# Dynamic and Stochastic Rational Behavior††thanks: This paper subsumes
“Nonparametric Analysis of Dynamic Random Utility Models.” The “ⓡ” symbol
indicates that the authors’ names are in certified random order, as described
by Ray and Robson (2018). We thank Roy Allen, Chris Chambers, Pierre-André
Chiappori, Mark Dean, Adam Dominiak, Laura Doval, Mikhail Freer, David
Freeman, Matt Kovach, Elliot Lipnowski, Paola Manzini, Krishna Pendakur, Matt
Polisson, John Quah, Jörg Stoye, Jesse Shapiro, Tomasz Strzalecki, and Levent
Ülkü for useful discussions and encouragement. Plávala acknowledges support
from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation,
project numbers 447948357 and 440958198), the Sino-German Center for Research
Promotion (Project M-0294), the ERC (Consolidator Grant 683107/TempoQ), the
German Ministry of Education and Research (Project QuKuK, BMBF Grant No.
16KIS1618K), and the Alexander von Humboldt Foundation. Aguiar thanks USFQ
School of Economics for kindly hosting him during the writing of this paper.

Nail Kashaev ⓡ Victor H. Aguiar ⓡ Martin Plávala ⓡ Charles Gauthier Kashaev:
Department of Economics, University of Western Ontario<EMAIL_ADDRESS>Aguiar: Department of Economics, University of Western Ontario;
<EMAIL_ADDRESS>Plávala: Naturwissenschaftlich-Technische Fakultät,
Universität Siegen<EMAIL_ADDRESS>Gauthier: ECARES, Université
Libre de Bruxelles<EMAIL_ADDRESS>
(This version August, 2023. First version: February, 2023)

###### Abstract

The (static) utility maximization model of Afriat (1967), which is the
standard in analysing choice behavior, is under scrutiny. We propose the
Dynamic Random Utility Model (DRUM) that is more flexible than the framework
of Afriat (1967) and more informative than the static Random Utility Model
(RUM) framework of McFadden and Richter (1990). Under DRUM, each decision-
maker randomly draws a utility function in each period and maximizes it
subject to a menu. DRUM allows for unrestricted time correlation and cross-
section heterogeneity in preferences. We characterize DRUM for situations when
panel data on choices and menus are available. DRUM is linked to a finite
mixture of deterministic behaviors that can be represented as a product of
static rationalizable behaviors. This link allows us to convert the
characterizations of the static RUM to its dynamic form. In an application, we
find that although the static utility maximization model fails to explain
population behavior, DRUM can explain it.

JEL classification numbers: C10, C33, D11, D12, D15.
Keywords: dynamic random utility, revealed preference.

## 1\. Introduction

A fundamental question in economics is whether decision makers exhibit
rational choice behavior. The traditional definition of rationality is
effectively equivalent to maximizing a utility function that is fixed in time.
However, the static utility maximization model is under empirical scrutiny.
Here, we study a notion of rationality in choice behavior that is stochastic
and dynamic—the Dynamic Random Utility Model (DRUM). Under DRUM, each consumer
or decision maker (DM) at each period maximizes the utility realized from a
stochastic utility process subject to a menu or budget.111From the point of
view of the DMs, utilities and budgets are known and deterministic.
Stochasticity appears due to unobserved heterogeneity from the point of view
of the observer. We provide a revealed preference characterization of DRUM for
situations in which the longitudinal distribution of choices or demands is
observed for a finite collection of menus or budgets in a finite time window.
This characterization does not place any parametric restriction on (i) the
form of utility functions, (ii) the correlation of utilities in time, and
(iii) the heterogeneity of utility in the cross-section. In an application, we
show DRUM can explain the behavior of a cross section of DMs when many of them
fail to be consistent with the static utility maximization model.

The two main frameworks available for analyzing consumer behavior are the
framework of static utility maximization based on Samuelson (1938) and Afriat
(1967) and the framework of random utility maximization (RUM) based on
McFadden and Richter (1990). DRUM addresses several empirical limitations of
these earlier models. In particular, the Samuelson-Afriat framework is under
scrutiny due to experimental and field evidence against it.222For examples in
regard to household consumption see Echenique et al. (2011) and Dean and
Martin (2016), and in regard to choices over portfolios with risk or
uncertainty see Choi et al. (2007, 2014), Ahn et al. (2014). The rationality
violations were originally thought to be small (Echenique et al., 2011, Choi
et al., 2007), but newer experimental datasets show these violations can be
severe (Brocas et al., 2019, Aguiar and Serrano, 2021, Halevy and Mayraz,
2022). There is evidence that failures of the Samuelson-Afriat framework are
driven by the stringent assumption of the stability of preferences over time.
For example, utility functions may change over time because of variability in
time of the neural computation of value (Kurtz-David et al., 2019), of
structural breaks (Cherchye et al., 2017), or of evolving risk aversion (Guiso
et al., 2018, Akesaka et al., 2021). DRUM allows preferences to change freely
in time. In contrast to the Samuelson-Afriat framework, RUM has found
reasonable success explaining repeated cross-sections of household choices
(Kawaguchi, 2017, Kitamura and Stoye, 2018). However, RUM cannot take
advantage of the longitudinal variation in choices available in many datasets,
and it may have limited empirical bite (Im and Rehbeck, 2021). By considering
a richer primitive, we simultaneously relax the assumption of stable
preferences over time implicit in the Samuelson-Afriat framework while
providing a more informative test of stochastic utility maximization than in
the McFadden-Richter framework.

Our first result is a mixture characterization of DRUM, which is analogous to
the RUM characterization in McFadden and Richter (1990). We exploit the fact
that DRUM is associated with a finite mixture of preference profiles in time.
We obtain results analogous to Kitamura and Stoye (2018) (henceforth, KS),
McFadden and Richter (1990), and Kawaguchi (2017) with a dynamic version of
the Axiom of Stochastic Revealed Preferences. This finite mixture
characterization lends itself to statistical testing using the results in KS.
This characterization can also be used for nonparametric counterfactual
analysis. In a Monte Carlo study, we show that the statistical test of KS
applied to our characterization of DRUM performs well in finite samples.

We show that the mixture representation of DRUM can be obtained using a
Kronecker product of the mixture representation of RUM in each
period.333Informally, the mixture representation of RUM can be represented as
a matrix whose columns are deterministic rational demand types. The analogous
matrix for DRUM is the Kronecker product of those RUM matrices. This
observation is vital to obtain (i) computational gains for testing because of
the modularity of the mixture representation; and (ii) a novel
characterization of DRUM using a recursive version of the Block and Marschak
(1960) inequalities (henceforth, BM inequalities). In a static setting,
Kitamura and Stoye (2018) were the first to observe that the empirical content
of RUM can be expressed as cone restrictions on observed data. In particular,
the Weyl-Minkowski theorem posits that a cone can be described equivalently by
a convex combination of its vertices ($\mathcal{V}$-representation) or by its
faces ($\mathcal{H}$-representation). Kitamura and Stoye (2018) note that the
$\mathcal{H}$-representation corresponds to what decision theorists would call
an axiomatic characterization of RUM.444Note that the BM inequalities are the
$\mathcal{H}$-representation of RUM in the finite abstract setup. We exploit
recent mathematical advancements in the analysis of the Kronecker products of
cones (due to Aubrun et al., 2021 and Aubrun et al., 2022) to provide an
axiomatic characterization of DRUM using the axiomatic characterization of
RUM. This paper is the first to bring this new mathematical tool to economics
and to show how it can be used in the DRUM setup and in structurally similar
models.555For example, models of bounded rationality, such as the model of
random consideration of Cattaneo et al. (2020) and its extension with
heterogeneous preferences in Kashaev and Aguiar (2022a), can be extended to a
dynamic setup in the spirit of DRUM.

The generalized Weyl-Minkowski theorem enables us to provide a full
characterization of DRUM via dynamic BM inequalities, which covers as a
special case the finite abstract setups of Li (2021) and of Chambers et al.
(2021) with full menu variation. Our characterization works for cases of
limited observability of menus and in the presence of a primitive order that
is respected by the support of the utility process. We also provide a novel
behavioral condition necessary for the consistency of the longitudinal
distribution of demand with DRUM, $\mathrm{D}$-monotonicity. It is also
sufficient in simple setups: (i) for any finite number of goods and two
budgets per period, and (ii) for two goods and any finite number of budgets
per period. $\mathrm{D}$-monotonicity is computationally simple to check and
provides a deeper understanding of the empirical content of DRUM. It restricts
the joint probability of choices in time beyond the RUM restrictions on
marginal distributions in each period. $\mathrm{D}$-monotonicity can be
thought of as a dynamic version of the Weak Axiom of Stochastic Revealed
Preference (see Bandyopadhyay et al., 1999, Hoderlein and Stoye, 2014) and a
stochastic version of the Weak Axiom of Revealed Preference (in time series)
by Samuelson (1938).

We synthesize the two main paradigms of nonparametric demand analysis:
Samuelson and Afriat and McFadden-Richter frameworks. The Samuelson-Afriat
framework requires observing a _time-series_ of choices and budgets for a
given consumer and assumes that the consumer maximizes the same utility
function in each period. When preferences are allowed to vary in time, there
are no empirical implications with only a time-series of choices. However,
when a _panel_ of choices is used, DRUM bounds the share of consumers or DMs
whose choices contain a revealed preference violation in the sense of Afriat
(1967). RUM instead requires observing a _cross-section_ of choices and
budgets from a population of consumers. The panel structure is ignored as
there is no time dimension. Hence, this approach misses the potential temporal
correlation of utilities. As a result, panels of choices over menus exist
that, when marginalized, are consistent with RUM but not with DRUM. In other
words, ignoring the time dimension of choices may lead to false positives when
testing DRUM. Importantly, our setup keeps the fundamental assumption in the
McFadden-Richter framework—the distribution of utilities does not depend on
the sequence of budgets or menus that the consumer faces in time.666This
assumption can be relaxed in the same spirit as Deb et al. (2021).

Our synthesis is advantageous because (i) it provides more informative bounds
on counterfactual choice due to the richer variation in the panel of choices,
(ii) it provides a theoretical justification for the validity of the RUM
framework when marginalizing choices, and (iii) it clarifies the role of
constant preferences across time in the Samuelson-Afriat framework.
Fortunately, our primitive with a longitudinal level of variation is readily
available in many consumption surveys, household scanner datasets, and
experimental datasets, as documented in Aguiar and Kashaev (2021).777In
practice, panels of choices are often pooled in the time dimension to create a
cross-section with enough budget variations to have empirical bite (Deb et
al., 2021, Kitamura and Stoye, 2018). In this case, we show that this approach
could lead to false rejections of DRUM due to ignoring the time labels of
budgets.

In our application, we find support for DRUM in a panel dataset collected by
Aguiar et al. (2023). In a large experimental dataset collected in Amazon
Mechanical Turk, a large cross section of DMs ($2135$ DMs) faced a sequence of
binary comparisons of lotteries. Although $8$ percent of DMs are inconsistent
with the static utility maximization model, we cannot reject the null
hypothesis that DRUM can explain the data.888A restriction of DRUM to Expected
Utility fails to explain the behavior of this sample of DMs. Monte Carlo
experiments mimicking the application setup provide evidence of high power of
our DRUM test in finite samples.

The DRUM framework is rich and extends well beyond the Samuelson-Afriat and
McFadden-Richter worlds. We therefore can cover all of the following special
cases: (i) consumption models of errors in the evaluation of utility (Kurtz-
David et al., 2019); (ii) dynamic random expected utility (defined in Frick et
al., 2019); (iii) static utility maximization in a population (without
measurement error) (Aguiar and Kashaev, 2021); (iv) dynamic utility
maximization in a population (Browning, 1989, Gauthier, 2018, Aguiar and
Kashaev, 2021); (v) changing utility or multiple-selves models (Cherchye et
al., 2017); and (vi) changing-taste modelled with a constant utility in time
with an additive shock (Adams et al., 2015).

Outline The paper is organized as follows. Section 2 introduces the setup.
Section 3 provides both a McFadden-Richter and a KS-type characterization of
DRUM. Section 4 provides a behavioral characterization of DRUM via linear
inequality constraints. Section 5 synthesizes the setups of Samuelson-Afriat
and McFadden-Richter. Section 6 provides results for our dynamic
counterfactual analysis. Section 7 provides an application to experimental
data. Section 8 contains the literature review. Section 9 concludes. All
proofs can be found in Appendix A. Appendix B provides Monte Carlo experiments
showcasing the finite sample properties of the statistical test of DRUM.

## 2\. Setup

We consider a time window $\mathcal{T}=\\{1,\cdots,T\\}$ with a finite
terminal period $T\geq 1$. Let $X^{t}$ be a nonempty finite choice set. We
endow $2^{X^{t}}\setminus\\{\emptyset\\}$ with some acyclic partial order
$>^{t}$. When $>^{t}$ is restricted to singletons, it induces an acyclic
partial order on $X^{t}$. We will abuse notation and write $x>^{t}y$ instead
of $\\{x\\}>^{t}\\{y\\}$ in this case. In each $t\in\mathcal{T}$, there are
$J^{t}<\infty$ distinct menus denoted by

$B^{t}_{j}\in 2^{X^{t}}\setminus\\{\emptyset\\},\quad
j\in\mathcal{J}^{t}=\\{1,\dots,J^{t}\\}.$

Since $X^{t}$ is a finite set, we denote the $i$-th element of menu
$j\in\mathcal{J}^{t}$ as $x^{t}_{i|j}$. That is,
$B^{t}_{j}=\\{x^{t}_{i|j}\\}_{i\in\mathcal{I}^{t}_{j}}$, where
$\mathcal{I}^{t}_{j}=\\{1,2,\dots,I^{t}_{j}\\}$ and $I^{t}_{j}$ is the number
of elements in menu $j$.

Define a menu path as an ordered collection of indexes
$\mathbf{j}=(j_{t})_{t\in\mathcal{T}}$, $j_{t}\in\mathcal{J}^{t}$. Menu paths
encode menus that were faced by agents in different time periods. Let
$\mathbf{J}$ be the set of all _observed_ menu paths. Given
$\mathbf{j}\in\mathbf{J}$, a _choice path_ is an array of alternatives
$x_{\mathbf{i}|\mathbf{j}}=\left(x^{t}_{i_{t}|{j_{t}}}\right)_{t\in\mathcal{T}}$
for some collection of indexes
$\mathbf{i}=\left(i_{t}\right)_{t\in\mathcal{T}}$ such that
$i_{t}\in\mathcal{I}^{t}_{j_{t}}$ for all $t$. Similar to a menu path, a
choice path encodes the choices of a DM in a given sequence of menus that she
faced. The set of all possible choice path index sets $\mathbf{i}$, given a
menu path $\mathbf{j}$, is denoted by $\mathbf{I}_{\mathbf{j}}$.

Note that every $\mathbf{j}\in\mathbf{J}$ encodes the Cartesian product of
menus
$\times_{t\in\mathcal{T}}B^{t}_{j_{t}}\subseteq\times_{t\in\mathcal{T}}X^{t}$.
Then, for every $\mathbf{j}$, let $\rho_{\mathbf{j}}$ be a probability measure
on $\times_{t\in\mathcal{T}}B^{t}_{j_{t}}$. That is,
$\rho_{\mathbf{j}}\left(x_{\mathbf{i}|\mathbf{j}}\right)\geq 0$ for all
$\mathbf{i}\in\mathbf{I}_{\mathbf{j}}$ and
$\sum_{\mathbf{i}\in\mathbf{I}_{\mathbf{j}}}\rho_{\mathbf{j}}\left(x_{\mathbf{i}|\mathbf{j}}\right)=1$.
The primitive in our framework is the collection of all observed
$\rho_{\mathbf{j}}$, $\rho=(\rho_{\mathbf{j}})_{\mathbf{j}\in\boldsymbol{J}}$.
We call this collection a _dynamic stochastic choice function_.

Given $\rho$, we can define a Dynamic Random Utility Model (DRUM). Let $U^{t}$
denote the set of all utility functions that (i) map $X^{t}$ to
${\mathds{R}}$, (ii) are injective, and (ii) monotone on $>^{t}$ (i.e., if
$S,S^{\prime}\in 2^{X^{t}}\setminus\\{\emptyset\\}$ and $S>^{t}S^{\prime}$,
then $\max_{s\in S}{u^{t}(s)}>\max_{s\in S^{\prime}}u^{t}(s)$). Also let
$\mathcal{U}=\times_{t\in\mathcal{T}}U^{t}$ and $u=(u^{t})_{t\in\mathcal{T}}$
be an element of $\mathcal{U}$.

######  Definition 1 (DRUM).

The dynamic stochastic choice function $\rho$ is consistent with DRUM if there
exists a probability measure over $\mathcal{U}$, $\mu$, such that

$\rho_{\mathbf{j}}\left(x_{\mathbf{i}|\mathbf{j}}\right)=\int\prod_{t\in\mathcal{T}}\mathds{1}\left(\,\operatorname*{arg\,max}_{y\in
B_{j_{t}}^{t}}u^{t}(y)=x^{t}_{i_{t}|j_{t}}\,\right)d\mu(u)$

for all $\mathbf{i}\in\mathbf{I}_{\mathbf{j}}$ and $\mathbf{j}\in\mathbf{J}$.

When $T=1$, DRUM coincides with RUM, such that every agent maximizes her
utility function $u^{1}$ over a menu and the analyst observes the distribution
of consumers’ choices. DRUM extends RUM by introducing a time dimension with
an unrestricted preference correlation across time. The stochastic utility
process is captured by $\mu$. Similar to RUM, DRUM does not restrict
preference heterogeneity in cross-sections (i.e., across agents) and requires
$\mu$ not to depend on either the menu paths or the alternatives in the
consumption space. In contrast to RUM and DRUM, the Samuelson-Afriat
framework, , does not use variation in choices of agents in cross-sections
(i.e., it is directed to the individual-level data or time series of choices).
Thus, it does not restrict preferences of individuals in cross-sections. In
contrast to DRUM, however, the Samuelson-Afriat framework imposes a strict
restriction that preferences are perfectly correlated across time (i.e.
$u^{t}=u^{s}\>\mu-\mathrm{a.s.}$ for all $t,s\in\mathcal{T}$). We formalize
these connections between RUM, Afriat’s framework, and DRUM in Section 5.

