfully
retrieves this preference under limited attention by assigning a higher weight
to the individuals who choose lottery 5 (according to time monotonicity).

The difference in magnitude may come from three sources. First, the RAS
captures the attention-forming process with time monotonicity and provides
extra information about the stopping time. This information helps in providing
a more accurate restriction on attention and leads to a more precise
estimation of the preference distribution. Second, the RAS allows a
correlation between the attention and a preference that extracts further
information about preferences from the data sets. However, my estimation only
uses the part of the full data where individuals face a full menu. Therefore,
it explains only that part of the data and not the entire data with full menu
variation. It would be interesting to add menu variation to the RAS for a
fairer comparison of the two estimations.

The RAS was tested with the experimental data, and the resulting p-value of
0.245 indicates that the null hypothesis cannot be rejected. This nonrejection
means that the data are likely generated by the RAS, given the simulated
attention rule $U^{sim}$. However, while the null hypothesis is rejected, the
p-value is still quite low. This is because the test only provides evidence
against one specific attention rule $U^{sim}$, rather than the entire set of
possible attention rules. One drawback of this testing method is that
rejecting the null hypothesis does not necessarily mean that the entire model
is rejected. Instead, it only rejects the model’s compatibility with a
specific attention rule. But since the null cannot be rejected, the entire
model of the RAS cannot be rejected as well.

## 7\. Conclusions

Menu variation is a ’gold mine’ in limited attention models. However, it is
crucial for researchers to develop limited attention models when menu
variation is not available. In this paper, I develop a random attention span
model. The RAS is able to provide preference distribution under limited
attention with only time variation instead of menu variation. Moreover, it has
the potential to generalize the use of variation in identifying preferences.

The model offers sufficient conditions and an algorithm for rejecting untrue
preference orderings. I provide a characterization of the RAS. With the
characterization, the RAS is testable with the estimated attention rule. I
then estimate the preference distribution and test the RAS with experimental
data from Aguiar et al., (2023). When restricted to the constant relative risk
aversion (CRRA) preferences, the RAS estimation has a general alignment with
the CRRA-LA model (Aguiar et al.,, 2023) that uses menu variation. However,
the RAS identifies a significantly higher percentage of decision-makers who
prefer a more complex lottery. I also test the RAS model using methods in
Kitamura and Stoye, (2018), and the test fails to reject the null that the
choice data set is generated by the RAS.

These results are important and desirable because the RAS uses different
pieces of information compared to the CRRA-LA setup (Aguiar et al.,,
2023).101010RAS uses data with time variation when decision-makers are facing
the same menu, while CRRA-LA estimation uses data when decision-makers are
facing all kinds of menus but without considering the stopping time factor.
However, in the experimental data by Aguiar et al., (2023), 32 menus are
assigned to decision-makers; therefore, the number of observations used by the
RAS is much smaller than the observations used by the CRRA-LA model. This
smaller sample means that the RAS is able to offer estimates of the preference
distribution as well as the CRRA-LA with fewer observations. Moreover, menu
variations are observed much less than time variation, which gives RAS an
advantage in empirical estimation.

In summary, RAS is a new model for understanding consumer behavior under
limited attention when menu variation is not available. This is especially
important in digital contexts where attention spans are short and menus don’t
change often.

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##  Appendix A Robustness Check for CRRA-LA Estimation

In section 5, I assume that lottery O is the outside option and is always
ranked last. Yet this is not necessarily true. To check the robustness of the
estimation in section 5, I withdraw the outside option assumption and put
lottery O into regular ranking settings. Table 6 show the robustness check
result. Compared to Table 4, the results show very similar estimations on
preference distribution $\hat{\pi}$; these results indicate that the
estimation of $\hat{\pi}$ is pretty robust with the assumption of the outside
option. The robustness check shows that the estimation in section 6 has very
satisfactory robustness.

Table 6: Estimation Results Robustness Check Preference Order | $\hat{p}$ | Implied $\sigma$
---|---|---
$l_{1}\succ l_{4}\succ l_{3}\succ l_{5}\succ l_{2}$ | $0.25744$ | $[-1,0.2287)$
$l_{4}\succ l_{1}\succ l_{5}\succ l_{3}\succ l_{2}$ | $0.00000$ | $(0.2287,0.2606)$
$l_{4}\succ l_{5}\succ l_{1}\succ l_{3}\succ l_{2}$ | $0.00000$ | $(0.2606,0.2728)$
$l_{5}\succ l_{4}\succ l_{2}\succ l_{3}\succ l_{1}$ | $0.32321$ | $(0.2728,0.2832)$
$l_{5}\succ l_{2}\succ l_{4}\succ l_{3}\succ l_{1}$ | $0.00000$ | $(0.2832,0.3001)$
$l_{2}\succ l_{5}\succ l_{3}\succ l_{4}\succ l_{1}$ | $0.41935$ | $(0.3001,1]$

