# Irrational random utility models∗

Daniele Caliari† & Henrik Petri‡

###### Abstract.

We show that the set of aggregate choices of a population of rational
decision-makers - random utility models (RUMs) - can be represented by a
population of irrational ones if, and only if, their preferences are
sufficiently uncorrelated. We call this representation: Irrational RUM. We
then show that almost all RUMs can be represented by a population in which at
least some decision-makers are irrational and that under specific conditions
their irrational behavior is unconstrained.

$\dagger$ WZB Berlin. E-mail<EMAIL_ADDRESS>
$\ddagger$ University of Gothenburg. E-mail<EMAIL_ADDRESS>
∗ We would like to thank the following for helpful comments and suggestions:
Dilip Ravindran, Jose Apesteguia, Victor Aguiar, Tilman Fries, Marco Mariotti,
Jörg Stoye, and Ferdinand Vieider.

Keywords: Stochastic choice, random utility models, rationality.

JEL Classification: D00, D90, D91.

##  1\. Introduction

The random utility model (RUM) is the most renowned model of stochastic choice
and it is the leading notion of stochastic rationality in economics (McFadden
& Richter (1990), McFadden (2006)): choice probabilities satisfy the RUM
hypothesis if they result from the aggregation of the choices of rational -
preference maximizing - decision-makers. These aggregate choices are normally
referred to as _stochastically rational_.

A starting point of the present paper is the observation that aggregate
choices may be stochastically rational even if all decision-makers are
irrational, i.e. they violate the tenets of rationality.111In this paper,
rationality equates the maximization of transitive and complete preferences
which is equivalent to both Sen’s property $\alpha$ and the weak axiom of
revealed preference (WARP); see Arrow (1959) and Sen (1971). The intuition
behind this observation is simple. If the irrationalities of decision-makers
are sufficiently uncorrelated, they will cancel out, and aggregate data will
appear _as if_ stochastically rational.222This intuition is not new. Becker
(1962) famously pointed to it while Grandmont (1992) noticed that ”Wald,
Hicks, Arrow, Hahn, and quite a few others” conjectured that enough
”heterogeneity” should yield a nicely behaved aggregate demand. We call the
subset of RUMs that display this property ” _irrational_ ” [I-RUM].

Our first main result is a characterization of I-RUMs using bounds on the
”correlation” between the preferences of the rational decision-makers that may
represent the RUM. We build on the intuition above and show that a population
of rational decision-makers is also represented by a population of irrational
ones whenever their preferences are sufficiently uncorrelated. Our second main
result focuses on the RUMs that can be represented by a population in which at
least some decision-makers are irrational, we call them ” _partially
irrational_ ” [pI-RUM]. We show that almost all RUMs are partially irrational,
and in fact, that all RUMs with full support (e.g. Logit model) are.333A RUM
with full support has strictly positive probability mass on every preference.
In a nutshell, considering irrational decision-makers can have severe
consequences for the interpretation of stochastically rational data, in fact,
_so severe as to undermine the very (rational) essence of the RUM hypothesis._

A relevant implication of our results relates to the falsifiability of the
rational interpretation of the RUM hypothesis; i.e. even if rationality is
violated by all decision-makers the RUM hypothesis may fail to be
rejected.444The problem of falsifiability of the rationality hypothesis was
noticed, within a theory of demand framework, by Blundell et al. (2003):
”revealed preference tests are unlikely to reject the integrability conditions
[_aggregate rationality_] for such data but it is not clear that we would wish
to characterize them as the outcome of a ’rational’ procedure.” In the second
part of the paper, we discuss this interpretation in more detail by focusing
on some properties of pi-RUMs. First, we show that, under the RUM hypothesis,
there is always a fraction of decision-makers whose behavior is unconstrained,
i.e. for this fraction of decision-makers rationality can be violated in the
most extreme sense (while aggregate data is still consistent with RUM). This
result is deliberately demanding from an irrationality perspective implying
that such a fraction of irrational decision-makers may be small. We,
therefore, go on to show that the fraction of irrational decision-makers that
are compatible with the RUM hypothesis is larger the less constrained is the
set of possible irrational behaviors. These results have important
consequences for empirical applications where irrational behavior is unlikely
to be extreme (i.e. more constrained) implying that the issue of the non-
falsifiability is likely to be widespread.

We illustrate our main results and intuitions through one simple example.

######  Example 1 (Irrational RUMs).

An analyst observes the aggregate choice probabilities in all non-empty menus
of a set of three alternatives - ”chicken” ($c$), ”steak” ($s$), and ”frogs’
legs” ($f$) - from a population where 1/2 of the decision-makers have
preference $c\hskip 1.49994ptP_{1}\hskip 1.49994pts\hskip 1.49994ptP_{1}\hskip
1.49994ptf$ and 1/2 have preference $s\hskip 1.49994ptP_{2}\hskip
1.49994ptc\hskip 1.49994ptP_{2}\hskip 1.49994ptf$. The first and second column
in the table below displays the choices of these rational decision-makers from
each menu with $P_{1}$’s choices highlighted in red and $P_{2}$’s in blue.

Menu | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\max(A,P_{1})}$ | $\max(A,P_{2})$ | $c_{1}(A)$ | $c_{2}(A)$
---|---|---|---|---
$csf$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}s}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}s}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c}$
$cs$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}s}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}s}$
$cf$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c}$
$sf$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}s}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}s}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}s}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}s}$

We now provide a different interpretation of the aggregate data from an
irrational perspective. Consider the Luce & Raiffa’s dinner example (Luce &
Raiffa, 1957) in which a decision-maker chooses chicken when only steak is
available, but switches to steak whenever also frogs’ legs are available even
if he dislikes frogs’ legs. Imagine now another decision-maker with the same
behavior but who chooses steak when only chicken is available and switches to
chicken in the presence of frogs’ legs. These well-known irrational behaviors
are summarized by the choices $c_{1}$ and $c_{2}$ in the table. Note that
$c_{1}$ and $c_{2}$ agree with $P_{1}$ except in one menu where choices are
obtained by ”switching” the choices of $P_{1}$ and $P_{2}$. A population in
which 1/2 of the decision-makers choose according to $c_{1}$ and $c_{2}$ will
induce the same aggregate data as the original population of rational
decision-makers.

Although the RUM in the above example has an irrational representation, if the
probability distribution on the preferences $P_{1}$ and $P_{2}$ is perturbed
only slightly then an irrational representation no longer exists. To see this,
suppose that 51% of the rational decision-makers have preference $P_{1}$ (so
prefer $c$ to $s$ and $f$). Then, in an irrational representation, 51% of the
irrational decision-makers must choose $c$ when $s$ is available. However, an
irrational decision-maker who chooses $c$ from the menu $csf$ must either
choose $s$ from $cs$ or $f$ from $cf$ (to induce a violation of rationality).
Since $51\%$ of the irrational decision-makers choose $c$ from $csf$ and $f$
is never chosen in $cf$, this implies that $51\%$ of the irrational decision
makers must choose $s$ when $c$ available! A contradiction. In our main
theorem, we characterize the exact conditions on the aggregate data under
which irrational representations exist.

Example 1 questions, at a fundamental level, the non-falsifiability of the
rational interpretation of the RUM hypothesis. A population of decision-makers
with a uniform distribution on $c_{1}$ and $c_{2}$ fails to reject the RUM
hypothesis even though all decision-makers are irrational. However, an
irrational representation exists only if the irrational behaviors (described
by $c_{1}$ and $c_{2}$) are sufficiently uncorrelated. Below, we modify the
opening example to show that the behaviors of irrational decision-makers can
instead be unrestricted (and highly correlated) and still be aligned with the
RUM hypothesis.

######  Example 1 (continued).

Consider a population where a fraction of decision-makers are rational and,
differently from above, have a uniform distribution on the set of preferences
on $\\{c,s,f\\}$; while the remaining ones are irrational. Simple calculations
show that whenever at least 6/7 of the decision-makers are rational then the
behavior of the remaining irrational ones is unconstrained w.r.t. stochastic
rationality.555To see this, note that regularity is a necessary and sufficient
condition for RUM with only three alternatives (Block & Marschak, 1960).
Suppose $\mu$ is the proportion of decision-makers who behave rationally with
a uniform distribution on the set of preferences, and $1-\mu$ are those who
behave irrationally. Assuming all irrational decision-makers make the same
mistake, regularity is satisfied whenever $1/2\mu-1/3\mu+\mu\geq 1$, or
equivalently, $\mu\geq 6/7$.

This bound may seem extreme at first glance, however, it assumes extreme
scenarios such as those in which every irrational decision-maker makes the
same mistake. To see how the bound changes with the set of potential
irrational behaviors, if $c_{1}$ is at most twice as likely to appear as
$c_{2}$, the fraction of irrational decision-makers compatible with the RUM
hypothesis increases from 1/7 to 1/3, while, as shown before, if $c_{1}$ and
$c_{2}$ are equally probable this fraction becomes one.666Let $\mu$ be the
proportion of decision-makers who switch to steak when frogs’ legs are
available. This implies that $1/2\mu$ decision-makers have the opposite
irrational behavior and $1-3/2\mu$ decision-makers are rational with uniform
preferences. The same inequality as in the previous footnote yields $\mu\leq
2/9$, or equivalently, a proportion of rational decision-makers greater or
equal to $2/3$.

###  1.1. Related Literature

Our study of irrational behavior within the RUM contributes to a longstanding
literature on stochastic choice. Most closely, our framework aligns with some
recent papers that also study probability distributions on choice functions
and their aggregation. Dardanoni et al. (2020) and Dardanoni et al. (2023)
refer to this approach as ”mixture choice functions” while Filiz-Ozbay &
Masatlioglu (2023) as ”random choice models”. In this paper, we will adopt
this second nomenclature. The focus of these papers is, however, different
from ours as they tackle the well-known identification issues of stochastic
choice models. Dardanoni et al. (2020) provide conditions under which an
analyst who only observes aggregate choice probabilities can identify
cognitive parameters under specific models and restrictive assumptions
(preference homogeneity or known distribution of preferences). Dardanoni et
al. (2023) refine these results by assuming that the analyst observes each
decision-maker’s choices and show that this allows the identification of both
cognitive parameters and preferences. Filiz-Ozbay & Masatlioglu (2023),
instead, study the aggregation of a population of decision-makers who are
possibly boundedly rational under the assumption that the collection of their
choices is progressive - a generalization to the well-known single-crossing
property (Apesteguia et al., 2017). The authors show that this restriction on
the set of choice functions allows the identification of the unique sources of
heterogeneity in the data.777See also Petri (2023) for a discussion of these
issues (in a setting of multivalued choice).

We also contribute to the longstanding literature on the characterizations of
the RUM (Block & Marschak (1960), Falmagne (1978), Barberá & Pattanaik (1986),
McFadden & Richter (1990), Gilboa (1990), Fiorini (2004), McFadden (2006),
Stoye (2019)), and its special cases such as single-crossing RUM (Apesteguia
et al., 2017), and Dual RUM (Manzini & Mariotti, 2018); as well as the study
of its non-identifiability issues (Turansick (2022), Suleymanov (2023)). More
specifically, our work is closely related to Manzini & Mariotti (2018) as Dual
RUMs will be crucial in our characterization of both I-RUMs and pI-RUMs, and
to Turansick (2022), Suleymanov (2023) as by considering irrational decision-
makers our results extend further the non-identifiability issues of the RUM.