Some examples of datasets in which a dynamic stochastic choice function is
(partially) observed are: (i) household longitudinal survey datasets, (ii)
scanner datasets, and (iii) experimental datasets with panels of choice. In
survey datasets (e.g., Encuesta de Presupuestos Familiares in Spain and
Progresa Household Survey in Mexico, see Deb et al., 2021, Aguiar and Kashaev,
2021), information about household purchases is usually collected several
times a year. For a given time period, budget variation across households is
driven by spatial or regional price variation (Aguiar and Kashaev, 2021).
Scanner datasets (e.g., Nielsen homescan data, see Gauthier, 2018) contain
information about weekly purchases of consumers. Budget variation in this case
is driven by price variation across stores in each time period (Gauthier,
2021). In experimental settings, subjects often face few budget paths drawn at
random from a common set of budgets (e.g., experiments on preferences over
giving, as in Porter and Adams, 2016). In our empirical application, we have a
panel of choices over different menus of lotteries from an experimental data
set collected by Aguiar et al. (2023), (see also McCausland et al. (2020) for
a panel of choices in discrete choice.999In that paper they study static RUM
in an individual setup using the time-series to estimate individual stochastic
choice.)

###  2.1. Preview of the Results: Binary Menus Example

We illustrate the setting and our main result with an example. We start with
the static RUM setup for a choice set $X^{t}=\\{x,y,z\\}$, where one observes
only binary menus. The order $>^{t}$ is assumed to be empty. Following
McFadden and Richter (1990), we describe RUM as a finite mixture of
deterministic types captured by a matrix $A^{t}$, as displayed in Table 1.
Each column of $A^{t}$ in Table 1 corresponds to a deterministic rational type
$r_{i}^{t}$ (e.g., $r_{1}^{t}$ represents a strict rational order over $X^{t}$
such that $xr_{1}^{t}yr_{1}^{t}z$). Note that there is some utility function
$u_{1}^{t}$ such that $xr_{1}^{t}yr_{1}^{t}z$ if and only if
$u_{1}^{t}(x)>u_{1}^{t}(y)>u_{1}^{t}(z)$. Each row in Table 1 corresponds to a
choice from a binary menu for each rational order. An element of $A^{t}$ that
corresponds to type $r^{t}_{i}$ and pair
$x^{\prime},\\{x^{\prime},y^{\prime}\\}$ is equal to $1$ if
$x^{\prime}r^{t}_{i}y^{\prime}$, and zero otherwise.

| $r_{1}^{t}$ | $r_{2}^{t}$ | $r_{3}^{t}$ | $r_{4}^{t}$ | $r_{5}^{t}$ | $r_{6}^{t}$
---|---|---|---|---|---|---
$x,\\{x,y\\}$ | $1$ | $1$ | - | - | 1 | -
$y,\\{x,y\\}$ | - | - | $1$ | $1$ | - | $1$
$x,\\{x,z\\}$ | $1$ | $1$ | $1$ | - | - | -
$z,\\{x,z\\}$ | - | - | - | $1$ | $1$ | $1$
$y,\\{y,z\\}$ | $1$ | - | $1$ | $1$ | - | -
$z,\\{y,z\\}$ | - | $1$ | - | - | $1$ | $1$
Table 1: The matrix $A^{t}$ for binary menus. $``-"$ corresponds to zero.

Now consider two periods in which the choice set remains the same in time. DMs
face (sequentially) two menus. Thus, menu paths are of the form
$\\{x^{\prime},y^{\prime}\\},\\{x^{\prime\prime},y^{\prime\prime}\\}$ for all
$x^{\prime},y^{\prime},x^{\prime\prime},y^{\prime\prime}\in X^{t}$. Given a
menu path, DMs choose a choice path (e.g., the ordered tuple
$(x,\\{x,y\\};y,\\{y,z\\})$ indicating the choices from menu path
$\\{x,y\\},\\{y,z\\}$). Our primitive, or data set, is the collection of the
joint probabilities of choice paths for each menu path, $\rho$. We show that
$\rho$ is consistent with DRUM (up to rearrangement) if and only if
$\rho=(A^{1}\otimes A^{2})\nu$ for some vector $\nu\geq 0$ such that
$\sum_{i}{\nu_{i}}=1$, where $\otimes$ is the Kronecker product.101010The
result of the Kronecker product of two matrices $A$ and $B$ is a block matrix
$A\otimes B=\left(\begin{array}[]{cccc}A_{1,1}B&A_{1,2}B&\dots&A_{1,n_{A}}B\\\
A_{2,1}B&A_{2,2}B&\dots&A_{2,n_{A}}B\\\ \dots&\dots&\dots&\dots\\\
A_{k_{A},1}B&A_{k_{A},1}B&\dots&A_{k_{A},n_{A}}B\end{array}\right),$ where $A$
is of the size $k_{A}$ by $n_{A}$. The vector $\nu$ is a distribution over
dynamic profiles of deterministic rational types or columns of $A^{1}\otimes
A^{2}$ (e.g., a dynamic preference profile $(r_{1}^{1},r_{6}^{2})$ is such
that preferences change from $xr_{1}^{1}yr_{1}^{1}z$ to
$zr_{6}^{2}yr_{6}^{2}x$). This representation of DRUM as a mixture of
deterministic dynamic rational types is called the
$\mathcal{V}$-representation. The recursive structure of the
$\mathcal{V}$-representation makes DRUM modular with the consequent
computational gains. In addition, when $\mathcal{T}=\\{1\\}$, the consistency
of $\rho$ with RUM is equivalent to the following triangle condition

$\rho_{\\{x^{\prime},y^{\prime}\\}}(x^{\prime})+\rho_{\\{y^{\prime},z^{\prime}\\}}(y^{\prime})-\rho_{\\{x^{\prime},z^{\prime}\\}}(x^{\prime})\geq
0$

for all $x^{\prime},y^{\prime},z^{\prime}\in X^{t}$.111111The triangle
conditions are equivalent to RUM with binary menus when $|X^{t}|\leq 5$
(Dridi, 1980). The triangle conditions can be summarized in a matrix $H^{t}$,
as displayed in Table 2. The triangle conditions can then be stated as
$H^{t}\rho\geq 0$ for the static case. This is called the
$\mathcal{H}$-representation of RUM.

$x,\\{x,y\\}$ | $y,\\{x,y\\}$ | $x,\\{x,z\\}$ | $z,\\{x,z\\}$ | $y,\\{y,z\\}$ | $z,\\{y,z\\}$
---|---|---|---|---|---
1 | - | -1 | - | 1 | -
-1 | - | 1 | - | - | 1
- | 1 | 1 | - | -1 | -
- | -1 | - | 1 | 1 | -
1 | - | - | 1 | - | -1
- | 1 | - | -1 | - | 1
Table 2: The matrix $H^{t}$ for binary menus.$``-"$ corresponds to zero.

Our results demonstrate that one can use this characterization of RUM to
obtain testable conditions for DRUM in the form of linear inequalities by
computing $(H^{1}\otimes H^{2})\rho\geq 0$. (We later show that, with more
work, we can also obtain the full characterization of DRUM in this
environment.) We call these conditions the _dynamic triangle conditions_. A
dynamic triangle condition can be expressed recursively in the binary setup as
follows:

$\displaystyle\mathrm{D}^{\triangle,1}_{z,x,y}((z,\\{x,z\\};x,\\{x,z\\}))=\mathrm{D}^{\triangle,2}_{x,z,y}((z,\\{x,z\\};x,\\{x,z\\})$
$\displaystyle+\mathrm{D}^{\triangle,2}_{x,z,y}((x,\\{x,y\\};x,\\{x,z\\})-\mathrm{D}^{\triangle,2}_{x,z,y}((z,\\{y,z\\};x,\\{x,z\\})\geq
0,$

where an instance of the dynamic triangle condition is

$\displaystyle\mathrm{D}^{\triangle,2}_{x,z,y}((z,\\{x,z\\};x,\\{x,z\\}))$
$\displaystyle=\rho_{\\{x,z\\},\\{x,z\\}}(z,x)+\rho_{\\{x,z\\},\\{y,z\\}}(z,z)-\rho_{\\{x,z\\},\\{x,y\\}}(z,x)\geq
0.$

This instance means that, when the choice in period $1$ is held constant, the
triangle inequality holds for $\rho$ for a triple of choices in period $2$.
Thus, the dynamic triangle condition recursively applies the triangle
inequality in the first period to the triangle inequality in the second
period.

Our results use the recursive version of the Weyl-Minkowski theorem to show
how this insight —how to derive the $\mathcal{V}$\- or
$\mathcal{H}$-representation of DRUM from its one-time $\mathcal{V}$\- or
$\mathcal{H}$-representations—can be generalized to obtain the
characterization of DRUM for any finite time window and any collection of
menus (i.e, beyond binary menus), whether the choice set is discrete (e.g.,
Li, 2021, Chambers et al., 2021) or continuous (e.g., the demand setup of KS).

###  2.2. Finite Abstract Setup

We consider a nonempty, finite, grand choice set $X^{t}$ in each
$t\in\mathcal{T}$ with an empty (hence, acyclic) order $>^{t}$, and assume
that the observed menus in each $t$ are all possible subsets of $X^{t}$ with
cardinality at least $2$. This setup is a generalization of Li (2021), which
assumes that $|X^{t}|\leq 3$, and Chambers et al. (2021), which in effect
assumes that $T=2$.

###  2.3. Demand Setup

Let $X^{*}\subseteq{\mathds{R}}^{K}_{+}$ be the consumption space with finite
$K\geq 2$ goods.121212${\mathds{R}}^{K}_{+}$ denotes the set of component-wise
nonnegative elements of the $K$-dimensional Euclidean space
${\mathds{R}}^{K}$. In each $t\in\mathcal{T}$, there are $J^{t}<\infty$
distinct budgets denoted by

$B^{*,t}_{j}=\left\\{y\in X^{*}\>:\>p_{j,t}^{\prime}y=w_{j,t}\right\\},\quad
j\in\mathcal{J}^{t}=\\{1,\dots,J^{t}\\},$

where $p_{j,t}\in{\mathds{R}}^{K}_{++}$ is the vector of prices and
$w_{j,t}>0$ is the expenditure level.

In a way that is similar to the general setup,
$\mathbf{j}=(j_{t})_{t\in\mathcal{T}}$, $j_{t}\in\mathcal{J}^{t}$ encodes
budgets that were faced by DMs in different time periods, referred to as
_budget paths_. Let $\mathbf{J}$ be a set of all observed budget paths.

For every $\mathbf{j}\in\mathbf{J}$, let $\mathrm{P}_{\mathbf{j}}$ be a
probability measure on the set of all Borel measurable subsets of
$\times_{t\in\mathcal{T}}X^{*}$. The primitive in the demand framework is the
collection of all observed $\mathrm{P}_{\mathbf{j}}$,
$\mathrm{P}=(\mathrm{P}_{\mathbf{j}})_{\mathbf{j}\in\boldsymbol{J}}$. We call
this collection a _dynamic stochastic demand system_.

Given $\mathrm{P}$, we can define a Dynamic Random Demand Model (DRDM). Let
$U^{*}$ denote the set of all continuous, strictly concave, and monotone
utility functions that map $X^{*}$ to ${\mathds{R}}$; let
$\mathcal{U}^{*}=\times_{t\in\mathcal{T}}U^{*}$, and let
$u^{*}=(u^{*t})_{t\in\mathcal{T}}\in\mathcal{U}^{*}$.

######  Definition 2 (DRDM).

The dynamic stochastic demand $\mathrm{P}$ is consistent with DRUM if there
exists a probability measure over $\mathcal{U}^{*}$, $\mu^{*}$, such that

$\mathrm{P}_{\mathbf{j}}\left(\left(O^{t}\right)_{t\in\mathcal{T}}\right)=\int\prod_{t\in\mathcal{T}}\mathds{1}\left(\,\operatorname*{arg\,max}_{y\in
B_{j_{t}}^{t}}u^{*t}(y)\in O^{t}\,\right)d\mu^{*}(u^{*})$

for all $\mathbf{j}\in\mathbf{J}$ and for all Borel measurable $O^{t}\subseteq
X^{*}$, $t\in\mathcal{T}$.

We next show that DRDM is empirically equivalent to DRUM provided that we
appropriately specify the choice set and primitive order. The monotonicity of
the utility functions generates choices on the budget hyperplane. In the RUM
demand setting, KS and Kawaguchi (2017) showed that to establish that
$\mathrm{P}$ is consistent with DRUM not all possible Borel sets need to be
checked. Stochastic rationalizability by RUM depends only on the probability
of certain regions of the budget hyperplanes called patches.

For any $t\in\mathcal{T}$ and $j\in\mathcal{J}^{t}$, let
$\\{x^{t}_{i|j}\\}_{i\in\mathcal{I}^{t}_{j}}$,
$\mathcal{I}^{t}_{j}=\\{1,\dots,I^{t}_{j}\\}$, denote a finite partition of
$B^{*,t}_{j}$ (where each element of the partition is indexed by $i$).

######  Definition 3 (Patches).

For every $t\in\mathcal{T}$, let
$\bigcup_{j\in\mathcal{J}^{t}}\\{x^{t}_{i|j}\\}$ be the coarsest partition of
$\bigcup_{j\in\mathcal{J}^{t}}B^{*,t}_{j}$ such that

$x^{t}_{i|j}\bigcap B^{*,t}_{j^{\prime}}\in\\{x^{t}_{i|j},\emptyset\\}$

for any $j,j^{\prime}\in\mathcal{J}^{t}$ and $i\in\mathcal{I}^{t}_{j}$. A set
$x^{t}_{i|j}$ is called a patch. If $x^{t}_{i|j}\subseteq
B^{*,t}_{j^{\prime}}$ for some $i$ and $j\neq j^{\prime}$, then $x^{t}_{i|j}$
is called an _intersection patch_.

By definition, patches can only be strictly above, strictly below, or on
budget hyperplanes. A typical patch belongs to one budget hyperplane. However,
intersection patches always belong to several budget hyperplanes. The case for
one time period, in which $K=2$ goods and $J^{t}=2$ budgets, is depicted in
Figure 1. Note that by definition $\\{x^{t}_{i|j}\\}$ is a partition of
$B^{*,t}_{j}$, and $I^{t}_{j}$ is the number of patches that form budget
$B^{*,t}_{j}$.

The (discretized) choice set is

$X^{t}=\bigcup_{i_{t}\in I^{t}_{j},j\in\mathcal{J}^{t}}\\{x_{i_{t}|j_{t}}\\}.$

The primitive order $>^{t}$ is given by $S^{\prime}>^{t}S$ for
$S,S^{\prime}\in 2^{X^{t}}\setminus\\{\emptyset\\}$ whenever, for any
$x_{i|j}\in S$ and $y\in x_{i|j}$, there exist $x_{i^{\prime}|j^{\prime}}\in
S^{\prime}$ and $y^{\prime}\in x_{i^{\prime}|j^{\prime}}$ such that
$y^{\prime}>y$, where $>$ is the strict vector order on $X^{*}$. We define a
_menu_ as the collection of patches from the same budget hyperplane

$B_{j_{t}}^{t}=\left\\{x_{i_{t}|j_{t}}\right\\}_{i_{t}\in I^{t}_{j}}.$

Henceforth, we refer to a menu or budget interchangeably.

$y_{1}$$y_{2}$$x^{t}_{3|1}$$x^{t}_{1|2}$$x^{t}_{2|1}$$x^{t}_{1|1}$$x^{t}_{2|2}$$B^{*,t}_{1}$$B^{*,t}_{2}$
Figure 1: Patches for the case with $K=2$ goods and $J^{t}=2$ budgets. The
only intersection patch is $x^{t}_{3|1}$, which is the intersection of
$B^{t}_{1}$ and $B^{t}_{2}$.

Let

$\rho\left(x_{\mathbf{i}|\mathbf{j}}\right)=\mathrm{P}_{\mathbf{j}}\left(x_{\mathbf{i}|\mathbf{j}}\right)$

denote the fraction of agents who pick from a choice path
$x_{\mathbf{i}|\mathbf{j}}$ given a budget path $\mathbf{j}$.

The main building block of our demand framework is the dynamic stochastic
choice function

$\rho=\left(\rho\left(x_{\mathbf{i}|\mathbf{j}}\right)\right)_{\mathbf{j}\in\mathbf{J},\mathbf{i}\in\mathbf{I}_{\mathbf{j}}}.$

The vector $\rho$ represents the distribution over finitely many patches and
contains all the necessary information needed to determine whether
$\mathrm{P}$ is consistent with DRDM and, in this discretized setup,
consistent with DRUM.

######  Lemma 1.

The following are equivalent:

  1. (i)

$P$ is consistent with DRDM.

  2. (ii)

$\rho$ is consistent with DRUM.

The proof of Lemma 1 follows from Kitamura and Stoye (2018) and Kawaguchi
(2017). Next we provide some parametric examples of DRUM in this domain.

######  Example 1 (Dynamic Random Cobb-Douglas Utility).

Let $K=2$ and $u^{t}(y_{1},y_{2})=y_{1}^{\alpha_{t}}y_{2}^{(1-\alpha_{t})}$.
The utility parameter $\alpha_{t}$ is random and such that
$\alpha_{t}=\max\\{\min\\{\alpha_{t-1}+\epsilon_{t},1\\},0\\}$, where
$(\epsilon_{t})_{t\in\mathcal{T}}$ are independent and identically distributed
mean-zero random innovations with variance $\sigma^{2}$. The dynamic
stochastic demand generated by this utility function is consistent with DRUM
as long as $(\alpha_{t})_{t\in\mathcal{T}}$ is independent of prices and
income.

######  Example 2 (Based on Adams et al., 2015).

For a deterministic utility $v:X^{*}\to{\mathds{R}}$, the random utility at
time $t\in\mathcal{T}$ is given by $u^{t}(x)=v(x)+\alpha_{t}^{\prime}x$, where
$\alpha_{t}$ is the random vector supported on ${\mathds{R}}^{K}$. The dynamic
stochastic demand generated by this utility function is consistent with DRUM
if $\alpha_{t}$ is independent of prices and income.

In the two examples above, as well as in the examples in Section 3 below, we
maintain the assumption that the distribution of preferences does not depend
on the budget. This assumption is satisfied in experimental setups such as the
ones in Porter and Adams (2016), McCausland et al. (2020), and Aguiar et al.
(2023). That said, it may not be realistic in other setups, such as when
saving is possible. This exogeneity assumption is relaxed in Section 2.4.