##  Appendix B Omitted Details About Example 3

For $S=\\{y_{1},y_{2},y_{3}\\}$, let $s=(s_{y})_{y\in S}\in\mathcal{U}$ be a
random search index with a continuous c.d.f. $F_{(s)}$ and
$\mathbf{u}=\left(\mathbf{u}_{y}\right)_{y\in S}\in\mathcal{U}$ be a fixed
utility. The search index represents how soon an item will be considered, the
larger the index that an item has, the sooner this item will be considered.
So, for any $y_{s}$ and $y_{z}$, $y_{s}$ appears in the search before $y_{z}$
if and only if $s_{y_{s}}\geq s_{y_{z}}$. The distribution of the search
indexes is independent from the preference orderings.

The fixed utility has the following ranking: $u_{y1}>u_{y2}>u_{y3}$, which
means that $y_{1}\succ y_{2}\succ y_{3}$.

$\tau(t)\in R$ denotes a threshold conditional on stopping time t. And for any
$t^{\prime}>t$, $\tau(t^{\prime})$ has a first-order stochastic dominance over
$\tau(t)$. We can view this as the people who stay longer are drawn from a
distribution where they have higher thresholds.

Next I am going to show that in this search model with satisfaction, the
accumulated attention does not increase with t for all consideration sets as
long as the threshold $\tau(t)$ does not decrease during t.

Suppose for a given preference ordering $\succ:y_{1}\succ y_{2}\succ y_{3}$.
For any consideration set A with cardinality $|A|=N$, denote $y_{A_{1}}\succ
y_{A_{2}}\succ...\succ y_{A_{N}}$, where $A_{i}$ represents the ith element in
consideration set A.

1\. For any $|A|=1$, the accumulated attention conditional on preference
ordering $\succ$ in this framework equals to: $\sum^{A}_{B\subseteq
A}\mu(B\mid t)=\mu(A\mid t)=Pr(u_{y_{A_{1}}}\geq\tau(t))\times
Pr(s_{y_{A_{1}}}\geq s_{y},\forall y\in S-y_{A_{1}})$

2\. For any $|A|=2$, the accumulated attention conditional on preference
ordering $\succ$ in this framework equals:

$\begin{split}\sum^{A}_{B\subseteq A}\mu(B\mid
t)=&\mu(\\{y_{A_{1}},y_{A_{2}}\\}\mid t)+\mu(\\{y_{A_{1}}\\}\mid
t)+\mu(\\{y_{A_{2}}\\}\mid t)\\\
=&Pr(u_{y_{A_{1}}}\geq\tau(t),u_{y_{A_{2}}}<\tau(t))\times
Pr(s_{y_{A_{2}}}\geq s_{y_{A_{1}}}\geq s_{y},\forall y\in
S-y_{A_{1}}-s_{y_{A_{2}}})\\\ &+Pr(u_{y_{A_{2}}}\geq\tau(t))\times
Pr(s_{y_{A_{2}}}\geq s_{y},\forall y\in S-y_{A_{2}})\\\
&+Pr(u_{y_{A_{1}}}\geq\tau(t))\times Pr(s_{y_{A_{1}}}\geq s_{y},\forall y\in
S-y_{A_{1}})\\\ =&Pr(u_{y_{A_{1}}}\geq\tau(t),u_{y_{A_{2}}}<\tau(t))\times
Pr(s_{y_{A_{2}}}\geq s_{y_{A_{1}}}\geq s_{y},\forall y\in
S-y_{A_{1}}-s_{y_{A_{2}}})\\\ &+Pr(u_{y_{A_{2}}}\geq\tau(t))\times
Pr(s_{y_{A_{2}}}\geq s_{y},\forall y\in S-y_{A_{2}})\\\
&+Pr(u_{y_{A_{1}}}\geq\tau(t),u_{y_{A_{2}}}<\tau(t))\times
Pr(s_{y_{A_{1}}}\geq s_{y},\forall y\in S-y_{A_{1}})\\\
&+Pr(u_{y_{A_{1}}}\geq\tau(t),u_{y_{A_{2}}}\geq\tau(t))\times
Pr(s_{y_{A_{1}}}\geq s_{y},\forall y\in S-y_{A_{1}})\end{split}$

Since the preference ordering is fixed, and $A_{N}$ by definition is the
lowest ranked item in consideration set A, then:

$Pr(u_{y_{A_{N}}}\geq\tau(t))=Pr(u_{y_{i}}\geq\tau(t),\forall y_{i}\in A)$

So

$\begin{split}\sum^{A}_{B\subseteq A}\mu(B\mid
t)=&Pr(u_{y_{A_{1}}}\geq\tau(t),u_{y_{A_{2}}}<\tau(t))\times
Pr(s_{y_{A_{2}}}\geq s_{y_{A_{1}}}\geq s_{y},\forall y\in
S-y_{A_{1}}-s_{y_{A_{2}}})\\\ &+Pr(u_{y_{A_{2}}}\geq\tau(t))\times
Pr(s_{y_{A_{2}}}\geq s_{y},\forall y\in S-y_{A_{2}})\\\
&+Pr(u_{y_{A_{1}}}\geq\tau(t),u_{y_{A_{2}}}<\tau(t))\times
Pr(s_{y_{A_{1}}}\geq s_{y},\forall y\in S-y_{A_{1}})\\\
&+Pr(u_{y_{A_{1}}}\geq\tau(t),u_{y_{A_{2}}}\geq\tau(t))\times
Pr(s_{y_{A_{1}}}\geq s_{y},\forall y\in S-y_{A_{1}})\\\
=&Pr(u_{y_{A_{1}}}\geq\tau(t),u_{y_{A_{2}}}<\tau(t))\times
Pr(s_{y_{A_{2}}}\geq s_{y_{A_{1}}}\geq s_{y},\forall y\in
S-y_{A_{1}}-s_{y_{A_{2}}})\\\
&+Pr(u_{y_{A_{1}}}\geq\tau(t),u_{y_{A_{2}}}\geq\tau(t))\times
Pr(s_{y_{A_{2}}}\geq s_{y},\forall y\in S-y_{A_{2}})\\\
&+Pr(u_{y_{A_{1}}}\geq\tau(t),u_{y_{A_{2}}}<\tau(t))\times
Pr(s_{y_{A_{1}}}\geq s_{y},\forall y\in S-y_{A_{1}})\\\
&+Pr(u_{y_{A_{1}}}\geq\tau(t),u_{y_{A_{2}}}\geq\tau(t))\times
Pr(s_{y_{A_{1}}}\geq s_{y},\forall y\in S-y_{A_{1}})\\\
=&Pr(u_{y_{A_{1}}}\geq\tau(t),u_{y_{A_{2}}}<\tau(t))\times[Pr(s_{y_{A_{2}}}\geq
s_{y_{A_{1}}}\geq s_{y},\forall y\in S-y_{A_{1}}-s_{y_{A_{2}}})\\\
&+Pr(s_{y_{A_{1}}}\geq s_{y},\forall y\in S-y_{A_{1}})]\\\
&+Pr(u_{y_{A_{1}}}\geq\tau(t),u_{y_{A_{2}}}\geq\tau(t))\times[Pr(s_{y_{A_{2}}}\geq
s_{y},\forall y\in S-y_{A_{2}})\\\ &+Pr(s_{y_{A_{1}}}\geq s_{y},\forall y\in
S-y_{A_{1}})]\\\ \end{split}$

Since

$Pr(u_{y_{A_{1}}}\geq\tau(t),u_{y_{A_{2}}}<\tau(t))+Pr(u_{y_{A_{1}}}\geq\tau(t),u_{y_{A_{2}}}\geq\tau(t))=Pr(u_{y_{A_{1}}}\geq\tau(t)),$

We have

$\begin{split}\sum^{A}_{B\subseteq A}\mu(B\mid
t)=&Pr(u_{y_{A_{1}}}\geq\tau(t))\times[Pr(s_{y_{A_{2}}}\geq s_{y_{A_{1}}}\geq
s_{y},\forall y\in S-y_{A_{1}}-s_{y_{A_{2}}})+\\\ &Pr(s_{y_{A_{1}}}\geq
s_{y},\forall y\in S-y_{A_{1}})]\\\
&+Pr(u_{y_{A_{2}}}\geq\tau(t))\times[Pr(s_{y_{A_{2}}}\geq s_{y},\forall y\in
S-y_{A_{2}})\\\ &-Pr(s_{y_{A_{2}}}\geq s_{y_{A_{1}}}\geq s_{y},\forall y\in
S-y_{A_{1}}-s_{y_{A_{2}}})]\end{split}$

In general, for all $A\not=S$ and $|A|=N$, the accumulated attention can be
constructed as:

$\sum^{A}_{B\subseteq A}\mu(B\mid
t)=\sum_{i=1}^{N}Pr(u_{y_{A_{i}}}\geq\tau(t))\times P_{y_{A_{i}}}$

where $P_{y_{A_{i}}}$ represents the probability of some certain search
ordering happening specifically correlating to $y_{A_{i}}$.