Identifiability and falsifiability under the RUM hypothesis are tightly
connected issues as they both leverage the tension between aggregate and
individual behavior which is an old problem within the literature on the
theory of demand. It was first recognized by Becker (1962) who pointed out
that the aggregation of erratic consumers may lead to a well-behaved aggregate
demand function. The connection between heterogeneity and aggregate behavior
was then investigated more in-depth by Grandmont (1992) who, without relying
on individual rationality, provided sufficient conditions for the distribution
of individual demand functions that guarantee a well-behaved aggregation.
These issues have been long recognized within the literature of demand
estimation that often incorporates tests of individual rationality, e.g. see
Blundell et al. (2003). In this tradition, two relevant papers for us are
Hoderlein & Stoye (2014), and Hoderlein & Stoye (2015). The authors ask
broadly what knowledge of the ”joint distribution of demand”, i.e. individual
demand functions, the analyst may have by knowing only the marginal
distribution of demand, i.e. aggregate choice probabilities. Their setting is
therefore similar to ours and, to the best of our knowledge, they are the
first to introduce Fréchet bounds to bound the fraction of (ir)rational
consumers, i.e. they prove, within the context of the theory of demand, the
necessity of these bounds. Our results abstract from the framework of the
theory of demand to provide necessary and sufficient conditions for stochastic
rationality to coexist with individual irrationality. We believe our
abstraction could lead to the application of these intuitions to broader
contexts where different definitions of (ir)rationality may apply.

The view of our results from the lens of falsifiability of the rational
interpretation of the RUM hypothesis is instead relatively novel. Im & Rehbeck
(2022) focus on a restrictive setting of two consumption goods and two
observations to show that unless more than half of the population is
irrational, the population overall could be stochastically rationalizable.
They refer to this notion as ”false acceptance of stochastic
rationalizability”. In this paper, we generalize their observation by
characterizing the exact conditions under which stochastic rationalizable
choices have an irrational representation. In our final discussion section, we
borrow some of their observations to show, within the framework of the theory
of demand, that considering only rational representations of RUMs may lead to
welfare misjudgments.

Finally, our results contribute to the literature on tests of the RUM
hypothesis: Kitamura & Stoye (2018), McCausland et al. (2020), Aguiar &
Kashaev (2021), Smeulders et al. (2021), Deb et al. (2023). More specifically,
we discuss our results in the framework of Kitamura & Stoye (2018). The
authors provide a statistical test for the RUM hypothesis based on the
distance between the observed choice probabilities and the set of RUMs which
implies that whenever the distance is zero, i.e. the choice probabilities are
stochastically rational, the RUM hypothesis is accepted. This observation is a
direct consequence of the restriction of the set of choice functions to the
rational ones. In weakening this assumption, our results show that the sole
focus on aggregate choice probabilities often does not provide any evidence
about individual behavior and that if evidence of individual rationality
exists they often do not depend on whether aggregate choice probabilities are
rational or not.

The paper is organized as follows: in section 2, we introduce some preliminary
notions; in section 3 we define I-RUMs, pI-RUMs, and introduce our correlation
bounds; in section 4 we state and discuss our characterization theorems; in
section 5, we discuss the falsifiability of the rational interpretation of
RUMs; finally, in section 6, we discuss some paradoxical implications of our
results. In the appendix, we first discuss some preliminary notions and
auxiliary results (appendix A), then we prove our main theorem (appendix B)
and, finally, we provide other proofs that are omitted from the main text
(appendix C).

##  2\. Preliminaries

We denote by $X$ a finite set of alternatives with $|X|=N$. A subset
$A\subseteq X$ is called a _menu_.888With a little abuse of notation, we use
the multiplicative notation $ab$, $abc$ for menus $\\{a,b\\},\\{a,b,c\\}$. Let
$\mathcal{A}$ denote the collection of all nonempty menus of $X$ with
cardinality greater than two, and let $|\mathcal{A}|=K$.

The empirical primitive is a _stochastic choice function_ , i.e. a map
$\rho:X\times\mathcal{A}\to[0,1]$ such that i) $\sum_{a\in A}\rho(a,A)=1$ for
all $A\in\mathcal{A}$ and ii) $\rho(a,A)=0$ for all $a\in X\setminus A$.

Our decision-makers choose once from several menus. Their choices are
described by a _choice function_ , i.e. a map $c:\mathcal{A}\to X$ such that
$c(A)\in A$ for all $A\in\mathcal{A}$. We are particularly interested in
choices that result from the act of preference maximization. Let $P$ be a
strict linear order, henceforth ”preference”, we denote by $c_{P}$ a choice
function that is rationalizable by a preference and refer to it as _rational_
, i.e. $c_{P}(A)=\max(P,A)=\\{a\in A:aPb\,\,\text{for all}\,\,b\in A\\}$ for
all $A\in\mathcal{A}$. If a choice function is not rationalizable we will
instead refer to it as _irrational_. We let $\mathcal{C}$ denote the
collection of all choice functions and $\mathcal{P}$ the set of rationalizable
choice functions. Note that there is a one-to-one correspondence between the
set of rational choice functions and the set of preferences, therefore,
throughout the paper we refer to them as $c_{P}$ or $P$ interchangeably.
Finally, and most importantly, we do not assume that the analyst observes the
individual choice functions even if, especially in experimental settings, this
is often the case.

A _random choice model (RCM)_ is a probability distribution $\mu$ on
$\mathcal{C}$.999We borrow the term ”Random Choice Model” (RCM) from Filiz-
Ozbay & Masatlioglu (2023). The support of $\mu$, denoted $\text{supp}(\mu)$,
is the set of choice functions with strictly positive probability, i.e.
$\\{c\in\mathcal{C}:\mu(c)>0\\}$. The RCM stochastic choice function is
$\rho_{\mu}(a,A)=\mu(\mathcal{C}(a,A))$ for all $a\in A$ and $A\subseteq X$,
where $\mathcal{C}(a,A)=\\{c\in\mathcal{C}:a=c(A)\\}$. A _random utility model
(RUM)_ is a probability distribution $\mu$ with support contained in
$\mathcal{P}$ (hence it is an RCM). Shortly, we write that $\rho$ is a RUM if
there is a distribution $\mu$ such that $\rho=\rho_{\mu}$ and
$\text{supp}(\mu)\subseteq\mathcal{P}$.101010We adopt the same convention
throughout the paper to refer to the set of empirical primitives that has a
specific representation with the class of the representation itself. Finally,
we say that a RUM is a _Dual RUM_ if the cardinality of its support is less
than or equal to two (i.e. if $|\text{supp}(\mu)|\leq 2$), and that a RUM has
full-support if $\text{supp}(\mu)=\mathcal{P}$. Note that the standard
definition of RUM as a probability distribution on preferences (Block &
Marschak, 1960) is here substituted by a probability distribution on rational
choice functions (McFadden & Richter (1990), Kitamura & Stoye (2018), Stoye
(2019)). Given the one-to-one correspondence between the two sets the two
definitions are equivalent from the perspective of the aggregate choice
probabilities. However, they are conceptually different. The requirement
$\text{supp}(\mu)\subseteq\mathcal{P}$ implies that each decision-maker is
rational throughout all choices, while as shown in Example 1, the same $\rho$
may also be represented by irrational decision-makers.

##  3\. Irrational random utility models

A stochastic choice function is a RUM if a probability distribution on a set
of rational choice functions can describe it. However, even if aggregate
choices are stochastically rational, they may hide a population of completely
irrational decision-makers questioning the rational foundations of the RUM.
This motivates us to study RUMs that result from the choices of a population
of irrational decision-makers.

######  Definition 3.1.

A stochastic choice function $\rho$ is an _Irrational RUM (I-RUM)_ if it is a
RUM and if there is a probability distribution $\mu$ on
$\mathcal{C}\setminus\mathcal{P}$ such that $\rho=\rho_{\mu}$.

A weaker concept is that of a RUM that has support on at least one irrational
choice function.

######  Definition 3.2.

A stochastic choice function $\rho$ is a _partially Irrational RUM (pI-RUM)_
if it is a RUM and if there is a probability distribution $\mu$ on
$\mathcal{C}$ such that $\rho=\rho_{\mu}$ and $\mu(c)>0$ for at least one
$c\in\mathcal{C}\setminus\mathcal{P}$.

The characterization of the set of I-RUMs is based on the intuition that only
a RUM with a probability distribution that is sufficiently spread out can be
an I-RUM. We develop our intuition within our opening example.

######  Example 1 (continued).

We would like to understand when a RUM $\mu$ with support on $P_{1}$ and
$P_{2}$ (as in example 1) has an irrational representation. An irrational
decision-maker who chooses alternative $c$ from the grand set $csf$ must
either choose $s$ from $cs$ or $f$ from $cf$ (to violate Sen’s property
$\alpha$). As a result, an I-RUM exists only if
$\rho(s,cs)+\rho(f,cf)\geq\rho(c,csf)$. Otherwise, every RCM representation of
$\mu$ would assign a strictly positive probability to a rational choice
function in which $c$ is always chosen (from all menus that contain $c$).
Since in our example, $f$ is never chosen, this simple observation is
sufficient to show that a RUM with support $P_{1}$ and $P_{2}$ is an I-RUM
only if
$\mu(P_{2})=\rho(s,cs)=\rho(s,cs)+\rho(f,cf)\geq\rho(c,csf)=\mu(P_{1})$.
Symmetric reasoning implies that $\mu$ is an I-RUM only if
$\mu(P_{1})=\rho(c,cs)=\rho(c,cs)+\rho(f,sf)\geq\rho(s,csf)=\mu(P_{2})$.
Hence, it follows that $\mu$ is an I-RUM only if $\mu(P_{1})=\mu(P_{2})=1/2$.

Menu | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\max(A,P_{1})}$ | $\max(A,P_{2})$ | $\max(A,P_{3})$
---|---|---|---
$csf$ | $c$ | $s$ | $f$
$cs$ | $c$ | $s$ | $s$
$cf$ | $c$ | $c$ | $f$
$sf$ | $s$ | $s$ | $f$

To better understand the mechanism underlying our characterization of I-RUMs
we next look at RUMs $\mu$ with support on preferences $P_{1}$ and $P_{3}$.
Applying the same reasoning as above, one can see that $\mu$ is an I-RUM only
if $\mu(P_{3})+\mu(P_{3})=\rho(s,cs)+\rho(f,cf)\geq\rho(c,csf)=\mu(P_{1})$
implying that $\mu(P_{1})\leq 2/3$. And, similarly that
$\mu(P_{1})+\mu(P_{1})=\rho(s,fs)+\rho(c,cf)\geq\rho(f,csf)=\mu(P_{3})$
implying that $\mu(P_{1})\geq 1/3$. We thus see that $\mu$ is an I-RUM only if
$\mu(P_{1})\in[1/3,2/3]$ (or equivalently $\mu(P_{3})\in[1/3,2/3]$). It is
also straightforward to check that all RUMs with $\mu(P_{1})\in[1/3,2/3]$ are
I-RUMs.

Note that the preferences $P_{1}$ and $P_{3}$ are less ”correlated” than those
of $P_{1}$ and $P_{2}$. They are less correlated in the sense that $P_{1}$ and
$P_{2}$ make the same choices in two menus, whereas $P_{1}$ and $P_{3}$ only
make the same choices in one menu. This observation supports the idea that a
less correlated set of preferences imply that a higher proportion of RUMs with
support on this set have an irrational representation.