###  2.4. Endogenous Expenditure in the Demand Setup

In the demand setup we assumed that budgets are exogenously given. Here, we
relax the exogeneity assumption by extending the results of Deb et al. (2021)
to our setup. Our new model will cover the classical consumption smoothing
problem with income uncertainty (Browning, 1989). As we did with DRDM, we can
define a Dynamic Random Augmented Demand Model (DRADM). Let $V$ denote the set
of all continuous, strictly concave, and monotone augmented utility functions
that map $X^{*}\times{\mathds{R}}_{-}$ to ${\mathds{R}}$, and let
$\mathcal{V}=\times_{t\in\mathcal{T}}V$ be the Cartesian product of $T$
repetitions of $V$.

######  Definition 4 (DRADM).

A dynamic stochastic demand $\mathrm{P}$ is consistent with DRADM if there
exists a probability measure over $\mathcal{V}$, $\eta$, such that

$\mathrm{P}_{\mathbf{j}}\left(\left(O^{t}\right)_{t\in\mathcal{T}}\right)=\int\prod_{t\in\mathcal{T}}\mathds{1}\left(\,\operatorname*{arg\,max}_{y\in
X^{*}}v^{t}(y,-p^{\prime}_{j,t}y)\in O^{t}\,\right)d\eta(v)$

for all $\mathbf{j}\in\mathbf{J}$ and for all Borel measurable $O^{t}\subseteq
X^{*}$, $t\in\mathcal{T}$, where $v=(v^{t})_{t\in\mathcal{T}}$.

While DRDM is an extension of RUM to a dynamic setting (i.e., DRDM and RUM
coincide when $T=1$), DRADM is a dynamic extension of the Random Augmented
Utility Model of Deb et al. (2021).

######  Example 3 (Consumption Smoothing with Income Uncertainty).

Consider a consumer with random income stream $y=(y_{t})_{t\in\mathcal{T}}$
who maximizes the expected flow of instantaneous, concave, locally
nonsatiated, and continuous utilities, $u$, and does so subject to the budget
constraints, discount factor $\delta$, history of incomes captured by the
information set $I_{t}$, and the initial level of savings $s_{0}$. That is, at
every time period $\tau$ the consumer solves

$\displaystyle\max_{\left\\{c_{\tau}(\cdot),s_{\tau}(\cdot)\right\\}_{\tau=t,\dots,T}}\mathds{E}\left[\sum_{\tau=t}^{T}\delta^{\tau-t}u(c_{\tau}(y))\Big{|}I_{\tau}\right]$

subject to

$p_{\tau}^{\prime}c_{\tau}(y)+s_{t}(y)=y_{\tau}+(1+r_{\tau})s_{\tau-1}(y).$

The sequences of consumption policy functions
$\left(c_{t}(\cdot)\right)_{t\in\mathcal{T}}$ and saving policy functions
$\left(s_{t}(\cdot)\right)_{t\in\mathcal{T}}$ fully describe the consumption
and saving decisions, respectively, of the consumer. In addition, we restrict
these functions to depend only on the income history. That is, for all $t$ it
is the case that $c_{t}(y^{\prime})=c_{t}(y)$ and $s_{t}(y^{\prime})=s_{t}(y)$
for all $y$ and $y^{\prime}$ such that $y^{\prime}_{\tau}=y_{\tau}$ for all
$\tau\leq t$. The Bellman equation for this problem is

$W_{t-1}(s_{t-1})=\max_{c}\left[u(c)+\delta\mathds{E}\left[W_{t}(y_{t}+(1+r_{t})s_{t-1}-p_{t}^{\prime}c)\Big{|}I_{t}\right]\right],$

where $W_{t}$ is the value function at time period $t$. Thus, one can define
the state-dependent utility function as

$\hat{v}^{t}(x,s_{t-1})=u(x)+\delta\mathds{E}\left[W_{t}(y_{t}+(1+r_{t})s_{t-1}(y)-p_{t}^{\prime}c)\Big{|}I_{t}\right].$

Any correlation in income across time would generate a correlation between
$\\{\hat{v}^{t}\\}_{t\in\mathcal{T}}$. One can define the augmented utility
function as

$v^{t}(x,-p^{\prime}x)=u(x)+\delta\mathds{E}\left[W_{t}(y_{t}+(1+r_{t})s_{t-1}(y)-p_{t}^{\prime}c)\Big{|}I_{t}\right].$

Notice that the utility $\hat{v}^{t}$ depends on $s_{t-1}$ only through the
contemporaneous expenditure $p^{\prime}x$. If one assumes that different
individuals have different $u$, $\delta$, and $y$ such that their joint
distribution does not depend on prices, then this setup is a particular case
of DRADM.

In Example 3, the random augmented utility stochastic process is independent
of prices because prices are determined exogenously by supply and demand
forces. Next, we characterize DRADM by using the fact that consistency with
DRADM is equivalent to consistency with DRDM for a normalized budget path. A
normalized budget path has the same price path $(p_{j,t})_{t\in\mathcal{T}}$
and income equal to $1$. Using these normalized budgets, we can define patches
as before to obtain

$\rho\left(x_{\mathbf{i}|\mathbf{j}}\right)=\mathrm{P}_{\mathbf{j}}\left(\left\\{y^{t}\in
X^{*}\>:\>y^{t}/p^{\prime}_{j,t}y^{t}\in
x_{i|j}^{t}\right\\}_{t\in\mathcal{T}}\right),$

for all $\mathbf{i}\in\mathbf{I}_{\mathbf{j}}$, $\mathbf{j}\in\mathbf{J}$. As
before, the stochastic choice function that corresponds to $\mathrm{P}$ is

$\rho=\left(\rho(x_{\mathbf{i},\mathbf{j}})\right)_{\mathbf{i}\in\mathbf{I}_{\mathbf{j}},\mathbf{j}\in\mathbf{J}}.$

Similarly to DRDM, we rule out intersection patches. The choice set $X^{t}$
and the partial order $>^{t}$ are defined analogously to the definition of
DRDM.

######  Lemma 2.

The following are equivalent:

  1. (i)

$\mathrm{P}$ is consistent with DRADM.

  2. (ii)

$\rho$ is consistent with DRUM.

The proof of Lemma 2 is omitted because it is analogous to the results in Deb
et al. (2021).

## 3\. Characterization of DRUM

Here we provide a characterization of rationalizability by DRUM when $\rho$ is
observed (i.e., estimable). The main result in this section is an analogue of
the results in McFadden and Richter (1990) and KS for RUM. Given the finite
choice set, let a preference profile be $\mathbf{r}=(r^{t})_{\in\mathcal{T}}$,
where $r^{t}$ is a linear order defined on the finite set of alternatives
available at time $t$, $X^{t}$. We restrict these linear orders to be
extensions of $>^{t}$ (i.e., for $S,S^{\prime}\in
2^{X^{t}}\setminus\\{\emptyset\\}$ and $S>^{t}S^{\prime}$, there is some $x\in
S$ such that $x$ is preferred to $y$, $xr^{t}y$, for all $y\in S^{\prime}$).
Recall that $\mathbf{i}$ encodes choices in each time period. Given
$\mathbf{r}$, we can encode choices in different time periods and menus in a
vector $a_{\mathbf{r}}$ as

$a_{\mathbf{r}}=\left(a_{\mathbf{r},\mathbf{i},\mathbf{j}}\right)_{\mathbf{j}\in\mathbf{J},\mathbf{i}\in\mathbf{I}_{\mathbf{j}}},$

with $a_{\mathbf{r},\mathbf{i},\mathbf{j}}=1$ if the alternative
$x^{t}_{i_{t}|j_{t}}$ is the best item available in $B^{t}_{j_{t}}$ according
to $r_{t}$ for all $t\in\mathcal{T}$, and
$a_{\mathbf{r},\mathbf{i},\mathbf{j}}=0$ otherwise. Denote $\mathcal{R}^{t}$
as the set of (strict) rational preferences in a given time period
$t\in\mathcal{T}$. The set of dynamic rational preference profiles
$\mathcal{R}$ is the set of all preference profiles $\mathbf{r}$ for which
there exists $u_{r}=(u^{t}_{r})_{t\in\mathcal{T}}\in\mathcal{U}$ such that

$a_{\mathbf{r},\mathbf{i},\mathbf{j}}=1\quad\iff\quad\forall
t\in\mathcal{T},\>\operatorname*{arg\,max}_{x\in
B^{t}_{j_{t}}}u^{t}_{r}(x)=x_{i_{t}|j_{t}}.$

We form the matrix $A_{T}$ by stacking the column vectors $a_{\mathbf{r}}$ for
all preference profiles $\mathbf{r}\in\mathcal{R}$. The dimension of this
matrix is $d_{\rho}\times\left\lvert\mathcal{R}\right\rvert$, where $d_{\rho}$
is the length of vector $\rho$. This matrix will be used to provide a
characterization of DRUM that is amenable to statistical testing.

The next axiom is the analogue of the axiom in McFadden and Richter (1990) for
(static) stochastic revealed preferences (Border, 2007) and will provide a
different characterization of DRUM.

######  Definition 5 (Axiom of Dynamic Stochastic Revealed Preference, ADSRP).

A stochastic choice function $\rho$ satisfies ADSRP if for every finite
sequence of pairs of menu and choice paths (including repetitions), $k$,
$\\{(\mathbf{i}_{k},\mathbf{j}_{k})\\}$ such that
$\mathbf{j}_{k}\in\mathbf{J}$ and
$\mathbf{i}_{k}\in\mathbf{I}_{\mathbf{j}_{k}}$, it follows that

$\sum_{k}\rho\left(x_{\mathbf{i}_{k}|\mathbf{j}_{k}}\right)\leq\max_{\mathbf{r}\in\mathcal{R}}\sum_{k}a_{\mathbf{r},\mathbf{i}_{k},\mathbf{j}_{k}}.$

The next theorem provides a full characterization of DRUM. Let

$\Delta^{L}=\left\\{y\in{\mathds{R}}_{+}^{L+1}\>:\>\sum_{l=1}^{L+1}y_{l}=1\right\\}$

denote the $L$-dimensional simplex.

######  Theorem 1.

The following are equivalent:

  1. (i)

$\rho$ is consistent with DRUM.

  2. (ii)

There exists $\nu\in\Delta^{|\mathcal{R}|-1}$ such that $\rho=A_{T}\nu$.

  3. (iii)

There exists $\nu\in{\mathds{R}}^{|\mathcal{R}|}_{+}$ such that
$\rho=A_{T}\nu$.

  4. (iv)

$\rho$ satisfies ADSRP.

The proof of Theorem 1 is analogous to the proofs for RUM in McFadden and
Richter (1990), McFadden (2005), KS, and Kawaguchi (2017). Theorem 1 (iii) is
amenable to statistical testing using the test developed in KS. However, the
number of columns in $A_{T}$ grows exponentially with $T$. Thus, naively,
testing DRUM may seem impossible for relatively small $T$ even if one uses the
tools of Smeulders et al. (2021). The next lemma shows that the computational
complexity of computing $A_{T}$, $T\geq 1$ does not grow that much relative to
the computation complexity of computing $A_{1}$.

######  Lemma 3.

Let $A^{t}$ be a matrix constructed under the assumption that
$\mathcal{T}=\\{t\\}$. That is, $A^{t}$ is the matrix that encodes static
rational types at time $t$. Then $A_{T}=\otimes_{t\in\mathcal{T}}A^{t}$ up to
a permutation of its rows.

###### Proof.

Note that the $k$-th and the $l$-th columns of $A_{1}$ and $A_{2}$,
$a^{1}_{k}$ and $a^{2}_{l}$, encode the choices of particular types of
consumers at time $t=1$ and $t=2$ (i.e., their choices in each menu at $t=1$
and $t=2$). Since there are no restrictions across $t$ on these deterministic
types, we can generate the $(k,l)$-type, $a^{1}_{k}\otimes a^{2}_{l}$, that
encodes what is picked in pairs of menus such that each menu is taken from two
different time periods. Next, if we take some column from $A^{3}$, we can
repeat the above step and obtain a composite type for three time periods.
Repeating this exercise $T$ times for all possible combinations of columns
will lead to a matrix that is equal to $A_{T}$ up to a permutation of rows. ∎

Lemma 3 substantially simplifies the computation of $A_{T}$ given that one can
use the methods in KS and Smeulders et al. (2021) to construct $A^{t}$. In
instances in which the menu structure is such that $A^{t}=A^{s}$ for $t\neq
s$, significant computational savings are achieved. Note that $A^{t}=A^{s}$
can occur without the observed menus in $t$ and the observed menus in $s$
being the same. In fact, in the demand setting, $A^{t}$ depends only on the
intersection structure induced by the budgets and not on the specific prices
(see examples in the next section). Lemma 3 also allows exploiting sparsity
because the Kronecker product propagates any zero entry in $A_{t}$. The
Kronecker product structure of the mixture representation of DRUM shows its
structure is modular. Indeed, the structure of $A_{T}$ is built from its
static components. This property allows one to parallelize the computation of
$A_{T}$.131313Since each factor of the Kronecker product can be computed
independently, we can parallelize along the time dimension. This modularity is
exploited to obtain a recursive characterization of DRUM.

Unfortunately, the DRUM characterization in Theorem 1 does not provide an
intuitive understanding of the behavioral implications of DRUM. In the next
sections, we provide just such an intuitive characterization of DRUM. This
characterization demonstrates that DRUM provides additional implications
relative to RUM in longitudinal data. That is, we show that requiring
consistency with (static) RUM for all conditional and marginal probabilities
is not enough to guarantee consistency with DRUM. In fact, the new conditions
will affect the joint distribution $\rho$.

## 4\. Axiomatic Characterization of DRUM via Linear Inequality Restrictions

In order to understand the axiomatic structure of DRUM, we provide a
characterization of many of its special cases via linear inequalities. We also
provide a way to obtain a general axiomatic characterization of DRUM via
linear inequalities when its static counterpart is known. First, we need some
preliminary mathematical results.

###

Theorem 1 (iii) states that to test whether $\rho$ is consistent with DRUM it
is enough to check whether it belongs to the convex cone

$\left\\{A_{T}v\>:\>v\geq 0\right\\}.$

This is called the $\mathcal{V}$-representation of the cone. The Weyl-
Minkowski theorem states that there exists an equivalent representation of the
cone (the $\mathcal{H}$-representation) via some matrix $B_{T}$:

$\left\\{z\>:\>H_{T}z\geq 0\right\\}.$

The $\mathcal{V}$-representation of the cone associated with DRUM provides an
interpretation of the former as the observed distribution over choices is a
finite mixture of deterministic types (KS, Smeulders et al.,
2021).141414Kitamura and Stoye (2018) were the first to notice, that in the
static case, checking whether a stochastic demand is consistent with RUM
amounts to checking whether its vector representation belongs to a convex
cone. They also introduced the Weyl-Minkowski theorem to the study of RUM in
economics. Unfortunately, the $\mathcal{V}$-representation does not give any
direct restrictions on the observed $\rho$. As a result, the analyst can
hardly use the $\mathcal{V}$-representation to arrive at any helpful intuition
about the empirical content of DRUM. In contrast, the
$\mathcal{H}$-representation can deliver direct and sometimes intuitive
restrictions on the data (see Section 4.1 or the dynamic BM inequalities in
Section 4.2).

In theory, if one possesses $A_{T}$, one can obtain $H_{T}$. Unfortunately, as
noted in KS, the construction of $H_{T}$ from $A_{T}$ is a nontrivial task
that becomes computationally burdensome even for moderate $T$ since the number
of columns of $A_{T}$ grows exponentially with $T$. In Lemma 3, we showed that
one could use the recursive structure of $A_{T}$ to simplify its construction
substantially. In this section, we show that the same intuition carries over
to the construction of $H_{T}$: one can move from the
$\mathcal{H}$-representation of RUM to the $\mathcal{H}$-representation of
DRUM with only a small computational costs. Our next result generalizes the
Weyl-Minkowski theorem in a direction that is useful for our recursive setup.
Henceforth, we assume that all cones are finite-dimensional and are subsets of
a Euclidean vector space.

######  Proposition 1.

If

$\left\\{K^{t}v\>:\>v\geq 0\right\\}=\left\\{z\>:\>L^{t}z\geq 0\right\\}$

for all $t\in\mathcal{T}$, then

$\left\\{\left(\otimes_{t\in\mathcal{T}}K^{t}\right)v\>:\>v\geq
0\right\\}\subseteq\left\\{z\>:\>\left(\otimes_{t\in\mathcal{T}}L^{t}\right)z\geq
0\right\\}.$

Proposition 1 is a direct extension of Theorem A in Aubrun et al. (2021) and
Theorem $7.15$ in de Bruyn (2020) to more than two time periods.151515We thank
Chris Chambers for pointing out an error in the proof of a previous version of
this result. Note that if $K^{t}$ represents a model that can be expressed as
mixtures of deterministic behavior (i.e., columns of $K^{t}$), then
Proposition 1 allows one to easily construct testable conditions of the
dynamic extensions of this model using its one-time
$\mathcal{H}$-representations.

Next, we provide not only testable conditions but a full equivalence of the
recursive $\mathcal{H}$-representation via Kronecker products and the
corresponding $\mathcal{V}$-representation. To do this, first define the
Kronecker power for a matrix $C$, $C^{\otimes_{k}}=\otimes_{j=1}^{k}C$ for any
integer $k\geq 1$.161616For $k=0$, the Kronecker power is equal to scalar $1$.
We say that the cone $\left\\{Cv\>:\>v\geq 0\right\\}$ is proper if $C$ is
full row rank, the cone is closed, and any line in the vector space that
contains the cone is not in the cone. For any $\phi^{t}$ in the interior of
$\\{L^{t\prime}v\>:\>v\geq 0\\}$ (e.g., the case in which $\phi^{t}$ is a
strict convex combination of columns of $L^{t\prime}$) and any $k\geq 1$,
define the projection map as follows:

$\gamma_{k}^{\phi^{t}}=\frac{1}{k}\sum_{j=1}^{k}\phi^{t,\otimes(j-1)}\otimes
I^{t}\otimes\phi^{t,\otimes(k-j)},$

where $I^{t}$ is the identity matrix in the vector space containing the cone
$\left\\{K^{t}v\>:\>v\geq 0\right\\}$. Define

$\Gamma^{\boldsymbol{\phi}}_{\mathbf{k}}=I^{1}\otimes\left(\otimes_{t\in\mathcal{T}\setminus\\{1\\}}\gamma_{k_{t}}^{\phi^{t}}\right)$

for a given collection of static operators $\gamma^{\phi^{t}}_{k_{t}}$,
$t\in\mathcal{T}\setminus\\{1\\}$.