And when $A=S$,

$\sum^{A}_{B\subseteq A}\mu(B\mid t)=1.$

Since $\tau(t^{\prime})$ has a first-order stochastic dominance over $\tau(t)$
for all $t^{\prime}>t$, $Pr(u_{y_{A_{i}}}\geq\tau(t))\geq
Pr(u_{y_{A_{i}}}\geq\tau(t^{\prime}))$ for all items in the consideration set
A. Therefore, the accumulated attention probability does not increase with t
for all strict subsets of the full menu and equals one if $A=S$. Recall the
monotonicity assumption in time:

For any $t<t^{\prime}$, $A\subset S$,

$\sum_{B\subseteq A}^{A}\mu(B\mid t)\geqslant\sum_{B\subseteq
A}^{A}\mu\left(B\mid t^{\prime}\right)$

and

$\sum_{B\subseteq S}^{S}\mu(B\mid t)=\sum_{B\subseteq S}^{S}\mu\left(B\mid
t^{\prime}\right)$

The search model with satisficing behavior satisfies the monotonicity in time
as long as the conditional threshold $\tau(t^{\prime})$ has a first-order
stochastic dominance over $\tau(t)$ for all $t^{\prime}>t$.

##  Appendix C Time Monotonicity and Drifting Diffusion Model

The drifting diffusion model (DDM) is a popular model that uses stopping time.
While both the RAS and the DDM use stopping time, the two models have quite
different angles from which they use stopping time.

In a classic DDM for value-based decisions, there are two items to choose
from, denoted by $\\{a,b\\}$. Decision-makers must choose one item that they
prefer, and their stopping time is noted. The DDM model assumes that decision-
makers collect noisy information from both alternatives until they reach some
threshold $\lambda>0$ at which they make a choice. The mean information
accumulation is denoted by $v(a)$ and $v(b)$, respectively. If $v(a)>v(b)$,
then $a$ is preferred over $b$; otherwise, $b$ is preferred over $a$.
Decision-makers are assumed to face instantaneous independent white noise
fluctuations as modeled by uncorrelated Wiener processes $W_{a}$ and $W_{b}$
during the information accumulation. Evidence accumulation in favor of $a$ and
$b$ is then represented by the two Brownian motions with drift $V_{a}(t)\equiv
v(a)t+\sigma W_{a}(t)$ and $V_{b}(t)\equiv v(b)t+\sigma W_{b}(t)$. Thus, the
net evidence is:

$Z_{a,b}(t)\equiv V_{a}(t)-V_{b}(t)=[v(a)-v(b)]t+\sigma\sqrt{2}W(t)$

With a threshold $\lambda$, a decision-maker stops evidence accumulation and
makes their decision if $Z_{a,b}(t)$ reaches either $\lambda$ or $-\lambda$.
The time a decision-maker first reaches either $\lambda$ or $-\lambda$ is
called a stopping time in the DDM.

$\mathrm{DT}_{a,b}\equiv\min\left\\{t:Z_{a,b}(t)=\lambda\text{ or
}Z_{a,b}(t)=-\lambda\right\\}$

The decision-maker will choose $a$ if $Z_{a,b}(t)$ reaches $\lambda$ first,
and will choose b otherwise.

$\mathrm{DO}_{a,b}\equiv\begin{cases}a&\text{ if
}Z_{a,b}\left(\mathrm{DT}_{a,b}\right)=\lambda\\\ b&\text{ if
}Z_{a,b}\left(\mathrm{DT}_{a,b}\right)=-\lambda\end{cases}$

There are fundamental differences in the underlying assumptions of the RAS and
the DDM. The DDM has only two alternatives, and one alternative is chosen if
the net evidence (with noisy evidence) is above a particular threshold. In
contrast, the RAS has multiple alternatives, and the decision-maker will
choose the most preferred option if it is in their consideration set.

The combination of noise and a threshold in the DDM leads to the possibility
of mistakes if the threshold is very low or if there is an unusually high
level of noise. However, in the RAS, decision-makers are assumed to make the
best choice among the alternatives that they actually consider. In the RAS,
the assumption of time monotonicity is exogenous to decision-makers’
attention, conditional on time, although the stopping time at the decision is
endogenous.

In general, in random attention models, researchers rule out the possibility
of mistakes, and the choices decision-makers make are based only on their
preferences and the consideration set at the time they make the decision. This
difference leads the DDM and the RAS to have different focuses of study. The
DDM focuses on choice accuracy, while RAS focuses on preference distribution.

I incorporate the idea of random attention into the DDM to illustrate the
conceptual difference in the following example.

######  Example 10.

(DDM with Random Attention). Suppose a menu has 2 items $S=\\{a,b\\}$. The
decision-makers must choose one item they prefer, and the stopping time is
collected. Decision-makers collect noisy information from both alternatives
until they reach some threshold $\lambda>0$ and then make the choice.

Decision-makers have random attention and are following time monotonicity. At
each time point, the probability that a decision-maker only sees item $a$ is
$u(a|t)$ and the probability that a decision-maker only sees item $b$ is
$u(b|t)$. A decision-maker’s probability of seeing both $a$ and $b$ is
$u(a,b|t)$.