In the remainder of this section, we formalize the intuition of Example 1. To
do so, we note that in RCM the relationship between $\rho$ and $\mu$ is
analogous to the one between joint and marginal distributions and, as a
result, it is governed by the well-known Fréchet bounds (see Appendix A.3 for
a discussion on the Fréchet bounds and their application to our framework).
Our bounds will come as a variation of the Fréchet bounds and we will denote
them as ”correlation bounds” to highlight the important role of correlation
between preferences of rational decision-makers to characterize I-RUMs. In
Appendix A.3, we also motivate the use of the word ”correlation” with a more
mathematical intuition.

We introduce a few more pieces of notation to simplify the exposition. Denote
by $x_{P}(N-1),x_{P}(N)$ the worst two alternatives according to a preference
$P$, and for each preference $P$ let $\mathcal{A}(P)=\\{A\subseteq X:|A|\geq
2\\}\setminus\\{\\{x_{P}(N-1),x_{P}(N)\\}\\}$, and note that
$|\mathcal{A}(P)|=K-1$.111111The exclusion of the worst two alternatives can
be explained from the perspective of rationality, i.e. the weak axiom of
revealed preference. These alternatives are only chosen within
$\\{x_{P}(N-1),x_{P}(N)\\}$ and therefore will never induce violations of
rationality. We next define for each preference $P$:

$\mathbb{C}^{\rho}_{P}=\frac{1}{K-2}\sum_{A\in\mathcal{A}(P)}\rho(c_{P}(A),A).$

The quantity $\mathbb{C}^{\rho}_{P}$ represents the probability mass in $\rho$
that ”correlates” with a rational choice function $c_{P}$. In other words, it
measures positively the correlation between the choices of the decision-makers
in the population w.r.t. to a specific preference. Given our Example 1, the
correlation bounds that will characterize I-RUMs will come as an upper bound
on the correlation between the choices of the decision-makers.

######  Definition 3.3 (Correlation Bounds).

A stochastic choice function $\rho$ satisfies the _Correlation bounds_ if for
all preferences $P$ it holds that

$\mathbb{C}^{\rho}_{P}\leq 1.$

The definition of correlation bounds does not rely on the stochastic choice
function being a RUM and, interestingly for us, if this is the case it can be
simplified and rewritten in the space of the preferences. For each $P$ define
the numbers
$n(P,P^{\prime})=|\\{A\in\mathcal{A}(P):c_{P}(A)=c_{P^{\prime}}(A)\\}|$ for
all $P^{\prime}$. These numbers represent the correlations between the choices
of rational decision-makers.

######  Lemma 3.4.

Let $\rho$ be a RUM with distribution $\mu$ and let $P$ be a preference. Then

$\mathbb{C}^{\rho}_{P}=\frac{1}{K-2}\sum_{P^{\prime}\in\mathcal{P}}\mu(P^{\prime})n(P,P^{\prime}).$

###### Proof.

The proof follows by noting that

$\sum_{A\in\mathcal{A}(P)}\rho(c_{P}(A),A)=\sum_{A\in\mathcal{A}(P)}\sum_{P^{\prime}\in\mathcal{P}}[\mu(P^{\prime})\mathbb{1}\\{c_{P}(A)=c_{P^{\prime}}(A)\\}]=$
$\sum_{P^{\prime}\in\mathcal{P}}\mu(P^{\prime})\sum_{A\in\mathcal{A}(P)}\mathbb{1}\\{c_{P}(A)=c_{P^{\prime}}(A)\\}=$
$\sum_{P^{\prime}\in\mathcal{P}}\mu(P^{\prime})n(P,P^{\prime}).$

The first equality follows since $\rho$ is a RUM. The second equality follows
by changing the order of summation and the final equality follows by
definition of the numbers $n(P,P^{\prime})$. ∎

##  4\. Characterization

In this section, we state and prove our characterization results. The
structure is as follows. First, in subsection 4.1 we characterize the set of
I-RUMs showing that any RUM that exhibits a sufficiently low degree of
correlation is an I-RUM. Second, in subsection 4.2, we characterize pI-RUMs
showing, instead, that almost any RUM is a pI-RUM and that all RUMs with full
support are pI-RUMs.

###  4.1. Characterization of I-RUMs

We are now ready to state our characterization of I-RUMs and provide a proof
outline.

######  Theorem 4.1.

Let $\rho$ be a RUM, then it is an I-RUM if and only if it satisfies the
correlation bounds.

_Proof (sketch)._ We delegate the full proof to Appendix B. The necessity of
the correlation bounds follows naturally from the interpretation of the
correlation bounds as variations of the Fréchet bounds. The sufficiency part
is instead the more interesting, and here we provide a sketch of the
arguments. First, we note that the set of RUMs that satisfy the correlation
bounds is convex. By Caratheodory’s theorem, we know that any point in a
convex set can be written as a convex combination of its extreme points,
therefore, if we can show that each extreme point of this set is an I-RUM, the
proof is complete. To show this, we proceed through three lemmas. The first
lemma shows that every Dual RUM that satisfies the correlation bounds with
equality is an I-RUM, while the final two lemmas show that each extreme point
of the set of I-RUMs is a Dual RUM that satisfies the correlation bounds with
equality.

The geometric intuition behind the proof can be explained as follows. The
extreme points of the convex set of RUMs are the RUMs with support on _single_
preferences. However, it is clear that any such RUM violates the correlation
bounds, and hence cannot be an extreme point of the convex set of I-RUMs.
Hence, a natural conjecture, which turns out to be true, is that the
collection of Dual RUMs, with support on _two_ preferences, constitute the
extreme points of this set. A minor modification of our opening example
illustrates the geometric structure of the set of I-RUMs.

######  Example 1 (continued).

Theorem 4.1 provides the visualization of the set of I-RUMs related to RUMs
with support $P_{1}$, $P_{2}$, and $P_{3}$ as convex combinations of the
respective irrational Dual RUMs. In this simple example, the numbers
$n(P_{1},P_{2})=1$, and $n(P_{1},P_{3})=n(P_{2},P_{3})=0$ contain all the
information needed. To illustrate, the Dual RUM with support $P_{1},P_{2}$ has
the following correlation bounds:

$\mathbb{C}_{P_{1}}^{\rho}=\frac{1}{2}[3\mu(P_{1})+\mu(P_{2})]\leq 1$
$\mathbb{C}_{P_{2}}^{\rho}=\frac{1}{2}[\mu(P_{1})+3\mu(P_{2})]\leq 1$

which imply $\mu(P_{1})=\mu(P_{2})=\frac{1}{2}$. Instead, the Dual RUM with
support $P_{1},P_{3}$ has the following correlation bounds:

$\mathbb{C}_{P_{1}}^{\rho}=\frac{1}{2}[3\mu(P_{1})+0\mu(P_{3})]\leq 1$
$\mathbb{C}_{P_{3}}^{\rho}=\frac{1}{2}[0\mu(P_{1})+3\mu(P_{3})]\leq 1$

which imply $\mu(P_{1})\in[\frac{1}{3},\frac{2}{3}]$.

Menu | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\max(A,P_{1})}$ | $\max(A,P_{2})$ | $\max(A,P_{3})$
---|---|---|---
$csf$ | $c$ | $s$ | $f$
$cs$ | $c$ | $s$ | $s$
$cf$ | $c$ | $c$ | $f$
$sf$ | $s$ | $s$ | $f$

$\displaystyle I-RUMs$$\displaystyle P_{2}$$\displaystyle Dual\
RUMs$$\displaystyle Dual\ RUMs$$\displaystyle Dual\ RUMs$$\displaystyle
P_{1}$$\displaystyle P_{3}$

We conclude section 4.1, by providing two corollaries that even more
intuitively convey the idea that I-RUMs are RUMs with sufficiently
uncorrelated preferences. The first corollary (which follows directly from
theorem 4.1) shows that if a preference $P$ in the support of a RUM has a
”sufficiently high” probability mass then the RUM cannot be an I-RUM.

######  Corollary 4.2.

Let $\rho$ be a RUM with distribution $\mu$. If $\mu(P)>\frac{K-2}{K-1}$ for
some preference $P$ then $\rho$ is not an I-RUM.

The second corollary, instead, shows that any RUM with sufficiently spread-out
probability on the preferences is an I-RUM.

######  Corollary 4.3.

Let $\rho$ be a RUM with distribution $\mu$. If $\mu(P)\leq\frac{1}{4}$ for
all $P\in\mathcal{P}$ then $\rho$ is an I-RUM.

###### Proof.

We prove that $\rho$ satisfies the correlation bounds. Let $P$ be a preference
and $Q$ be the preference that agrees with $P$ on the ranking of all
alternatives except their worst two alternatives. We rewrite the correlation
bounds as in lemma 3.4 and note that we have:

$(K-2)\mathbb{C}^{\rho}_{P}=\sum_{P^{\prime}\in\mathcal{P}}\mu(P^{\prime})n(P,P^{\prime})=$
$\mu(P)(K-1)+\mu(Q)(K-1)+\sum_{P^{\prime}\in\mathcal{P}\setminus\\{P,Q\\}}\mu(P^{\prime})n(P,P^{\prime})\leq$
$[\mu(P)+\mu(Q)](K-1)+\sum_{P^{\prime}\in\mathcal{P}\setminus\\{P,Q\\}}\mu(P^{\prime})(K-3)=$
$[\mu(P)+\mu(Q)](K-1)+[1-\mu(P)-\mu(Q)](K-3)\leq K-2$

where the first (in)equality follows by lemma 3.4, the second (in)equality
since $n(P,P)=n(P,Q)=K-1$, the third (in)equality follows since
$n(P,{P^{\prime}})\leq K-3$ for all
$P^{\prime}\in\mathcal{P}\setminus\\{P,Q\\}$, the fourth in(equality) since
$\sum_{P^{\prime}\in\mathcal{P}\setminus\\{P,Q\\}}\mu(P^{\prime})=1-\mu(P)-\mu(Q)$
and the final inequality follows since $\mu(P)+\mu(Q)\leq\frac{1}{2}$.∎

###  4.2. Characterization of pI-RUMs

We next turn to the characterization of pI-RUMs, i.e. these are the RUMs that
have support on at least one irrational choice function.

######  Theorem 4.4.

Let $\rho$ be a RUM, then the following statements are equivalent

  1. (1)

$\rho$ is a pI-RUM,

  2. (2)

$\rho$ is a non-trivial convex combination of an I-RUM and a Dual RUM,

  3. (3)

there are menus $A,B\in\mathcal{A}$ and alternatives $a,b\in A\cap B$ such
that $a,b$ are chosen from both $A,B$ with strictly positive probability.

We delegate the proof of theorem 4.4 to the Appendix C. Most of the
implications are straightforward. The only difficult step is the implication
(1) $\Rightarrow$ (2). To prove that every pI-RUM is a non-trivial convex
combination of an I-RUM and a Dual RUM we show that every RUM can be written
as a convex combination (not necessarily non-trivial) of an I-RUM and a Dual
RUM. This result is recorded as proposition 4.5 below.

######  Proposition 4.5.

Every RUM can be written as a convex combination of an I-RUM and a Dual RUM.