######  Theorem 2.

Suppose $K^{t}$ is proper for all $t\in\mathcal{T}$. Then

$\left\\{\left(\otimes_{t\in\mathcal{T}}K^{t}\right)v\>:\>v\geq
0\right\\}=\bigcap_{k_{1}=1,k_{2}\cdots,k_{T}\geq
1}\left\\{\Gamma^{\boldsymbol{\phi}\prime}_{\mathbf{k}}z\>:\>\left(\otimes_{t\in\mathcal{T}}L^{t,\otimes_{k_{t}}}\right)z\geq
0\right\\}.$

Moreover, $K^{t}$ does not have full column rank for at most one
$t\in\mathcal{T}$ if and only if

$\left\\{\left(\otimes_{t\in\mathcal{T}}K^{t}\right)v\>:\>v\geq
0\right\\}=\left\\{z\>:\>\left(\otimes_{t\in\mathcal{T}}L^{t}\right)z\geq
0\right\\}.$

For $T=2$, Theorem 2 is proved in Aubrun et al. (2022). It easily extends to
more than two time periods due to the associativity of the Kronecker
product.171717See also Remark 1 on page 9 of Aubrun et al. (2022). The
“moreover” part of our Theorem 2 above is established in Theorem A and
Corollary $4$ in Aubrun et al. (2021). The first part of Theorem 2 essentially
provides an approximation result that allows us to obtain the
$\mathcal{H}$-representation of a dynamic model via its one-time counterparts
by using extensions of the model to $k_{t}$ time periods. For some cones the
number of extensions can be infinite but in practice we can use a finite
number of extensions with the knowledge that in the limit this produces an
exact characterization. The simplest case of the previous result happens under
additional full column rank requirements.

###

Proposition 1 gives the necessary conditions for building the
$\mathcal{H}$-representation of DRUM from its static components (i.e., where
$K^{t}=A^{t}$ and $L^{t}=H^{t}$, where $H^{t}$ is the matrix from the
$\mathcal{H}$-representation of a cone generated by $A^{t}$). We show that
despite the fact that $A^{t}$ does not generate a proper cone (because it is
never of full row rank), we can use Theorem 2 to obtain sufficient conditions.
The row rank is not full because of the “adding-up” constraint—one alternative
has to be picked from _every_ menu. Hence, the sum of all rows belonging to
the same menu will give the row of ones. In the running example with binary
menus, with matrix $A^{t}$ given by Table 1, the sum of the first two rows is
equal to the sum of the third and the fourth rows and is equal to the sum of
the last two rows. However, Theorem 2 can still be used to obtain the
characterization of DRUM as Theorem 3 demonstrates. To formalize this,
consider the following submatrix of $A^{t}$, $t\in\mathcal{T}$: from every
menu except the first one, pick the last alternative and remove the
corresponding row from $A^{t}$. Let $A^{t*}$ denote the resulting matrix.
Observe that when $\mathcal{T}=\\{t\\}$, $\rho$ is consistent with DRUM if and
only if $\rho^{*}$, defined analogously to $A^{t*}$, is such that
$A^{t*}\nu^{*}=\rho^{*}$ for some $\nu^{*}\geq 0$. In particular, given the
simplex constraints on a given $\rho$, we can safely drop the adding-up
constraints in matrix $H^{t}$ from now on when computing the
$\mathcal{H}$-representation of $A^{t}$. The reason is that the adding-up
constraints are guaranteed to hold. Before stating the next theorem, we need
to introduce a key behavioral condition implied by DRUM.

######  Definition 6 (Stability).

We say that $\rho$ is _stable_ if
$\sum_{i\in\mathcal{I}^{t}_{j}}\rho\left(x_{\mathbf{i}|\mathbf{j}}\right)$ is
the same for all $j\in\mathcal{J}^{t}$, for any $t\in\mathcal{T}$ and
$x_{\mathbf{i}|\mathbf{j}}$.

Stability means that the marginal distribution of choices at any $t$ does not
depend on the menu in any other $t^{\prime}\neq t$. Under stability, the
marginal distribution of choices will not change due to either the menu the
consumers faced in the past or the menus the consumers will face in the
future. Recall that we have assumed that the stochastic utility process does
not depend on the budgets. This condition is an implication of that
assumption.181818Stability and the simplex constraints on $\rho$ are formally
defining a vector subspace within which $A^{t}$ is associated with a proper
cone. Stability was first defined in Strzalecki (2021). Chambers et al. (2021)
call this condition “marginality” in their domain.

We also need a notion of uniqueness of RUM and DRUM that is directly connected
to the requirement of full column rank in Theorem 2.

######  Definition 7 (Uniqueness).

We say that $A^{t}$ generates a unique RUM when the system $\rho=A^{t}\nu$ has
a unique solution for all stochastic choice functions $\rho$. Also, we say
that DRUM (associated with matrix $\otimes_{t\in\mathcal{T}}A^{t}$) satisfies
uniqueness when for all $t\in\mathcal{T}\setminus{\\{t^{\prime}\\}}$, $A^{t}$
generates a unique RUM for a given period $t^{\prime}$.

Set $\phi^{*,t}$ to be the average of all columns of $H^{t\prime}$. In other
words, $\phi^{*,t}$ represents a testable linear inequality of static RUM such
that for $\mathcal{T}=\\{t\\}$, $\phi^{*,t\prime}\rho\geq 0$. Define
implicitly the associated operator
$\Gamma^{\boldsymbol{\phi}^{*}\prime}_{\mathbf{k}}$ as well.

######  Theorem 3.

Assume that $A^{t*}$ is full row rank for all $t\in\mathcal{T}$. Then $\rho$
is consistent with DRUM if and only if $\rho$ is stable and

$\displaystyle\rho\in\bigcap_{k_{1}=1,k_{2},\cdots,k_{T}\geq
1}\left\\{\Gamma^{\boldsymbol{\phi}^{*}\prime}_{\mathbf{k}}z\>:\>\left(\otimes_{t\in\mathcal{T}}H^{t,\otimes_{k_{t}}}\right)z\geq
0\right\\}.$ (1)

Moreover, $\rho$ is consistent with a unique DRUM if and only if $\rho$ is
stable and $(\otimes_{t\in\mathcal{T}}H^{t})\rho\geq 0$.

The condition that $A^{t*}$ is full row rank for all $t\in\mathcal{T}$ is
satisfied in all the examples we are aware of. Importantly, the condition
holds for the abstract setups of Li (2021) and of Chambers et al. (2021), as
proved by Dogan and Yildiz (2022). It holds as well in the finite abstract
setup with limited menu variation, as proven in Saito (2017). We conjecture
that this condition is true in many other settings and have verified it to be
true in the demand setup with up to six budgets per period with a maximal
intersection pattern, and with $K\in\\{2,3,4,5\\}$.191919Note that the fact
that the full row rank condition is the only condition that needs to be
verified in each case as the cone associated with RUM is closed, and any line
in the vector space that contains the cone is not in the cone.

Note that stability is a set of equality restrictions on $\rho$. Since any
equality restriction can be represented as two inequality restrictions,
Theorem 3 allows us to recursively obtain the $\mathcal{H}$-representation of
DRUM from the $\mathcal{H}$-representation of its static components for any
time window, for the unique DRUM case. In other words, one just needs to
derive the $\mathcal{H}$-representation of a (unique) RUM and then convert it
to the dynamic setting and add the constraints implied by stability. This
delivers a substantial gain over the direct computation of the
$\mathcal{H}$-representation since the existing numerical algorithms
transforming $\mathcal{V}$-representations to $\mathcal{H}$-representations
are known to work only for small and moderate-size problems. That is, the
computational complexity of the dynamic problem is only bounded by the
computational complexity of the static one.

To better interpret our axiomatization of DRUM, note that for $T=2$,

$\otimes_{t=1}^{T}H^{t}\rho=\text{vec}\left(H^{2}RH^{1\prime}\right),$

where $\text{vec}(C)$ is the vector obtained by stacking columns of matrix $C$
and $R$ is the matrix such that $\text{vec}\left(R\right)=\rho$. Since every
column of $R$ corresponds to a subvector of $\rho$ with a fixed choice in
period $t=1$, in the case of unique RUM, $V_{2}=H^{2}R$ captures the
distributions over static preferences types in $t=2$ conditional on choices in
$t=1$. The requirement that

$\otimes_{t=1}^{T}H^{t}\rho=\text{vec}(H^{2}RH^{1\prime})=\text{vec}(V_{2}H^{1\prime})\geq
0$

is effectively equivalent to requiring that $V_{2}^{\prime}$ can be written as
a mixture over linear orders at $t=1$. In other words, in the symmetric case
where $H^{1}=H^{2}$, we just recursively apply the same RUM restrictions to
linear combinations of $\rho$. The necessity of these conditions for DRUM is
not surprising because its separable structure in time (i.e., DRUM describes
DMs maximizing a sequence of random utilities in time). In turns out, these
conditions (together with stability) are sufficient when DRUM is unique.

When DRUM is not unique, then additional restrictions implied by DRUM can
emerge. In particular, the separability of DRUM does not translate to its
axiomatic structure. The reason is that for unique DRUM, each period RUM
(except one) is associated with a convex cone geometrically equivalent to a
simplex. The cone associated with the $\mathcal{H}-$representation is, in this
case, the Kronecker product of the faces of the different simplices. The
vertex of such a cone coincides precisely with the collection of deterministic
preference profiles that characterize the unique DRUM. In contrast, when
uniqueness fails for more than one time period, then the cones associated with
RUM in each period are no longer equivalent to simplices, in that case, the
Kronecker product of the facets of this cone may have vertices that are not
consistent with a deterministic preference profile, in addition to the vertex
associated with the $\mathcal{V}$-representation of DRUM. We will call these
additional vertices _nonrational preference profiles_. DRUM further restricts
behavior beyond the recursive application of the static conditions such that
we rule out these nonrational preference profiles. The way we can eliminate
these additional restrictions is by creating a _thought experiment_ where we
replicate one time period to obtain additional _experiments_ where we hope
extensions of the nonrational preference profiles are not longer in the cone
produced by the Kronecker product of the facets of the RUM cones. Then we
project this extension back to the original time window, and the projection is
chosen in such a way as to keep the projection to the interior of the cone. In
other words, each of these replications weakens the effect of nonrational
preference profiles allowing us to isolate the DRUM types.

Remarkably, since Theorem 2 provides a necessary and sufficient condition, if
DRUM is not unique, then there is no hope that the
$\mathcal{H}$-representation of DRUM is just a Kronecker product of the static
matrices associated with the $\mathcal{H}$-representation of static RUM. That
is why the problem of providing a full characterization of DRUM is a hard
problem.

For the case of the nonunique DRUM, we still obtain the recursive linear
inequalities, implied by the Kronecker product of the static RUM inequalities,
as necessary conditions for consistency with DRUM. But to obtain a sufficient
condition, we have to do more work. Nevertheless, our result provides an
explicit way to compute the additional restrictions on $\rho$ consistent with
DRUM. Indeed, applying the results in Aubrun et al. (2022), we obtain
equation/condition (1) that describes a decreasing sequence of outer
approximations to the convex cone associated with DRUM. Doherty et al. (2004)
shows that using these outer approximations can do a good job approximating
some cones of interest for finite $k$.

The general characterization of (static) RUM for our demand setup via
$\mathcal{H}$-representation is yet to be discovered (Stoye, 2019). Only
special cases are fully solved: the case of two budgets (Hoderlein and Stoye,
2014), and the case of three goods and three budgets (KS). This stands in
contrast with the abstract setup solved in Block and Marschak (1960) and
Falmagne (1978). Fortunately, the BM inequalities can be modified in our
discretized setup, as we will see below, to deal with the demand setup.
Nevertheless, our result implies that once the generic
$\mathcal{H}$-representation of RUM becomes available, the analogous DRUM
characterization will also become available.

We want to highlight that our result bears a conceptual resemblance to Debreu
and Scarf (1963) result on the core convergence to the competitive economy
allocation. In that result, as in ours, there is a gap between the core and
the competitive economy allocation. Using replicas of the economy, Debreu and
Scarf (1963) can eliminate that gap. We replicate one of the periods until we
eliminate the gap between the axiomatic structure of DRUM and the recursive
application of the static axioms of RUM. Similarly to this classical result of
Debreu and Scarf (1963), Theorem 3 shows how one can shrink the gap by
diminishing the relative importance of the nonrational preference profiles in
the cone associated with the Kronecker product of the facets (or axioms) of
static RUM.

###

In our running example, $A^{t}$ is not full column rank. That means that the
dynamic triangle conditions (and stability) are necessary but not sufficient
conditions for DRUM. In our running example, for $T=2$, we will now obtain
explicitly all components of equation (1), and explain how they provide
testable implications for DRUM that become sufficient as we take large enough
$k$. For $k=1$, we conclude that if $\rho$ is consistent with DRUM, then it
satisfies $\otimes_{t\in\mathcal{T}}H^{t}\rho\geq 0$ or the dynamic triangle
conditions. We then focus on the second extension of the theory for $k=2$.
This means that we consider three virtual periods, $T^{v,3}=3$. Let
$\mathbf{J}^{v}$ be the set of all menu paths in the virtual time window.
Define $\rho^{v}$ as a dynamic stochastic choice function on $\mathbf{J}^{v}$.

To simplify the exposition and to obtain a reduction of dimensionality, we
will use the fact that $A^{t}$, in Table 1, can be simplified without loss of
generality by computing $A^{t*}$ as the submatrix with all rows of $A^{t}$
except for row $4$ and row $6$. Due to the simplex constraints of DRUM these
rows are redundant. We can then obtain the $\mathcal{H}$-representation matrix
$H^{t*}$ from Table 2 by deleting columns $4$ and $6$, and rows $2$, $4$, $5$,
and $6$. We have to retain the nonnegativity constraints that are not
associated with the deleted rows in $A^{t}$. Since we are imposing the simplex
and stability constraints on $\rho$, we can simplify equation (1) to work with
the reduced $H^{t*}$ and the reduced $\rho^{*}$, of which the latter is
equivalent to the subarray of $\rho$ obtained after we delete all entries with
a choice path containing one of the deleted rows of $A^{t}$. (We define
$\rho^{*,v}$ in an analogous way to $\rho^{*}$.)

We set $\phi^{*,2}$ as the average of all the triangle conditions in Table 2
and the nonnegativity constraints:

$\phi^{*,2}=\frac{1}{6}\left(\begin{array}[]{cccccccccccc}2&2&1&1\end{array}\right)^{\prime}.$

Recall that $I^{t}$ is a diagonal matrix of dimension $4$ for all
$t\in\mathcal{T}$. The projection mapping $\gamma_{2}^{\phi^{*,2}}$ is given
by a matrix of size $4\times 16$ whose rows with nonnegative entries add up to
$1$:

$\left(\begin{array}[]{cccccccccccccccc}\frac{1}{3}&\frac{1}{6}&\frac{1}{12}&\frac{1}{12}&\frac{1}{6}&0&0&0&\frac{1}{12}&0&0&0&\frac{1}{12}&0&0&0\\\
0&\frac{1}{6}&0&0&\frac{1}{6}&\frac{1}{3}&\frac{1}{12}&\frac{1}{12}&0&\frac{1}{12}&0&0&0&\frac{1}{12}&0&0\\\
0&0&\frac{1}{6}&0&0&0&\frac{1}{6}&0&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&\frac{1}{12}&0&0&\frac{1}{12}&0\\\
0&0&0&\frac{1}{6}&0&0&0&\frac{1}{6}&0&0&0&\frac{1}{12}&\frac{1}{6}&\frac{1}{6}&\frac{1}{12}&\frac{1}{6}\\\
\end{array}\right).$

The matrix $\gamma_{2}^{\phi^{*,2}}$ can be interpreted as a linear operator
taking weighted averages. In fact, the associated matrix
$\Gamma^{\phi^{*,2}\prime}_{2}$ reduces any $\rho^{v}$ of size $64\times 1$ to
a vector of size $16\times 1$, that is the size of $\rho^{*}$. Recall that we
defined the matrix $H^{t}$ by Table 2 and nonnegativity constraints. If $\rho$
is consistent with DRUM, then there is some virtual $\rho^{v}$ such that the
following will be satisfied:

$\displaystyle\rho^{*}=\Gamma^{\phi^{*,2}\prime}_{2}\rho^{*,v}$
$\displaystyle\otimes_{t=1}^{3}H^{*,t}\rho^{*,v}\geq 0.$

It should be evident that the dynamic triangle inequality conditions are
implied by the previous conditions, but new emergent conditions also appear.
In particular, we write down explicitly the first entries of the $\rho^{*}$
vector (with each entry representing the probability of a choice path) in
terms of the $\rho^{*,v}$ vector (the remaining entries can be computed easily
by the reader):

$\displaystyle\rho^{*}_{1}$
$\displaystyle=\frac{\rho_{1}^{*,v}}{3}+\frac{\rho_{2}^{*,v}}{6}+\frac{\rho_{3}^{*,v}}{12}+\frac{\rho_{4}^{*,v}}{12}+\frac{\rho_{5}^{*,v}}{6}+\frac{\rho_{9}^{*,v}}{12}+\frac{\rho_{13}^{*,v}}{12}$
$\displaystyle\rho^{*}_{2}$
$\displaystyle=\frac{\rho_{2}^{*,v}}{6}+\frac{\rho_{5}^{*,v}}{6}+\frac{\rho_{6}^{*,v}}{3}+\frac{\rho_{7}^{*,v}}{12}+\frac{\rho_{8}^{*,v}}{12}+\frac{\rho_{10}^{*,v}}{12}+\frac{\rho_{14}^{*,v}}{12}.$

These two equations illustrate the fact that $\rho_{1}$ and $\rho_{2}$ become
connected or dependent through $\rho^{*,v}$ entries $\rho_{2}^{*,v}$ and
$\rho_{5}^{*,v}$. In addition, we can obtain new inequalities explicitly from
these relations. Behaviorally, these additional conditions state that when
$\rho$ is consistent with DRUM, then $\rho$ must be the result of projecting,
back to $T=2$, a $\rho^{v}$ that is consistent with the dynamic triangle
conditions, in a larger time window. The projection mapping
$\Gamma^{\phi^{*,2}\prime}_{2}$ can be interpreted as a projection device
based on the average of testable implications of the static case $\phi^{*,2}$.
This device asks what the choices of the virtual DMs associated with
$\rho^{v}$ in the actual time window are. We can make analogous statements for
any $k\geq 3$.