According to time monotonicity, the probability of considering only one item
decreases as time passes, and the probability of considering both items
increases. That is: For any $t<t^{\prime}$,

$u(a|t)\geqslant u(a|t^{\prime})$ $u(b|t)\geqslant u(b|t^{\prime})$

and

$u(a,b|t)\leqslant u(a,b|t^{\prime})$

If the mean information accumulations are $v(a)$ and $v(b)$ ($a\succ b$ if
$v(a)>v(b)$ and $b\succ a$ if $v(b)>v(a)$), then decision-makers are assumed
to face instantaneous independent white noise fluctuations modeled by
uncorrelated Wiener processes $W_{a}$ and $W_{b}$ during the information
accumulations.

Decision-makers cannot start evidence accumulation of an item until that item
is in their consideration set. If time is discrete, then
$V_{a}(1)=(1-u(b|t=1))v(a)+\sigma w_{a}(1)$, where $(1-u(b|t=1))$ is the
probability that the decision-maker takes $a$ into consideration so they can
accumulate evidence for $a$. $V_{a}(2)=V_{a}(1)+(1-u(b|t=2))v(a)+\sigma
w_{a}(2)$ and thus $V_{a}(t)=v(a)(t-\sum_{0}^{t}u(b|t))+\sigma W_{a}(t)$.

If time is continuous, the evidence accumulation in favor of a and b is then
represented by the two Brownian motions with drift $V_{a}(t)\equiv
v(a)(t-\int_{0}^{t}u(b|t)dt)+\sigma W_{a}(t)$ and $V_{b}(t)\equiv
v(b)(t-\int_{0}^{t}u(a|t)dt)+\sigma W_{b}(t)$. Thus, the net evidence is:

$Z_{a,b}(t)\equiv
V_{a}(t)-V_{b}(t)=[v(a)-v(b)]t+v(b)\int_{0}^{t}u(a|t)dt-v(a)\int_{0}^{t}u(b|t)dt+\sigma\sqrt{2}W(t)$

In the DDM with random attention above, the attention rule enters the model
exogenously. Moreover, if the probability of noticing an item is zero, then it
is impossible that the net evidence reaches the threshold which is in favor of
this item. This impossibility means decision-makers will not pick an item they
did not notice at all. And if $t\rightarrow+\infty$, the net evidence
converges to the item that the decision-maker prefers. While the DDM with
random attention is unable to either predict choices or identify the
preference anywhere in between due to the noise and the "satisfactory-like"
threshold.

##  Appendix D Proof for Proposition 1

###### Proof of Proposition 1.

The first step is to prove that (i) implies (ii). For a fixed ranking
$\succ=\\{i_{1}\succ i_{2}\succ i_{3}\succ...\succ i_{n}\\}$, I consider from
the lowest ordered item $i_{n}$, according to the RAS, the probability of
choosing the last ordered item should be the probability of only considering
that particular item:

$\pi(i_{n}\mid t,\succ)=\mu(\\{i_{n}\\}\mid t,\succ)$ $\pi(i_{n}\mid
t^{\prime},\succ)=\mu(\\{i_{n}\\}\mid t^{\prime},\succ)$

For second last ranked item $i_{n-1}$

$\pi(i_{n-1}\mid t,\succ)=\mu(\\{i_{n-1}\\}\mid
t,\succ)+\mu(\\{i_{n-1},i_{n}\\}\mid t,\succ)$ $\pi(i_{n-1}\mid
t^{\prime},\succ)=\mu(\\{i_{n-1}\\}\mid
t^{\prime},\succ)+\mu(\\{i_{n-1},i_{n}\\}\mid t^{\prime},\succ)$

and so on.

Therefore,

$\sum_{i_{x}\in A_{y}}\pi(i_{x}\mid t,\succ)=\sum_{B\subseteq
A_{y}}^{A_{y}}\mu(B\mid t,\succ)$

According to monotonicity in time, it directly follows that

$\sum_{i_{x}\in A_{y}}\pi\left(i_{x}\mid
t^{\prime},\succ\right)\leqslant\sum_{i_{x}\in A_{y}}\pi(i_{x}\mid t,\succ)$

The second step is to prove that (ii) implies (i). The proof is done by
construction. Suppose we see a sequence $(i_{1},i_{2},i_{3},...,i_{n})$ such
that for all y, $t<t^{\prime}$,

$\sum_{i_{x}\in A_{y}}\pi\left(i_{x}\mid
t^{\prime},\succ\right)\leqslant\sum_{i_{x}\in A_{y}}\pi(i_{x}\mid t,\succ)$

First, I claim that this is the ranking sequence $\succ=\\{i_{1}\succ
i_{2}\succ i_{3}\succ...\succ i_{n}\\}$. One can construct an attention rule
such that for any k and t,