Our characterization of pI-RUMs shows that almost any RUM is partially
irrational. Indeed, condition (3) characterizing pI-RUMs is very weak. It
holds whenever there is a menu $B$ with cardinality strictly greater than two,
and alternatives $a,b\in B$ such that $\rho(a,B),\rho(b,B)>0$. Thus, even if
aggregate choices are consistent with RUM unless very restrictive conditions
hold, the same choices can always be explained by a population of decision-
makers where at least one of them is irrational. An immediate corollary to
theorem 4.4 shows that every full-support RUM is a pI-RUM, hence that some of
the most influential RUMs such that the Logit model can always be represented
by a population of partially irrational decision-makers.

######  Corollary 4.6.

Every RUM with full support is a pI-RUM.

##  5\. Non-falsifiability of rationality under RUM

Theorem 4.1 and 4.4 characterize the cases where the RUM hypothesis may not
convey information regarding individual behavior. I-RUMs are examples of
aggregate data from a population when the rationality assumption fails
everywhere but the RUM hypothesis is not rejected while pI-RUMs show that
almost any RUM is compatible with some failure of rationality. We encode this
implication of our results as the non-falsifiability of the rational
interpretation of the RUM hypothesis, i.e. ”all decision-makers are rational”.

In this section, we investigate this problem in more detail. Specifically, we
show that the RUM hypothesis not only fails to be rejected when a fraction of
decision-makers are irrational but also that the irrational behavior of these
decision-makers may be unconstrained. Further, we complement this result by
showing that the fraction of irrational decision-makers compatible with the
RUM hypothesis is monotonically related to the set of potential irrational
behaviors.

The overall idea of the next results is to counterweight a $\rho$ that is not
a RUM, i.e. it has negative BM polynomials, with a $\rho^{*}$ that is a RUM
with full support, i.e. it has strictly positive BM polynomials (everywhere).
In Appendix A.2, we discuss the standard characterization of RUMs with BM
polynomials, as well as, some preliminary results that play a role in the
proofs of the results in this section.

######  Proposition 5.1.

Let $\rho^{*}$ be a RUM with full support and $\mathtt{M}$ a collection of
RCMs. Then there is a $\bar{\alpha}(\mathtt{M})\in(0,1)$ such that:

  1. (1)

if $1>\alpha\geq\bar{\alpha}(\mathtt{M})$ then for all $\mu\in\mathtt{M}$ it
holds that $\alpha\rho^{*}+(1-\alpha)\rho_{\mu}$ is a RUM.

  2. (2)

if $0<\alpha<\bar{\alpha}(\mathtt{M})$ then there is a $\mu\in\mathtt{M}$ such
that $\alpha\rho^{*}+(1-\alpha)\rho_{\mu}$ is not a RUM.

Proposition 5.1 shows that, within a pi-RUM, whenever the rational decision-
makers are represented by a RUM with full support, then there exists a lower
bound on the fraction $\bar{\alpha}(\mathtt{M})$ of rational decision-makers
such that the RUM hypothesis is satisfied irrespective of the behavior of the
remaining decision-makers. Recalling our opening example, if $\rho^{*}$ is a
RUM with full support on $X=\\{c,s,f\\}$ such that $\mu(P)=\mu(P^{\prime})$
for all $P,P^{\prime}\in\mathcal{P}$, and $\mathtt{M}$ is the collection of
all RCMs, then $\bar{\alpha}(\mathtt{M})=6/7$. Proposition 5.1 also shows that
the lower bound is tight, i.e. whenever there are less than
$\bar{\alpha}(\mathtt{M})$ rational decision-makers then there is a group of
$1-\bar{\alpha}(\mathtt{M})$ irrational decision-makers that will induce a
rejection of the RUM hypothesis. Again, if $\alpha<6/7$ then if $\mu$ is the
RCM where all decision-makers choose $c$ from $csf$ and $s$ from $cs$, then
$\alpha\rho^{*}+(1-\alpha)\rho_{\mu}$ is not a RUM.

In our opening example, we also show that the types and distribution of
irrational behaviors have implications on $\bar{\alpha}(\mathtt{M})$. The
following corollary, which follows immediately from proposition 5.1 formalizes
these intuitions.

######  Corollary 5.2.

Let $\mathtt{M},\mathtt{M}^{\prime}$ be two collections of RCMs, and
$\rho^{*}$ a RUM with full support. Then
$\mathtt{M}^{\prime}\subseteq\mathtt{M}$ implies
$\bar{\alpha}(\mathtt{M}^{\prime})\leq\bar{\alpha}(\mathtt{M})$.

##  6\. Discussion: a paradox of stochastic rationality

In this section, we discuss the tension between the notions of stochastic and
individual rationality which result in the following paradox. On the one hand,
there are stochastically rationalizable choices with an irrational
representation. On the other hand, there are stochastic choice functions that
are not stochastically rational but that lack an irrational representation,
i.e. they cannot be represented by an RCM with support outside the set of
rational choice functions (see example 2 below). The paradoxical conclusion
that emerges is thus that some non-rationalizable stochastic choice functions
may need more rational choice functions than some rationalizable ones do.

######  Example 2.

Suppose we observe the choices over all non-empty menus of a set of three
alternatives $\\{a,b,c\\}$ from a population of decision-makers. Let $P$ be
the preference $aPbPc$ and $c_{P}$ the corresponding choice function. Let $c$
be the non-rationalizable choice function defined by
$c(abc)=c(ac)=a,c(ab)=b,c(bc)=b$. Suppose that $\mu$ is an RCM with
$\mu(c_{P})=\frac{2}{3}$ and $\mu(c)=\frac{1}{3}$. The aggregated choice
probabilities are then $\rho(a,abc)=\rho(a,ac)=\rho(b,bc)=1$ and
$\rho(a,ab)=\frac{2}{3}$ and it is clear that $\rho$ is not a RUM stochastic
choice function (since it violates regularity, i.e. $\rho(a,abc)>\rho(a,ab)$).
Also, note that $\rho$ violates the correlation bounds since

$\rho(a,abc)+\rho(a,ab)+\rho(a,ac)=1+\frac{2}{3}+1=2+\frac{2}{3}>2$

This paradoxical conclusion is even more striking if we observe that (i) the
boundary points of the set of I-RUMs may be also boundary points of the set of
RUMs, i.e. there is a stochastic choice function that is an I-RUM and an
arbitrarily close stochastic choice function that is not a RUM but does not
have an irrational representation, and (ii) the interior points of I-RUMs are
also interior points of RUMs. For example, let
$\rho(a,abc)=\rho(a,ab)=\frac{1}{2}$, and $\rho(a,ac)=\rho(b,bc)=1$. This is
the $\rho$ represented in example 1, hence, an I-RUM. Note that, a minor
perturbation $\varepsilon>0$ such that
$\rho^{\varepsilon}(a,abc)=\frac{1}{2}+\varepsilon$ implies both that
$\rho^{\varepsilon}$ is not a RUM because regularity is violated and also that
$\rho^{\varepsilon}$ cannot be represented by all irrational decision-makers
because the correlation bounds are violated. Instead, the stochastic choice
function $\rho(a,A)=\frac{1}{|A|}$ for all $a\in A$ and $A\subseteq X$ is in
the interior of both the set of RUMs and I-RUMs, namely any perturbed
stochastic choice function $\rho^{\varepsilon}$ is a RUM, and at the same
time, any $\rho^{\varepsilon}$ can be represented by a population of all
irrational decision-makers.

In other words, our discussion highlights the following trade-off. On the one
hand stochastic choice functions with concentrated probability mass that are
not RUMs are surely for one part the result of the choices of irrational
decision-makers, but also for another part surely the result of the choices of
rational decision-makers. On the other hand, stochastic choice functions with
spread out probability mass may be the result of the choices of all rational
decision-makers, but at the same time, may also be the result of the choices
of all irrational decision-makers.

###  6.1. An example within the theory of demand

We conclude the section by discussing this takeaway within the Theory of
Demand using a simple example by Kitamura & Stoye (2018). We do not claim
novelty for the intuitions that follow as, for example, among others Im &
Rehbeck (2022) pointed out similar facts. However, we would like to re-
interpret this example, in which correlation bounds are necessary and
sufficient (Matzkin, 2007), within our framework to provide some further
insights. Figure 1 displays the potential choices of decision-makers from two
budget sets: $\mathcal{B}_{1},\mathcal{B}_{2}$.

Figure 1. Example from Kitamura & Stoye (2018)

Assuming Walras’ Law, $\pi_{i|j}$ denotes the share of the decision-makers who
choose in the segment $i$ from the budget set $\mathcal{B}_{j}$. Ignoring the
intersection, the choice probabilities in this example are described by the
vector $(\pi_{1|1},\pi_{2|1},\pi_{1|2},\pi_{2|2})$. There are four possible
pairs of choices that the decision-makers can make and only one of them is
irrational, i.e. $\pi_{1|1},\pi_{1|2}$. By similar reasoning as in our Example
1, this setting does not admit I-RUMs, but only pI-RUMs. Nonetheless, our
discussion of the paradox of stochastic rationality easily follows.

First, suppose we observe the stochastically rational choice probabilities
$(\pi_{1|1},\pi_{2|1},\pi_{1|2},\pi_{2|2})=(0.5,0.5,0.5,0.5)$. This RUM is a
pI-RUM where the fraction of irrational decision-makers is at most 50%. The
contingency tables below display two possible scenarios in which either 0% or
50% of the decision-makers are irrational. One can also note that not only
this RUM is a pi-RUM, but there is only one representation in which every
decision-maker is rational, while there is an infinity of representations in
which at least some decision-makers are irrational. In the contingency tables,
we color the irrational choices in red and the rational ones in blue.

$\left[\begin{array}[]{c| cc | c}&\pi_{1|1}&\pi_{2|1}&\\\ \hline\cr\pi_{1|2}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0.5}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}&0.5\\\ \pi_{2|2}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0.5}&0.5\\\ \hline\cr&0.5&0.5&1\end{array}\right]\left[\begin{array}[]{c| cc | c}&\pi_{1|1}&\pi_{2|1}&\\\ \hline\cr\pi_{1|2}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0.5}&0.5\\\ \pi_{2|2}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0.5}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}&0.5\\\ \hline\cr&0.5&0.5&1\end{array}\right]$

Imagine now to observe the following non-stochastically rational choice
probabilities: $(\pi_{1|1},\pi_{2|1},\pi_{1|2},\pi_{2|2})=(1,0,0.1,0.9)$. A
rapid inspection will reveal that the fraction of irrational decision-makers
is both at least and at most 10% and that the unique RCM that produces the
choice probabilities is:

$\left[\begin{array}[]{c| cc | c}&\pi_{1|1}&\pi_{2|1}&\\\ \hline\cr\pi_{1|2}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0.1}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}&0.1\\\ \pi_{2|2}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0.9}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}&0.9\\\ \hline\cr&1&0&1\end{array}\right]$

This simple example reveals a couple of interesting facts. First, statistical
tests that focus solely on the distance between the observed aggregate choice
probabilities and the set of RUMs may lose relevant information regarding the
rationality of the underlying behavior of the decision-makers. Second, as
noticed by Im & Rehbeck (2022), assuming individual rationality implies the
identification of 50% of the population with a high (resp. low) marginal rate
of substitution between the two goods while relaxing this assumption opens to
the possibility (identification of the joint distribution is now lost) that
100% of the individuals has a mild marginal rate of substitution with 50% of
them being rational and 50% irrational. Clearly, welfare conclusions may
differ substantially between these two scenarios.

##  7\. Concluding remarks

In the last decades, economics has welcomed several critiques of its rational
foundations. Being located within the literature on individual decision-
making, our paper fits within this tendency as we have provided a
characterization of the set of populations that are rational at an aggregate
level and can be also represented as irrational ones at an individual level.

