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Quantile Restrictions, Revealed Rankings, and the Limits of Multinomial Choice
\title{Quantile Restrictions, Revealed Rankings, and the Limits of Multinomial Choice}
\author{Tatiana Komarova\thanks{Faculty of Economics, University of Cambridge. [email removed].}}
\date{August 14, 2026}
\maketitle
\begin{abstract}
This paper analyzes when choice probabilities reveal rankings of deterministic utility indices in semiparametric discrete choice models. It begins with binary choice, where quantile thresholds guarantee ranking recovery, and shows that such thresholds can arise either from behavioral departures from utility maximization (e.g., limited attention) under exchangeable unobservables, or from non-exchangeable unobservables under standard utility maximization. These behavioral and distributional routes are then extended to multinomial choice. Under limited attention, balance restrictions on attention probabilities yield global linear ranking partitions which are robust to the distribution of unobservables and, given sufficiently rich joint variation in the differences of utility indices, are also necessary. Absent the required attention restrictions, opposite rankings can produce overlapping probability images. Under non-exchangeable unobservables, a comparable distribution-uniform partition generally need not exist. \\
Holding the distribution fixed, however, ranking recovery remains possible via an injective nonlinear map from normalized utility differences to choice probabilities under both behavioral and distributional extensions. Together, the results distinguish distribution-robust global ranking partitions from ranking recovery with a fixed distribution of unobservables and clarify the limits of extending binary quantile restrictions to multinomial choice.
\vspace{0.05in}
\begin{description}
\item[Keywords:] Binary choice; Multinomial choice; Quantile restrictions; Probability simplex; Ranking; Ranking ambiguity; Consideration sets; Non-exchangeability
\end{description}
\end{abstract}
\setstretch{1.35}
\newpage
\section{Introduction}
Semiparametric discrete choice models often obtain identifying content from weak restrictions on the conditional distribution of unobservables rather than from a fully parametric specification. A classic example is the binary response model studied by \citet{Manski1985,Manski1988}, $Y=1(x\beta+\varepsilon\geq 0)$, together with the conditional median restriction $Med(\varepsilon| x)=0$. Under this restriction, observed choice probabilities reveal the sign of the deterministic utility index as in
\begin{equation}
\label{medianECON}
P(Y=1| x)\geq (\leq)\frac12
\quad\Longleftrightarrow\quad
x\beta\geq (\leq)0.
\end{equation}
More generally, if
$Q_\tau(\varepsilon| x)=0$,
then
\begin{equation}
\label{quantilenECON}
P(Y=1| x)\geq (\leq)1-\tau
\quad\Longleftrightarrow\quad
x\beta\geq (\leq)0.
\end{equation}
The econometric content of this representation is well known. Geometrically, in \eqref{quantilenECON} a single threshold $1-\tau$ divides the one-dimensional probability simplex into two regions corresponding to the two possible rankings of the deterministic utilities. That is, once the location of $P(Y=1| x)$ relative to the threshold is known, the ranking is known as well.
This paper analyzes whether an analogous idea can be extended to multinomial choice. With $J>2$ alternatives, the vector of choice probabilities lies in a $(J-1)$-dimensional simplex and there is no longer a single probability threshold. I use the term \emph{``quantile'' multinomial choice} to signify the property that a specified probability domain
can be partitioned into regions such that membership in a
region reveals a unique ranking of the deterministic utility indices. In the strongest results below this domain is the ambient probability simplex itself. In some weaker results, it is the common structural probability image generated by the fixed conditional distribution of unobservables. The ``quantile'' terminology is not meant to be associated with a conventional multivariate quantile notion.
A weaker property than a full ``quantile '' model is \emph{ranking recovery} when each attainable
probability vector has a unique ordinal interpretation, even when the attainable probability set covers only part of the ambient probability domain.
The extension from two to several alternatives is not automatic. With two alternatives, all relevant variation operates along a single comparison. With three or more alternatives, the probability of choosing $j$ rather than $k$ can also be affected by what happens to other alternatives. As illustrated in the paper, generally probability vectors generated by different rankings of the deterministic indices can overlap. In other words, the same observed probability vector obtained under some $x$ and $\widetilde{x}$ may then be compatible with both $x_j\beta_j>x_k\beta_k$ and $\widetilde{x}_j\beta_j<\widetilde{x}_k\beta_k$. This failure of ordinal recovery can be understood as \emph{ranking ambiguity}. One of the central questions of this paper is what restrictions on the decision making or on the distribution of utility unobservables prevent such ambiguity.
I approach this question by first returning to the binary model and asking what economic structures can generate the quantile representation \eqref{quantilenECON}. In the classical random utility interpretation underlying the median model, agents maximize latent utility and the two utility unobservables are conditionally exchangeable.\footnote{Original \citet{Manski1975} restriction was conditional i.i.d. but \citet{GoereeHoltPalfrey2005} and later \citet{Fox2007} note that conditional exchangeability suffices for the rank order result. In binary choice, exchangeability also implies a zero median of the error difference.} Most semiparametric work takes the median or quantile restriction as a primitive. I look into rationalizing quantile restrictions in discrete choice models along two different routes. Either of them can generate representation \eqref{quantilenECON}. A \emph{behavioral route} maintains conditional exchangeability of the latent utility unobservables but allows decision making to differ from utility maximization. The decision rule may reflect limited consideration, noisy perception, or other stochastic behavior. A \emph{distributional route} maintains standard utility maximization but allows the joint distribution of the utility unobservables to be non-exchangeable.
I then carry these two routes into multinomial choice. The natural benchmark is the classical random utility model of \citet{Manski1975}. Under utility maximization with conditionally i.i.d. unobservables and some further standard \citet{Manski1975} conditions,
\begin{equation}
\label{Manskiranorder} P(Y=j|\mathbf x)\geq P(Y=k|\mathbf x)
\quad\Longleftrightarrow\quad
x_j\beta_j\geq x_k\beta_k,\footnote{Again, \citet{GoereeHoltPalfrey2005} and \citet{Fox2007} show that conditional exchangeability is enough to ensure this ranking relationship.}
\end{equation}
which I will refer to as Manski's rank order property. Pairwise equalities of choice probabilities, therefore, describe hyperplanes that split the probability simplex into regions corresponding to unique complete rankings of the deterministic utility indices. I use this classical case as the multinomial ``median'' benchmark and ask how much of its ranking geometry survives under limited attention and under non-exchangeable unobservables in latent utilities.
The paper makes three main substantive contributions and develops a
framework for distinguishing the robustness of the resulting ranking
claims.
The first main contribution is to give economic foundations for binary non-median (and median as a special case) probability thresholds. In line with the behavioral route mentioned above,
I describe general stochastic decision rules that give \eqref{quantilenECON} under conditionally exchangeable unobservables and go beyond standard utility maximization. These include limited attention, noisy perception, and random choice, with exogenous consideration sets providing a particularly transparent foundation for both median and non-median cases. For the second distributional route, I show how non-median thresholds can instead arise from asymmetry in the joint distribution of latent utility unobservables with agents still maximizing utility.
The second main contribution is to study ranking recovery and stronger ``quantile'' partitionings in multinomial models and to do so at different levels capturing different robustness properties. A \emph{Behavioral environment} level specifies a decision rule and an admissible class of conditional distributions of unobservables. A \emph{Structural environment} additionally fixes one such conditional distribution.\footnote{The main text also mentions an \emph{empirical specification} level, which also fixes the marginal distribution of the covariates, but does not study findings at this level as it requires a case-by-case analysis.} In addition to these two levels, the paper also studies which results apply for a fixed $\boldsymbol\beta$ or uniformly over all admissible $\boldsymbol\beta$.
The sharpest results arise in the behavioral route when looking at the exogenous consideration sets. Limited attention changes the geometry of the probability simplex since choice probabilities now combine utility comparisons across different menus. Nevertheless, particular restrictions on attention probabilities can restore linear pairwise ranking rules, generalizing \citet{Manski1975} partitioning. These are restrictive balancing conditions on attention probabilities and they become increasingly more restrictive as $J$ increases. When these conditions are satisfied, the conclusions are strong as the partitioning is not only (i) linear, (ii) fully determined by attention probabilities and (iii) applies to any absolutely continuous conditionally exchangeable distribution of unobservables, but is also (iv) uniform over the admissible index parameters $\boldsymbol{\beta}$. Thus, this is the case when ``quantile'' multinomial choice is obtained, which in particular implies the ranking recovery property for attainable probabilities. But when the balancing conditions on attention probabilities fail, the probability images of opposite pairwise rankings can overlap, producing ranking ambiguity. To my knowledge, this paper provides the first characterization of the exogenous attention structures that support distribution-robust linear ordinal recovery that includes asymmetric structures generating nonstandard probability boundaries,\footnote{That is, different from \citet{Manski1975} boundaries $\{p_k=p_{\ell}\}$.} together with corresponding necessity and failure results.
The distributional route gives a different conclusion. If the
conditional distribution of the utility unobservables is allowed to
vary over an admissible class of absolutely continuous, potentially
non-exchangeable distributions, there is in general no probability
partition that is common across that class.
The third main contribution is to show that ranking recovery and ``quantile'' partitionings can be restored under weaker robustness requirements of the \emph{Structural environment} level. In both the limited attention model and the model with non-exchangeable unobservables, a common location-scale structure induces a map from normalized deterministic utility index differences into choice probabilities. Under the conditions developed in the paper, this map is injective, so the probabilities again identify the ranking of the deterministic utility indices, ensuring ranking recovery property on the attainable set of probability vectors. The map is generally nonlinear. Under limited attention it depends on the attention probabilities, while under utility maximization with non-exchangeable unobservables it depends on the fixed distribution of the unobservables. Importantly, it does not depend on the parameter vector $\boldsymbol{\beta}$. This map is best understood as giving a common structural ranking partition for the given \emph{Structural environment}. The fact that the map is common to the environment supports an interpretation of this case as \emph{Structural environment} ``quantile'' multinomial choice when admissible values of index parameter $ \boldsymbol{\beta} $ jointly cover the maximal probability domain permitted by this location-scale structure.
Overall, a ``quantile'' interpretation of multinomial choice requires enough structure to generate a global ranking partition of the relevant probability domain. Whether that partition is over the ambient simplex and robust to the distribution of unobservables, or whether it is specific to a given \emph{Structural environment} is a separate robustness question.
There is also a fundamental asymmetry between binary and multinomial cases. In binary choice, behavioral and distributional distortions act on a single margin and a quantile representation can survive under much weaker conditions than an analogous ``quantile'' representation in multinomial choice.
The paper contributes to several related literatures. It builds directly on the semiparametric discrete choice framework of \citet{Manski1975,Manski1985,Manski1988}. It is also related to work on binary choice under quantile restrictions. \citet{Kordas2006} studies smoothed binary regression quantiles, while \citet{Volgushev2020} develops asymptotic theory for binary response quantile processes with linear quantile restrictions. That literature largely takes the quantile restriction as an econometric primitive. This paper instead asks which behavioral and distributional structures can generate the probability threshold and what remains of its ordinal interpretation when choice becomes multinomial. \citet{Matzkin1993} studies nonparametric identification of multinomial choice models with flexible deterministic utility functions and weak distributional restrictions, but preserves the classical pairwise ranking geometry of \citet{Manski1975}.
The paper is also related to the literature on consideration sets. \cite{Fox2007} also studies rank ordering in multinomial choice models with unobserved random consideration sets. His result preserves Manski's rank order property \eqref{Manskiranorder} by imposing restrictions on covariate-dependent consideration probabilities that essentially force the consideration mechanism to reinforce the Manski ranking \eqref{Manskiranorder}. In particular, in binary choice his assumptions force the model to preserve the classical Manski median threshold. In contrast, this paper holds attention probabilities fixed and characterizes the balance conditions under which ordinal recovery nevertheless emerges from the attention structure itself, allowing the relevant separators to differ from the standard comparison $\{p_k=p_{\ell}\}$.