###  4.1. The Simple Setup: two budgets per time period

Here we illustrate our main results in the demand environment with two budgets
in each time period $B^{*,t}_{1}$ and $B^{*,t}_{2}$ such that $B^{*,t}_{1}\cap
B^{*,t}_{2}\neq\emptyset$ and $w_{1,t}/p_{1,t,K}>w_{2,t}/p_{2,t,K}$ for all
$t\in\mathcal{T}$. To simplify the analysis, we assume that the intersection
patches are picked with probability zero. Thus, in each time period there are
four patches $x^{t}_{1|1},x^{t}_{2|1},x^{t}_{1|2}$, and $x^{t}_{2|2}$ (see
Figure 2 for a graphical representation of the case with $K=2$
goods).202020Formally, $x^{t}_{1|1}=\\{y\in
B^{*,t}_{1}\>:\>p^{\prime}_{2,t}y>w_{2,t}\\}$, $x^{t}_{2|1}=\\{y\in
B^{*,t}_{1}\>:\>p^{\prime}_{2,t}y<w_{2,t}\\}$, $x^{t}_{1|2}=\\{y\in
B^{t}_{2}\>:\>p^{\prime}_{1,t}y<w_{1,t}\\}$, and $x^{t}_{2|2}=\\{y\in
B^{*,t}_{2}\>:\>p^{\prime}_{1,t}y>w_{1,t}\\}$. We call choice path
configurations implied by these four patches the _simple setup_ choice paths.
An example of a budget path for $T=2$ is $(2,1)$ (i.e. $B^{1}_{2}$ and
$B^{2}_{1}$), and an example of a choice path in this budget path is
$\left(x_{1|2}^{1},x_{1|1}^{2}\right)$.

In this setup, there are three rational demand types per time period that are
described in Table 3.212121We use the convenient notation developed in Im and
Rehbeck (2021). Each demand type $\theta^{t}_{i,j}$ picks the $i$-th patch in
menu $B_{1}^{t}$ and the $j$-th patch in menu $B^{t}_{2}$ at time $t$.

$y_{1}$$y_{2}$$x^{1}_{1|2}$$x^{1}_{2|1}$$x^{1}_{1|1}$$x^{1}_{2|2}$$B^{*,1}_{1}$$B^{*,1}_{2}$

$y_{1}$$y_{2}$$x^{2}_{1|2}$$x^{2}_{2|1}$$x^{2}_{1|1}$$x^{2}_{2|2}$$B^{*,2}_{1}$$B^{*,2}_{2}$

Figure 2: Simple setup for $K=2$ goods and no intersection patches.  Type/Budget | $B^{t}_{1}$ | $B^{t}_{2}$
---|---|---
$\theta^{t}_{1,1}$ | $x^{t}_{1|1}$ | $x^{t}_{1|2}$
$\theta^{t}_{1,2}$ | $x^{t}_{1|1}$ | $x^{t}_{2|2}$
$\theta^{t}_{2,2}$ | $x^{t}_{2|1}$ | $x^{t}_{2|2}$
Table 3: Choices of 3 rational types in menus $B^{t}_{1}$ and $B^{t}_{2}$ at
time $t$.

We can now write down the associated $A_{T}$ matrix. Since there are two
intersecting budgets in every time period, $A^{t}=A^{t^{\prime}}$ for all
$t,t^{\prime}\in\mathcal{T}$. Thus, by Lemma 3, we can compute the matrix
$A^{t}$ for one period. We display the matrix $A^{t}$ in Table 4. Note that it
is easy to verify that $A^{t}$ has full column rank; the DRUM associated with
$\otimes_{t\in\mathcal{T}}A^{t}$ is thus unique according to our definition.
This allows us to obtain necessary and sufficient conditions explicitly. Using
$A^{t}$, we can write down the matrix $A_{T}$ for any $T$ (e.g., see Table 5
for $T=2$ or $A_{T}=A^{1}\otimes A^{2}$).

| $\theta^{t}_{1,1}$ | $\theta^{t}_{1,2}$ | $\theta^{t}_{2,2}$
---|---|---|---
$x^{t}_{1|1}$ | $1$ | $1$ | -
$x^{t}_{2|1}$ | - | - | $1$
$x^{t}_{1|2}$ | $1$ | - | -
$x^{t}_{2|2}$ | - | $1$ | $1$
Table 4: The matrix $A^{t}$ for 2 budgets. $``-^{\prime\prime}$ corresponds to zero. | $(\theta^{1}_{1,1},\theta^{2}_{1,1})$ | $(\theta^{1}_{1,1},\theta^{2}_{1,2})$ | $(\theta^{1}_{1,1},\theta^{2}_{2,2})$ | $(\theta^{1}_{1,2},\theta^{2}_{1,1})$ | $(\theta^{1}_{1,2},\theta^{2}_{1,2})$ | $(\theta^{1}_{1,2},\theta^{2}_{2,2})$ | $(\theta^{1}_{2,2},\theta^{2}_{1,1})$ | $(\theta^{1}_{2,2},\theta^{2}_{1,2})$ | $(\theta^{1}_{2,2},\theta^{2}_{2,2})$
---|---|---|---|---|---|---|---|---|---
$\left(x^{1}_{1|1},x^{2}_{1|1}\right)$ | $1$ | $1$ | - | 1 | 1 | - | - | - | -
$\left(x^{1}_{1|1},x^{2}_{2|1}\right)$ | - | - | 1 | - | - | 1 | - | - | -
$\left(x^{1}_{1|1},x^{2}_{1|2}\right)$ | $1$ | - | - | 1 | - | - | - | - | -
$\left(x^{1}_{1|1},x^{2}_{2|2}\right)$ | - | 1 | 1 | - | 1 | 1 | - | - | -
$\left(x^{1}_{2|1},x^{2}_{1|1}\right)$ | - | - | - | - | - | - | 1 | 1 | -
$\left(x^{1}_{2|1},x^{2}_{2|1}\right)$ | - | - | - | - | - | - | - | - | 1
$\left(x^{1}_{2|1},x^{2}_{1|2}\right)$ | - | - | - | - | - | - | 1 | - | -
$\left(x^{1}_{2|1},x^{2}_{2|2}\right)$ | - | - | - | - | - | - | - | 1 | 1
$\left(x^{1}_{1|2},x^{2}_{1|1}\right)$ | $1$ | 1 | - | - | - | - | - | - | -
$\left(x^{1}_{1|2},x^{2}_{2|1}\right)$ | - | - | 1 | - | - | - | - | - | -
$\left(x^{1}_{1|2},x^{2}_{1|2}\right)$ | 1 | - | - | - | - | - | - | - | -
$\left(x^{1}_{1|2},x^{2}_{2|2}\right)$ | - | 1 | 1 | - | - | - | - | - | -
$\left(x^{1}_{2|2},x^{2}_{1|1}\right)$ | - | - | - | 1 | 1 | - | 1 | 1 | -
$\left(x^{1}_{2|2},x^{2}_{2|1}\right)$ | - | - | - | - | - | 1 | - | - | 1
$\left(x^{1}_{2|2},x^{2}_{1|2}\right)$ | - | - | - | 1 | - | - | 1 | - | -
$\left(x^{1}_{2|2},x^{2}_{2|2}\right)$ | - | - | - | - | 1 | 1 | - | 1 | 1
Table 5: The matrix $A_{T}$ for 2 time periods with 2 budgets per period.
$``-^{\prime\prime}$ corresponds to zero.

####  $\mathrm{D}$-monotonicity

We introduce a new behavioral restriction on $\rho$ that, together with
stability, characterizes DRUM in the simple setup. We first introduce a static
notion of _dominance_ among patches.

######  Definition 8 (Patch-Revealed Dominance).

We say that patch $x^{t}_{i|j}$ is revealed dominant to
$x^{t}_{i^{\prime}|j^{\prime}}$ if
$x^{t}_{i_{t}|j_{t}}>^{t}x^{t}_{i^{\prime}_{t}|j^{\prime}_{t}}$.

Patch-revealed dominance requires that all elements in the dominant patch are
directly revealed preferred (in Afriat’s sense) to the dominated patch, and
that all the elements of the dominated patch are not directly revealed
preferred to the elements of the dominant patch. We can visualize this
ordering in Figure 2, where $x^{1}_{1|1}>^{t}x^{1}_{1|2}$ and
$x^{2}_{2|2}>^{t}x^{2}_{2|1}$. Let
$x_{\mathbf{i}|\mathbf{j}}\downarrow_{t}x^{t}_{i^{\prime}_{t}|j^{\prime}_{t}}$
denote a choice path where the $t$-th component of $x_{\mathbf{i}|\mathbf{j}}$
was replaced by $x^{t}_{i^{\prime}_{t}|j^{\prime}_{t}}$. We show that if
$\rho$ is consistent with DRUM and if
$x^{t}_{i^{\prime}_{t}|j^{\prime}_{t}}>^{t}x^{t}_{i_{t}|j_{t}}$, then

$\rho\left(x_{\mathbf{i}|\mathbf{j}}\downarrow_{t}x^{t}_{i^{\prime}_{t}|j^{\prime}_{t}}\right)\geq\rho\left(x_{\mathbf{i}|\mathbf{j}}\right).$

We illustrate the necessity for the simple case where $T=1$; in that case
$A_{T}\nu=\rho$ can be rewritten as

$\displaystyle\nu_{1}+\nu_{2}$
$\displaystyle=\rho(x^{1}_{1|1}),\quad\quad\nu_{3}=\rho(x^{1}_{2|1}),$
$\displaystyle\nu_{1}$
$\displaystyle=\rho(x^{1}_{1|2}),\quad\quad\nu_{2}+\nu_{3}=\rho(x^{1}_{2|2}).$

Then the following two inequalities, consistent with monotonicity, have to be
satisfied:

$\displaystyle 0\leq\nu_{2}=\rho(x^{1}_{1|1})-\rho(x^{1}_{1|2}),$
$\displaystyle 0\leq\nu_{2}=\rho(x^{1}_{2|2})-\rho(x^{1}_{2|1}).$

In fact, we can write down the $\mathcal{H}$-representation of static RUM for
the simple setup using matrix $H^{2,t}$ as defined in Table 6, capturing this
monotonicity condition such that $H^{2,t}\rho\geq 0$.

$x^{t}_{1|1}$ | $x^{t}_{2|1}$ | $x^{t}_{1|2}$ | $x^{t}_{2|2}$
---|---|---|---
1 | - | -1 | -
1 | - | - | -
- | 1 | - | -
- | - | 1 | -
Table 6: The $\mathcal{H}$-representation of RUM for $2$ goods and $2$
budgets.

For $T\geq 2$, DRUM implies that $\rho$ satisfies dynamic monotonicity. For
illustrative purposes, set $T=2$. Then we get the new condition by exploiting
the recursive structure of the matrix $A_{T}$, as shown here:

$\rho=A_{T}\nu=A^{1}\otimes
A^{2}\nu=\left(\begin{array}[]{ccc}A^{1}&A^{1}&0\\\ 0&0&A^{1}\\\ A^{1}&0&0\\\
0&A^{1}&A^{1}\end{array}\right)\left(\begin{array}[]{c}\nu^{1}_{1}\\\
\nu^{1}_{2}\\\
\nu^{1}_{3}\end{array}\right)=\left(\begin{array}[]{c}A^{1}(\nu^{1}_{1}+\nu^{1}_{2})\\\
A^{1}\nu^{1}_{3}\\\ A^{1}\nu^{1}_{1}\\\
A^{1}(\nu^{1}_{2}+\nu^{1}_{3})\end{array}\right).$

We can next derive the following system of equations:

$\displaystyle A^{1}\nu_{1}=\left[\rho^{1}_{1|1}-\rho^{1}_{1|2}\right],$
$\displaystyle A^{1}\nu_{2}=\left[\rho^{1}_{2|2}-\rho^{1}_{2|1}\right],$

where $\rho^{1}_{i|j}$ is a vector of all probabilities that correspond to all
choice paths that contain patch $x^{1}_{i|j}$. For example,

$\rho^{1}_{1|1}=\left(\begin{array}[]{c}\rho\left(\left(x^{1}_{1|1},x^{2}_{1|1}\right)\right)\\\
\rho\left(\left(x^{1}_{1|1},x^{2}_{2|1}\right)\right)\\\
\rho\left(\left(x^{1}_{1|1},x^{2}_{1|2}\right)\right)\\\
\rho\left(\left(x^{1}_{1|1},x^{2}_{2|2}\right)\right).\end{array}\right)$

Repeating the argument for $T=1$, from
$A^{1}\nu_{1}=\left[\rho^{1}_{1|1}-\rho^{1}_{1|2}\right]$, we derive the
following:

$\displaystyle
0\leq\left[\rho\left(\left(x^{1}_{1|1},x^{2}_{1|1}\right)\right)-\rho\left(\left(x^{1}_{1|2},x^{2}_{1|1}\right)\right)\right]-\left[\rho\left(\left(x^{1}_{1|1},x^{2}_{1|2}\right)\right)-\rho\left(\left(x^{1}_{1|2},x^{2}_{1|2}\right)\right)\right].$
(2)

The following inequalities have to hold under DRUM:

$\displaystyle\rho\left(\left(x^{1}_{1|1},x^{2}_{1|1}\right)\right)-\rho\left(\left(x^{1}_{1|2},x^{2}_{1|1}\right)\right)\geq
0,$
$\displaystyle\rho\left(\left(x^{1}_{1|1},x^{2}_{1|2}\right)\right)-\rho\left(\left(x^{1}_{1|2},x^{2}_{1|2}\right)\right)\geq
0.$

Thus, Inequality (2) shown just above imposes a restriction on how the
distribution over patches can grow. In particular, it implies that the
increase in probability caused by switching from patch $x^{1}_{1|2}$ to the
dominant patch $x^{1}_{1|1}$ is bigger if the patch in the second time period,
$x^{2}_{1|1}$, dominates $x^{2}_{1|2}$. In other words, there is some form of
complementarity between dominant patches in different time periods. In order
to generalize the above arguments for an arbitrary but finite time window, we
would need to work with higher-order differences (that are in fact the
Kronecker products of the static monotonicity conditions). To do this, we next
introduce the difference operator.

######  Definition 9 (Difference operator).

For any $t$, $x^{t}_{i^{\prime}_{t}|i^{\prime}_{t}}$, and
$x_{\mathbf{i}|\mathbf{j}}$, let
$\mathrm{D}\left(x^{t}_{i^{\prime}_{t}|j^{\prime}_{t}}\right)[\cdot]$ be a
linear operator such that

$\mathrm{D}\left(x^{t}_{i^{\prime}_{t}|j^{\prime}_{t}}\right)\left[f(x_{\mathbf{i}|\mathbf{j}})\right]=f\left(x_{\mathbf{i}|\mathbf{j}}\downarrow_{t}x^{t}_{i^{\prime}_{t}|j^{\prime}_{t}}\right)-f\left(x_{\mathbf{i}|\mathbf{j}}\right)$

for any $f$ that maps choice paths to reals.

The $\mathrm{D}$-operator applied to $\rho$ calculates the difference in
$\rho$ when only one patch in a choice path was replaced. When the operator is
applied twice to two different time periods, it computes the difference in
differences. That is, for $t_{1}\neq t_{2}$, we have the following equalities:

$\displaystyle\mathrm{D}\left(x^{t_{2}}_{i^{\prime}_{t_{2}}|j^{\prime}_{t_{2}}}\right)\mathrm{D}\left(x^{t_{1}}_{i^{\prime}_{t_{1}}|j^{\prime}_{t_{1}}}\right)\left[f\left(x_{\mathbf{i}|\mathbf{j}}\right)\right]=\mathrm{D}\left(x^{t_{2}}_{i^{\prime}_{t_{2}}|j^{\prime}_{t_{2}}}\right)\left[f\left(x_{\mathbf{i}|\mathbf{j}}\downarrow_{t_{1}}x^{t_{1}}_{i^{\prime}_{t_{1}}|j^{\prime}_{t_{1}}}\right)-f\left(x_{\mathbf{i}|\mathbf{j}}\right)\right]=$
$\displaystyle\mathrm{D}\left(x^{t_{2}}_{i^{\prime}_{t_{2}}|j^{\prime}_{t_{2}}}\right)\left[f\left(x_{\mathbf{i}|\mathbf{j}}\downarrow_{t_{1}}x^{t_{1}}_{i^{\prime}_{t_{1}}|j^{\prime}_{t_{1}}}\right)\right]-\mathrm{D}\left(x^{t_{2}}_{i^{\prime}_{t_{2}}|j^{\prime}_{t_{2}}}\right)\left[f\left(x_{\mathbf{i}|\mathbf{j}}\right)\right]=$
$\displaystyle\left[f\left(x_{\mathbf{i}|\mathbf{j}}\downarrow_{t_{1}}x^{t_{1}}_{i^{\prime}_{t_{1}}|j^{\prime}_{t_{1}}}\downarrow_{t_{2}}x^{t_{2}}_{i^{\prime}_{t_{2}}|j^{\prime}_{t_{2}}}\right)-f\left(x_{\mathbf{i}|\mathbf{j}}\downarrow_{t_{1}}x^{t_{1}}_{i^{\prime}_{t_{1}}|j^{\prime}_{t_{1}}}\right)\right]-\left[f\left(x_{\mathbf{i}|\mathbf{j}}\downarrow_{t_{2}}x^{t_{2}}_{i^{\prime}_{t_{2}}|j^{\prime}_{t_{2}}}\right)-f\left(x_{\mathbf{i}|\mathbf{j}}\right)\right],$

where the second equality uses the linearity of
$\mathrm{D}\left(x^{t_{2}}_{i^{\prime}_{t_{2}}|j^{\prime}_{t_{2}}}\right)$.

Similarly, we can apply $\mathrm{D}$ any $K$ number of times. Let

$\boldsymbol{\mathcal{T}}=\left\\{\left(t_{k}\right)_{k=1}^{K}\>:\>t_{k}\in\mathcal{T},\>t_{k}<t_{k+1},K\leq
T\right\\}$

be a collection of all possible increasing sequences of length at most $T$.
For any $\mathbf{t}\in\boldsymbol{\mathcal{T}}$ and any
$x^{\mathbf{t}}_{\mathbf{i}^{\prime}|\mathbf{j}^{\prime}}=\left(x^{t}_{i^{\prime}_{t}|j^{\prime}_{t}}\right)_{t\in\mathbf{t}}$,
we define

$\mathrm{D}\left(x^{\mathbf{t}}_{\mathbf{i}^{\prime}|\mathbf{j}^{\prime}}\right)\left[f\left(x_{\mathbf{i}|\mathbf{j}}\right)\right]=\mathrm{D}\left(x^{t_{K}}_{i^{\prime}_{t_{K}}|j^{\prime}_{t_{K}}}\right)\dots\mathrm{D}\left(x^{t_{2}}_{i^{\prime}_{t_{2}}|j^{\prime}_{t_{2}}}\right)\mathrm{D}\left(x^{t_{1}}_{i^{\prime}_{t_{1}}|j^{\prime}_{t_{1}}}\right)\left[f\left(x_{\mathbf{i}|\mathbf{j}}\right)\right],$

where $K$ is the length of $\mathbf{t}$.