In these concluding remarks, we touch on a different and more positive
viewpoint of our paper borrowed from Becker (1962): ”Economic theory is much
more compatible with irrational behavior than had been previously suspected.”
In our characterization results, we not only show that rational aggregate
choices may be individually fully irrational but also that the aggregation of
individually irrational behaviors may well be stochastically rational and that
individuals with correlated mistakes can never be translated into rational
aggregate behavior. These observations resonate within the framework of
standard economic theory where market efficiency is built upon uncorrelated
errors of individual investors.

##  Appendix A Auxiliary results

###  A.1. Preliminaries on convex analysis

A set $X\subset\mathbb{R}^{d}$ is convex if for all $x,y\in X$ and
$\alpha\in(0,1)$ it holds that $\alpha x+(1-\alpha)y\in X$. The convex hull of
a set $Y\subseteq\mathbb{R}^{d}$ is the intersection of all convex sets
containing $Y$ and is denoted $\text{conv}(Y)$, or equivalently the set of all
points that can be obtained as convex combinations of points from $Y$. A point
$x\in X\subseteq\mathbb{R}^{d}$ is called an _extreme point_ of $X$ if there
is no $\alpha\in(0,1)$ and $y,z\in X\setminus\\{x\\}$ with $y\neq z$ such that
$x=\alpha y+(1-\alpha)z$. In words, a point $x\in X$ is an extreme point if it
cannot be written as a non-trivial convex combination of two other distinct
points in $X$. Denote the set of extreme point of a convex set $X$ by
$\text{ext}(X)$. The next result is a basic result from convex analysis
(Rockafellar, 1970, p.155) and will be employed when showing sufficiency of
the Fréchet bounds in theorem 4.1.

######  Lemma A.1.

_(Carathéodory’s theorem)_ Every point $x$ in a convex set
$X\subseteq\mathbb{R}^{d}$ can be written as a convex combination of the
extreme points of $X$. I.e. $\text{conv}(\text{ext}(X))=X$.

###  A.2. Preliminaries on stochastic rationality and Block-Marschak
polynomials

The well-known characterization of the RUM is based on the Block-Marschak-
polynomials (BM-polynomials). Define for all $A\subseteq X$ and $a\in X$ the
BM-polynomial at $a$ in $A$ by:

$\text{BM}_{\rho}(a,A)=\sum_{A\subseteq B\subseteq X}(-1)^{|B\setminus
A|}\rho(a,B)$

######  Theorem A.2 (Block & Marschak (1960), Falmagne (1978), Barberá &
Pattanaik (1986), Monderer (1992), Fiorini (2004)).

$\rho$ is a RUM if and only if $\text{BM}_{\rho}(a,A)\geq 0$ for all
$A\subseteq X$ and $a\in X$.

We now state two facts about BM-polynomials. The first shows that the BM
polynomials act as a convex operator on the set of stochastic choice
functions.

######  Lemma A.3.

Let $\alpha\in(0,1)$ and $\rho,\rho^{\prime}$ be stochastic choice functions
then

$\text{BM}_{\alpha\rho+(1-\alpha)\rho^{\prime}}(a,A)=\alpha\text{BM}_{\rho}(a,A)+(1-\alpha)\text{BM}_{\rho^{\prime}}(a,A)$

for all $a\in A$ and $A\subseteq X$.

###### Proof.

The proof is simple and follows by expanding the definition of a BM
polynomial. We have that

$\text{BM}_{\alpha\rho+(1-\alpha)\rho^{\prime}}(a,A)=\sum_{A\subseteq
B\subseteq X}(-1)^{|B\setminus
A|}\left[\alpha\rho(a,B)+(1-\alpha)\rho^{\prime}(a,B)\right]=$
$\alpha\sum_{A\subseteq B\subseteq X}(-1)^{|B\setminus
A|}\rho(a,B)+(1-\alpha)\sum_{A\subseteq B\subseteq X}(-1)^{|B\setminus
A|}\rho^{\prime}(a,B)=$
$\alpha\text{BM}_{\rho}(a,A)+(1-\alpha)\text{BM}_{\rho^{\prime}}(a,A).$

∎

The second shows that RUMs with full support are characterized by strictly
positive BM-polynomials.

######  Lemma A.4.

A RUM is full support if and only if $\text{BM}_{\rho_{\mu}}(a,A)>0$ for all
$a\in A$ and $A\subseteq X$.

###### Proof.

This result follows from the discussion in (Turansick, 2022, p.7) ∎

###  A.3. Preliminaries on Fréchet bounds

The Fréchet bounds govern the relationship between joint and marginal
distributions. Specifically, they come as bounds on the joint distribution
given the marginals. In a general setting, the Fréchet bounds are obtained as
follows.

######  Lemma A.5.

Let $\Omega$ be a sample space and $P:\Omega\to[0,1]$ a probability measure.
For any finite collection of events $E_{1},...,E_{K}\subseteq\Omega$ it then
holds that

$\max\\{0,\sum_{k=1}^{K}P(E_{k})-(K-1)\\}\leq P(\cap_{k=1}^{K}E_{k}).$

The Fréchet bounds follows from the basic laws of probability and an induction
argument.121212More specifically, they follow by noting that for any two
events $E_{1},E_{2}$ it holds that $P(E_{1}\cap
E_{2})=P(E_{1})+P(E_{2})-P(E_{1}\cup E_{2})$ which implies that $P(E_{1}\cap
E_{2})=P(E_{1})+P(E_{2})-P(E_{1}\cup E_{2})\geq P(E_{1})+P(E_{2})-1$ and since
also $P(E_{1}\cap E_{2})\geq 0$ we have $P(E_{1}\cap
E_{2})\geq\max\\{0,P(E_{1})+P(E_{2})-1\\}$. An induction argument then
completes the proof. In RCMs, $\mu$ is the joint distribution while
$\rho_{\mu}$ is the marginal distribution, and the operation
$\rho(c(A),A)=\mu(\\{c^{\prime}\in\mathbb{C}:c^{\prime}(A)=c(A)\\})$ is
nothing more than a marginalization. The following result is hence a direct
corollary to lemma A.5.

######  Lemma A.6.

Let $\rho$ be a stochastic choice function, $c$ be a choice function, and
$\mathcal{B}$ be an arbitrary collection of choice sets. It then holds that

$\max\\{0,\sum_{A\in\mathcal{B}}\rho(c(A),A)-(|\mathcal{B}|-1)\\}\leq\mu(\cap_{A\in\mathcal{B}}\\{c^{\prime}\in\mathcal{C}:c^{\prime}(A)=c(A)\\})$

for all RCMs $\mu$ such that $\rho=\rho_{\mu}$.

It should be noted that the lefthand side of the above inequality is
independent of the RCM $\mu$ that represents $\rho$. For our purposes, it is
also instructive to consider the special case of lemma A.6 when $\mathcal{B}$
is the collection of all non-empty and non-singleton menus of $X$, i.e. when
$\mathcal{B}=\\{A\subseteq X:|A|\geq 2\\}$ and $K=|\mathcal{B}|$. Then, since
$\cap_{A\in\mathcal{B}}\\{c^{\prime}\in\mathcal{C}:c^{\prime}(A)=c(A)\\}=\\{c\\}$
we have the following result.

######  Lemma A.7.

Let $c$ be a choice function then it holds that

$\max\\{0,\sum_{A\in\mathcal{B}}\rho(c(A),A)-(K-1)\\}\leq\mu(c)$

for all RCMs $\mu$ such that $\rho=\rho_{\mu}$.

By this formulation of the Fréchet bounds it is clear that if the left-hand
side of the inequality is strictly positive, then any $\mu$ that represents
$\rho$ must assign (strictly) positive probability to the choice function on
the right-hand side of the inequality. In particular, if $c$ is rationalizable
then any RCM $\mu$ must assign positive probability to $c$, implying that
$\rho_{\mu}$ is not an I-RUM. Equivalently, if a RUM has an irregular
representation, so that $\mu(P)=0$ for all preferences $P$, then lemma A.7
implies the following bounds.

Weak correlation bounds. A stochastic choice function $\rho$ satisfies the
_weak correlation bounds_ if for all preferences $P$ it holds that

$\frac{1}{K-1}\sum_{A\subseteq X:|A|\geq 2}\rho(c_{P}(A),A)\leq 1.$

Although Lemma A.7 implies that the weak correlation bounds are necessary for
an irrational representation they are not quite sufficient. The following
example illustrates.

######  Example 6.

Let $\mu(aP_{1}bP_{1}c)=\mu(aP_{2}cP_{2}b)=0.4$,
$\mu(bP_{3}cP_{3}a)=\mu(cP_{4}bP_{4}a)=0.1$. The following $p_{\mu}$ arises:
$\rho_{\mu}(a,X)=\rho_{\mu}(a,ab)=\rho_{\mu}(a,ac)=0.8$,
$\rho_{\mu}(b,bc)=0.5$. The two choice functions $c_{P_{1}},c_{P_{2}}$ related
to $P_{1},P_{2}$ are salient and they both satisfy the weak correlation
bounds.

$\sum_{A\subseteq X:|A|\geq 2}\rho(c_{P_{1}}(A),A)=\sum_{A\subseteq X:|A|\geq
2}\rho(c_{P_{2}}(A),A)=2.9<K-1=3$

However, one can easily see that if $\mu^{\prime}$ is an RCM with
$\mu^{\prime}(c_{P_{1}})=\mu^{\prime}(c_{P_{2}})=0$ then $\rho$ cannot be
represented by $\mu^{\prime}$ (i.e. $\rho\neq\rho_{\mu^{\prime}}$).

The problem in the above example is that $P_{1}$ and $P_{2}$ are close to
being the same preference since they agree on the choices from all menus
except $\\{b,c\\}$ (which contains the worst two alternatives of both orders).
Moreover, a large fraction of the overall probability is assigned to $P_{1}$
and $P_{2}$ (i.e. probability $0.8$) and hence the stochastic choice function
$\rho_{\mu}$ exhibits a high degree of correlation. It turns out that a slight
variation of these bounds, namely the correlation bounds of section 3, are
sufficient for an irrational representation of RUM.

Before providing the proof of theorem 4.1, we would like to briefly motivate
the use of the word ”correlation” which intends a measure of concordance
between the choices of the decision-makers in the population w.r.t. to a
specific choice function. To see this, note that, w.r.t to a choice function
$c$, we can split the probability mass in $\rho$ between the one that
”correlates” and ”uncorrelates” with $c$.

$\frac{1}{K-1}\bigg{[}\underbrace{\sum_{A\subseteq X:|A|\geq
2}\rho(c(A),A)}_{\text{"correlation"
$(\hat{\mathbb{C}}^{\rho}_{c})$}}+\underbrace{\sum_{A\subseteq X:|A|\geq
2}\rho(A\setminus c(A),A)}_{\text{"uncorrelation"
$(\hat{\mathbb{U}}^{\rho}_{c})$}}\bigg{]}=1$

Finally, we note that
$\hat{\mathbb{C}}^{\rho}_{c},\hat{\mathbb{U}}^{\rho}_{c}\in[0,1]$ and the
equation above implies
$\hat{\mathbb{C}}^{\rho}_{c}-\hat{\mathbb{U}}^{\rho}_{c}\in[-1,1]$. An
intuitive ”correlation” interpretation comes after noticing the relationship
with the notion of Kendall rank correlation as $\hat{\mathbb{C}}^{\rho}_{c}$
is the probability mass on the choices that are concordant with $c$ while
$\hat{\mathbb{U}}^{\rho}_{c}$ is the probability mass on the choices that are
discordant with $c$, and $\frac{1}{K-1}$ is the normalization factor.