$\mu(A_{k}\mid t,\succ)=\pi(a_{k}\mid t,\succ)$

and the probability of paying attention to any attention set other than this
lower contour set form $A_{i}$, is zero. Then the RAS representation survives
as follows:

For any k

$\begin{split}\pi(a_{k}\mid t,\succ)&=\sum_{A\subseteq S}1(a\text{ is
}-\succ\text{ best in }A)\cdot\mu(A\mid t,\succ)\\\ &=\mu(A_{k}\mid
t,\succ)+\sum_{A\subseteq S/A_{k}}1(a\text{ is }-\succ\text{ best in
}A)\cdot\mu(A\mid t,\succ)\\\ &=\mu(A_{k}\mid t,\succ)+0\\\ &=\mu(A_{k}\mid
t,\succ)\end{split}$

Therefore if (ii) happens in the stochastic dataset, then there exists an
attention rule matrix that satisfies the RAS representation. ∎

##  Appendix E Proof for Theorem 2

###### Proof of Theorem 2.

The direction from (i) to (ii) follows directly from the definition of RAS
representation. If the stochastic choice dataset admits RAS representation
with time monotonicity, then the true attention rule U should follow the time
monotonicity, and each element in U and P should be non-negative and less than
one. Which yields

$U\cdot A\cdot P=\Pi$

The direction from (ii) to (i) is as follows. Since $V$ fits the monotonicity
inequality restriction. Simply claim the $V$ and $P$ are the true attention
rule following time monotonicity and preference distribution. The feature a)
ensures V can be a probability measure for attention rule and $p^{\prime}$ can
be the preference distribution. According to equation 1,

$V\cdot A\cdot P=\Pi$

is a RAS representation.

∎

##  Appendix F Hit-and-Run Algorithm

This paper uses "Hit-and-Run" algorithm (Aguiar and Kashaev,, 2021) to
simulate attention rule matrix $U$, which satisfies time monotonicity.

For each preference ordering $\succ_{i}$, first generate a vector of
probabilities that decision-makers are paying attention to each consideration
set at the beginning $t=0$.

$U(S,\succ_{i},t=0)=\begin{bmatrix}\mu(\\{a\\}\mid
t=0,\succ_{i})&\mu(\\{b\\}\mid t=0,\succ_{i})&...&\mu(S\mid t=0,\succ_{i})\\\
\end{bmatrix}$

Then the vector of accumulative attention can be calculated

$V(S,\succ_{i},t=0)=\begin{bmatrix}\alpha(\\{a\\}\mid
t=0,\succ_{i})&\alpha(\\{b\\}\mid t=0,\succ_{i})&...&\alpha(S\mid
t=0,\succ_{i})\\\ \end{bmatrix}$

where $\alpha(A\mid t,\succ_{i})=\sum_{B\subseteq A}^{A}\mu(B\mid
t,\succ_{i})$.

In the choice dataset (Aguiar et al.,, 2023), the outside option lottery o
always stays in the menu. At $t=0$, I only observe decision-makers choosing
the outside option. Therefore in the empirical estimation, I assume that the
outside option is always considered and only the outside option can be
considered at $t=0$. Therefore there are 32 possible consideration sets (given
that there are 5 lotteries and 1 outside option). This yields
$\mu(\\{l_{o}\\}\mid t=0,\succ_{i})$, and therefore

$U(S,\succ_{i},t=0)=\begin{bmatrix}1&0&0&...&0&0\\\ \end{bmatrix}_{1\times
32}$

According to time monotonicity, for any $t<t^{\prime}$, $A\subset S$,

$\alpha(A\mid t,\succ_{i})\geqslant\alpha(A\mid t^{\prime},\succ_{i})$

and

$\alpha(S\mid t,\succ_{i})=1$

This means an attention rule at $t=1$ satisfies time monotonicity only if the
accumulative attention vector $V(S,\succ_{i},t=1)-V(S,\succ_{i},t=0)\leq 0$.

The idea behind the "hit-and-run" algorithm is (i) to pick some initial row
$U(S,\succ_{i},t=0)$; (ii) to construct a candidate row by moving along a
random direction within the constrained set on a randomly chosen distance;
(iii) to use a user-specified Monte-Carlo acceptance rule to assign to either
the initial point $U(S,\succ_{i},t=0)$ or the candidate point
$U(S,\succ_{i},t=1)$; (iv) to apply steps (ii) and (iii) to
$U(S,\succ_{i},t=1)$ to construct $U(S,\succ_{i},t=2)$; (v) to repeat until
the length of the chain reaches user chosen number (maximum of time periods).

To put it short, let $e^{t}=U(S,\succ_{i},t)$. Take some arbitrary $e^{t}$
that already satisfied time monotonicity. Let $\xi$ be a direction vector.
Thus, the candidate vector is

$e^{t+1}=e^{t}+\gamma\xi$

where $\gamma$ determines the scale of the perturbation $\gamma\xi$.