\citet{BarseghyanMolinariThirkettle2021} develop a discrete choice model with heterogeneous risk preferences and unobserved consideration sets and provide conditions for its identification. In comparing limited consideration with standard random utility models, they introduce a conditional analogue of Manski’s rank order property \eqref{Manskiranorder} (conditional on latent risk preference type) and show that it implies a generalized dominance restriction on observed choice probabilities, a restriction that limited consideration can violate. An earlier working paper \citet{BarseghyanMolinariThirkettle2019} also studies a Random Consideration Level (RCL) model and shows that it preserves the conditional Manski's rank order property. RCL imposes a strong symmetry restriction, where conditional on consideration set size, all sets are equally likely, so swapping any two alternatives leaves the probability of the consideration set unchanged. This symmetry is a special case of the balance restrictions characterized here. The present paper allows asymmetric attention, characterizes the restrictions under which distribution-robust ordinal recovery remains possible, and derives the resulting—potentially non-Manski linear probability boundaries.
\citet{Allen2026} shows that, with bounded observable utility shifters, exogenous limited consideration can be observationally equivalent to full consideration, whereas this paper studies when it preserves ordinal recovery directly.
Other relevant consideration sets literature is \citet{MasatliogluNakajimaOzbay2012}, \citet{ManziniMariotti2014}, \citet{CattaneoMaMasatliogluSuleymanov2020}, and \citet{AguiarBoccardiKashaevKim2023}. The question here is different. Rather than treating consideration probabilities themselves as the principal object to be identified, I take the consideration structure as given and ask what ordinal information about deterministic utilities is encoded in the resulting choice probability geometry. In addition, the object of interest is not a single fixed preference ranking, but the ranking of deterministic utility indices as covariates vary. The relevant variation thus comes from the movement of the choice probabilities across covariate values and from whether the resulting probability images associated with different rankings remain separated. \citet{DardanoniManziniMariottiTyson2020} study recovery of cognitive heterogeneity from aggregate choice shares, showing that a single sufficiently large menu can suffice under homogeneous preferences.
Finally, the results at the \emph{Structural environment} level are
related to single-crossing arguments in
\citet{ApesteguiaBallesterLu2017} and
\citet{BarseghyanMolinariThirkettle2021}. Those papers use
single-crossing conditions in heterogeneous preferences or types as part
of their identification arguments. Here I do not impose a single-crossing ordering of
heterogeneous preferences as a primitive. Instead, for a fixed
\emph{Structural environment}, common location-scale restrictions
generate a generally multidimensional map from normalized deterministic
utility index differences into choice probabilities. Injectivity of
this map yields ranking recovery, while the range of attainable
normalized indices determines how much of the associated global ranking
partition is attained. Thus, both approaches exploit ordinal structure,
but in different spaces.
The rest of the paper is organized as follows. Section~\ref{sec:binary} studies binary choice and develops behavioral and distributional foundations for median and quantile threshold representations. Section~\ref{sec:multinomial} turns to multinomial choice, introduces the ranking-recovery framework and the levels of robustness, and studies limited attention and non-exchangeable utility unobservables. Section~\ref{sec:conclusion} concludes.
\section{Binary choice}
\label{sec:binary}
A standard starting point, including in \citet{Manski1985, Manski1988}, for a semiparametric binary choice model is the formulation
$$y=1(x\beta+\varepsilon \geq 0), \quad x \in \mathbb{R}^{1 \times k}, \quad \beta \in \mathbb{R}^{k \times 1},$$
with $x\beta$ describing the part of utility depending on the observed characteristics and potentially identifiable, whereas $\varepsilon$ describes the unobservable component driving it. The subsequent analysis of such a model relies on what is assumed about the relationship between the unobserved $\varepsilon$ and observed covariates $x$. A canonical restriction of that relationship comes from \citet{Manski1985, Manski1988} and relies on imposing the following constraint:
\begin{equation} \label{median} Med(\varepsilon|x)=0.
\end{equation}
This restriction still allows for rich dependencies on $x$ from the conditional distribution of $\varepsilon$ while at the same time providing an inferential context on utility indices $x \beta$ in terms of observed choice probabilities $P(Y=1|x)$, which I review in more detail below.
The motivation for the median restriction (\ref{median}) is standard and comes from \citet{Manski1975} random utility framework in which each option is associated with a latent utility and the agent chooses the option with the higher utility. In the binary case, let $j=0,1$ and define latent utilities as $U_j = x_j \beta_j + \varepsilon_j$, $x_j \in \mathbb{R}^{1 \times k_j}$, $\beta_j \in \mathbb{R}^{k_j \times 1}$ for $j=0,1$.
In the standard utility optimization the agent chooses option 1 if and only if $ U_1 \geq U_0 \iff \varepsilon_1 - \varepsilon_0 \geq -x_1\beta_1 + x_0\beta_0.$
Under the assumption that $(\varepsilon_0,\varepsilon_1)$ are i.i.d. conditional on $(x_0,x_1)$, the difference $\varepsilon_1 - \varepsilon_0$ has zero conditional median and the c.d.f of this difference conditional on $(x_0,x_1)$ is strictly increasing in the neighborhood of 0.
In what follows, define \(x\beta = x_1\beta_1 - x_0\beta_0\), where \(x\) collects all distinct covariates appearing in \(x_1\) or \(x_0\). The associated coefficient vector \(\beta\) assigns to each covariate the difference between its coefficients in \(x_1\beta_1\) and \(x_0\beta_0\), using \(0\) for absent covariates. Thus, \(x\) consolidates \((x_1,x_0)\) without redundancy. In particular, if \(x_1\) and \(x_0\) share no covariates, then
\(
x=(x_1,x_0)\), $\beta=(\beta_1^{\top},-\beta_0^{\top})^{\top}$.
Conditioning on \(x\) is therefore informationally equivalent to conditioning on \((x_1,x_0)\).
Denoting
$\varepsilon=\varepsilon_1-\varepsilon_0,$ we end up with the model
\begin{align}
\label{modelBC}
Y =1(x\beta+\varepsilon \geq 0), & \quad Med(\varepsilon|x)=0,\\
\text{c.d.f.} F_{\varepsilon|x}(\cdot|x) \text{ is strictly } & \text{increasing around 0}
\label{modelBC_part2}
\end{align}
(\ref{modelBC})-(\ref{modelBC_part2}) implies that observed choice probabilities \( P(Y=1 | x) \) are informative about the sign of the utility index in the following sense:
\[
P(Y=1 | x) \; \geq (\leq) \; P(Y=0 | x) \iff x\beta \geq (\leq) \; 0,
\]
or equivalently, \eqref{medianECON}.
Characterization (\ref{medianECON}) underlies both identification arguments and estimation methods in semiparametric binary choice models.
\citet{Manski1988} proposed a quantile-based semiparametric binary choice model:
\begin{equation}
\label{modelBCquantile}
Y = 1(x\beta + \varepsilon \geq 0), \quad Q_{\tau}(\varepsilon | x) = 0,
\end{equation}
plus the monotonicity of the conditional c.d.f. analogous to the above, but the quantile index \( \tau \in (0,1) \) need not equal \( 1/2 \). Choice probabilities are then informative as \eqref{quantilenECON} holds. Behavioral or distributional motivations for \eqref{modelBCquantile} or \eqref{quantilenECON} have received very little attention in the literature.
\begin{remark}\label{remark1}
Unlike \citet{Manski1985,Manski1988}, who work directly with \(U_1-U_0\), I keep \(U_1\) and \(U_0\) separate, allowing asymmetry between \(\varepsilon_1\) and \(\varepsilon_0\) and more general pre-choice evaluation of alternatives.
\end{remark}
\begin{remark}\label{remark2}
Since our focus is not identification of utility indices, the linear indices \(x_j\beta_j\), \(j=0,1\), could be replaced by general functions \(\phi_j(x_j)\), with \(x\beta\) replaced by \(\phi_1(x_1)-\phi_0(x_0)\). We maintain linearity for comparability with \citet{Manski1975,Manski1985,Manski1988}.
\end{remark}
\begin{remark}\label{remark3}
By the standard random utility paradigm, this paper means a setting in which the agent chooses the option with the highest realized latent utility.
\end{remark}
\subsection{Behavioral route: General theorem (any $\tau$)}
\label{sec:median}
We start by imposing the following assumption, which permits more general conditional dependence between \(\varepsilon_1\) and \(\varepsilon_0\) than the conditional i.i.d. structure in \citet{Manski1975}.
\begin{assumption}[Distribution of $(\varepsilon_1,\varepsilon_0)|x$] \label{assn:binary_distributionMED} Suppose the following hold:
\begin{enumerate}
\item[(a)] \(\varepsilon_1\) and \(\varepsilon_0\) are identically distributed conditional on \(x\), with \(\varepsilon_j| x\) absolutely continuous with respect to Lebesgue measure.
\item[(b)] The copula $C_x(u_1,u_0) :=C(u_1,u_0 |x)$ that describes the dependence between $\varepsilon_1$ and $\varepsilon_0$ conditional on $x$ is exchangeable (that is, $C_x(u,v)=C_x(v,u)$) and absolutely continuous.
\item[(c)] The support of the joint distribution of $(\varepsilon_1,\varepsilon_0)|x$ is convex and has a non-empty interior.
\end{enumerate}
\end{assumption}
Thus, the unobservables may be conditionally correlated, with dependence restricted to exchangeable copulas. This nests the conditional independence copula of \citet{Manski1975}.
\begin{remark}
Throughout, the aim is to derive conditions that are \emph{generic}, in the sense that they do not rely on particular features of the conditional distribution of the unobservables given $x$, nor on the marginal distribution of $x$ or on the value of $\beta$. This perspective is especially important when discussing necessity.
To that end, the paper abstracts from specifications in which restrictions on the joint support of $(x,\varepsilon)$ limit the range of the index $x\beta$ relative to the support of $\varepsilon$. Such cases can be viewed as non-generic, as they can typically be resolved by enriching the model by, for instance, expanding the support of $x$ or adjusting $\beta$ so that $x\beta$ spans the relevant range of $\varepsilon$.
The necessity results below are therefore stated for this enriched class, which we call \emph{generic models}.\footnote{In much of the classic literature, the model is effectively specified in this enriched form: \(\varepsilon\mid x\) has full support on \(\mathbb{R}\), and \(x\beta\) varies over \(\mathbb{R}\) with \(x\). These assumptions are mainly for analytical convenience and can be relaxed in some settings.} In the multinomial models of Section~\ref{sec:multinomial}, this idea is formalized through different layers of analysis but doing so here would be unnecessarily cumbersome.
\end{remark}
The theorem below characterizes the behavioral conditions under which
\eqref{quantilenECON} holds for any \(\tau \in (0,1)\) under Assumption~\ref{assn:binary_distributionMED}, while allowing departures from utility maximization. It nests the median specification~\eqref{medianECON} as the special case \(\tau=1/2\), again allowing deviations from the standard random utility paradigm.
\begin{theorem}[General theorem for any $\tau \in (0,1)$] \label{prop:median}
Let $U_j = x_j\beta_j+\varepsilon_j$, $j=0,1$, and let the distribution of $(\varepsilon_1, \varepsilon_0) $ satisfy Assumption \ref{assn:binary_distributionMED}.
\textbf{A.} The choice behavior of an economic agent can be represented as in (\ref{quantilenECON}) if the following conditions are satisfied:
\begin{enumerate}
\item[(1)] $P(Y=1| U_1 \geq U_0,x) -P(Y=1|U_0 > U_1,x) >0$ a.e.;
\item[(2)] $P(Y=1| U_1 \geq U_0,x) - P(Y=0|U_0 > U_1,x) = 1-2\tau$ a.e.
\end{enumerate}
Equivalently, Condition (2) can be written as $P(Y=1 | U_1 \ge U_0, x) + P(Y=1 | U_0 > U_1, x) = 2(1-\tau)$ a.e.
\textbf{B.} If the choice behavior of an economic agent can be represented as in (\ref{quantilenECON}) and condition (2) holds, then Condition (1) is necessary away from index indifference:
\begin{equation}
\label{condition1_weaker}
P(Y=1 | U_1 \ge U_0, x) - P(Y=1 | U_0 > U_1, x) > 0
\quad \text{a.e. on }
\{x:x\beta\neq0\}
\end{equation}
\textbf{C.} Condition (2) cannot be omitted from the distribution-free sufficiency result in Part A. More precisely, for every \(\tau\in(0,1)\) and every \(\beta \neq 0\), there exists a generic model satisfying Assumption 1 and the strict version of Condition (1), but violating Condition (2) on a set of positive measure, for which \eqref{quantilenECON} fails.