##  Appendix B Proof of theorem 4.1

###  B.1. The necessity of the Correlation bounds

The necessity of the Correlation bounds follows by an application of the
Fréchet bounds in lemma A.6. Recall that $\mathcal{A}(P)=\\{A\subseteq
X:|A|\geq 2\\}\setminus\\{\\{x_{P}(N-1),x_{P}(N)\\}\\}$ and
$|\mathcal{A}(P)|=K-1$ for all preferences $P$. The following lemma is a
special case of lemma A.6 and it follows by noting that for each preference
$P$ we have
$\cap_{A\in\mathcal{A}(P)}\\{c\in\mathcal{C}:c(A)=c_{P}(A)\\}=\\{c_{P},c_{Q}\\}$
where $Q$ agrees with $P$ on all the menus in $\mathcal{A}(P)$.

######  Lemma B.1.

Let $\rho$ be a stochastic choice function, $P$ be a preference, and $Q$ be a
preference that agrees with $P$ on all menus in $\mathcal{A}(P)$. It then
holds that

$\max\\{0,\sum_{A\in\mathcal{A}(P)}\rho(c_{P}(A),A)-(K-2)\\}\leq\mu(\\{c_{P},c_{Q}\\})$

for all RCMs $\mu$ such that $\rho=\rho_{\mu}$.

Since, any I-RUM $\mu$ puts zero probability on all the rational choice
functions, i.e. $\mu(c_{P})=0$ for all preferences $P$, lemma B.1 implies that
any I-RUM must satisfy the following inequalities for all preferences $P$:

$\mathbb{C}^{\rho}_{P}=\frac{1}{K-2}\sum_{A\in\mathcal{A}(P)}\rho(c_{P}(A),A)\leq
1,$

which are exactly the correlation bounds. The preceding discussion thus
implies that the correlation bounds are necessary for an I-RUM representation.

###  B.2. The sufficiency of the Correlation bounds

We refer to the main text for an exposition and sketch of the general idea
behind the proof. Let $\rho$ be a RUM with distribution $\mu$. We say that
$\rho$ _satisfies the Correlation bounds with strict inequality_ if
$\mathbb{C}^{\rho}_{P}<1$ for all preferences $P$. Similarly we say that
$\rho$ _satisfies the Correlation bounds with equality_ if
$\mathbb{C}^{\rho}_{P}=1$ for some preference $P$ and
$\mathbb{C}^{\rho}_{P^{\prime}}\leq 1$ for all preferences $P^{\prime}$.

######  Lemma B.2.

The subset of RUMs that satisfy the Correlation bounds is a convex set. I.e.
the set of all RUMs $\rho$ such that

$\mathbb{C}^{\rho}_{P}\leq 1$

for all preferences $P$ is convex.

###### Proof.

Let $\rho^{\prime},\rho^{\prime\prime}$ be RUMs with distributions
$\mu^{\prime},\mu^{\prime\prime}$ that satisfy the Correlation bounds and let
$\alpha\in(0,1)$. We claim that
$\rho=\alpha\rho^{\prime}+(1-\alpha)\rho^{\prime\prime}$ satisfies the
Correlation bounds. For all preferences $P$ we have
$\mu(P)=\alpha\mu^{\prime}(P)+(1-\alpha)\mu^{\prime\prime}(P)$. Hence

$(K-2)\mathbb{C}^{\rho}_{P}=\sum_{P^{\prime}\in\mathcal{P}}\mu(P^{\prime})n(P,P^{\prime})=\sum_{P^{\prime}\in\mathcal{P}}\left[\alpha\mu^{\prime}(P)+(1-\alpha)\mu^{\prime\prime}(P)\right]n(P,P^{\prime})$
$=\alpha\sum_{P^{\prime}\in\mathcal{P}}\mu^{\prime}(P^{\prime})n(P,P^{\prime})+(1-\alpha)\sum_{P^{\prime}\in\mathcal{P}}\mu^{\prime\prime}(P^{\prime})n(P,P^{\prime})=$
$=(K-2)\left[\alpha\mathbb{C}^{\rho^{\prime}}_{P}+(1-\alpha)\mathbb{C}^{\rho^{\prime\prime}}_{P}\right]\leq
K-2$

where the last inequality holds since
$\mathbb{C}^{\rho^{\prime}}_{P},\mathbb{C}^{\rho^{\prime\prime}}_{P}\leq 1$
for all preferences $P$.

∎

######  Lemma B.3.

Every Dual RUM that satisfies the Correlation bounds with equality is an
I-RUM.

###### Proof.

Let $\mu$ be a dual RUM with support on $\\{P_{1},P_{2}\\}$. Assume that
$\mu(P_{1})\leq\mu(P_{2})$ and that
$\mu(P_{1})n(P_{2},P_{1})+\mu(P_{2})n(P_{2},P_{2})=K-2$. Rearranging the
latter equality it follows that
$\mu(P_{2})=\frac{K-2-n(P_{2},P_{1})}{K-1-n(P_{2},P_{1})}$. Let
$k\in\mathbb{N}$ be such that $k\mu(P_{1})=\mu(P_{2})$. I.e.
$k=K-2-n(P_{2},P_{1})$. Order the elements of
$\\{A\in\mathcal{A}(P_{2}):c_{P_{2}}(A)\neq c_{P_{1}}(A)\\}$ (which has
cardinality $k+1$) as $A_{1},...,A_{k+1}$. For each $i\in\\{1,...,k+1\\}$
define a choice function $c^{\prime}_{i}$ by

$c^{\prime}_{i}(A)=\begin{cases}c_{P_{1}}(A)&\text{if $A=A_{i}$}\\\
c_{P_{2}}(A)&\text{if $A\neq A_{i}$}\\\ \end{cases}$

We next show that each $c^{\prime}_{i}$ is irrational. Since
$A_{i}\neq\\{x_{P_{2}}(N-1),x_{P_{2}}(N)\\}$, it follows that $x_{P_{2}}(K)\in
A_{i}$ for some $K<N-1$.131313For a preference $P$ and for each
$i\in\\{1,...,n\\}$ we define $x_{P}(i)$ as the alternative ranked at position
$i$ according to $P$, i.e. $|\\{x\in X:xPx_{P}(i)\\}|=i-1$. Let $K^{*}$ be the
smallest such $K$. If $A_{i}=\\{x_{P_{2}}(K^{*}),...,x_{P_{2}}(N)\\}$. Then
$c^{\prime}_{i}(A_{i})=c_{P_{1}}(A_{i})\neq c_{P_{2}}(A_{i})=x_{P_{2}}(K^{*})$
but
$c^{\prime}_{i}(\\{x_{P_{2}}(K^{*}),x_{P_{2}}(K^{*}+1)\\})=c_{P_{2}}(\\{x_{P_{2}}(K^{*}),x_{P_{2}}(K^{*}+1)\\})=x_{P_{2}}(K^{*})$.
A violation of Sen’s property $\alpha$. If
$A_{i}\subset\\{x_{P_{2}}(K^{*}),...,x_{P_{2}}(N)\\}$ then
$c^{\prime}_{i}(\\{x_{P_{2}}(K^{*}),...,x_{P_{2}}(N)\\})=c_{P_{2}}(\\{x_{P_{2}}(K^{*}),...,x_{P_{2}}(N)\\})=x_{P_{2}}(K^{*})$
and $c^{\prime}_{i}(A_{i})=c_{P_{1}}(A_{i})\neq
c_{P_{2}}(A_{i})=x_{P_{2}}(K^{*})$. A violation of Sen’s property $\alpha$.

Let $\mu^{\prime}$ be the uniform distribution on
$c^{\prime}_{1},...,c^{\prime}_{k+1}$, i.e.

$\mu^{\prime}(c^{\prime}_{i})=\frac{1}{k+1}$

for all $i\in\\{1,...,k+1\\}$. To show that $\mu$ is an I-RUM, it remains to
show that the stochastic choices generated by $\mu$ and $\mu^{\prime}$ are the
same. There are two cases.

Case 1: If $A\neq A_{i}$ for all $i\in\\{1,...,k+1\\}$ then
$c^{\prime}_{i}(A)=c_{P_{2}}(A)=c_{P_{1}}(A)$ for all $i\in\\{1,...,k+1\\}$
and it is clear that the stochastic choices generated by $\mu$ and
$\mu^{\prime}$ are the same (i.e. that $\rho_{\mu}=\rho_{\mu^{\prime}}$).

Case 2: If $A=A_{i}$ for some $i\in\\{1,...,k\\}$. Then
$c^{\prime}_{i}(A)=c_{P_{1}}(A)$ and $c^{\prime}_{j}(A)=c_{P_{2}}(A)$ for all
$j\neq i$. It hence follows that

$\sum_{j=1}^{n}\mu^{\prime}(c^{\prime}_{j})\mathbb{1}\\{a=c^{\prime}_{j}(A)\\}=\frac{1}{k+1}\mathbb{1}\\{a=c^{\prime}_{i}(A)\\}+\sum_{j\neq
i}^{n}\frac{1}{k+1}\mathbb{1}\\{a=c^{\prime}_{j}(A)\\}=$

$=\frac{1}{k+1}\mathbb{1}\\{a=c_{P_{1}}(A)\\}+\frac{k}{k+1}\mathbb{1}\\{a=c_{P_{2}}(A)\\}$

$=\mu(P_{1})\mathbb{1}\\{a=c_{P_{1}}(A)\\}+\mu(P_{2})\mathbb{1}\\{a=c_{P_{2}}(A)\\},$

where the last equality follows since $k\mu(P_{1})=\mu(P_{2})$ implies that
$\mu(P_{1})=\frac{1}{k+1}$ and $\mu(P_{2})=\frac{k}{k+1}$. ∎

######  Lemma B.4.

Let $\mu$ be a RUM such that the correlation bounds are satisfied with
equality. Then there are Dual RUMs $(\hat{\mu_{i}})_{i=1}^{n}$ such that

  1. (1)

$\mu$ is a convex combination of the $(\hat{\mu_{i}})_{i=1}^{n}$, i.e. there
are weights $\delta_{i}$ s.t. $\mu=\sum_{k=1}^{n}\delta_{i}\hat{\mu_{i}}$,

  2. (2)

each Dual RUM $\hat{\mu_{i}}$ satisfies the correlation bounds with equality.

###### Proof.

Order the set of preferences as $\\{P_{0},...,P_{n}\\}$. Let $\mu$ be a RUM
that satisfies the correlation bounds with equality. W.l.o.g. assume that the
correlation bound is satisfied with equality at $P_{0}$ (i.e. we have
$K-2=\sum\limits_{i=0}^{n}n(P_{0},P_{i})\mu(P)$). We next construct a series
of Dual RUMs that satisfies (1) and (2) in the statement of the lemma.