######  Definition 10 ($\mathrm{D}$-monotonicity).

We say that $\rho$ is $\mathrm{D}$-monotone if for any
$\mathbf{t}\in\boldsymbol{\mathcal{T}}$,
$x^{\mathbf{t}}_{\mathbf{i}^{\prime}|\mathbf{j}^{\prime}}$, and any
$x_{\mathbf{i}|\mathbf{j}}$ such that
$x^{t}_{i^{\prime}_{t}|j^{\prime}_{t}}>^{t}x^{t}_{i_{t}|j_{t}}$ for all
$t\in\mathbf{t}$, it is the case that

$\mathrm{D}\left(x^{\mathbf{t}}_{\mathbf{i}^{\prime}|\mathbf{j}^{\prime}}\right)\left[\rho\left(x_{\mathbf{i}|\mathbf{j}}\right)\right]\geq
0.$

$\mathrm{D}$-monotonicity is the generalization to our dynamic setup of the
Weak Axiom of Stochastic Revealed Preference (WASRP) introduced by
Bandyopadhyay et al. (1999). Since the set of utility functions we are
considering is monotone, our condition coincides with the stochastic
substitutability condition in Bandyopadhyay et al. (2004) when
$\mathcal{T}=\\{t\\}$. Whenever the domain of choices is incomplete, Dasgupta
and Pattanaik (2007) shows that WASRP is sufficient but not necessary for
regularity. This observation translates to the dynamic case as well.222222In
that regard, $\mathrm{D}$-monotonicity is not implied by any of the conditions
derived in Li (2021) or Chambers et al. (2021) that require complete menu
variation and use generalizations of the static regularity conditions for the
dynamic or correlated case. We emphasize that $\mathrm{D}$-monotonicity
imposes stronger restrictions on the data than WASRP does. We also note that
the strength of these restrictions increases as the time window expands (see
our Monte Carlo experiments in Appendix B). As such, $\mathrm{D}$-monotonicity
can be used to derive informative counterfactual bounds on the longitudinal
distribution of out-of-sample demand with sufficiently long panels. Given
these observations, we are ready to present the main result of this section.

######  Theorem 4.

For the simple setup, the following are equivalent:

  1. (i)

$\rho$ is consistent with DRUM.

  2. (ii)

$\rho$ is $\mathrm{D}$-monotone and stable.

  3. (iii)

$\rho$ is stable and $\otimes_{t\in\mathcal{T}}H^{2,t}\rho\geq 0$.

The fact that $(i)$ implies $(ii)$ is easy to verify. The converse is proved
constructively. The equivalence of $(iii)$ and $(i)$ is a corollary of Theorem
3. Importantly, in that case, we can set $k_{t}=1$ for all $t\in\mathcal{T}$.
This is because $A^{t}$ in the simple setup is full column rank and we can use
Theorem 2.

######  Corollary 1.

For the simple setup if $\rho=A_{T}\nu=A_{T}\nu^{\prime}$ for some
$\nu,\nu^{\prime}\in\Delta^{|\mathcal{R}-1|}$, then $\nu=\nu^{\prime}$.

Stability and $\mathrm{D}$-monotonicity are logically independent as we
demonstrate in the next counterexample to DRUM.232323Another example of a
$\rho$ that fails both conditions of the simple setup is discussed in Section
5.

######  Example 4 (Violation of $\mathrm{D}$-monotonicity).

Consider the stochastic demand presented in Table 7. It satisfies stability.
However, it fails to satisfy $\mathrm{D}$-monotonicity because
$\rho\left(\left(x^{1}_{1|2},x^{2}_{1|1}\right)\right)-\rho\left(\left(x^{1}_{1|2},x^{2}_{1|2}\right)\right)=-\frac{1}{4}$
and $x^{2}_{1|1}>^{t}x^{2}_{1|2}$.

| $x^{2}_{1|1}$ | $x^{2}_{2|1}$ | $x^{2}_{1|2}$ | $x^{2}_{2|2}$
---|---|---|---|---
$x^{1}_{1|1}$ | $3/4$ | - | $3/4$ | -
$x^{1}_{2|1}$ | - | $1/4$ | $1/4$ | -
$x^{1}_{1|2}$ | - | $1/4$ | $1/4$ |
$x^{1}_{2|2}$ | $3/4$ | - | $3/4$ | -
Table 7: Matrix representation of $\rho$ for $T=2$ that violates
$\mathrm{D}$-monotonicity, but satisfies simple stability. “-” corresponds to
zero

#### Generalization of $\mathrm{D}$-monotonicity for the demand setup with
three goods and three budgets per period

Our results can be used to construct a set of necessary and sufficient
conditions for DRUM in the demand setup on the basis of the
$\mathcal{H}$-representation of (static, $\mathcal{T}=\\{t\\}$) DRDM for the
case of three goods and three budgets.242424In each period there are $3$
budgets with maximal intersections as in Example $3.2$ in KS. The
$\mathcal{V}$-representation in this case is given by matrix $A^{t}$ in Table
8. The $\mathcal{H}$-representation, $H^{3,t}$, is displayed in Table 9
(without the nonnegativity constraints). We note that $H^{3,t}=H^{3,s}$ for
any $s,t\in\mathcal{T}$. We can then establish the following direct
implication of Theorem 3 since $A^{t}$ in this case is such that $A^{t*}$ is
full row rank.

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | - | - | - | - | - | - | - | - | - | - | - | - | - | $x^{t}_{1|1}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
- | - | - | - | - | - | - | - | - | - | - | - | 1 | 1 | 1 | 1 | 1 | - | - | - | - | - | - | - | - | $x^{t}_{2|1}$
- | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | 1 | 1 | 1 | 1 | 1 | - | - | - | $x^{t}_{3|1}$
- | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | 1 | 1 | 1 | $x^{t}_{4|1}$
1 | 1 | 1 | 1 | - | - | - | - | - | - | - | - | 1 | 1 | 1 | 1 | - | 1 | 1 | - | - | - | 1 | 1 | - | $x^{t}_{1|2}$
- | - | - | - | 1 | 1 | 1 | 1 | - | - | - | - | - | - | - | - | - | - | - | 1 | - | - | - | - | - | $x^{t}_{2|2}$
- | - | - | - | - | - | - | - | 1 | 1 | - | - | - | - | - | - | 1 | - | - | - | 1 | - | - | - | 1 | $x^{t}_{3|2}$
- | - | - | - | - | - | - | - | - | - | 1 | 1 | - | - | - | - | - | - | - | - | - | 1 | - | - | - | $x^{t}_{4|2}$
1 | - | - | - | 1 | - | - | - | 1 | - | 1 | - | 1 | - | - | - | 1 | 1 | - | 1 | 1 | 1 | 1 | - | 1 | $x^{t}_{1|3}$
- | 1 | - | - | - | 1 | - | - | - | - | - | - | - | 1 | - | - | - | - | 1 | - | - | - | - | 1 | - | $x^{t}_{2|3}$
- | - | 1 | - | - | - | 1 | - | - | 1 | - | 1 | - | - | 1 | - | - | - | - | - | - | - | - | - | - | $x^{t}_{3|3}$
- | - | - | 1 | - | - | - | 1 | - | - | - | - | - | - | - | 1 | - | - | - | - | - | - | - | - | - | $x^{t}_{4|3}$
Table 8: $A^{t}$ for $3$ goods and $3$ budgets. “-” corresponds to zero. $x^{t}_{1|1}$ | $x^{t}_{2|1}$ | $x^{t}_{3|1}$ | $x^{t}_{4|1}$ | $x^{t}_{1|2}$ | $x^{t}_{2|2}$ | $x^{t}_{3|2}$ | $x^{t}_{4|2}$ | $x^{t}_{1|3}$ | $x^{t}_{2|3}$ | $x^{t}_{3|3}$ | $x^{t}_{4|3}$
---|---|---|---|---|---|---|---|---|---|---|---
- | - | - | -1 | - | - | - | -1 | 1 | 1 | 1 | -
- | - | - | -1 | 1 | - | - | - | 1 | - | - | -
1 | - | - | - | 1 | - | - | - | - | - | - | -1
1 | - | - | - | - | - | - | -1 | 1 | - | - | -
- | -1 | - | -1 | 1 | - | 1 | - | - | - | - | -
- | - | - | - | 1 | 1 | - | - | - | -1 | - | -1
- | - | -1 | -1 | - | - | - | - | 1 | 1 | - | -
Table 9: The $\mathcal{H}$-representation of RUM for $3$ goods and $3$ budgets
excluding nonnegativity.

######  Corollary 2.

For the demand setup ($K=3$, $J^{t}=3$ for all $t\in\mathcal{T}$), the
following are equivalent:

  1. (i)

$\rho$ is consistent with DRUM.

  2. (ii)

$\rho$ is stable and

$\rho\in\bigcap_{k_{1},\cdots,k_{T}\geq
1}\left\\{\Gamma^{\boldsymbol{\phi}^{*}\prime}_{\mathbf{k}}z\>:\>\left(\otimes_{t\in\mathcal{T}}H^{3,t,\otimes_{k_{t}}}\right)z\geq
0\right\\}.$

For $k=1$ we obtain necessary conditions in the form of
$(\otimes_{t\in\mathcal{T}}H^{3,t})\rho\geq 0$. The last three rows of the
matrix displayed in Table 9 and their Kronecker product in time are consistent
with $\mathrm{D}$-monotonicity.252525Note that if
$\mathrm{P}=(\mathrm{P}_{\mathbf{j}})_{\mathbf{j}\in\mathbf{J}}$ is consistent
with DRDM, then the stochastic demand system consisting of 2 different budget
paths $(\mathrm{P}_{\mathbf{j}},\mathrm{P}_{\mathbf{j}^{\prime}})$ would also
be consistent with DRDM. Moreover, note that
$(\mathrm{P}_{\mathbf{j}},\mathrm{P}_{\mathbf{j}^{\prime}})$ form the simple
setup since at every time period there are exactly two budgets. Thus,
$\mathds{D}$-monotonicity is a simple necessary condition for DRUM. The rest
of the conditions produce new testable implications. For instance, for $T=2$,
the following conditions are implied by monotonicity in row $4$:

$\displaystyle
D^{*1}(x_{i|j}^{2})=\left[\rho\left(\left(x^{1}_{1|2},x_{i|j}^{2}\right)\right)+\rho\left(\left(x^{1}_{3|2},x_{i|j}^{2}\right)\right)-\rho\left(\left(x^{1}_{2|1},x_{i|j}^{2}\right)\right)-\rho\left(\left(x^{1}_{4|1},x_{i|j}^{2}\right)\right)\right]\geq
0.$

The interaction of row $3$, a triangle condition, and row $4$, a monotonicity
condition, gives the implication:

$\displaystyle D^{*1}(x_{1|1}^{2})+D^{*1}(x_{1|2}^{2})-D^{*1}(x_{4|3}^{2})\geq
0.$

Obtaining sufficient conditions in terms of an $H^{3,t}$ matrix would require
more work because $A^{t}$ is no longer full column rank in this case.
Nevertheless, we can obtain sufficient conditions explicitly from the
knowledge of $H^{3,t}$ alone. We set $\phi^{*,t}$ as the average of all
conditions implied by $H^{3,t}$. The rest of the computations for any $k\geq
2$ can be done in the same way as in the running example for binary menus.

###  4.2. Characterization of DRUM via Recursive Block Marschak Inequalities

Here, we provide a characterization, via linear inequalities, based on the BM
inequalities for the case of limited menu variability and with monotonicity of
utilities.

Recall that $\mathcal{J}^{t}$ denotes the set of observed menus at time $t$.
Let $\bar{\mathcal{J}}^{t}=\\{1,2,\dots,2^{\left\lvert
X^{t}\right\rvert}-1\\}$ be the “virtual” set of menus such that there is a
one-to-one mapping between $j_{t}\in\bar{\mathcal{J}}^{t}$ and a nonempty
subset of $X^{t}$ (i.e., the virtual data set has full menu variation). We
also assume that this mapping is such that the first $\mathcal{J}^{t}$ indexes
correspond to observed menus $B^{t}_{j}$. That is $\mathcal{J}^{t}$ is the set
of all observed menus and $\bar{\mathcal{J}}^{t}\setminus\mathcal{J}^{t}$ is
the set of all “virtual menus” that were not observed in the data. Using this
extended definition of menus, we can define a set of all menu paths
$\bar{\mathbf{J}}$ (including the observed ones). Note that the set of
observed menu paths $\mathbf{J}$ is a subset of $\bar{\mathbf{J}}$. Finally,
as before, given a menu path $\mathbf{j}\in\bar{\mathbf{J}}$, let $\bar{\rho}$
be a distribution over choice paths $\mathbf{i}$ in $\mathbf{j}$. That is,
$\bar{\rho}\left(x_{\mathbf{i}|\mathbf{j}}\right)\geq 0$ and
$\sum_{\mathbf{i}\in\mathbf{I}_{\mathbf{j}}}\bar{\rho}\left(x_{\mathbf{i}|\mathbf{j}}\right)=1$
for all $\mathbf{j}\in\bar{\mathbf{J}}$. Recall that the observed $\rho$ is
defined as

$\rho=\left(\rho\left(x_{\mathbf{i}|\mathbf{j}}\right)\right)_{\mathbf{j}\in\mathbf{J},\mathbf{i}\in\mathbf{I}_{\mathbf{j}}}.$

Similarly, the extended stochastic choice function is defined by

$\bar{\rho}=\left(\bar{\rho}\left(x_{\mathbf{i}|\mathbf{j}}\right)\right)_{\mathbf{j}\in\bar{\mathbf{J}},\mathbf{i}\in\mathbf{I}_{\mathbf{j}}}.$

Next we define some properties of $\bar{\rho}$.

######  Definition 11.

We say that $\bar{\rho}$ _agrees_ with $\rho$ if they coincide on observed
menu paths. That is,
$\bar{\rho}\left(x_{\mathbf{i}|\mathbf{j}}\right)=\rho\left(x_{\mathbf{i}|\mathbf{j}}\right)$
for all $\mathbf{j}\in\mathbf{J}$ and $\mathbf{i}\in\mathbf{I}_{\mathbf{j}}$.

If $\bar{\rho}$ agrees with $\rho$, then it is an extension of the latter to
virtual menu paths.

######  Definition 12 (Increasing Utility (IU) Consistency).

We say that $\bar{\rho}$ is _IU-consistent_ if
$\bar{\rho}\left(x_{\mathbf{i}|\mathbf{j}}\right)=0$ whenever there exists
$t\in\mathcal{T}$ and $I^{\prime}_{t}\subseteq\mathcal{I}^{t}_{j_{t}}$ such
that $\cup_{i^{\prime}_{t}\in
I^{\prime}_{t}}\\{x_{i^{\prime}_{t}|j_{t}}\\}>^{t}\\{x_{i_{t}|j_{t}}\\}$.

IU-consistency captures the empirical content of the monotonicity of the
utility functions with respect to $>^{t}$.

######  Definition 13 (BM inequalities).

We say that $\bar{\rho}$ satisfies the BM inequalities if for all
$t\in\mathcal{T}$, $\mathbf{j}\in\bar{\mathbf{J}}$, and
$\mathbf{i}\in\mathbf{I}_{\mathbf{j}}$

$\mathds{B}^{t}(\mathbf{i},\mathbf{j})=\sum_{j_{t}^{\prime}:B^{t}_{j_{t}}\subseteq
B^{t}_{j_{t}^{\prime}}}(-1)^{\left\lvert B^{t}_{j_{t}^{\prime}}\setminus
B^{t}_{j_{t}}\right\rvert}\bar{\rho}\left(x_{\mathbf{i}|\mathbf{j^{\prime}}}\right)\geq
0.$

Note that the BM inequalities are linear in $\bar{\rho}$. Hence, we can
construct matrix $\bar{H}^{t}$ with elements in $\\{-1,0,1\\}$ such that each
row of $\bar{H}^{t}$ corresponds to one BM inequality.

We are now ready to state the two main results of this section: first, a BM
characterization of RUM for our setup, and second, an analogous
characterization for DRUM.

######  Theorem 5.

Let $\mathcal{T}=\\{t\\}$. For a given $\rho$, the following are equivalent:

  1. (i)

$\rho$ is consistent with RUM.

  2. (ii)

There exists $\bar{\rho}$ that agrees with $\rho$, is IU-consistent, and
satisfies the BM inequalities.

  3. (iii)

There exists $\bar{\rho}$ that agrees with $\rho$, is IU-consistent, and is
such that $\bar{H}^{t}\bar{\rho}\geq 0$.

The BM inequalities provide the $\mathcal{H}$-representation of RUM for the
static case. For this extended setup with $\bar{\rho}$, we say that
$\bar{A}^{t}$ generates a unique RUM when the system
$\bar{\rho}=\bar{A}^{t}\nu$ has a unique solution for all choice functions
$\bar{\rho}$. We also say that DRUM (associated to matrix
$\otimes_{t\in\mathcal{T}}\bar{A}^{t}$) satisfies uniqueness when, for all
$t\in\mathcal{T}\setminus{\\{t^{\prime}\\}}$, $\bar{A}^{t}$ generates a unique
RUM for an arbitrary period $t^{\prime}$. We can obtain the modified
$\mathcal{H}$-representation, $\bar{H}^{t}$, for the unique RUM for
$\mathcal{T}=\\{t\\}$ from Turansick (2022). The rows of
$\bar{H}^{t}$correspond to: (i) BM inequalities, (ii) nonnegativity
constraints, and (iii) the uniqueness restriction in Theorem $1$ in Turansick
(2022).262626The conditions in Turansick (2022) are equivalent to having some
BM polynomials to be equal to zero. This can be expressed in our setup by
adding to the $\mathcal{H}$-representation the row corresponding to the
relevant BM polynomial and with the same row multiplied by $-1$. This
guarantees that the relevant BM polynomial will be zero.

######  Theorem 6.

The following are equivalent:

  1. (i)

$\rho$ is consistent with DRUM satisfying uniqueness.