\end{theorem}
Condition (1) of Theorem \ref{prop:median} ensures that the agent is more likely to choose
option~1 when $U_1 \geq U_0$ than when $U_0 > U_1$. It rules out choice rules so perverse that higher utility of an option
reduces the probability of choosing it. This guarantees the expected monotonic relationship since as the index $x\beta$ crosses zero from below, $P(Y=1|x)$ crosses the threshold $1-\tau$ from below.\footnote{This monotonicity condition is distinct from the single-crossing property in random utility models (\citet{ApesteguiaBallesterLu2017}). Single-crossing imposes an ordering on preference heterogeneity such that rankings over alternatives change at most once across types, whereas the present condition is a much weaker reduced-form restriction on choice probabilities. }
Condition (2) of Theorem \ref{prop:median} relaxes the standard random utility paradigm implication
$P(Y=1 \mid U_1 \geq U_0, x)
=
P(Y=0 \mid U_0 > U_1, x)
=1.$ Under Assumption \ref{assn:binary_distributionMED}, this implication would yield the median condition \eqref{median}. Theorem \ref{prop:median} instead allows choice reversals relative to this benchmark, even when \(\tau=1/2\), provided they are independent of which alternative has higher realized utility. For \(\tau\neq 1/2\), the theorem gives the first behavioral characterization in the literature of the representation \eqref{quantilenECON}.\footnote{If Condition (2) holds but Condition (1) fails, monotonicity may reverse, yielding the unintended implication \(P(Y=1| x)\leq 1-\tau \iff x\beta\geq 0\).}
Part A in Theorem \ref{prop:median} is about sufficiency, whereas parts B and C establish near-necessity. Part B cannot be strengthened to include $x$ with $x\beta=0$. Part C shows that if Condition (1) holds strictly, then violation of
Condition (2) generically invalidates \eqref{quantilenECON}.
Note that Theorem \ref{prop:median} does not require choices to depend only on \(U_1-U_0\). Choices may depend on the levels of \(U_1\) and \(U_0\) as well, provided the theorem's two conditions hold with examples of such behavioral models given below. Before presenting examples, let's summarize some implications of model (\ref{quantilenECON}) on the choice probabilities.
\begin{prop}\label{prop:implications}
Consider the setting in Theorem \ref{prop:median} and suppose conditions (1) and (2) there hold. Then
$P(Y=1|U_1<U_0,x) \leq P(Y=1|x) \leq P(Y=1|U_1\geq U_0,x)$ a.e., and
\begin{itemize}
\item[(i)] for $\tau>1/2$,
$P(Y=1|x) \leq 2(1-\tau)$;
\item[(ii)] for $\tau<1/2$, $P(Y=1|x) \geq 2(1-\tau)-1$.
\end{itemize}
\end{prop}
Thus, Proposition~\ref{prop:implications} yields a testable implication: when $\tau \neq 1/2 $, the choice probability $P(Y=1| x)$ is bounded strictly
away from at least one boundary. This is driven by the fact that under conditions of Proposition~\ref{prop:implications},
$P(Y=1|U_1 \geq U_0,x)+P(Y=1|U_1 < U_0,x)=2(1-\tau) \neq 1.$
In the median model $P(Y=1|x)$ can approach 1 and 0 arbitrarily closely on sets on positive (but decreasing) measure of $x$.
We now give some examples of behavioral models that go beyond the traditional random utility framework while still admitting the representation ~(\ref{medianECON}) based on Theorem \ref{prop:median}.
\begin{example}[Mixture of random choice and utility maximization, $\tau=1/2$]\label{ex:randchoice}
For an economic agent characterized by $x$, with probability $r(x) \in (0,1)$, the choice between 1 and 0 is made by tossing a fair coin, and with probability $1-r(x)$, the choice is made on the basis of comparing latent utilities as in the standard random utility paradigm. Then $P(Y=1| U_1\geq U_0,x) = r(x)/2 + (1-r(x)) = 1-r(x)/2$ and
$P(Y=1| U_0>U_1,x) = r(x)/2$.
Condition (1) holds since $1-r(x)/2 - r(x)/2 = 1-r(x)>0$. Condition (2) holds for $\tau=1/2$ since $P(Y=1|U_1 \geq U_0, x)$ and $P(Y=1|U_1 < U_0, x)$ sum to 1.\end{example}
\begin{example}[Noisy Perception Model, $\tau=1/2$]\label{ex:noisy} Suppose an agent does not know their $U_j$ and instead perceives their noisy versions $\xi\cdot U_j$ where $\xi$ is independent of
$(U_0,U_1)$ conditional on $x$ with $P(\xi\geq 0\mid x) > P(\xi<0\mid x)$\footnote{Equivalently, $P(\xi \geq 0 | x)>1/2$. This is not a strong requirement especially taking into account that $\xi$ centered around 1 would give us the disturbed latent utilities centered around true $U_1,U_0$.} ($\xi$ and $x$ can be dependent). Agent's choice is made by maximizing these noisy versions. Hence, $P(Y=1|U_1 \geq U_0, x) = P( \xi \cdot U_1 \geq \xi \cdot U_0 |U_1 \geq U_0, x) = P(\xi \geq 0 | x)$ and $P(Y=1|U_1 < U_0, x) = P(\xi < 0 | x)$. Condition (1) in Theorem \ref{prop:median} is guaranteed by $P(\xi \geq 0 | x) - P(\xi < 0 | x)>0$. Condition (2) holds too a.e. for $\tau=1/2$ as $P(Y=1|U_1 \geq U_0, x)$ and $P(Y=1|U_1 < U_0, x)$ sum to 1. $\blacksquare$
\end{example}
Neither of these two examples give behavioral models that lead to \eqref{quantilenECON} with $\tau \neq 1/2$. Therefore, next the paper focuses on more canonical behavioral models that maintain Assumption \ref{assn:binary_distributionMED} and deliver conditions (1) and (2) of Theorem \ref{prop:median} for $\tau \neq 1/2$.
\subsection{Consideration sets as the canonical behavioral model}
As it turns out, models with consideration sets provide the canonical behavioral implementation of
Theorem~\ref{prop:median} for $\tau \neq 1/2$.
\vskip 0.05in
\noindent \emph{Independent consideration set formation} Suppose the collection of possible consideration sets is
$\{\{0\}, \{1\}, \{0,1\}\}$, with probabilities
$\gamma_0$, $\gamma_1$, and
$1-\gamma_0-\gamma_1 \in (0,1)$, respectively. These consideration sets are
formed independently of $(U_0,U_1,x)$. When the agent faces the set
$\{0,1\}$, she chooses the alternative with the higher latent utility. The model
can be written as $Y = z_1z_2 + (1-z_1)\mathbf{1}\{U_1 \geq U_0\}$, where $z_1$ and $z_2$ are independent binary random variables, also independent
of $(U_0,U_1,x)$, with $P(z_1=1)=\gamma_0+\gamma_1$,
$P(z_2=1)=\frac{\gamma_1}{\gamma_0+\gamma_1}$. Here, $z_1$ indicates that the consideration set is a singleton, while $z_2$
indicates that this singleton is $\{1\}$. Thus, independence of $z_1$ and
$z_2$ from $(U_0,U_1,x)$ captures the exogeneity of consideration set formation. Namely,
which alternatives enter the consideration set is completely independent of the
other variables in the model.
Then
$P(Y=1|x)=\gamma_1 +(1-\gamma_0 -\gamma_1) (1-F_{\varepsilon|x} (-x\beta)) $
and, under Assumption \ref{assn:binary_distributionMED},
$$x\beta \; \, \geq (\leq) \; \, 0 \iff P(Y=1|x) \; \, \geq (\leq) \; \, \frac{1}{2}+\frac{1}{2}(\gamma_1-\gamma_0).$$
If $\gamma_0=\gamma_1$ then we in fact obtain a model with the median characterization \eqref{medianECON} which is special case of the model in Example \ref{ex:randchoice}. If $\gamma_0\neq \gamma_1$ we obtain a quantile characterization (\ref{quantilenECON}) for $\tau=1/2(1-\gamma_1+\gamma_0)\neq 1/2$. The asymmetry in consideration probabilities $\gamma_1 - \gamma_0$ directly
controls the quantile index.\footnote{An equivalent approach would be to rely on Theorem \ref{prop:median} and show that
$P(Y=1|U_1 \geq U_0,x) -P(Y=1|U_1<U_0,x) = 1-\gamma_0-\gamma_1>0 \; a.e.$,
and that
$P(Y=1|U_1 \geq U_0,x) = 1-\gamma_0$, $P(Y=0|U_0 > U_1,x) =1-\gamma_1,$
which creates a non-median formulation for $\gamma_0\neq \gamma_1$.}
The consideration sets model presented here has testable implications beyond Proposition \ref{prop:implications}, including when $\gamma_1 = \gamma_0$. Namely, $P(Y=1|x) \in [\gamma_1,1-\gamma_0]$, so choice probabilities are
bounded away from zero if $\gamma_1>0$, from one if $\gamma_0>0$, and
from both boundaries if both are positive. Since
$\gamma_0+\gamma_1>0$ under our maintained conditions, they are always
bounded away from at least one boundary. Hence, if $P(Y=1| x)$ lies
arbitrarily close to both zero and one with positive probability over
$x$, this model is ruled out.
When $\gamma_0=0$, we have $z_2=1$ a.e., and the only consideration sets are $\{1\}$ and $\{0,1\}$. This case captures a predisposition toward option 1, with $\gamma_1$ measuring its strength. Equivalently, it can represent probabilistic conformity to a social norm being option 1.
There are other models that may result in (\ref{quantilenECON}) and are in essence similar to the consideration sets models. Example \ref{ex:otherquantile} illustrates that.
\begin{example}[Elements of heuristic decision making]
\label{ex:otherquantile} Suppose an agent's decision involves evaluation of an aspect $w$ in the first step which is independent of $(U_0,U_1,x)$ in the following way. If $w$ is above $\overline{w}$, then 1 is chosen. If $w$ is below $\underline{w}<\overline{w}$, then 0 is chosen.
If $w \in (\underline{w},\overline{w})$ then the decision is made on the basis of which latent utility is greater.
Let $\gamma_1=P(w\geq \overline{w})$, $\gamma_0=P(w\leq \underline{w})$ and $\gamma_0+\gamma_1 \in [0,1)$. Then
$$x\beta \; \, \geq (\leq) \; \, 0 \iff P(Y=1|x) \; \,\geq (\leq) \; \, \frac{1}{2} +\frac{1}{2}(\gamma_1-\gamma_0). \blacksquare$$
\end{example}
\vskip 0.05in
\noindent \textit{Heterogeneous consideration set formation} What if we weakened the assumption on $z_1, z_2$ by allowing them to be correlated with $x$ but maintained their independence from $(U_0,U_1)$ and from each other given $x$? Denoting $P(z_1=1|x)=\gamma_0(x)+\gamma_1(x)$, $P(z_2=1|x)=\frac{\gamma_1(x)}{\gamma_0(x)+\gamma_1(x)}$, we obtain
$P(Y=1|x)=\gamma_1(x) +(1-\gamma_0(x) -\gamma_1(x)) (1-F_{\varepsilon|x} (-x\beta)) $,
and under Assumption \ref{assn:binary_distributionMED},
$$x\beta \; \geq (\leq) \; 0 \iff P(Y=1|x) \; \geq (\leq) \; \frac{1}{2}+\frac{1}{2}(\gamma_1(x)-\gamma_0(x)).$$
Thus, models with heterogeneous consideration set formation (where $\gamma_0$ and
$\gamma_1$ depend on $x$) can also generate \eqref{quantilenECON}, provided
$\gamma_1(x) - \gamma_0(x) = \Delta$ a.e.\ for some constant $\Delta \in (-1,1)$. If in this condition $\Delta>0$, then $\gamma_1(x)$ cannot approach 0 arbitrarily closely as it is bounded away from 0 by $\Delta$, therefore $P(Y=1|x)$ potentially can approach 1 arbitrarily
closely but not 0. If $\Delta<0$, then $\gamma_0(x)$ cannot approach 0 arbitrarily closely (it has to be greater or equal than $-\Delta$) therefore $P(Y=1|x)$ potentially can approach 0 arbitrarily
closely but not 1. To summarize, when this heterogeneous choice formation model gives (\ref{quantilenECON}) model for $\tau \neq 1/2$, it weakens the testable implications of independent consideration set formation. Probabilities now need only be bounded away from one boundary, as in Proposition \ref{prop:implications}.