For each $i\in\\{1,...,n\\}$ we define a dual RUM $\hat{\mu}_{i}$ with support
on $\\{P_{0},P_{i}\\}$ and such that the correlation bounds are satisfied with
equality

$\hat{\mu}_{i}(P_{0})=\frac{K-2-n(P_{0},P_{i})}{K-1-n(P_{0},P_{i})}$
$\hat{\mu}_{i}(P_{i})=\frac{1}{K-1-n(P_{0},P_{i})}$

The weights $(\delta_{i})_{i=1}^{n}$ on the Dual RUMs are then defined for
each $i\in\\{1,...,n\\}$ as follows:

$\delta_{i}=\mu(P_{i})[K-1-n(P_{0},P_{i})]$

We show that $\sum_{i=1}^{n}\delta_{i}=1$, i.e. that the $\delta_{i}$ are
indeed weights (that $\delta_{i}\geq 0$ is obvious). We have that

$\sum_{i=1}^{n}\delta_{i}=\sum_{i=1}^{n}\mu(P_{i})[K-1-n(P_{0},P_{i})]=$

$[K-1][1-\mu(P_{0})]-\sum_{i=1}^{n}n(P_{0},P_{i})\mu(P_{i})=$

$1+K-2-\sum_{i=0}^{n}n(P_{0},P_{i})\mu(P_{i})=1,$

where the first equality follows by definition of the weights $\delta_{i}$,
the second equality follows since
$\sum_{i=1}^{n}\mu(P_{i})[K-1]=[K-1][1-\mu(P_{0})]$, the third equality
follows by noting that $n(P_{0},P_{0})\mu(P_{0})=[K-1]\mu(P_{0})$ and
rearranging, and the final equality follows since the correlation bound holds
with equality at $P_{0}$.

It remains to show that $\mu$ is a convex combination of the Dual RUMs
$(\hat{\mu}_{i})_{i=1}^{n}$ defined above. It is clear that for all
$i\in\\{1,...,n\\}$ it holds that

$\mu(P_{i})=\delta_{i}\hat{\mu}_{i}(P_{i})=\sum_{j=1}^{n}\delta_{j}\hat{\mu}_{j}(P_{i}),$

where the first equality follows by construction of $\delta_{i}$ and
$\hat{\mu}_{i}$, and the second equality follows since
$\hat{\mu}_{j}(P_{i})=0$ for all $j\neq i$. Finally, we note that

$\mu(P_{0})=1-\sum_{i=1}^{n}\mu(P_{i})=\sum_{i=1}^{n}[\delta_{i}-\delta_{i}\hat{\mu}_{i}(P_{i})]=$
$\sum_{i=1}^{n}\delta_{i}[1-\hat{\mu}_{i}(P_{i})]=\sum_{i=1}^{n}\delta_{i}\hat{\mu}_{i}(P_{0}).$

∎

######  Lemma B.5.

Let $\rho$ be a RUM with distribution $\mu$ (so $\rho=\rho_{\mu}$) that
satisfies the correlation bounds with strict inequality then $\rho$ is not an
extreme point.

###### Proof.

Let $\mu$ be a RUM $\mu$ that satisfies the correlation bounds with strict
inequality (i.e. $\mathbb{C}^{\rho}_{P}<1$ for all preferences $P$). Clearly,
$\mu$ has support of cardinality greater or equal than two. Let $P_{1},P_{2}$
be two (distinct) preferences in the support of $\mu$. Let $\alpha$ be a
distribution with support on $P_{1},P_{2}$ such that the Dual RUM $\rho^{d}$,
defined for all $a\in A\subseteq X$ by

$\rho^{d}(a,A)=\alpha(P_{1})\mathbb{1}\\{a=\max(P_{1},A)\\}+\alpha(P_{2})\mathbb{1}\\{a=\max(P_{2},A)\\},$

satisfies the correlation bounds with equality (it could also be constructed
to satisfy the correlation bounds with strict inequality, it is immaterial for
the argument). For each $\beta\in(0,1)$ such that
$\mu(P_{i})-\beta\alpha(P_{i})>0$ for all $i\in\\{1,2\\}$ (or equivalently
$\beta<\min\\{\frac{\mu(P_{1})}{\alpha(P_{1})},\frac{\mu(P_{2})}{\alpha(P_{2})}\\}$)
define a RUM $\mu^{*}_{\beta}$ by

$\mu^{*}_{\beta}(P)=\begin{cases}\frac{\mu(P)-\beta\alpha(P)}{1-\beta}&\text{if
$P\in\\{P_{1},P_{2}\\}$}\\\
\frac{\mu(P)}{1-\beta}&\text{otherwise.}\end{cases}$

Let $\rho^{*}(\beta)=\rho_{\mu^{*}_{\beta}}$, i.e. $\rho^{*}(\beta)$ is the
RUM stochastic choice function with distribution $\mu^{*}_{\beta}$. For each
preference $P$ define a function $f_{P}$ by

$f_{P}(\beta)=\mathbb{P}^{\rho^{*}(\beta)}_{c_{P}}$

for all
$\beta\in[0,\min\\{\frac{\mu(P_{1})}{\alpha(P_{1})},\frac{\mu(P_{2})}{\alpha(P_{2})}\\})$.
First note that
$f_{P}(0)=\mathbb{C}^{\rho^{*}(0)}_{c_{P}}=\mathbb{C}^{\rho}_{c_{P}}$ for all
preferences $P$. Hence, since the correlation bounds are satisfied with strict
inequality we have $f_{P}(0)<1$ for all preferences $P$. Define a (vector-
valued) function $f(\beta):=(f_{P}(\beta))_{P\in\mathcal{P}}$. Since each
$f_{P}$ is a continuous function of $\beta$ it follows that $f$ is a
continuous function of $\beta$ and hence there is a $\hat{\beta}>0$ such that
$f_{P}(\beta)<1$ for all $0\leq\beta<\hat{\beta}$ and for all preferences $P$.
Let $\beta$ be such that
$0<\beta<\hat{\beta}<\min\\{\frac{\mu(P_{1})}{\alpha(P_{1})},\frac{\mu(P_{2})}{\alpha(P_{2})}\\}$.
We then have that $\rho=\beta\rho^{d}+(1-\beta)\rho^{*}(\beta)$ and since both
$\rho^{d}$ and $\rho^{*}(\beta)$ satisfy the correlation bounds, it follows
that $\rho$ is not an extreme point (as it can be written as a non-trivial
convex combination of two other RUMs that satisfy the correlation bounds). ∎

##  Appendix C Other proofs omitted from main text

###  C.1. Proof of Theorem 4.4

For each menu $A$ define a relation $\sim_{A}$ by $a\sim_{A}b$ if and only if
$\rho(a,A)>0$ and $\rho(b,A)>0$. We will use these relations in the proof of
theorem 4.4 below.

###### Proof.

We show that (1) $\Rightarrow$ (2) $\Rightarrow$ (3) $\Rightarrow$ (1).

(1) $\Rightarrow$ (2): By proposition 4.5 it follows that any RUM (hence any
partial I-RUM) with support consisting of three or more preferences is a non-
trivial convex combination of an I-RUM and a dual RUM. If $\rho$ is a partial
I-RUM with support consisting of at most two preferences, then either $\rho$
is an I-RUM in which case it is a convex combination of itself (a dual RUM)
and an I-RUM. Otherwise, $\rho$ is a convex combination of a preference (a
degenerate dual RUM) and an irrational choice function.

(2) $\Rightarrow$ (3): It is clear that if (2) holds then $\rho_{\mu}$ is a
partial I-RUM. Suppose that (3) is not true. We claim that there for every
menu $B\subseteq X$ with $|B|\geq 3$ is a $c(B)$ with $\rho(c(B),B)=1$. If not
then there are $a,b\in B$ with $\rho(a,B)>0$ and $\rho(b,B)>0$ and regularity
of RUM141414I.e. every RUM stochastic choice function satisfies regularity
which states that $\rho(a,B)\geq\rho(a,A)$ for all $a\in B\subseteq A$. then
implies that $\rho(a,ab)>0$ and $\rho(b,ab)>0$. Hence $a\sim_{B}b$ and
$a\sim_{ab}b$. A contradiction.

The previous paragraph implies that there is exactly one binary subset
$\\{a,b\\}$ with $\rho(a,ab)>0$ and $\rho(b,ab)>0$. Assume, there is a binary
subset $\\{c,d\\}\neq\\{a,b\\}$ with $\rho(c,cd)>0$ and $\rho(d,cd)>0$. By the
previous paragraph there is an $x\in\\{a,b,c,d\\}$ with $\rho(x,abcd)=1$.
Assume w.l.o.g. that $x=c$, then $1=\rho(c,abcd)\leq\rho(c,cd)$, implying that
$\rho(d,cd)=0$. A contradiction.

(3) $\Rightarrow$ (1): Let $\rho$ be a stochastic choice function such that
$\rho=\rho_{\mu}$ for some RUM $\mu$ and assume that there are
$A,B\in\mathcal{A}$ and alternatives $a,b\in A\cap B$ with $a\sim_{A}b$ and
$a\sim_{B}b$. We construct a partial I-RUM that represents $\rho$. Since there
are $a,b\in A\cap B\subseteq X$ with $\rho(a,A)>0,\rho(b,A)>0$ and
$\rho(a,B)>0,\rho(b,B)>0$, there is a $B\subseteq X$ with $|B|\geq 3$ and
$a,b\in B$ such that $\rho(a,B)>0$ and $\rho(b,B)>0$. Let $P_{1}$ be a
preference in the support of $\mu$ such that $\max(P_{1},B)=a$ and $P_{2}$ a
preference in the support of $\mu$ such that $\max(P_{2},B)=b$. Assume
w.l.o.g. that $\mu(P_{1})\leq\mu(P_{2})$. Construct choice functions
$c^{\prime}_{1},c^{\prime}_{2}$ such that

$c^{\prime}_{i}(A)=\begin{cases}\max(P_{i},A)&\text{if $A\neq B$}\\\
\max(P_{j},A)&\text{if $A=B$}\\\ \end{cases}$

It is clear that both $c_{1}$ and $c_{2}$ are irrational. Let $\mu^{\prime}$
be an RCM such that $\mu^{\prime}(c_{P})=\mu(c_{P})$ for all
$P\in\mathcal{P}\setminus\\{P_{1},P_{2}\\}$ and
$\mu^{\prime}(c^{\prime}_{1})=\mu(c_{P_{1}})$ and
$\mu^{\prime}(c^{\prime}_{2})=\mu(c_{P_{1}})$ and
$\mu^{\prime}(c_{P_{2}})=\mu(c_{P_{2}})-\mu(c_{P_{1}})$. It is straightforward
to check that $\mu^{\prime}$ represents $\rho$, i.e. that
$\rho=\rho_{\mu^{\prime}}$ and hence it follows that $\rho$ is a partial
I-RUM. ∎

###  C.2. Proof of proposition 4.5

###### Proof.

We prove the claim by induction on $k$ where $k$ is the cardinality of the
support of the RUM.

_Base cases ( $k=1,2$)_. These cases are trivial as RUMs with support of
cardinality smaller than two already satisfy the conclusion of the claim.