  2. (ii)

$\rho$ is IU-consistent, stable, and satisfies
$(\otimes_{t\in\mathcal{T}}\bar{H}^{t})\bar{\rho}\geq 0$.

The result is a direct consequence of Theorem 3 and Theorem 5. Theorem 6 is
not exactly a $\mathcal{H}$-representation of DRUM; however, it becomes one
when all menus in $\bar{\mathbf{J}}$ are observed, as is assumed in Chambers
et al. (2021) and in Li (2021). Note, moreover, that our proof can be used in
the environments of Chambers et al. (2021) and Li (2021) in the finite
abstract setting because the primitive order $>^{t}$ can be empty. Notice that
the results in Li (2021) are a special case of Theorem 6 because when the
choice set is of cardinality $3$ or less, RUM is unique (Fishburn, 1998,
Turansick, 2022). Our result also generalizes Theorem $3$ in Chambers et al.
(2021), which is equivalent to the special case $T=2$.

We generalize the BM inequalities for the case of unobserved menus. Even if,
for our primitive, this recursive characterization of DRUM is not an
$\mathcal{H}$-representation of DRUM, this characterization has several
advantages: (i) it avoids the computation of matrix $A_{T}$ associated with
the $\mathcal{V}$-representation, which can be computationally burdensome
(Kitamura and Stoye, 2019 note that computing $A^{t}$ is NP hard); (ii) it
provides further intuition about the additional empirical bite of DRUM in
comparison to RUM; and (iii) it means DRUM can be tested with a linear
program.272727For the statistical problem, we can use tools in KS and Fang et
al. (2023).

Note that one cannot weaken the uniqueness assumption at all. In fact, the
result fails to be true when $A^{t}$ is associated with nonunique static RUM
for more than two periods. In that case, we have to go back to Theorem 7.

We set $\phi^{*,t}$ for any $t\in\mathcal{T}$ as the average of rows of
$\bar{H}^{t}$. The typical entry of $\bar{H}^{t}$ corresponding to the BM
inequalities is:

$\bar{H}^{t}_{(i_{t},j_{t}),(i^{\prime}_{t},j^{\prime}_{t})}=(-1)^{|B^{t}_{j^{\prime}_{t}}\setminus
B^{t}_{j_{t}}|}\mathds{1}\left(\,x_{i_{t}|j_{t}}=x_{i^{\prime}_{t}|j^{\prime}_{t}},B^{t}_{j_{t}}\subseteq
B^{t}_{j^{\prime}_{t}},x_{i_{t}|j^{\prime}_{t}}\in
B^{t}_{j^{\prime}_{t}}\,\right).$

Hence,
$\phi^{*,t}_{i_{t}\prime,j_{t}\prime}=\sum_{i_{t},j_{t}}\bar{H}^{t}_{(i_{t},j_{t}),(i^{\prime}_{t},j^{\prime}_{t})}$.
As a direct application of Theorem 3, we can now establish the following
result.

######  Theorem 7.

The following are equivalent:

  1. (i)

$\rho$ is consistent with DRUM.

  2. (ii)

There exists $\bar{\rho}$ that agrees with $\rho$, is IU-consistent, stable,
and satisfies

$\bar{\rho}\in\bigcap_{k_{1},\cdots,k_{T}\geq
1}\left\\{\Gamma^{\boldsymbol{\phi}^{*}\prime}_{\mathbf{k}}z\>:\>\left(\otimes_{t\in\mathcal{T}}\bar{H}^{t,\otimes_{k_{t}}}\right)z\geq
0\right\\}.$

To understand the intuition behind the $\mathcal{H}$-representation of DRUM,
we focus on a necessary condition that it implies. We define a new set of
inequalities we call DRUM-BM.

######  Definition 14 (DRUM-BM inequalities).

We say that $\bar{\rho}$ satisfies the DRUM-BM inequalities if, for all
$t\in\mathcal{T}$, $\mathbf{j}\in\bar{\mathbf{J}}$, and
$\mathbf{i}\in\mathbf{I}_{\mathbf{j}}$, we have that
$\mathds{B}_{t}(\mathbf{i},\mathbf{j})\geq 0$, where
$\mathds{B}_{T}(\mathbf{i},\mathbf{j})=\mathds{B}^{T}(\mathbf{i},\mathbf{j})$
and

$\mathds{B}_{t}(\mathbf{i},\mathbf{j})=\sum_{j_{t}^{\prime}:B^{t}_{j_{t}}\subseteq
B^{t}_{j_{t}^{\prime}}}(-1)^{\left\lvert B^{t}_{j^{\prime}_{t}}\setminus
B^{t}_{j_{t}}\right\rvert}\mathds{B}_{t+1}(\mathbf{i},\mathbf{j^{\prime}})$

for all $t\in\mathcal{T}\setminus\\{T\\}$.

Now we establish the following result.

######  Corollary 3.

If $\rho$ is consistent with DRUM, then $\bar{\rho}$ satisfies the DRUM-BM
inequalities.

The DRUM-BM inequalities are as intuitive as the BM inequalities for the
static case but they are not enough for DRUM. This is because these conditions
interact with the hierarchical theory extensions for the nonunique DRUM case,
giving rise to emergent conditions.

## 5\. Relationship with the Samuelson-Afriat and the McFadden-Richter
frameworks

In this section, we study the implications of DRUM for time series and cross-
sections. First, we look at a time series with the assumption of constant
utility across time periods as in the Samuelson-Afriat framework. In this
case, DRUM implies that the (deterministic) Strong Axiom of Revealed
Preference (SARP) has to hold in time series. Second, we study cross-sections,
like the ones described in McFadden and Richter (1990) and McFadden (2005)
that are obtained by _marginalizing_ or _pooling_ panels. Marginalization and
pooling correspond to the empirical practices of using cross-sections that
correspond to one or many time periods, respectively. We show that if $\rho$
is consistent with DRUM, then any marginal distribution derived from it is
rationalizable by RUM. At the same time, not every DRUM-consistent panel is
RUM-rationalizable when pooled. Importantly, marginal consistency with RUM is
not sufficient for consistency with DRUM.

###  5.1. The Samuelson-Afriat framework

DRUM has no testable implications for a time series without further
restrictions. That is, if we observe $\rho_{\mathbf{j}}$ for a single budget
path $\mathbf{j}$, then there are no testable restrictions of DRUM. (We need
at least two observed budget paths to test DRUM.) However, in the Samuelson-
Afriat framework, one needs only a time-series of choices from budgets in
order to test utility maximization. The reason for this is that the Samuelson-
Afriat framework makes an additional assumption on the stochastic process,
namely, that $\mu$ is such that $u^{t}=u^{s}$ $\mu-\mathrm{a.s.}$ for all
$t,s\in\mathcal{T}$. We call this restriction _constancy_ of the stochastic
utility process. Under this restriction, the testable implications of DRUM in
a time series are re-established. We need some preliminaries to formalize this
intuition. To simplify the exposition, all the results in this section are for
the demand setup.

######  Definition 15 (Strong Axiom of Revealed Path Dominance, SARPD).

For a given $\mathbf{j}\in\mathbf{J}$, we say that
$\rho_{\mathbf{j}}=(\rho(x_{\mathbf{i}|\mathbf{j}}))_{\mathbf{i}\in\mathbf{I}_{\mathbf{j}}}$
satisfies SARPD if

$\rho\left(x_{\mathbf{i}|\mathbf{j}}\right)=0$

whenever there is a finite set of patches from $x_{\mathbf{i}|\mathbf{j}}$,
$\left\\{x^{t_{n}}_{i_{t_{n}}|j_{t_{n}}}\right\\}_{n=1}^{N}$, such that
$x^{t_{1}}_{i_{t_{1}}|j_{t_{1}}}\succeq^{*}x^{t_{2}}_{i_{t_{2}}|j_{t_{2}}}\succeq^{*}\dots\succeq^{*}x^{t_{N}}_{i_{t_{N}}|j_{t_{N}}}$
and
$x^{t_{N}}_{i_{t_{N}}|j_{t_{N}}}\succeq^{*}x^{t_{1}}_{i_{t_{1}}|j_{t_{1}}}$(where
$x^{t}_{i_{t}|j_{t}}\succeq^{*}x^{s}_{i_{s}|j_{s}}$ whenever there are $x\in
x^{t}_{i_{t}|j_{t}}$ and $y\in x^{s}_{i_{s}|j_{s}}$ such that
$p_{j_{t}}^{\prime}(x-y)\geq 0$).

SARPD requires that the probability of observing a choice path that contains
consumption bundles that form a revealed preference cycle is zero. It is
analogous to the Strong Axiom of Revealed Preferences (SARP) in the Samuelson-
Afriat framework. Using SARPD, we can establish the following result:

######  Proposition 2.

If $\rho$ is rationalized by DRUM with $\mu$ that satisfies constancy, then
$\rho_{\mathbf{j}}$ satisfies SARPD for all $\mathbf{j}\in\mathbf{J}$.

Note that DRUM bounds above the probability of choice paths that contain a
revealed preference cycle. To see this, we consider again the simple setup
with $T=2$. There are two choice paths that contain a revealed preference
cycle: $\left(x_{1|2}^{1},x_{2|1}^{2}\right)$ and
$\left(x_{2|1}^{1},x_{1|2}^{2}\right)$. We focus on the first choice path
without loss of generality. Using $\mathrm{D}$-monotonicity, we know that

$\rho\left(\left(x_{1|2}^{1},x_{2|1}^{2}\right)\right)\leq\rho\left(\left(x_{1|1}^{1},x_{2|2}^{2}\right)\right).$

This means that the probability of a choice path that contains a violation of
SARP (i.e., it contains a revealed preference cycle), is bounded above by the
probability of a choice path that contains no such cycles. That is, DRUM
meaningfully restricts the probability of choice paths with revealed
preference cycles. This _endogenous_ bound on the probability of a choice path
that contains a revealed preference cycle has an important advantage with
respect to measures of deviations from rationality like the Critical Cost
Efficiency Index (Afriat, 1973). Indeed, in the literature that uses this
index, it is an open question how to set a threshold below which the level of
deviations from static utility maximization is deemed reasonable. In our
setup, we convert this problem into a population one and then bound
endogenously the fraction of consumers that have choices that involve revealed
preference cycles. Importantly, notice that if $\rho$ is degenerate taking
values on $\\{0,1\\}$ for a given budget path then $\mathrm{D}$-monotonicity
is equivalent to the Weak Axiom of Revealed Preference by Samuelson (1938) in
the Samuelson-Afriat framework. To see this, note that the probability of
choice paths with revealed preference cycles of size $2$ (i.e., violations of
the weak axiom) under the degeneracy of $\rho$ must be zero.

###  5.2. Marginal and Conditional Distributions

Given a budget path $\mathbf{j}$, let $\rho^{\textrm{c}}_{t,\mathbf{j}}$ and
$\rho^{\textrm{m}}_{t,\mathbf{j}}$ be the conditional and the marginal
distributions, respectively, over patches implied by $\rho_{\mathbf{j}}$. That
is,

$\displaystyle\rho^{\textrm{c}}_{t,\mathbf{j}}\left(x_{\mathbf{i}|\mathbf{j}}\right)$
$\displaystyle=\dfrac{\rho\left(x_{\mathbf{i}|\mathbf{j}}\right)}{\sum_{i\in\mathcal{I}^{t}_{j_{t}}}\rho\left(x_{\mathbf{i}|\mathbf{j}}\right)},$
$\displaystyle\rho^{\textrm{m}}_{t,\mathbf{j}}\left(x_{i_{t}|j_{t}}\right)$
$\displaystyle=\sum_{\tau\in\mathcal{T}\setminus\\{t\\}}\sum_{i\in\mathcal{I}^{\tau}_{j_{\tau}}}\rho\left(x_{\mathbf{i}|\mathbf{j}}\right),$

where the conditional distribution is defined only when
$\sum_{i\in\mathcal{I}^{t}_{j_{t}}}\rho\left(x_{i_{t}|j_{t}}\right)\neq 0$.
Given the marginal distribution of a budget path, we can also define the
slicing distribution as

$\rho^{\textrm{s}}_{t}\left(x_{i_{t}|j_{t}}\right)=\sum_{\mathbf{j}\in\mathbf{J}}\rho^{\textrm{m}}_{t,\mathbf{j}}\left(x_{i_{t}|j_{t}}\right)F(\mathbf{j}|j_{t}),$

where $F(\mathbf{j}|j_{t})$ is the conditional probability of observing the
budget path $\mathbf{j}$ conditional on the $t$-th budget being $j_{t}$ in the
data. The slicing distribution is a mixture of marginal distributions. It
corresponds to the situation in which the researcher only focuses on one
cross-section.

######  Proposition 3.

If $\rho$ is rationalized by DRUM, then $\rho^{\textrm{c}}_{t,\mathbf{j}}$,
$\rho^{\textrm{m}}_{t,\mathbf{j}}$, and $\rho^{\textrm{s}}_{t}$ are
rationalized by RUM for any $t\in\mathcal{T}$ and $\mathbf{j}\in\mathbf{J}$.

Proposition 3 means that if $\rho$ is consistent with DRUM, then the data are
consistent with RUM in any given cross-section (i.e., slice). In this sense,
the empirical implications of DRUM when an analyst has access to only a slice
of choices is the same as the empirical implications of RUM. However,
consistency of the marginal or slicing distributions does not exhaust the
empirical content of DRUM. This is illustrated in Example 5.

######  Example 5.

[Marginals are consistent with WASRP but not rationalized by DRUM] Consider
$\rho$ as presented in Table 10. This $\rho$ violates stability and
$\mathrm{D}$-monotonicity. So DRUM cannot possibly explain it. At the same
time its marginal probabilities at $t=1$ are consistent with the WASRP:
$\rho^{\textrm{m}}_{1,(2,1)}\left(x^{1}_{1|2}\right)=\frac{1}{2}$,
$\rho^{\textrm{m}}_{1,(1,1)}\left(x^{1}_{2|1}\right)=\frac{1}{2}$; and
$\rho^{\textrm{m}}_{1,(2,2)}\left(x^{1}_{1|2}\right)=\frac{2}{3}$,
$\rho^{\textrm{m}}_{1,(1,2)}\left(x^{1}_{2|1}\right)=\frac{1}{3}$. Thus, each
of these marginal distributions is consistent with RUM.282828Recall that WASRP
is the necessary and sufficient condition for marginal probabilities to be
rationalized by RUM in the sense of Proposition 3. Moreover, the slicing
distribution would satisfy
$\rho^{\textrm{s}}_{1}\left(x^{1}_{2|1}\right)=F((1,1)|1)\frac{1}{2}+F((1,2)|1)\frac{1}{3}$
and
$\rho^{\textrm{s}}_{1}\left(x^{1}_{1|2}\right)=F((2,1)|2)\frac{1}{2}+F((2,2)|2)\frac{2}{3}$.
As a result, depending on $F$,

$\rho^{\textrm{s}}_{1}\left(x^{1}_{1|2}\right)+\rho^{\textrm{s}}_{1}\left(x^{1}_{2|1}\right)\in\left[5/6,7/6\right].$

Thus, if, for example, all budget paths are observed with equal conditional
probabilities, then
$\rho^{\textrm{s}}_{1}\left(x^{1}_{1|2}\right)+\rho^{\textrm{s}}_{1}\left(x^{1}_{2|1}\right)=1$.
The slicing distribution, therefore, is also consistent with RUM.

| $x^{2}_{1|1}$ | $x^{2}_{2|1}$ | $x^{2}_{1|2}$ | $x^{2}_{2|2}$
---|---|---|---|---
$x^{1}_{1|1}$ | 1/6 | 1/3 | 2/3 | -
$x^{1}_{2|1}$ | 1/3 | 1/6 | 1/6 | 1/6
$x^{1}_{1|2}$ | 1/6 | 1/3 | 2/3 | -
$x^{1}_{2|2}$ | 1/3 | 1/6 | 1/6 | 1/6
Table 10: Matrix representation of $\rho$ that is consistent with RUM after
slicing, but is not consistent with DRUM

###  5.3. Pooling

In practice, and in the absence of panel variation, several years or time
periods of choices from budgets are pooled before testing for consistency with
RUM (KS,Deb et al., 2021). Here we explore a potential pitfall of this
practice. We show that when a panel dataset that is consistent with DRUM is
pooled, it may not be consistent with RUM. The spurious rejection of
rationality may be driven by the fact that pooling requires us to ignore time
labels and imposes the restriction that the distribution of preferences is
independent across time.

First, we formally define _pooling_. To simplify the exposition, assume that
$B_{j}^{*,t}\neq B_{j^{\prime}}^{*,t^{\prime}}$ for all
$t,t^{\prime}\in\mathcal{T}$, $j\in\mathcal{J}^{t}$, and
$j^{\prime}\in\mathcal{J}^{t^{\prime}}$. That is, there are no repeated
budgets across time and agents. Let $\mathcal{J}=\\{1,2,\dots,J\\}$, where
$J=\sum_{t\in\mathcal{T}}J^{t}$ is the total number of budgets.

######  Definition 16 (Pooled Patches).

Let
$\mathcal{X}=\bigcup_{t\in\mathcal{T}}\bigcup_{j\in\mathcal{J}^{t}}\\{\xi^{t}_{k|j}\\}$
be the coarsest partition of
$\bigcup_{t\in\mathcal{T}}\bigcup_{j\in\mathcal{J}^{t}}B^{*,t}_{j}$ such that
$\xi^{t}_{k|j}\bigcap B^{*,t}_{j^{\prime}}\in\\{\xi^{t}_{k|j},\emptyset\\}$
for any $j,j^{\prime}$, and $k$.

The pooled patches $\\{\xi^{t}_{k|j}\\}$ partition every $x^{t}_{i|j}$ since
$B^{*,t}_{j}$ may now intersect with budgets from different periods (see
Figure 3).

$y_{1}$$y_{2}$$x^{1}_{1|1}$$B^{*,1}_{1}$

$y_{1}$$y_{2}$$x^{2}_{1|1}$$B^{*,2}_{1}$

$y_{1}$$y_{2}$$\xi^{2}_{1|1}$$\xi^{1}_{2|1}$$\xi^{1}_{1|1}$$\xi^{2}_{2|1}$$B^{*,1}_{1}$$B^{*,2}_{1}$

Figure 3: $K=2$ goods, $T=2$ time periods, one budget per time period. The
first and the second picture depict patches in 2 time periods. The third
picture depicts new patches that arise after pooling the data.