The question now is whether there are other foundations that lead to (\ref{quantilenECON}), particularly those where the choice probabilities could approach both natural probability boundaries 0 and 1 arbitrarily closely for $\tau \neq 1/2$. This is addressed in the next section.
\subsection{Distributional route: Non-exchangeable distribution of $(\varepsilon_0,\varepsilon_1|x)$}
We now show that \eqref{quantilenECON} can also be rationalized within the
standard random utility paradigm by allowing the joint distribution of
$(\varepsilon_0,\varepsilon_1)$ to be asymmetric. The following theorem covers
both non-exchangeable copulas and unequal marginal distributions. Although it
allows for behavior beyond the standard random utility model, our main interest
is the standard case, where the role of distributional asymmetries in the
unobservables is especially transparent.
\begin{theorem}\label{prop:copula}
Let $U_j=x_j\beta_j+\varepsilon_j$, $j=0,1$, and let the joint distribution of $(\varepsilon_0, \varepsilon_1)|x $ be absolutely continuous, have convex support with nonempty interior and let the support of $\varepsilon_1- \varepsilon_0|x $ contain $0$ in its interior. Then:
\textbf{A.}
The choice behavior can be represented as in (\ref{quantilenECON}) if the following conditions hold:
\begin{enumerate}
\item[(1')] Condition (1) in Theorem \ref{prop:median} holds;
\item[(2')] $P(\varepsilon_1 \geq \varepsilon_0|x) \cdot P(Y=1|U_1\geq U_0,x) + P(\varepsilon_1 < \varepsilon_0|x) \cdot P(Y=1|U_1 < U_0,x)=1-\tau$ a.e.
\end{enumerate}
\textbf{B.} If the choice behavior of an economic agent can be represented as in (\ref{quantilenECON}) and condition (2') holds, then Condition (1') is necessary away from index indifference, as in
\eqref{condition1_weaker}.
\textbf{C.}
Condition (2') cannot be omitted from the distribution-free sufficiency
result in Part \textbf{A}. More precisely, for every fixed
$\beta\neq 0$, there exists a generic model satisfying the maintained
assumptions and the strict version of Condition (1'), but violating
Condition (2') on a set of positive $P_X$-measure, for which
\eqref{quantilenECON} fails.
\end{theorem}
Condition (2') in Theorem \ref{prop:copula} suggests that the covariates $x$ and the error dependence interact in a way that stabilizes the threshold for the choice probability. Analogously to Theorem~\ref{prop:median}, part \textbf{A} is about sufficiency, while parts \textbf{B} and \textbf{C} provide corresponding near-necessity results.
Condition (2') reduces to Condition (2) when the joint distribution of
$(\varepsilon_0,\varepsilon_1)$ is exchangeable, since
$P(\varepsilon_1\geq\varepsilon_0| x) = 1/2$ in that case.
Under the standard random utility paradigm
($P(Y=1|U_1\geq U_0,x) = 1$, $P(Y=1| U_1<U_0,x) = 0$),
Condition (2') becomes $P(\varepsilon_1\geq\varepsilon_0 | x) = 1-\tau$
a.e., which is the constancy across $x$ of the asymmetry of the joint unobservable distribution. When the marginals of $\varepsilon_0$ and $\varepsilon_1$ conditional on $x$
are identical (Assumption \ref{assn:binary_distributionMED}(a) holds) but
the copula $C_x$ is not exchangeable, this becomes
$P_{C_x}(v\geq u| x) = 1-\tau$ a.e., that is, the copula must have a fixed
degree of asymmetry across $x$. Appendix C shows that this condition is satisfied for an explicit
parametric family of absolutely continuous copulas, and that any $\tau\in(0,1)$ is
attainable.
The key difference of the described distributional route from the behavioral route is that no boundary restriction
on $P(Y=1| x)$ emerges as choice probabilities can approach both zero and
one, as the copula non-exchangeability can accommodate any
probability level.
\section{Multinomial choice: More than two options}
\label{sec:multinomial}
In the binary case characterization \eqref{quantilenECON} effectively described the partitioning of the one-dimensional simplex $\Delta_1=\{p_0+p_1 =1, p_1 \in [0,1]\}$, more precisely its feasible subset $\{p_0+p_1 =1, p_1 \in [\max\{0,2(1-\tau)-1\}, \min\{2(1-\tau),1\}]\}$ implied by the testable restrictions in Proposition \ref{prop:implications}. With more than two options, the analysis inevitably becomes substantially richer and more complex. This section studies the extent to which the partition result of the binary case survives under more general behavioral and distributional assumptions.
\subsection{Review of \citet{Manski1975} multinomial model: ``median'' multinomial choice}
\label{sec:reviewManski}
I begin with the classical \citet{Manski1975} multinomial choice model. I consider it the ``median'' benchmark for multinomial choice, as the binary choice model \eqref{medianECON} under the median threshold arises as its special case.
The latent utility of an agent $i$ for the option $j$ is
\begin{equation}
\label{Umulti} U^*_{ij} = x_{ij}\beta_j+\varepsilon_{ij}, \quad x_{ij} \in \mathbb{R}^{1 \times k_j}, \quad \beta_j \in \mathbb{R}^{k_j \times 1}, \quad j=1, \ldots, J.
\end{equation}
On a general note, and not specifically in relation to \citet{Manski1975}, it is useful to distinguish the support of the covariates from the range of the deterministic utility indices. Even if each $x_j$ has full support, the vector $(x_1\beta_1,\ldots,x_J\beta_J)^{\top}$ need not span $\mathbb R^J$, e.g. when covariates are shared across alternatives. Moreover, since choices and rankings depend only on relative utilities, the relevant richness concerns the $J-1$ index differences rather than the $J$ index levels. Sufficient ranking results later in this paper apply on the index configurations that are attainable. This includes the special case in which the $J-1$ index differences range over all of $\mathbb R^{J-1}$, but does not require it. Additional richness of the joint index difference will be imposed only where it is needed for converse results or for conclusions that require every strict ordering region to be nonempty.
The assumption of i.i.d. of $\boldsymbol{\varepsilon}=(\varepsilon_{i1}, \ldots, \varepsilon_{iJ})^{\top}$ conditional on $\mathbf{x}_i=(x_{i1},\ldots,x_{iJ})^{\top}$ and the support of $\boldsymbol{\varepsilon}|\mathbf{x}_i$ being $\mathbb{R}^J$, under the standard random utility paradigm one obtains
\begin{equation}
\label{ECONmedianmulti} \forall \, j, k \qquad P(Y=j|\mathbf{x}_i) \geq P(Y=k|\mathbf{x}_i) \quad \iff \quad x_{ij}\beta_j \geq x_{ik}\beta_k.
\end{equation}
Relations \eqref{ECONmedianmulti} imply that $P(Y=j|\mathbf{x}_i) = P(Y=k|\mathbf{x}_i)$ $ \iff$ $x_{ij}\beta_j =x_{ik}\beta_k$. Note that the full $\mathbb{R}^J$ support assumption on the conditional distribution of $\boldsymbol{\varepsilon}|\mathbf{x}$ guarantees that the vector $\mathfrak{P}(\mathbf{x}):=(P(Y=1|\mathbf{x}),\ldots,P(Y=J|\mathbf{x}))^{\top}$ never hits the boundary of the $(J-1)$-dimensional simplex $\Delta_{J-1}$:
$$\Delta_{J-1}=\{\mathbf{p}=(p_1,\ldots,p_J): \sum_{j=1}^J p_j=1, p_j \geq 0, j=1, \ldots, J\}.$$
Without that full support assumption on the distribution of $\boldsymbol{\varepsilon}|\mathbf{x}$, the utility indices may reach the boundary of the support of the unobservables or even extend beyond it in which case inequality relationships between choice probabilities may become uninformative about the corresponding relationships between the associated utility indices.
Therefore, when working under the more general distributional assumptions, it would be natural to reformulate \eqref{ECONmedianmulti} by
excluding points in the simplex that are obtained when some index differences $x_j\beta_j-x_k\beta_k$ are either at the boundary or beyond the support of $\varepsilon_k-\varepsilon_j|\mathbf{x}$. This will be done in a technically proper way later in our discussion. For now, for expositional simplicity let us maintain that full support condition.
There are some observations worth noting here. We can define preference relations over the choice set either in terms of probabilities,
$$ j \stackrel{(p)}{\succeq} k \text{ by } i \quad \iff \quad P(Y=j|\mathbf{x}_i) \geq P(Y=k|\mathbf{x}_i),$$
or in terms of indices,
$$ j \stackrel{(index)}{\succeq} k \text{ by } i \quad \iff \quad x_{ij}\beta_j \geq x_{ik}\beta_k.$$
These two preference relations are equivalent. Moreover, they satisfy the standard axioms of preference relations, including reflexivity, completeness and transitivity.
Another implication of that is that we can create an ordering of the whole list of options by utilizing pairwise relations described above. Namely, take any permutation $(k_1,\ldots,k_J)$ of $(1,\ldots, J)$. Then
\begin{equation}\label{proference_all1} k_1 \stackrel{(index)}{\succeq} k_2 \ldots \stackrel{(index)}{\succeq} \ldots \stackrel{(index)}{\succeq} k_J \text{ by } i
\end{equation}
iff
\begin{equation}\label{proference_all2}k_{j-1} \stackrel{(p)}{\succeq} k_j \text{ by } i \text{ for any } j=2, \ldots, J.
\end{equation}
The following theorem essentially shows that \eqref{ECONmedianmulti} is consistent with
a much wider class of decision rules than standard utility maximization. Before that, let's formulate an assumption on unobservables.
\begin{assumption}[Distribution of $\boldsymbol{\varepsilon}|\mathbf{x}$] \label{assn:multi_distributionMED}
The joint distribution $\boldsymbol{\varepsilon}|\mathbf{x}$ is exchangeable and absolutely continuous and has a convex support on $\mathbb{R}^J$ with a non-empty interior.
\end{assumption}
In Theorem \ref{th:multimedian} below the maximum of latent utilities $U_j$, $j=1, \ldots, J$, is denoted as $U^{J:J}$.
\begin{theorem}
\label{th:multimedian} Let $U_j$ , $j=1, \ldots, J$, be given as in (\ref{Umulti}), and let the distribution of $\boldsymbol{\varepsilon}|\mathbf{x}$ satisfy Assumption \ref{assn:multi_distributionMED}. Then the behavior of an economic agent can be represented
as
{\small$$x_{j}\beta_j \geq x_{k}\beta_k \Rightarrow P(Y=j|\mathbf{x}) \geq P(Y=k|\mathbf{x}), \quad P(Y=j|\mathbf{x}) > P(Y=k|\mathbf{x}) \Rightarrow x_{j}\beta_j > x_{k}\beta_k$$}if the following conditions are satisfied:
\begin{enumerate}
\item[(1)] For any $m\neq j$, $P (Y = j|U_j = U^{J:J}, \mathbf{x}) - P (Y = j|U_m = U^{J:J}, \mathbf{x}) > 0$ a.e. in $\mathbf{x}$;
\item[(2)] $P (Y = j|U_j = U^{J:J}, \mathbf{x})$ does not depend on $j$ a.e. in $\mathbf{x}$;
\item[(3)] $P (Y = j|U_m = U^{J:J}, \mathbf{x})$, $m\neq j$, does not depend on $j$, $m$ a.e. in $\mathbf{x}$
\end{enumerate}
\end{theorem}
This theorem contains a modified version of \eqref{ECONmedianmulti}. This modification is needed to deal with the case of both $x_j\beta_j$ and $x_k \beta_k$ being outside the interior of the support of $\boldsymbol{\varepsilon}|\mathbf{x}$ and, thus, is only relevant when such support is bounded in some directions. For brevity and also due to multiplicity of options, I do not focus on near-necessity formulations (analogous to \textbf{B}
and \textbf{C} in theorems \ref{prop:median} and \ref{prop:copula}).