Sign constraints. I start with sign constraints to make sure the attention
rule vector is the distribution of attention: $\mu(A_{k}\mid t,\succ_{i})$ for
all $k\in K_{c}$, where $K_{c}$ is a set of indexes that correspond to
$\mu(A_{k}\mid t,\succ_{i})$ in $e^{t}$. Hence, the constraints take the form

$\gamma\xi_{k}\geq-e_{k}^{t},\quad\forall k\in K_{c}$

and

$\gamma\xi_{k}\leq 1-e_{k}^{t},\quad\forall k\in K_{c}$

Define $K_{+}=\left\\{k\in K_{c}:\xi_{k}>0\right\\}$, $K_{-}=\left\\{k\in
K_{c}:\xi_{k}<0\right\\}$ and $K_{0}=\left\\{k\in K_{c}:\xi_{k}=0\right\\}$.
Then the sign constraints are:

$\displaystyle\frac{1-e_{k}^{t}}{\xi_{k}}\geq\gamma\geq-\frac{e_{k}^{t}}{\xi_{k}},\quad\forall
k\in K_{+}$
$\displaystyle\frac{1-e_{k}^{t}}{\xi_{k}}\leq\gamma\leq-\frac{e_{k}^{t}}{\xi_{k}},\quad\forall
k\in K_{-}$

Note that the constraints that correspond to $k\in K_{0}$ are always satisfied
since $1\geq e_{k}^{t}\geq 0$. Thus, the sign constraints can be simplified to

$\min_{k\in K_{-}}-\frac{e_{k}^{t}}{\xi_{k}}\geq\gamma\geq\max_{k\in
K_{+}}-\frac{e_{k}^{t}}{\xi_{k}}$

and

$\min_{k\in K_{+}}\frac{1-e_{k}^{t}}{\xi_{k}}\geq\gamma\geq\max_{k\in
K_{-}}\frac{1-e_{k}^{t}}{\xi_{k}}$

Time monotonicity constraints. Denote

$\Psi(t)=\begin{bmatrix}\psi_{1}^{t}&\psi_{2}^{t}&...&\psi_{K}^{t}\\\
\end{bmatrix}$

where $K=||K_{c}||$ and $\psi_{k}^{t}=\sum_{A_{x}\subseteq A_{k}}\xi_{k}$.
This $\psi_{k}^{t}$ can be considered as accumulative movement direction for
the accumulative attention.

According to time monotonicity, for any $t<t^{\prime}$, $A\subset S$,

$\alpha(A\mid t,\succ_{i})\geqslant\alpha(A\mid t^{\prime},\succ_{i})$

Denote $\alpha_{k}^{t}=\alpha(A_{k}\mid t,\succ_{i})$, the time monotonicity
is

$\alpha_{k}^{t+1}-\alpha_{k}^{t}\leq 0,\quad\forall k\in K_{c}$

Moreover,

$\alpha_{k}^{t+1}=\alpha_{k}^{t}+\gamma\psi_{k}^{t},\quad\forall k\in K_{c}$

Therefore,

$\gamma\psi_{k}^{t}\leq 0,\quad\forall k\in K_{c}$

Define $K^{\psi}_{+}=\left\\{k\in K_{c}:\psi_{k}>0\right\\}$,
$K^{\psi}_{-}=\left\\{k\in K_{c}:\psi_{k}<0\right\\}$ and
$K^{\psi}_{0}=\left\\{k\in K_{c}:\psi_{k}=0\right\\}$. Therefore time
monotonicity indicates:

$\displaystyle\gamma\geq 0,\quad\forall k\in K^{\psi}_{-}$
$\displaystyle\gamma\leq 0,\quad\forall k\in K^{\psi}_{+}$

Note that the time monotonicity indicates $\gamma=0$ unless all elements in
the accumulative direction vector $\psi^{t}$ share the same sign. Therefore to
operate the simulation process, the direction vector $\xi^{t}$ has to satisfy
the condition such that all elements in accumulative direction $\psi^{t}$ have
the same sign. Computationally, one can restrict the domain of k, and then the
sign of $\gamma$ is automatically defined. Here I restrict that $k\in
K^{\psi}_{-}$, and therefore $\gamma\geq 0$.

And the restrictions is as follows:

$\displaystyle\min_{k}-\frac{e_{k}^{t}}{\xi_{k}}\geq\gamma\geq
0,\quad\forall\xi^{t}\leq 0$
$\displaystyle\min_{k}\frac{1-e_{k}^{t}}{\xi_{k}}\geq\gamma\geq
0,\quad\forall\xi^{t}\geq 0$

Therefore, the algorithm for each attention rule conditional on a preference
$\succ_{i}$ generation process is as follows:

(i) Pick some initial row $e^{t}$;

(ii) Generate a random direction vector $\xi^{t}$, which makes $\psi^{t}$
contain only elements that share the same sign or 0.