_Induction hypothesis and induction step._ As induction hypothesis assume that
the claim is true for all $1\leq k<m$ (where $m>2$). To prove the induction
step let $k=m$. Then the support of $\mu$ consists of preferences
$P_{1},...,P_{m}$. Define an equivalence relation $\sim$ on
$\\{P_{1},...,P_{m}\\}$ by $P_{i}\sim P_{j}$ if and only if $P_{i}$ and
$P_{j}$ rank all alternatives, except the worst two, the same. Order the
equivalence classes of $\sim$ as $E_{1},...,E_{n}$ where
$\mu(E_{1})\leq...\leq\mu(E_{n})$ and $2\leq n\leq m$. Let $P_{11}$ be a
preference in $E_{1}$ and $P_{22}$ a preference in $E_{2}$. Since $P_{11}$ and
$P_{22}$ belong to different equivalence classes there is a set of three
alternatives $\\{a,b,c\\}$ such that
$\max(P_{11},\\{a,b,c\\})\neq\max(P_{22},\\{a,b,c\\})$. Define a choice
function $c_{11}$ such that

$c_{11}(A)=\begin{cases}\max(P_{11},A)&\text{if $A\neq\\{a,b,c\\}$}\\\
\max(P_{22},A)&\text{if $A=\\{a,b,c\\}$}\\\ \end{cases}$

Similarly, define $c_{22}$ by

$c_{22}(A)=\begin{cases}\max(P_{22},A)&\text{if $A\neq\\{a,b,c\\}$}\\\
\max(P_{11},A)&\text{if $A=\\{a,b,c\\}$}\\\ \end{cases}$

It is straightforward to check that both of these choice functions are non-
rational (as they violate WARP). Consider the RCM $\mu^{\prime}$ defined by
$\mu^{\prime}(c_{11})=\mu(P_{11})$ and $\mu^{\prime}(c_{22})=\mu(P_{11})$ and
with $\mu^{\prime}(c_{P_{22}})=\mu(P_{22})-\mu(P_{11})$ and
$\mu^{\prime}(c_{P})=\mu(P)$ for all
$P\in\\{P_{1},...,P_{n}\\}\setminus\\{P_{11},P_{22}\\}$. It is straightforward
to check that $\rho_{\mu^{\prime}}(a,A)=\rho_{\mu}(a,A)$ for all $A\subseteq
X$ and $a\in A$. Next, we note that we can write $\mu^{\prime}$ as a convex
combination of $c_{11}$, $c_{22}$ and a RUM $\mu^{\prime\prime}$ with support
of cardinality less than or equal to $m-1$. But this later RUM
$\mu^{\prime\prime}$ can, by the induction hypothesis, be written as an RCM
with support consisting of at most two preferences. The claim now follows. ∎

###  C.3. Proof of proposition 5.1

###### Proof.

Let $\mathtt{SCF}$ be the collection of all stochastic choice functions.
Define a function $F:\mathtt{SCF}\to\mathbb{R}$ by

$F(\rho)=\min_{a\in A:A\subseteq X}\text{BM}_{\rho}(a,A).$

Viewing $\mathtt{SCF}$ as a convex subset of $\mathbb{R}^{d}$ (for suitable
$d$) it is straightforward to check that $F$ is a continuous function (to see
this, first note that $\text{BM}_{\rho}(a,A)$ is continuous for fixed $a\in A$
and $A\subseteq X$, and since the minimum of a finite number of continuous
functions is continuous it follows that $F$ is continuous). Let
$\mathtt{M}^{*}$ denote the closure of $\mathtt{M}$. Since $\mathtt{M}^{*}$ is
compact and $F$ is continuous there is a $M>0$ such that $F(\rho)\geq-M$ for
all $\rho\in\mathtt{M}^{*}$ (hence for all $\rho\in\mathtt{M}$). Since
$\rho_{\mu}$ is a full support RUM there is an $\beta>0$ such that
$\text{BM}_{\rho_{\mu}}(a,A)\geq\beta$ for all $a\in A$ and $A\subseteq X$.
Let $1>\bar{\alpha}>0$ be large enough such that
$\bar{\alpha}>\frac{M}{\beta+M}>0$ (such a number clearly exists). Let
$\alpha\geq\bar{\alpha}$. We need to prove that
$\rho^{*}=\alpha\rho_{\mu}+(1-\alpha)\rho$ is a RUM.

Let $\alpha\geq\bar{\alpha}$ and $\rho\in\mathtt{M}$. We need to prove that
$\rho^{*}=\alpha\rho_{\mu}+(1-\alpha)\rho$ is a RUM. It suffices to show that
the BM-polynomials of $\rho^{*}$ are non-negative. Let $a\in A$ and
$A\subseteq X$. There are two cases.

Case 1. Assume first that $\text{BM}_{\rho}(a,A)\geq 0$. By lemma A.3 we then
have

$\text{BM}_{\rho^{*}}(a,A)=\alpha\text{BM}_{\rho_{\mu}}(a,A)+(1-\alpha)\text{BM}_{\rho}(a,A)\geq
0$

Case 2. Assume next that $\text{BM}_{\rho}(a,A)<0$. Then

$\text{BM}_{\rho^{*}}(a,A)=\alpha\text{BM}_{\rho_{\mu}}(a,A)+(1-\alpha)\text{BM}_{\rho}(a,A)\geq$
$\bar{\alpha}\text{BM}_{\rho_{\mu}}(a,A)+(1-\bar{\alpha})\text{BM}_{\rho}(a,A)\geq$
$\bar{\alpha}\beta+(1-\bar{\alpha})(-M)=\bar{\alpha}(\beta+M)-M>0.$

The first (in)equality follows by lemma A.3 (i.e. convexity of the BM
operator). The second (in)equality follows since $\alpha\geq\bar{\alpha}$ and
$\text{BM}_{\rho_{\mu}}(a,A)>\text{BM}_{\rho}(a,A)$ and the third (in)equality
follows since $\text{BM}_{\rho_{\mu}}(a,A)\geq\beta$ and since
$\text{BM}_{\rho}(a,A)\geq F(\rho)\geq-M$. ∎

## References

  * Aguiar & Kashaev (2021) Aguiar, V. H., & Kashaev, N. (2021).  Stochastic revealed preferences with measurement error.  The Review of Economic Studies, 88(4), 2042–2093.
  * Apesteguia et al. (2017) Apesteguia, J., Ballester, M. A., & Lu, J. (2017).  Single-crossing random utility models.  Econometrica, 85(2), 661–674.
  * Arrow (1959) Arrow, K. J. (1959).  Rational choice functions and orderings.  Economica, 26(102), 121–127.
  * Barberá & Pattanaik (1986) Barberá, S., & Pattanaik, P. K. (1986).  Falmagne and the rationalizability of stochastic choices in terms of random orderings.  Econometrica, (pp. 707–715).
  * Becker (1962) Becker, G. S. (1962).  Irrational behaviour and economic theory.  Journal of Political Economy, (70), 1–13.
  * Block & Marschak (1960) Block, H., & Marschak, J. (1960).  Random orderings and stochastic theories of responses.  Contributions to Probability and Statistics, (Stanford University Press).
  * Blundell et al. (2003) Blundell, R. W., Browning, M., & Crawford, I. A. (2003).  Nonparametric engel curves and revealed preference.  Econometrica, 71(1), 205–240.
  * Dardanoni et al. (2023) Dardanoni, V., Manzini, P., Mariotti, M., Petri, H., & Tyson, C. J. (2023).  Mixture choice data: revealing preferences and cognition.  Journal of Political Economy, 131(3), 687–715.
  * Dardanoni et al. (2020) Dardanoni, V., Manzini, P., Mariotti, M., & Tyson, C. (2020).  Inferring cognitive heterogeneity from aggregate choices.  Econometrica, 88(3), 1269–1296.
  * Deb et al. (2023) Deb, R., Kitamura, Y., Quah, J. K., & Stoye, J. (2023).  Revealed price preference: theory and empirical analysis.  The Review of Economic Studies, 90(2), 707–743.
  * Falmagne (1978) Falmagne, J.-C. (1978).  A representation theorem for finite random scale systems.  Journal of Mathematical Psychology, 18(1), 52–72.
  * Filiz-Ozbay & Masatlioglu (2023) Filiz-Ozbay, E., & Masatlioglu, Y. (2023).  Progressive random choice.  Journal of Political Economy, 131(3), 716–750.
  * Fiorini (2004) Fiorini, S. (2004).  A short proof of a theorem of falmagne.  Journal of mathematical psychology, 48(1), 80–82.
  * Gilboa (1990) Gilboa, I. (1990).  A necessary but insufficient condition for the stochastic binary choice problem.  Journal of Mathematical Psychology, 34(4), 371–392.
  * Grandmont (1992) Grandmont, J.-M. (1992).  Transformations of the commodity space, behavioral heterogeneity, and the aggregation problem.  Journal of Economic Theory, 57(1), 1–35.
  * Hoderlein & Stoye (2014) Hoderlein, S., & Stoye, J. (2014).  Revealed preferences in a heterogeneous population.  Review of Economics and Statistics, 96(2), 197–213.
  * Hoderlein & Stoye (2015) Hoderlein, S., & Stoye, J. (2015).  Testing stochastic rationality and predicting stochastic demand: the case of two goods.  Economic Theory Bulletin, 3(2), 313–328.
  * Im & Rehbeck (2022) Im, C., & Rehbeck, J. (2022).  Non-rationalizable individuals and stochastic rationalizability.  Economics Letters, 219, 110786.
  * Kitamura & Stoye (2018) Kitamura, Y., & Stoye, J. (2018).  Nonparametric analysis of random utility models.  Econometrica, 86(6), 1883–1909.
  * Luce & Raiffa (1957) Luce, R., & Raiffa, H. (1957).  Introduction and critical survey.
  * Manzini & Mariotti (2018) Manzini, P., & Mariotti, M. (2018).  Dual random utility maximisation.  Journal of Economic Theory, 177, 162–182.
  * Matzkin (2007) Matzkin, R. L. (2007).  Heterogeneous choice.  Econometric Society Monographs, 43, 75.
  * McCausland et al. (2020) McCausland, W. J., Davis-Stober, C., Marley, A. A., Park, S., & Brown, N. (2020).  Testing the random utility hypothesis directly.  The Economic Journal, 130(625), 183–207.
  * McFadden & Richter (1990) McFadden, D., & Richter, M. K. (1990).  Stochastic rationality and revealed stochastic preference.  Preferences, Uncertainty, and Optimality, Essays in Honor of Leo Hurwicz, Westview Press: Boulder, CO, (pp. 161–186).
  * McFadden (2006) McFadden, D. L. (2006).  Revealed stochastic preference: a synthesis.  In Rationality and Equilibrium: A Symposium in Honor of Marcel K. Richter, (pp. 1–20). Springer.
  * Monderer (1992) Monderer, D. (1992).  The stochastic choice problem: A game-theoretic approach.  Journal of Mathematical Psychology, 36(4), 547–554.
  * Petri (2023) Petri, H. (2023).  Random (ordered) multivalued choice.  Working paper.
  * Rockafellar (1970) Rockafellar, R. T. (1970).  Convex Analysis.  Princeton: Princeton University Press.
URL https://doi.org/10.1515/9781400873173

  * Sen (1971) Sen, A. (1971).  Choice functions and revealed preference.  The Review of Economic Studies, 38(3), 307–317.
  * Smeulders et al. (2021) Smeulders, B., Cherchye, L., & De Rock, B. (2021).  Nonparametric analysis of random utility models: computational tools for statistical testing.  Econometrica, 89(1), 437–455.
  * Stoye (2019) Stoye, J. (2019).  Revealed stochastic preference: A one-paragraph proof and generalization.  Economics Letters, 177, 66–68.
  * Suleymanov (2023) Suleymanov, E. (2023).  Branching-independent random utility model.  Working paper.
  * Turansick (2022) Turansick, C. (2022).  Identification in the random utility model.  Journal of Economic Theory, 203.