\citet{Manski1975} is a special case of Theorem \ref{th:multimedian} where the joint conditional copula $C_{\mathbf{x}}$ is the independence copula and $P (Y = j|U_j = U^{J:J}, \mathbf{x})=1$, $P (Y = j|U_m = U^{J:J}, \mathbf{x})=0$, $m\neq j$, a.e.
Example \ref{ex:simple_multi} below is that of multinomial choice models that satisfy the sufficient conditions of Theorem \ref{th:multimedian} but go beyond the standard random utility paradigm.
\begin{example}
\label{ex:simple_multi}
Throughout these examples, we will denote $\mathbf{U}=(U_1,\ldots,U_J)^{\top}$.
1. \textit{Mistakes in the implementation}. With probability $\alpha(\mathbf{x})\in(1/J,1)$, the agent correctly
chooses the utility-maximizing option, otherwise she randomizes uniformly over the
nonmaximizing alternatives.
This model allows for mistakes in implementation even when the agent correctly identifies the utility-maximizing alternative.
Conditions (1)-(3) of Theorem \ref{th:multimedian} are satisfied as
$P(Y=j\mid U_j=U^{J:J},\mathbf{x})=\alpha(\mathbf{x})$, while for any $m\neq j$,
we have $P(Y=j\mid U_m=U^{J:J},\mathbf{x})=\frac{1-\alpha(\mathbf{x})}{J-1}$. Condition 1 holds because $\alpha(\mathbf{x})>\frac{1}{J}$ and, hence, $\alpha(\mathbf{x}) - (1-\alpha(\mathbf{x}))/(J-1) > 0$.
2. \textit{Random shortlist around the utility maximizer.}
Suppose the agent identifies the utility-maximizing alternative and forms
a shortlist of fixed size $r\in\{2,\ldots,J-1\}$ that always contains
that alternative. The remaining $r-1$ members are drawn uniformly without
replacement from the other $J-1$ alternatives. The agent then chooses
uniformly from the shortlist. If $U_j=U^{J:J}$, then $P(Y=j\mid U_j=U^{J:J},\mathbf{x})=\frac{1}{r}$. If instead $U_m=U^{J:J}$ for some $m\neq j$, alternative $j$ enters the
shortlist with probability $(r-1)/(J-1)$, and hence $P(Y=j\mid U_m=U^{J:J},\mathbf{x})
=
\frac{r-1}{r(J-1)}$. Conditions (2) and (3) of Theorem~\ref{th:multimedian} therefore hold,
while Condition (1) follows from $\frac{1}{r}-\frac{r-1}{r(J-1)}
= \frac{J-r}{r(J-1)}>0.$
\end{example}
\subsection{Geometric structure of the ``median'' multinomial model}
To make progress on what we can consider a ``quantile'' multinomial choice model, we give a geometric interpretation of the \citet{Manski1975} model reviewed in Section \ref{sec:reviewManski} as we associate it with a ``median'' benchmark for multinomial choice.
Since in \citet{Manski1975} an individual always makes a choice, the vector of choice probabilities $\mathfrak{P}(\mathbf{x}) = (P(Y=1 |\mathbf{x}), \ldots, P(Y=J|\mathbf{x}))^{\top}$ lies in the $(J-1)$-dimensional simplex $\Delta_{J-1}$, as already mentioned above. The geometry of \eqref{ECONmedianmulti} is striking.
First, under total indifference \begin{equation}
\label{totalindiff}
x_1\beta_1=\ldots =x_J
\beta_J
\end{equation} in the index space, such an index collection is mapped by the model to the barycentre $\pi^*:=(1/J \ldots,1/J)^{\top} \in \Delta_{J-1}$, as implied by exchangeability in Assumption \ref{assn:multi_distributionMED}.
Under pairwise indifference
{\small\begin{equation}
\label{partialindiff}
x_{k_1}\beta_{k_1}=x_{k_2}\beta_{k_2}, \quad k_1 \neq k_2,
\end{equation}}model
\eqref{ECONmedianmulti} implies $P(Y=k_1|\mathbf{x}) = P(Y=k_2|\mathbf{x})$.
Pairwise indifference therefore maps into the hyperplane
$\{p_{k_1}=p_{k_2}\}$ in $\Delta_{J-1}$. Taken across all pairs,
these hyperplanes partition $\Delta_{J-1}$ into $J!$ regions
corresponding to the possible complete strict rankings of the indices.
The attainable probability set inherits the corresponding regions,
although some may be empty. All $J!$ regions are attained only when
the index differences vary sufficiently richly. The regions are pointed convex cones emanating from $\boldsymbol{\pi}^*$,
truncated at the simplex boundary (Figure~\ref{fig:simplex1} for $J=3$). Each of the 6 regions in Figure \ref{fig:simplex1} corresponds to one of the possible six preference relations on the whole set of options according to (\ref{proference_all1})-(\ref{proference_all2}). For example, the region of the simplex formed by $p_2\leq p_1$, $p_1\leq p_3$ corresponds to the case $3 \succeq 1 \succeq 2$ either in the index ordering $\stackrel{(index)}{\succeq}$ of probability ordering $\stackrel{(p)}{\succeq}$. This region has a common $1$-dimensional border with the region corresponding to $3 \succeq 2 \succeq 1$ and the region corresponding to $1 \succeq 3 \succeq 2$. The fact that the simplex is partitioned into six regions non-overlapping in their interior is tantamount to transitivity and completeness properties of the preferences defined through $\stackrel{p}{\succeq}$ on the set of three options.
In the general case of $J$ options, each of $J!$ regions collects all choice probabilities that can arise under a complete ordering $x_{k_1}\beta_{k_1} \geq x_{k_2}\beta_{k_2} \ldots \geq x_{k_J}\beta_{k_J}$ for a permutation $(k_1,\ldots,k_J)$ of elements in $\mathcal{J}$. Each such region shares a
$(J-2)$-dimensional boundary with exactly
$J-1$ neighboring regions, obtained by swapping the order of two adjacent elements in the sequence $x_{k_1}\beta_{k_1} \geq x_{k_2}\beta_{k_2} \ldots \geq x_{k_J}\beta_{k_J}$ while holding all other orderings fixed. This partition is what we call the ``median'' benchmark. A more general ``quantile'' version is a different global ranking partition of the ambient space, with a shifted reference point $\boldsymbol{\pi}^*\neq(1/J,\ldots,1/J)^{\top}$ and possibly non-hyperplanar boundaries.
\begin{figure}
\centering
\includegraphics[width=0.5\linewidth]{Simplex1new.png}
\caption{Illustration of the model with 3 options under \citet{Manski1975}. Each region corresponds to one complete utility ordering;
all share the vertex $\boldsymbol{\pi}^* = (1/3,1/3,1/3)^\top$.}
\label{fig:simplex1}
\end{figure}
\subsection{Two levels of analysis. Notation and preliminaries}
\label{sec:notation_preliminaries}
The analysis is conducted at two levels, which differ in what is held fixed
and what is allowed to vary. Let $\mathcal{B}_0=\{\|\boldsymbol{\beta}\|=1\}$ denote the admissible,
normalized parameter space for $\boldsymbol{\beta}
=
(\beta_1^{\top},\ldots,\beta_J^{\top})^{\top}$.
A \emph{Behavioral environment}, denoted by $B$, specifies a decision
rule, such as standard random utility or a consideration set model with
given attention probabilities, and an admissible class
$\mathcal{P}$ of conditional distributions of
$\boldsymbol{\varepsilon}$ given $\mathbf{x}$. Results at this level are uniform over the
conditional distribution of unobservables in $\mathcal{P}$. A \emph{Structural environment}, denoted by $M$, further fixes a conditional
distribution $\mathbb{P}_{\boldsymbol{\varepsilon}\mid\mathbf{x},\,
\mathbf{x}\in\mathbb{R}^{M_e}}
\in\mathcal{P}$. It can therefore be written as $M
=
\left(
B,
\mathbb{P}_{\boldsymbol{\varepsilon}\mid\mathbf{x},\,
\mathbf{x}\in\mathbb{R}^{M_e}}
\right)$. The conditional distribution of $\boldsymbol{\varepsilon}$ is specified for
every $\mathbf{x}\in\mathbb{R}^{M_e}$, where
$M_e$ is the effective dimension of $\mathbf{x}$ (thus, if e.g. the same covariate enters several indices, it appears only once in $\mathbf{x}$, and perfect linear combinations can be dealt with analogously). This allows conclusions that are not driven by restrictions on the covariate domain.\footnote{This echoes the paper's discussion of enriched support in the binary choice case. At the \emph{Structural environment} level, the support of $\mathbf{x}$ has already been maximally enriched.}
The class $\mathcal{P}$ may be defined by imposing suitable restrictions on
these conditional distributions (analogously to \cite{Manski1975}).
In principle, one could consider an \emph{empirical specification} level, which further fixes the marginal
distribution of $\mathbf{x}$ and is characterized by $\left(
M,\boldsymbol{\beta},\mathbb{P}_{\mathbf{x}}
\right)$.\footnote{If the support of $\mathbb{P}_{\mathbf{x}}$ is smaller than
$\mathbb{R}^{M_e}$, some of the conditional distributions specified by $\mathbb{P}_{\boldsymbol{\varepsilon}\mid\mathbf{x},\,
\mathbf{x}\in\mathbb{R}^{M_e}}$
are not used by the empirical specification.} I don't consider this level theoretically as it needs to be approached on a case-by-case basis.
For any $\boldsymbol{\beta}\in\mathcal{B}_0$, I call $(B,\boldsymbol{\beta})$
a \emph{fixed-$\boldsymbol{\beta}$ behavioral specification}, and $(M,\boldsymbol{\beta})$
a \emph{fixed-$\boldsymbol{\beta}$ structural model}. The scope of results with respect to $\boldsymbol{\beta}$
is a separate matter. A behavioral specification level or structural model
result may hold e.g. for a fixed $\boldsymbol{\beta}$, or generically over
$\boldsymbol{\beta}$, or uniformly over
$\boldsymbol{\beta}\in\mathcal{B}_0$. Uniformity over $\boldsymbol{\beta}$ is particularly attractive as it means that the same probability based
ranking implication applies for every
$\boldsymbol{\beta}\in\mathcal{B}_0$. This is stronger than establishing,
separately for each $\boldsymbol{\beta}$, a ranking implication that may
depend on the particular parameter value.
The result of \citet{Manski1975} is at the \emph{Behavioral environment} level and
has the uniformity-in-$\beta$ scope. Given that in empirical practice $\boldsymbol{\beta}\in\mathcal{B}_0$ is not known, the uniform-in-$\boldsymbol{\beta}$ results may therefore prove most valuable for parameter identification.\footnote{If the deterministic component of latent utility were by a more
general function $\phi_j(x_j)$ rather than $x_j\beta_j$, the analogous
distinction would be between a behavioral environment that leaves
$(\phi_1,\ldots,\phi_J)$ unspecified and a behavioral specification that
fixes those functions. Uniformity over the deterministic utility functions
would then require the same probability based ranking implication to hold
throughout the admissible class of such functions.}
Section~\ref{sec:binary} results for binary choice are also at the \emph{Behavioral environment} level and uniform in $\beta$.
\vskip 0.05in
\noindent
\underline{\textit{Simplex subsets of interest.}}
Fix a structural environment $M$ and a parameter value
$\boldsymbol{\beta}\in\mathcal{B}_0$. For the fixed-$\boldsymbol{\beta}$
structural model $(M,\boldsymbol{\beta})$, let $\mathbf{p}(\mathbf{x})
=
\left(
P(Y=1|\mathbf{x}),
\ldots,
P(Y=J|\mathbf{x})
\right)^{\top}
\in\Delta_{J-1}$
denote the conditional choice-probability vector generated at
$\mathbf{x}$. To simplify notation, we suppress the dependence of
$\mathbf{p}(\mathbf{x})$ on $M$ and $\boldsymbol{\beta}$ whenever the
fixed structural model is clear from context.