(iii) Given the direction vector $\xi^{t}$, the range for the scale of the
perturbation $\gamma$ can be determined:

$\displaystyle\min_{k}-\frac{e_{k}^{t}}{\xi_{k}}\geq\gamma\geq
0,\quad\forall\xi^{t}\leq 0$
$\displaystyle\min_{k}\frac{1-e_{k}^{t}}{\xi_{k}}\geq\gamma\geq
0,\quad\forall\xi^{t}\geq 0$

(iv) Randomly pick $\gamma$ within the range, and get the next row
$e^{t+1}=e^{t}+\gamma\xi^{t}$

(v) Repeat until the length of the chain reaches the number of maximum time
periods.

(vi) The attention rule conditional on a preference $\succ_{i}$ is generated.
Denoted as $U(\succ_{i})$

The attention rule under heterogeneous preference is simply the stack of all
simulated conditional attention rules:

$U^{sim}=[U(\succ_{1}),U(\succ_{2}),...,U(\succ_{n})]$

##  Appendix G A Simulation Exercise

To show that simulation can cover the true attention rule and preference
distribution if the number of simulations is large enough. I take the approach
of simulation on attention rules. Specifically, I simulate a matrix of
attention rules $U^{nsim}$ which satisfies time monotonicity. For example, in
three items and two time periods case, the simulated attention matrix is as
follows:

$U^{nsim}=\begin{bmatrix}\mu^{nsim}(a\mid t_{1},\succ_{1})&\mu^{nsim}(b\mid
t_{1},\succ_{1})&...&\mu^{nsim}(a,b,c\mid t_{1},\succ_{6})\\\ \mu^{nsim}(a\mid
t_{2},\succ_{1})&\mu^{nsim}(b\mid t_{2},\succ_{1})&...&\mu^{nsim}(a,b,c\mid
t_{2},\succ_{6})\\\ \end{bmatrix}$

Each roll contains all attention rules conditional on 6 preference orderings
(since there are 3 items) and different time periods ($t_{1}$ for roll 1 and
$t_{2}$ for roll 2 in this case).

Denote the observed stochastic dataset as:

$\Pi(S)=\begin{bmatrix}\Pi(t_{1}\mid S)\\\ \Pi(t_{2}\mid S)\\\ \end{bmatrix}$

Therefore the RAS representation can be stated as follows:

$\Pi(S)=U*A*P^{diag}$

where A is the projection matrix transforming attention rule U to choice
probability conditional on preference, and $P^{diag}$ is the block diagonal
matrix of preference distribution $P$.

This section performs a test showing with a large enough simulation on the
attention rule, the true parameter will be captured.

There are 3 time periods and 3 items. And in each period, the probabilities of
choosing each item are recorded.I set
$P^{*}=\begin{bmatrix}1/6&1/6&1/6&1/6&1/6&1/6)\end{bmatrix}$.

I randomly pick a truth attention rule $U^{*}$ and have the choice dataset
$\Pi^{*}$ and they stay constant across all the simulation procedures.

The testing consists of two layers of simulation, I call it the inner layer
(with N times simulation) and the outer layer (with T times of simulation).

For each outer layer $k=1,2,...,T$, I simulate N times of attention rules
$U_{k,i}$, where $i=1,2,...,N$. Define $d(U_{k,i}*A*P^{diag},\Pi(S))$ as the
Euclidean distance between the simulated $U_{k,i}*A*P^{diag}$ and the
stochastic dataset $\Pi(S)$.

Define $d_{k}$ as the shortest distance between the simulation and the dataset
across the inner layer simulations:

$d_{k}=Inf_{i}d(U_{k,i}*A*P^{diag},\Pi(S))$

We check if the distance between the truth and simulated estimates are close
enough $d_{k}\leq\delta$, here I pick $\delta=0.05$. If yes, then denote
$c_{k}=1$, else $c_{k}=0$.

The estimate of probability that $P$ lies in the range of estimated
distribution $\hat{P_{k}}$ is $\eta=\frac{\sum^{T}_{k=1}c_{k}}{T}$.

I pick $T=10000$, $N=10,100,1000,10000$. Here I denote the probability that
$P$ lies in the range of estimated distribution $\hat{P_{k}}$ when N=10 as
$\eta_{10}$.

The following table is the result:

$\begin{array}[]{cccc}\eta_{10}&\eta_{100}&\eta_{1000}&\eta_{10000}\\\
0.4528&0.7963&0.9021&0.9891\end{array}$

Theoretically, and N goes to infinity, $\eta$ should converge to 1. Meaning
that if we have a large enough simulation of attention rules, it will cover
the true attention distribution.

This test also shows that with a large number of simulated attention rules,
one can capture a range of preference distributions by simply solving the
linear equation based on each simulated attention rule.