Given these new patches, we can define the pooled stochastic function
$\rho^{\text{pool}}\left(\xi^{t}_{k|j}\right)$ as the probability of observing
someone picking from patch $\xi^{t}_{k|j}$. Next, we construct an example in
which $\rho$ is rationalizable by DRUM but the corresponding
$\rho^{\text{pool}}$ is not consistent with RUM (in the sense of Proposition
3). Consider the setting with $K=2$ goods and $T=2$ time periods. In each time
period $t$, there is only one budget $B_{1}^{*,t}$. Assume that
$B_{1}^{*,1}\neq B_{1}^{*,2}$ and $B_{1}^{*,1}\cup B_{1}^{*,2}\neq\emptyset$
(see Figure 3). Given that there is no budget variation for any given time
period, there is only one choice path $\left(x^{1}_{1|1},x^{2}_{1|1}\right)$.
Thus, the trivial $\rho\left(\left(x^{1}_{1|1},x^{2}_{1|1}\right)\right)=1$ is
rationalizable by DRUM. After pooling, since the budgets overlap, there are
four patches (we assume that there is no intersection patch). Since there is
only one choice path, DRUM does not impose any restrictions on the choice of
individuals on these two budgets. As a result, we can take $\nu^{1}$ and
$\nu^{2}$ from the DRUM definition such that
$\rho^{\text{pool}}\left(\xi^{2}_{1|1}\right)+\rho^{\text{pool}}\left(\xi^{1}_{2|1}\right)>1$.
This $\rho^{\text{pool}}$ cannot be consistent with RUM.

## 6\. Counterfactuals in the Demand Setup

This section shows how to conduct sharp counterfactual analyses within our
framework in the demand setup.292929 _Sharpness_ in this setting means that we
can compute the tightest set of parameters that are consistent with the
observed data and the model. The sharpness of our results follows from the
fact that we have a full characterization of DRUM.303030For early connections
between nonparametric counterfactuals and specification testing, see Varian
(1982, 1984). See Blundell et al. (2008), Norets and Tang (2014), Blundell et
al. (2014), Allen and Rehbeck (2019), Aguiar and Kashaev (2021), and Aguiar et
al. (2022) for recent examples in the analysis of demand, dynamic binary
choice, and production. To simplify the exposition, we focus on the simple
setup (i.e., two intersecting budgets per period) for which we possess the
$\mathcal{H}$-representation.

Given $\rho$ in the time window $\mathcal{T}$, we want to bound some known
function of counterfactual stochastic demands at the counterfactual time
$T+1$. We assume that consumers face a pair of prices $p_{1,T+1}$ and
$p_{2,T+1}$ that are known to the analyst at $T+1$. Let income in each period
be $1$. Denote the extended time window by
$\mathcal{T}^{\mathrm{c}}=\mathcal{T}\cup\\{T+1\\}$. Similarly, the extended
set of budget paths is denoted by $\mathbf{J}^{\mathrm{c}}$, and the extended
vector representation of stochastic demand is denoted by $\rho^{\mathrm{c}}$.

Let $y^{\mathrm{c}}_{j_{T+1}}$ denote the counterfactual random demand of a
consumer facing budget $j_{T+1}$ at time $T+1$. That is,

$y_{j_{T+1}}^{\mathrm{c}}=\operatorname*{arg\,max}_{y\in
B_{j_{T+1}}^{T+1}}u^{T+1}(y),$

where $u^{T+1}$ is a random utility function at time $T+1$.

######  Definition 17 (Counterfactual marginal and conditional demands).

Given $\rho$, $x_{\mathbf{i}|\mathbf{j}}$, and budget
$j_{T+1}\in\mathcal{J}^{T+1}$, the counterfactual conditional and marginal
demands $\rho^{*}\left(\cdot|j_{T+1},x_{\mathbf{i}|\mathbf{j}}\right)$ and
$\rho^{**}\left(\cdot|j_{T+1}\right)$ are distributions over patches of
$j_{T+1}$ such that

$\displaystyle\rho_{j_{T+1}}^{*}\left(x^{T+1}_{i_{T+1}|j_{T+1}}|x_{\mathbf{i}|\mathbf{j}}\right)$
$\displaystyle=\rho^{\mathrm{c}}\left(\left(x^{t}_{i_{t}|j_{t}}\right)_{t\in\mathcal{T}^{\mathrm{c}}}\right)\Big{/}\rho\left(x_{\mathbf{i}|\mathbf{j}}\right),$
$\displaystyle\rho_{j_{T+1}}^{**}\left(x^{T+1}_{i_{T+1}|j_{T+1}}\right)$
$\displaystyle=\sum_{x_{\mathbf{i}|\mathbf{j}}}\rho^{\mathrm{c}}\left(\left(x^{t}_{i_{t}|j_{t}}\right)_{t\in\mathcal{T}^{\mathrm{c}}}\right)$

for any $\rho^{\mathrm{c}}$ that satisfies $\mathrm{D}$-monotonicity and
stability, and is such that

$\rho\left(x_{\mathbf{i}|\mathbf{j}}\right)=\sum_{i_{T+1}\in\mathcal{I}^{T+1}_{j_{T+1}}}\rho^{\mathrm{c}}\left(\left(x^{t}_{i_{t}|j_{t}}\right)_{t\in\mathcal{T}^{\mathrm{c}}}\right).$

The counterfactual conditional and marginal distributions fully characterize
the choices of consumers in counterfactual situations, thus allowing us to
compute sharp bounds for the expectation of any function of $y^{\mathrm{c}}$.
For a given measurable function $g:X\to{\mathds{R}}$, let

$\displaystyle\underline{g}\left(x^{t}_{i_{t}|j_{t}}\right)=\inf_{y\in
x^{t}_{i_{t}|j_{t}}}g(y),\quad\quad\overline{g}\left(x^{t}_{i_{t}|j_{t}}\right)=\sup_{y\in
x^{t}_{i_{t}|j_{t}}}g(y).$

be the smallest and the largest values, respectively, $g$ can take over the
patch $x^{t}_{i_{t}|j_{t}}$.

######  Proposition 4.

Given $\rho$, $x_{\mathbf{i}|\mathbf{j}}$, and budget
$j_{T+1}\in\mathcal{J}^{T+1}$,

$\displaystyle{\scriptstyle\inf_{\rho_{j_{T+1}}^{*}}\sum_{i\in\mathcal{I}^{T+1}_{j_{T+1}}}\rho_{j_{T+1}}^{*}\left(x^{T+1}_{i|j_{T+1}}|x_{\mathbf{i}|\mathbf{j}}\right)\underline{g}\left(x^{T+1}_{i|j_{T+1}}\right)}$
$\displaystyle{\scriptstyle\leq\mathds{E}\left[g\left(y_{j_{T+1}}^{\mathrm{c}}\right)|x_{\mathbf{i}|\mathbf{j}}\right]\leq\sup_{\rho_{j_{T+1}}^{*}}\sum_{i\in\mathcal{I}^{T+1}_{j_{T+1}}}\rho_{j_{T+1}}^{*}\left(x^{T+1}_{i|j_{T+1}}|x_{\mathbf{i}|\mathbf{j}}\right)\overline{g}\left(x^{T+1}_{i|j_{T+1}}\right)},$
$\displaystyle{\scriptstyle\inf_{\rho_{j_{T+1}}^{**}}\sum_{i\in\mathcal{I}^{T+1}_{j_{T+1}}}\rho_{j_{T+1}}^{**}\left(x^{T+1}_{i|j_{T+1}}\right)\underline{g}\left(x^{T+1}_{i|j_{T+1}}\right)}$
$\displaystyle{\scriptstyle\leq\mathds{E}\left[g\left(y_{j_{T+1}}^{\mathrm{c}}\right)\right]\leq\sup_{\rho_{j_{T+1}}^{**}}\sum_{i\in\mathcal{I}^{T+1}_{j_{T+1}}}\rho_{j_{T+1}}^{**}\left(x^{T+1}_{i|j_{T+1}}\right)\overline{g}\left(x^{T+1}_{i|j_{T+1}}\right),}$

where the infimum and supremum are taken over all possible counterfactual
marginal and conditional distributions.

Note that our results are complementary to those of Kitamura and Stoye (2019)
that predict counterfactual stochastic demand for a new budget in a given
cross-section using static RUM. We can use their techniques here as well. The
main difference from their work is that we focus on the counterfactual
prediction in the time-dimension allowing dynamic preference change.

## 7\. Empirical Application: Binary Menus of Lotteries

We study a sample of experimental subjects studied in Aguiar et al. (2023)
that was surveyed in the MTurk platform between August $25$, $2018$ and
September $17$, $2018$. We find evidence that deterministic rationality fails
to explain the totality of the sample behavior, yet consistency with DRUM
cannot be rejected. Moreover, restricting the linear orders in DRUM to those
consistent with Expected Utility (Frick et al., 2019) cannot explain the
population behavior.

The sample contains $2135$ DMs. The grand choice set is the same across time
$X^{t}=\\{l_{1},l_{2},l_{3}\\}$. These lotteries are defined over the set of
prizes $Z=\\{0,10,30,50\\}$ in tokens. The lotteries are
$l_{1}=(1/2,0,0,0,1/2)$, $l_{2}=(0,1/2,1/2,0)$ and
$l_{3}=\frac{1}{2}l_{1}+\frac{1}{2}l_{2}$. The binary menus are
$\\{l_{1},l_{2}\\},\\{l_{1},l_{3}\\}$, and $\\{l_{2},l_{3}\\}$ at every
$t\in\mathcal{T}$. In the experiment, consumers face one of these three menus
uniformly at random and in uniformly random order. No DM faces the same binary
menu twice. That means that there are six menu paths. Note that the design
ensures that the probability of facing any of the six menu paths is uniform
across DMs. Payments are made at random for one of the choices made by the
DMs. (For details of the payment, recruitment, and sample demographics see
Aguiar et al., 2023.)

Concerns about limited consideration are not first-order here because these
binary menus were shown to DMs under a low-cost-of-attention treatment (see
Aguiar et al., 2023 for details). In addition, DMs already faced a task
focused on varying the attention cost, as described in Aguiar et al. (2023),
before facing the binary-menu task. This means DMs are familiar with the
lotteries when they face the binary menus. In addition, we do not have
concerns about measurement error due to the experimental design and discrete
choice nature of the data. This means we can focus on testing deterministic
rationality at the individual level versus DRUM without confounding due to
limited consideration and measurement error.

###

We test deterministic rationality at the individual level. In particular, we
check the Strong Axiom of Revealed Preference (SARP) which implies that it
cannot be the case that $l_{i}\succ^{*}l_{i}^{\prime}$,
$l_{i}^{\prime}\succ^{*}l_{i}^{\prime\prime}$, and
$l_{i}^{\prime\prime}\succ^{*}l_{i}$ with $l_{i}\succ^{*}l_{i}^{\prime}$
defined as $l_{i}$ is picked out of $\\{l_{i},l_{i}^{\prime}\\}$). We observe
that $92$ percent of DMs are consistent with SARP in our sample. Given that
the power of this experiment to detect violations of SARP is low, we consider
a $8$ percent rejection rate to represent a significant fraction of DMs.
Formally, the whole sample of DMs is not consistent with (static) utility
maximization.

###

We compute the sample analogue of $\rho$, $\hat{\rho}$, following the
methodology described in KS and Aguiar et al. (2023). We implement the
statistical test described in KS and Aguiar et al. (2023) to test the null
hypothesis that $\rho=A\nu$ for $\nu\geq 0$. $A$ is computed in two steps.
First, we compute the matrix $A^{t}$ whose columns correspond to all possible
linear orders on $X^{t}$ and whose rows correspond to all possible lotteries
from each menu. Observe that $A^{t}$ is a square matrix of dimension $6$ for
all $t\in\mathcal{T}$ such that $A_{T}=\otimes_{t=1}^{3}A^{t}$. In this
application, we do not observe all possible choice paths since, by design, DMs
never see repeated binary menus. Thus, $A$ is obtained by extracting the
submatrix of $A_{T}$ that corresponds to observed menu paths ($6$ menu paths
out of $27$). We cannot reject the null hypothesis of consistency with DRUM
($p$-value is $0.72$).313131The number of bootstrap samples used in every test
is 999. We choose the tuning parameter $\tau_{N}$ following KS. Our finding
confirms that a sample can be consistent with DRUM even when a significant
fraction of DMs is inconsistent with static utility maximization. We believe
our results are compatible with the findings of Kurtz-David et al. (2019)
which document that DMs fail deterministic rationality (in a different choice
domain) because they make mistakes in evaluating the utility of lotteries.
These mistakes can be interpreted as a form of dynamic random taste shocks.
Indeed, DMs may get fatigued or learn, so their mistakes are correlated in
time but with draws from the same distribution across choice paths, which are
hence consistent with DRUM. To alleviate the concerns about the finite sample
power of our test, we perform Monte Carlo experiments mimicking the setup of
this application and find evidence that our test has high power even in small
samples.

###

Kashaev and Aguiar (2022b) propose a methodology to test for the null
hypothesis of consistency with the Expected Utility version of RUM. Their
approach restricts the set of preference orders in the static test of RUM to
those that are further consistent with expected utility maximization. We can
apply the same idea to our dynamic setting to test the null hypothesis of
consistency with the Expected Utility version of DRUM. That is, we test DRUM
with the additional restriction that the set of utilities is consistent with
Expected Utility. We reject the null hypothesis ($p$-value is less than
$0.001$).

## 8\. Related Literature

DRUM was first introduced in Strzalecki (2021) for abstract discrete choice
domains. Frick et al. (2019) offered an axiomatic characterization with
decision trees and expected utility restrictions on stochastic utility
processes.323232Applying techniques from Kashaev and Aguiar (2022b), we
provide a KS-type characterization for special DRUM cases with expected
utility restrictions. However, as far as we know, our work is the first to
develop a BM-like characterization for DRUM. In finite abstract discrete
choice spaces, two partial characterizations exist when the primitive is the
joint distribution of choices across time and total menu variation.

Li (2021) provides an BM-like axiomatic DRUM characterization for any finite
number of periods, and full menu variation, but with no more than three
alternatives. Chambers et al. (2021) considers correlated choice: a joint
distribution of choice on a pair of menus faced by two different DMs or a
group. This model is mathematically equivalent to DRUM in the abstract domain,
characterizing DRUM for a special case of two periods, an abstract and finite
choice set, but with a uniqueness property for one DM’s choice (i.e., one
period has uniquely identified RUM). Our work subsumes and generalizes both Li
(2021) and Chambers et al. (2021). Moreover, our results go beyond both Li
(2021) and Chambers et al. (2021), and our general setup also includes
classical consumer choice domains with primitive orders, binary menus, and
general limited menu variation. Our DRUM respects this primitive order and
restricts utilities to be monotonic, while addressing limited observability of
menus and menu paths.

Our work also contributes to the random exponential discounting literature, as
in Browning (1989), by generalizing Deb et al. (2021)’s demand setup to a
dynamic context. Apesteguia et al. (2022) introduces a heterogeneous model
with exponential discounting and time separability, differing in choice
domains and being semiparametric, while our setup is nonparametric. Lu and
Saito (2018) examines exponential discounting with random discount factors and
stochastic choices over consumption streams in the first period.

Aguiar and Kashaev (2021) studies panel setups with first-order-conditions
approaches for some dynamic preferences, allowing for measurement error.
However, their setup doesn’t accommodate changing utility beyond discount
factors or marginal utility of income. Im and Rehbeck (2021) investigates the
McFadden-Richter framework and panel structure limitations, suggesting
individual static rationality checks, like the Samuelson-Afriat framework. We
generalize the Samuelson-Afriat framework, allowing utility changes over time
while exploiting panel structure for more empirical implications.

DRUM allows for consumption to be correlated in time, similar to rational
addiction in Becker and Murphy (1988) (henceforth, Habits as Durables–HAD).
However, the consumption correlation in DRUM is due solely to preference
correlation over time. Demuynck and Verriest (2013) provides a
characterization of the short-memory habits model of Becker and Murphy (1988)
which is similar to Afriat (1967)’s theorem, and applies it to a panel of
Spanish households’ choices. However, the HAD pass-rate is slightly above $50$
percent, subject to the same critique made for the traditional utility
maximization problem. This revealed-preference test of HAD imposes no
parametric restrictions on utilities besides monotonicity and concavity,
making it more general than parametric structural work on HAD. Theoretically,
the relationship between DRUM and HAD remains an open question since a RUM-
like characterization of a stochastic generalization of HAD is unavailable.

Dynamic Discrete Choice (DDC) models, surveyed in Aguirregabiria and Mira
(2010), are the most popular approach in studying discrete choice with
attribute variation. Frick et al. (2019) thoroughly explores the relationship
between DRUM and DDC in their domain. Much of this analysis carries over to
our domain. Specifically, in our setup with exogenously given menu paths, DDC
is nested by DRUM. However, our focus differs from most work on DDC in that we
emphasize nonparametric utilities, unrestricted heterogeneity, comparative
statics, and counterfactual predictions, rather than identification and
estimation in parametric settings. No axiomatization of DDC has yet been
arrived at, to our knowledge, except for the special case of independent and
identically distributed logit shocks in Fudenberg and Strzalecki (2015), so we
cannot fully compare DDC with DRUM.

## 9\. Conclusion

In this paper, we have fully characterized the Dynamic Random Utility Model
(DRUM), a new model of consumer behavior in which we observe a panel of
choices from budget paths. In contrast to the static utility maximization
framework, DRUM does not require the assumption that decision makers or
consumers keep their preferences stable over time. This generality is
essential because the static utility maximization framework often fails to
explain the behavior of individuals.

Our characterization works for any finite collection of choice paths in any
finite time window. The characterization can be applied directly to existing
panel consumption datasets using the statistical tools in Kitamura and Stoye
(2018). Moreover, our simple setup characterization showcases the fact that
DRUM implies a richer set of behavioral restrictions on the panel of choices
than does the Random Utility Model, alleviating some concerns about the
empirical bite of the latter in a richer domain. These features position DRUM
as in-between the Samuelson-Afriat framework and the McFadden-Richter
framework as a result of combining their strengths and reducing their
weaknesses.

We have also introduced to economics a generalization of the Weyl-Minkowski
theorem for cones. This result is the basis of a recursive characterization of
DRUM in the demand and abstract choice setups. This new mathematical result
will be helpful beyond DRUM to obtain analogous generalizations of bounded
rational models of stochastic choice.

## References

  * Adams et al. (2015) Adams, A., Blundell, R., Browning, M. and Crawford, I. (2015). Prices versus preferences: taste change and revealed preference. Tech. rep., IFS Working Papers.