Define the fixed-$\boldsymbol{\beta}$ feasible probability set at the \emph{Structural environment} level by
$\Delta^{M,\boldsymbol{\beta}}_{J-1}
=
\operatorname{Cl}
\left\{
\mathbf{p}(\mathbf{x})
:
\mathbf{x}\in\mathbb{R}^{M_e}
\right\}$. Thus, $\Delta^{M,\boldsymbol{\beta}}_{J-1}$ is the closure of the set of
probability vectors attainable under $(M,\boldsymbol{\beta})$ as the
covariates vary over the full covariate domain. The ultimate feasible probability set associated with the structural
environment $M$, allowing the preference parameter $\boldsymbol{\beta}$ to vary, is
$\Delta^{M,\cup_{\boldsymbol{\beta}}}_{J-1}
=
\operatorname{Cl}
\left(
\bigcup_{\boldsymbol{\beta}\in\mathcal{B}_0}
\Delta^{M,\boldsymbol{\beta}}_{J-1}
\right).$
This union collects the probability vectors attainable for at least one
admissible value of $\boldsymbol{\beta}$.
We can define the corresponding objects at the behavioral specification level with
$\Delta^{B,\boldsymbol{\beta}}_{J-1}
=
\operatorname{Cl}
\left(
\bigcup_{Q\in\mathcal{P}}
\Delta^{M_Q,\boldsymbol{\beta}}_{J-1}
\right)$
aggregating feasible probability vectors across all conditional
distributions of unobservables admitted by $B$, while holding $\boldsymbol{\beta}$ fixed. The corresponding ultimate feasible set is
$\Delta^{B,\cup_{\boldsymbol{\beta}}}_{J-1}
=
\operatorname{Cl}
\left(
\bigcup_{\boldsymbol{\beta}\in\mathcal{B}_0}
\Delta^{B,\boldsymbol{\beta}}_{J-1}
\right)$.
Under some behavioral environments, the sets $\Delta^{M,\boldsymbol{\beta}}_{J-1}$,
$\Delta^{B,\boldsymbol{\beta}}_{J-1}$, $\Delta^{M,\cup_{\boldsymbol{\beta}}}_{J-1}$,
$\Delta^{B,\cup_{\boldsymbol{\beta}}}_{J-1}$
may have affine dimension strictly smaller than $J-1$, even when the
conditional support of
$\boldsymbol{\varepsilon}|\mathbf{x}$ is convex and has nonempty interior
in $\mathbb{R}^{J}$. It will be clarified later in the paper under what conditions the relevant
feasible set has full affine dimension.
\vskip 0.05in
\noindent
\underline{\textit{Images of pairwise ordered utility indices.}} I study whether an observed probability vector (unambiguously) reveals the ordering of the underlying deterministic utility indices. Since this question can be posed either conditionally on a fixed value of $\boldsymbol{\beta}$ or uniformly across values of $\boldsymbol{\beta}$, the corresponding image sets need to be defined separately.
For any $k_1,k_2\in\mathcal{J}$ with $k_1\neq k_2$, and for any
\emph{fixed-$\boldsymbol{\beta}$ structural model} $(M,\boldsymbol{\beta})$, define
\begin{equation}
\label{eq:defPk1k2_Model}
\mathcal{P}^{+,M,\boldsymbol{\beta}}_{k_1,k_2}
=
\left\{
\mathbf{p}\in\Delta_{J-1}:
\exists\,\mathbf{x}\in\mathbb{R}^{M_e}
\text{ s.t. }
\mathbf{p}(\mathbf{x})=\mathbf{p}, \; \;
x_{k_1}\beta_{k_1}
>
x_{k_2}\beta_{k_2}
\right\},
\end{equation}
\begin{equation}
\label{eq:defIk1k2_Model}
\mathcal{I}^{M,\boldsymbol{\beta}}_{k_1,k_2}
=
\left\{
\mathbf{p}\in\Delta_{J-1}:
\exists\,\mathbf{x}\in\mathbb{R}^{M_e}
\text{ s.t. }
\mathbf{p}(\mathbf{x})=\mathbf{p}, \; \;
x_{k_1}\beta_{k_1}
=
x_{k_2}\beta_{k_2}
\right\}.
\end{equation}
Thus, $\mathcal{P}^{+,M,\boldsymbol{\beta}}_{k_1,k_2}$ collects the choice
probability vectors generated under the \emph{fixed-$\boldsymbol{\beta}$ structural
model} when
$k_1\stackrel{\mathrm{index}}{\succ}k_2$,
whereas $\mathcal{I}^{M,\boldsymbol{\beta}}_{k_1,k_2}$ collects those
generated under pairwise index indifference. When $\boldsymbol{\beta}$ is allowed to vary within the structural
environment $M$, define the ultimate ordering and indifference images by $\mathcal{P}^{+,M,\cup_{\boldsymbol{\beta}}}_{k_1,k_2}
=
\bigcup_{\boldsymbol{\beta}\in\mathcal{B}_0}
\mathcal{P}^{+,M,\boldsymbol{\beta}}_{k_1,k_2}$ and $\mathcal{I}^{M,\cup_{\boldsymbol{\beta}}}_{k_1,k_2}
=
\bigcup_{\boldsymbol{\beta}\in\mathcal{B}_0}
\mathcal{I}^{M,\boldsymbol{\beta}}_{k_1,k_2}$, respectively.
For a \emph{behavioral specification} level for a fixed
$\boldsymbol{\beta}\in\mathcal{B}_0$, define $\mathcal{P}^{+,B,\boldsymbol{\beta}}_{k_1,k_2}
=
\bigcup_{Q\in\mathcal{P}}
\mathcal{P}^{+,M_{Q},\boldsymbol{\beta}}_{k_1,k_2}$, $\mathcal{I}^{B,\boldsymbol{\beta}}_{k_1,k_2}
=
\bigcup_{Q\in\mathcal{P}}
\mathcal{I}^{M_{Q},\boldsymbol{\beta}}_{k_1,k_2}$.
These sets aggregate the fixed-$\boldsymbol{\beta}$ ordering and indifference
images across all structural models admitted by $B$. When $\boldsymbol{\beta}$ is allowed to vary within $\mathcal{B}_0$, define the ultimate behavioral specification images by $\mathcal{P}^{+,B,\cup_{\boldsymbol{\beta}}}_{k_1,k_2}
=
\bigcup_{\boldsymbol{\beta}\in\mathcal{B}_0}
\mathcal{P}^{+,B,\boldsymbol{\beta}}_{k_1,k_2}$,
$\mathcal{I}^{B,\cup_{\boldsymbol{\beta}}}_{k_1,k_2}
=
\bigcup_{\boldsymbol{\beta}\in\mathcal{B}_0}
\mathcal{I}^{B,\boldsymbol{\beta}}_{k_1,k_2}$.
\vskip 0.05in
\noindent \underline{\textit{Support of unobservables and relative interiors.}} If the \citet{Manski1975} setting has $\mathbb{R}^J$ as the
support of $\boldsymbol{\varepsilon}|\mathbf{x}$, and the $J-1$
index differences range over all of $\mathbb{R}^{J-1}$, then every
probability vector in $\operatorname*{\mathrm{relint}}(\Delta_{J-1})$ is
attainable, while no probability vector on $\partial\Delta_{J-1}$ is
attained. Since the feasible probability sets are defined as closures, it follows that $\Delta^{M,\cup_{\boldsymbol{\beta}}}_{J-1}
= \Delta^{B,\cup_{\boldsymbol{\beta}}}_{J-1}
= \Delta_{J-1}$.
More general properties of the support of
$\boldsymbol{\varepsilon}|\mathbf{x}$ (e.g., requiring only
that it be convex and have nonempty interior in $\mathbb{R}^J$) may
instead allow choice probabilities on the boundary of the relevant
feasible probability set to be attained. In such cases, distinct index
configurations may generate the same boundary probability vector,
limiting the ranking information contained in choice probabilities at
the boundary. For this reason, the ranking analysis in this paper focuses on the
relative interior of the relevant feasible probability set.
\vskip 0.05in
\noindent
\underline{\textit{Ranking recovery value of choice probabilities.}}
I distinguish two possible scopes for ranking recovery, which are
fixed-$\boldsymbol{\beta}$ recovery
and uniform recovery across $\boldsymbol{\beta}$. These scopes are distinct
from the level of analysis (\emph{Behavioral environment}, \emph{Structural environment}) introduced above.
A \emph{fixed-$\boldsymbol{\beta}$ recovery} result at the \emph{Structural environment} $M$ level requires the three sets $\mathcal{P}^{+,M,\boldsymbol{\beta}}_{k_1,k_2}$, $\mathcal{I}^{M,\boldsymbol{\beta}}_{k_1,k_2}$, $\mathcal{P}^{+,M,\boldsymbol{\beta}}_{k_2,k_1}$ to be pairwise disjoint in $\operatorname*{\mathrm{relint}}(\Delta^{M,\boldsymbol{\beta}}_{J-1})$. Equivalently, the sign of $x_{k_1}\beta_{k_1}-x_{k_2}\beta_{k_2}$ must be constant over every subset $S_{\mathbf{p}}=\left\{\mathbf{x}\in\mathbb{R}^{M_e}:\mathbf{p}(\mathbf{x})=\mathbf{p}\right\}.$ A result is \emph{generic in $\boldsymbol{\beta}$} if this fixed-$\boldsymbol{\beta}$
recovery property holds for every
$\boldsymbol{\beta}\in\mathcal{B}^{*}$, where
$\mathcal{B}^{*}$ is a full measure subset of
$\mathcal{B}_0$. By contrast, recovery \emph{uniform over $\boldsymbol{\beta}$} requires the
stronger condition that
$\mathcal{P}^{+,M,\cup_{\boldsymbol{\beta}}}_{k_1,k_2}$,
$\mathcal{I}^{M,\cup_{\boldsymbol{\beta}}}_{k_1,k_2}$,
$\mathcal{P}^{+,M,\cup_{\boldsymbol{\beta}}}_{k_2,k_1}$
be pairwise disjoint in $\operatorname*{\mathrm{relint}}
(
\Delta^{M,\cup_{\boldsymbol{\beta}}}_{J-1}
).$ A \emph{fixed-$\boldsymbol{\beta}$ recovery} at the \emph{Behavioral environment} level
requires $\mathcal{P}^{+,B,\boldsymbol{\beta}}_{k_1,k_2}$, $\mathcal{I}^{B,\boldsymbol{\beta}}_{k_1,k_2}$, $\mathcal{P}^{+,B,\boldsymbol{\beta}}_{k_2,k_1}$
to be pairwise disjoint within $\operatorname*{\mathrm{relint}}
(
\Delta^{B,\boldsymbol{\beta}}_{J-1}
).$
Fixed-$\boldsymbol{\beta}$ recovery, if it holds,
has the following implication at the \emph{Structural environment} (\emph{Behavioral environment}) level.
\begin{enumerate}
\item[\textit{Implication 1.}]
The strict ordering images do not overlap: $\mathcal{P}^{+,M,\boldsymbol{\beta}}_{k_1,k_2}
\cap
\mathcal{P}^{+,M,\boldsymbol{\beta}}_{k_2,k_1}
\cap
\operatorname*{\mathrm{relint}}
(
\Delta^{M,\boldsymbol{\beta}}_{J-1}
)
=
\varnothing$ $\left(\mathcal{P}^{+,B,\boldsymbol{\beta}}_{k_1,k_2}
\cap
\mathcal{P}^{+,B,\boldsymbol{\beta}}_{k_2,k_1}
\cap
\operatorname*{\mathrm{relint}}
(
\Delta^{B,\boldsymbol{\beta}}_{J-1})
=
\varnothing\right)$. Thus, an attainable probability vector $\mathbf{p}=\mathbf{p}(\mathbf x)=\mathbf{p}(\widetilde{\mathbf{x}})$ for a given $M$ (for any $M$ compliant with the \emph{behavioral environment} $B$) cannot imply both $x_{k_1}\beta_{k_1}>x_{k_2}\beta_{k_2}$
and
$\widetilde{x}_{k_1}\beta_{k_1}<\widetilde{x}_{k_2}\beta_{k_2}$.
\item[\textit{Implication 2}.]
Suppose fixed-$\boldsymbol{\beta}$ recovery holds for every distinct pair of
alternatives. For any permutation
$(k_1,\ldots,k_J)$ of $\mathcal{J}$, we have $\mathbf{p}
\in
\bigcap_{j=1}^{J-1}
\mathcal{P}^{+,M,\boldsymbol{\beta}}_{k_j,k_{j+1}}
\cap
\operatorname*{\mathrm{relint}}
(
\Delta^{M,\boldsymbol{\beta}}_{J-1})$
implies the complete strict ordering $x_{k_1}\beta_{k_1}
>
x_{k_2}\beta_{k_2}
>
\cdots
>
x_{k_J}\beta_{k_J}$ in the given \emph{Structural environment} $M$ $\bigl(\mathbf{p}
\in
\bigcap_{j=1}^{J-1}
\mathcal{P}^{+,B,\boldsymbol{\beta}}_{k_j,k_{j+1}}
\cap
\operatorname*{\mathrm{relint}}
(
\Delta^{B,\boldsymbol{\beta}}_{J-1}
)$
implies the complete strict ordering $x_{k_1}\beta_{k_1}
>
x_{k_2}\beta_{k_2}
>
\cdots
>
x_{k_J}\beta_{k_J}$ in any \emph{Structural environment} $M$ compliant with the \emph{Behavioral environment} $B$$\bigr)$.
\\
Regions corresponding to different complete strict orderings are pairwise
disjoint.
\item[\textit{Implication 3}.]
For an attainable probability vector $\mathbf{p}$, define $k_1
\succ_{\mathbf{p}}^{M,\boldsymbol{\beta}}
k_2
\quad\Longleftrightarrow\quad
\mathbf{p}
\in
\mathcal{P}^{+,M,\boldsymbol{\beta}}_{k_1,k_2}.$
If fixed-$\boldsymbol{\beta}$ recovery holds for every pair, then
$
\mathbf{p}
\in
\mathcal{P}^{+,M,\boldsymbol{\beta}}_{k_1,k_2}
\cap
\mathcal{P}^{+,M,\boldsymbol{\beta}}_{k_2,k_3}
\quad \Rightarrow \quad
\mathbf{p}
\in
\mathcal{P}^{+,M,\boldsymbol{\beta}}_{k_1,k_3}.
$
The recovered relation $\succ_{\mathbf{p}}^{M,\boldsymbol{\beta}}$ is therefore transitive (the relation recovered from $\mathbf{p}$ coincides with the numerical
ordering of the deterministic utility indices).
An analogous implication holds at the \emph{Behavioral environment} level
after replacing the structural environment objects by their
behavioral environment counterparts. \\
\end{enumerate}
Uniform recovery across $\boldsymbol{\beta}$ rules out cross-$\boldsymbol{\beta}$ ambiguity and ensures the same probability
vector cannot be generated with $k_1$ ranked above $k_2$ under one admissible
value of $\boldsymbol{\beta}$ and with $k_2$ ranked above $k_1$ under another.
The analogous definition applies at the \emph{Behavioral environment} level.
\vskip 0.05in
\noindent \underline{\textit{Back to \citet{Manski1975}.}} This was discussed earlier. I revisit it here using the new notation. Under full support of $\mathbb{P}_{\boldsymbol{\varepsilon}|\mathbf{x}, \mathbf{x} \in \mathbb{R}^{M_e}}$, it holds that $\operatorname*{\mathrm{relint}}(\Delta^{B,\boldsymbol{\beta}}_{J-1})=\operatorname*{\mathrm{relint}}(\Delta^{M,\boldsymbol{\beta}}_{J-1})=\operatorname*{\mathrm{relint}}(\Delta^{B, \cup_{\boldsymbol{\beta}}}_{J-1})=\operatorname*{\mathrm{relint}}(\Delta^{M, \cup_{\boldsymbol{\beta}}}_{J-1})=\operatorname*{\mathrm{relint}}(\Delta_{J-1})$. Ranking recovery holds uniformly over $\boldsymbol{\beta}$ for any pair $k_1,k_2 \in \mathcal{J}$, $k_1 \neq k_2$, and $\mathcal{P}^{+,B, \cup_{\boldsymbol{\beta}}}_{k_1,k_2} \cap \operatorname*{\mathrm{relint}}(\Delta^{B, \cup_{\boldsymbol{\beta}}}_{J-1}) = \operatorname*{\mathrm{relint}}(\Delta^{B, \cup_{\boldsymbol{\beta}}}_{J-1}) \cap \{p_{k_1} > p_{k_2}\}$, $\mathcal{I}^{B, \cup_{\boldsymbol{\beta}}}_{k_1,k_2} \cap \operatorname*{\mathrm{relint}}(\Delta^{B, \cup_{\boldsymbol{\beta}}}_{J-1}) = \operatorname*{\mathrm{relint}}(\Delta^{B, \cup_{\boldsymbol{\beta}}}_{J-1}) \cap \{p_{k_1} = p_{k_2}\}$.
\vskip 0.05in
The remainder of the analysis examines how far the global simplex partitioning and ranking recovery properties of the \citet{Manski1975} benchmark survive the paper's behavioral and distributional extensions. Table~\ref{tab:multinomial_roadmap} summarizes the main results across robustness levels.
\begin{table}[tbp]
\centering
\caption{Roadmap of multinomial ranking results}
\label{tab:multinomial_roadmap}
\footnotesize
\setlength{\tabcolsep}{3pt}
\begin{tabular}{@{}p{0.17\textwidth}p{0.21\textwidth}p{0.37\textwidth}p{0.19\textwidth}@{}}
\hline
\textbf{Robustness level}
&
\textbf{Restriction}
&
\textbf{Ranking result}
&
\textbf{$\boldsymbol\beta$ scope}
\\
\hline
\multicolumn{4}{@{}l}{\textit{Behavioral route: limited attention with exchangeable unobservables}}
\\[0.15em]
Behavioral environment
&
Rank-one balance
&
Common linear pairwise ranking rules. Under sufficiently rich joint
index-difference variation, the rank conditions are also necessary for
the corresponding robust representation. For $J\geq4$, more general
linear separators using additional probability coordinates are
characterized in the Online Appendix.
&
The ranking rule is common to all admissible $\boldsymbol\beta$; the
attainable ranking regions need not be.
\\[0.45em]
Behavioral environment
&
Rank conditions fail
&
Opposite pairwise rankings may generate overlapping probability images;
with sufficiently rich index difference variation, the corresponding
robust linear representation fails.
&
The failure concerns robustness across admissible conditional unobservables
distributions; uniformity over $\boldsymbol\beta$ is a separate requirement.
\\[0.45em]
Structural environment
&
Common location-scale structure
&
A common injective, generally nonlinear map $\Psi$ recovers the normalized
indices, and hence all pairwise rankings, from every attainable probability
vector.
&
$\Psi$ is common across $\boldsymbol\beta$; changing $\boldsymbol\beta$
changes only the attainable normalized-index set
$\mathcal D_{\boldsymbol\beta}$ and hence its probability-space trace.
\\
\hline
\multicolumn{4}{@{}l}{\textit{Distributional route: utility maximization with non-exchangeable unobservables}}
\\[0.15em]
Behavioral environment
&
General admissible distribution class
&
No generic probability-space ranking rule is common across the admissible
unobservables distributions.
&
In general there is no distribution- and $\boldsymbol\beta$-uniform
ranking partition.
\\[0.45em]
Structural environment
&
Common location--scale structure
&
A common injective, generally nonlinear map $\Psi$ restores ranking
recovery, although its geometry may be asymmetric.
&
$\Psi$ is common across $\boldsymbol\beta$; the attainable ranking regions
and their geometry depend on $\mathcal D_{\boldsymbol\beta}$.
\\
\hline
\end{tabular}
\end{table}
\subsection{ Consideration sets: the behavioral route. ``Quantile'' version?}
\subsubsection{Behavioral environment}
I start by formulating a set of basic assumptions on the consideration set process.
\begin{assumption}
\label{assn:indCS} The consideration set formation is independent of $(\mathbf{x}, \boldsymbol{\varepsilon})$.
\end{assumption}
Under Assumption~\ref{assn:indCS}, denote by $\gamma_A$ the
probability of the consideration set $A\subseteq \mathcal{J}$ and write
$P(Y=j|\mathbf{x}) = \sum_{A\in\mathcal{A}:j\in A}\gamma_A P(Y=j| A,\mathbf{x})$,
where $\mathcal{A}$ denotes the collection of all consideration sets and $P(Y = j | A, \mathbf{x})$ denotes the probability of choosing $j$ when considering a given set $A$. Let's refer to consideration sets that arise with positive probability, that is, sets $A$ such that $\gamma_A>0$, as \textit{active} consideration sets. Recall that an analogous independence assumption was also our starting point in the binary choice setting.
\begin{assumption}\label{assn:noemptyCS} We suppose that $\gamma_{\varnothing}=0$\footnote{Some existing work allows for empty consideration sets by interpreting this event as leading to a default or no-choice outcome (see e.g. \citet{ManziniMariotti2014}, \citet{Horan2019}). We could allow $\gamma_{\varnothing}>0$ and specify the corresponding choice rule, but doing so would not substantively affect our results. It would only introduce additional notation and case distinctions. For this reason, we set $\gamma_{\varnothing}=0$ throughout.} and every element $j \in \mathcal{J}$ belongs to some active consideration set.
\end{assumption}
\begin{assumption}[Standard random utility paradigm within a consideration set] \label{assn:rationalCS} Given a consideration set $A$, an agent $i$ chooses an option $j$ from $A$ with the highest latent utility $U_{ij}^*=x_{ij}\beta_j+\varepsilon_{ij}$.
\end{assumption}
Assumptions \ref{assn:noemptyCS} and \ref{assn:rationalCS} guarantee that an economic agent always makes a choice so $\sum_{j=1}^J P(Y=j|\mathbf{x})=1$ for all $\mathbf{x}$. Thus, under these assumptions we once again can look at the probability simplex $\Delta_{J-1}$ to characterize preference relations in terms of points from this simplex. The second part of Assumption \ref{assn:noemptyCS} ensures that every alternative can, in principle, be chosen with a positive probability.
Finally, I give a definition of connectedness of elements in a given consideration sets structure. Its importance is discussed later.
\begin{definition}[Consideration set structure connectivity]
\label{def:cs_connected}
Consider $J$ options and exogenous consideration set structure for $\mathcal{J}$ described by probabilities $\{\gamma_{A}\}_{A \in \mathcal{A}}$. Define the set of active non-singleton consideration sets as
$\mathcal{A}_{\geq 2} = \{ A \in \mathcal{A} : |A| \geq 2, \gamma_A > 0 \}.$
We will say that \textit{options $\ell$ and $h$ are connected} in the structure if either $\ell=h$, or there exists a sequence of sets $ A_1, A_2, \ldots, A_m \in \mathcal{A}_{\geq 2} $ such that:
$ \ell \in A_1 $,
$ h \in A_m $,
$ A_k \cap A_{k+1} \neq \emptyset $ for $ k = 1, 2, \ldots, m-1 $.
This sequence forms a path from $ \ell $ to $ h $.
We will say that the \textit{consideration set structure for $\mathcal{J}$ is connected} if any two options in $\mathcal{J}$ are connected.
\end{definition}
The connectivity condition ensures that in our model all pairs of candidates $ (\ell, h) $ can be compared probabilistically, either directly (if they share an active consideration set) or indirectly (through a chain of active consideration sets). Effectively, the connectedness will guarantee that utilities propagate through overlapping sets.
When $J=3$, connectedness of the whole structure is equivalent to requiring that each option $j$ belongs to at least one active non-singleton consideration set, that is, to some set in $\mathcal{A}_{\geq 2}$. Hence, connectedness can fail only if some option appears only as a singleton consideration set. For $J\geq 4$, failures of connectedness can take more varied forms. For example, if $J=4$ and the only active non-singleton consideration sets are $A_1=\{1,2\}$ and $A_2=\{3,4\}$, then the structure is not connected since options $1$ and $2$ are isolated from options $3$ and $4$ in the decision-making process. If, however, there is also an active consideration set $A_3=\{1,3\}$, then connectedness is restored. In that case, for instance, options $2$ and $4$ are connected through the path $\{1,2\} \to \{1,3\} \to \{3,4\}$.
The knowledge of $\{\gamma_A\}_{A \in \mathcal{A}}$ would, of course, reveal the nature of consideration set structure connectivity. But even without such knowledge subsets of connected elements can be identified from choice probabilities, as a consequence of Proposition \ref{prop:notconnected} below.
\begin{prop}[Testable implications for the lack of connectivity] \label{prop:notconnected}
Suppose Assumptions \ref{assn:indCS}--\ref{assn:rationalCS} hold. If the consideration set structure for $\mathcal{J}$ is not connected in the sense of Definition \ref{def:cs_connected}, then there exists a proper subset $B\subset \mathcal{J}$ such that $P(Y \in B|\mathbf{x}) $ is a.e. constant in $\mathbf{x}$ for any model with the given \textit{Behavioral environment}, with value in $(0,1)$.
\end{prop}
Proposition \ref{prop:notconnected} implies that $ \mathcal{J} $ admits a partition into subsets $ B_1, \dots, B_M $ ($ M \geq 1 $) where elements are connected if and only if they belong to the same $ B_m $. In case all the elements in $\mathcal{J}$ are connected, we have $M=1$ with $B_1=\mathcal{J}$.
From now on, by \textit{Behavioral environment} throughout the consideration set discussion we will mean (i) the consideration set paradigm that satisfies Assumptions \ref{assn:indCS}- \ref{assn:rationalCS} with a given collection $\{\gamma_A\}_{A \in \mathcal{A}}$\footnote{This collection may be unknown to an econometrician.} ; (ii) family $\mathcal{P}$ of conditional distributions $\mathbb{P}_{\boldsymbol{\varepsilon}|\mathbf{x}, \mathbf{x} \in \mathbb{R}^{M_e}}$ that collects all joint distributions that satisfy Assumption \ref{assn:multi_distributionMED}. Note that the absolute continuity of the joint distribution in Assumption \ref{assn:multi_distributionMED} guarantees that ties $U^*_j=U^*_{k}$ for $k \neq j$ happen with probability 0 and, thus, can effectively be ignored in our derivations.
\vskip 0.1in
\noindent \textit{Probability simplex.} The form and affine and topological dimensions of $\Delta^{B, \cup_{\beta}}_{J-1}$ depend on how elements are connected within the consideration set structure.
Using the partitioning of $\mathcal{J}$ into $B_1$, \ldots, $B_M$, $M\geq 1$, as discussed after Proposition \ref{prop:notconnected},
{\small$$\Delta^{B, \cup_{\boldsymbol{\beta}}}_{J-1} = \bigcap_{m=1}^M \left\{\mathbf{p} \in \Delta_{J-1}: \; \sum_{\widetilde{A} \in \mathcal{A}:\widetilde{A} \subseteq S } \gamma_{\widetilde{A}} \leq \sum_{k \in S} p_k \leq \sum_{\widetilde{A} \in \mathcal{A}:\widetilde{A} \cap S \neq \emptyset } \gamma_{\widetilde{A}} \qquad \forall S \subseteq B_m\right\}.$$}If any two elements in $\mathcal{J}$ are connected in the sense of Definition \ref{def:cs_connected}, then
$\Delta^{B, \cup_{\boldsymbol{\beta}}}_{J-1} $ has affine and topological dimensions equal to $J-1$.
Given the \textit{Behavioral environment}, under Assumption \ref{assn:multi_distributionMED} the total indifference relation \eqref{totalindiff} is mapped into a single point $\boldsymbol{\pi}^*=(\pi_1^*, \ldots, \pi_J^*)$ such that
\begin{equation}
\label{pistar}
\pi_j^* = \sum_{A: j \in A} \frac{\gamma_A}{|A|}, \quad j=1, \ldots, J.
\end{equation}
This does not depend on the connectivity of elements in the consideration set structure.
\subsubsection{Partitioning by linear hyperplanes (Behavioral environment level)}
Let's now ask when the model delivers a ``quantile'' partition of the ambient probability domain $\operatorname*{\mathrm{relint}}(\Delta^{B,\cup_{\boldsymbol{\beta}} }_{J-1})$ by linear hyperplanes at the \textit{Behavioral environment} level and uniformly over $\boldsymbol{\beta}$. When looking at the attainable choice probability vectors, this is effectively asking when
$\mathcal{I}^{B, \cup_{\boldsymbol{\beta}}}_{k_1,k_2}$ is in a linear hyperplane in $\operatorname*{\mathrm{relint}}(\Delta^{B,\cup_{\boldsymbol{\beta}} }_{J-1})$ and $\mathcal{P}^{+,B, \cup_{\boldsymbol{\beta}}}_{k_1,k_2} \cap \mathcal{P}^{+,B, \cup_{\boldsymbol{\beta}}}_{k_2,k_1} \cap \operatorname*{\mathrm{relint}}(\Delta^{B,\cup_{\boldsymbol{\beta}} }_{J-1}) = \varnothing$ for any $k_1 \neq k_2$.
For interpretability I first focus on $J=3$ and then give the extension to $J\geq 4$ in
Appendix B. For $J=3$, \eqref{pistar} becomes $\pi_j^* = \gamma_{\{1,2,3\}}/3+\gamma_{\{j,k\}}/2+\gamma_{\{j,h\}}/2+\gamma_{\{j\}}$, $k \neq j, k\neq h, h\neq j.$
For the sufficient direction in Theorem \ref{th:linearJ3} below, no additional assumptions on the range obtained by the index vector are required. For the converse, suppose that, as $\mathbf{x}$ and $\boldsymbol{\beta}$ vary over their stated domains, configurations satisfying $x_{k_1}\beta_{k_1}=x_{k_2}\beta_{k_2}$ allow the remaining index to vary arbitrarily relative to their common value.
\begin{theorem} \label{th:linearJ3} Suppose Assumptions \ref{assn:multi_distributionMED}--\ref{assn:rationalCS} hold and $J=3$. Let elements $k_1, k_2$, $k_1 \neq k_2$, in $\mathcal{J}$ be connected in the sense of Definition \ref{def:cs_connected}. Then:
\textbf{A.} If the rank of
\(
C_{k_1,k_2} = \begin{pmatrix}
\gamma_{\{1,2,3\}} & \gamma_{\{k_1,k_3\}} \\
\gamma_{\{1,2,3\}} & \gamma_{\{k_2,k_3\}}
\end{pmatrix}
\)
is strictly less than $2$, then $\mathcal{I}^{B,\cup_{\boldsymbol{\beta}}}_{k_1,k_2}\cap\operatorname*{\mathrm{relint}}(\Delta^{B,\cup_{\boldsymbol{\beta}}}_2)$
is contained in a linear hyperplane of dimension at most $J-2=1$. Under the additional index variation condition above,
the converse also holds.
\textbf{B.} When $\mathrm{rank}(C_{k_1,k_2})=1$, the indifference set $\mathcal{I}^{B,\cup_{\boldsymbol{\beta}}}_{k_1,k_2}
\cap
\operatorname*{\mathrm{relint}}\!\left(\Delta^{B,\cup_{\boldsymbol{\beta}}}_2\right)$ coincides with the intersection of
$\operatorname*{\mathrm{relint}}(\Delta^{B,\cup_{\boldsymbol{\beta}}}_2)$ and the linear hyperplane
{\small$$(\gamma_{\{1,2,3\}}+\gamma_{\{k_2,k_3\}})
(p_{k_1}-\pi^*_{k_1})
=
(\gamma_{\{1,2,3\}}+\gamma_{\{k_1,k_3\}})
(p_{k_2}-\pi^*_{k_2}).$$}
Set
$\mathcal{P}^{+,B, \cup_{\boldsymbol{\beta}}}_{k_1,k_2}\cap \operatorname*{\mathrm{relint}}(\Delta^{B, \cup_{\boldsymbol{\beta}}}_{2})$ coincides with
\begin{multline*} \left\{\mathbf{p} \in \operatorname*{\mathrm{relint}}(\Delta^{B,\cup_{\boldsymbol{\beta}}}_{2}): (\gamma_{\{1,2,3\}}+\gamma_{\{k_2,k_3\}})(p_{k_1}-\pi^*_{k_1}) > (\gamma_{\{1,2,3\}}+\gamma_{\{k_1,k_3\}}) (p_{k_2}-\pi^*_{k_2})\right\}.
\end{multline*}
\textbf{C.} When $\mathrm{rank}(C_{k_1,k_2})=0$, then
$k_3\in\mathcal{J}\setminus\{k_1,k_2\}$ is not connected to
$k_1$ or $k_2$, and
{\small
$$\mathcal{I}^{B,\cup_{\boldsymbol{\beta}}}_{k_1,k_2}
=
\left\{
\mathbf{p}\in\Delta^{B,\cup_{\boldsymbol{\beta}}}_{2}:
p_{k_1}-\pi^*_{k_1}=0,\;
p_{k_2}-\pi^*_{k_2}=0,\;
p_{k_3}-\pi^*_{k_3}=0
\right\},$$}
{\small$$\mathcal{P}^{+,B,\cup_{\boldsymbol{\beta}}}_{k_1,k_2}
\cap
\operatorname*{\mathrm{relint}}(\Delta^{B,\cup_{\boldsymbol{\beta}}}_{2})
=
\left\{
\mathbf{p}\in\operatorname*{\mathrm{relint}}(\Delta^{B,\cup_{\boldsymbol{\beta}}}_{2}):
p_{k_1}-\pi^*_{k_1}>0,\;
p_{k_2}-\pi^*_{k_2}<0,\;
p_{k_3}=\pi^*_{k_3}
\right\}.$$}
\end{theorem}
When the two index differences range over all of $\mathbb{R}^2$, the dimensions of the sets in the theorem can be stated more sharply. In part \textbf{B} (part \textbf{C}),
$\mathcal{I}^{B,\cup_{\boldsymbol{\beta}}}_{k_1,k_2}
\cap\operatorname*{\mathrm{relint}}(\Delta^{B,\cup_{\boldsymbol{\beta}}}_{2})$
has dimension $1$ ($0$ in part \textbf{C}), while
$\mathcal{P}^{+,B,\cup_{\boldsymbol{\beta}}}_{k_1,k_2}
\cap\operatorname*{\mathrm{relint}}(\Delta^{B,\cup_{\boldsymbol{\beta}}}_{2})$
has dimension $2$ ($1$ in part \textbf{C}).
When $\gamma_{\{1,2,3\}}>0$, condition $\mathrm{rank}(C_{k_1,k_2}) = 1$ is equivalent to
$\gamma_{\{k_1,k_3\}} = \gamma_{\{k_2,k_3\}}$.\footnote{$\gamma_{\{k_1\}}$, $\gamma_{\{k_2\}}$, $\gamma_{\{k_3\}}$ can all be different.}
One can think of this as attention neutrality of $k_3$ with respect to the pair
$(k_1,k_2)$. It characterizes
when the consideration process does not ``take sides'' between the competing
pair. When $\gamma_{\{1,2,3\}}=0$, the two pair menu probabilities may differ, and $C_{k_1,k_2}$
has rank one provided at least one of them is positive. If both are zero, the matrix has rank zero.
Corollary \ref{cor:linearJ3allpairs} next shows that the three pairwise linear separators induce six geometric cells in $\Delta_2$, and that their attainable traces recover the corresponding strict utility rankings.
\begin{corollary}[``Quantile'' linear partitioning for $J=3$ and ranking recovery]\label{cor:linearJ3allpairs}
Suppose Assumptions~\ref{assn:multi_distributionMED}-\ref{assn:rationalCS} hold and $J=3$.
Let $\mathcal J$ be connected in the sense of
Definition~\ref{def:cs_connected}, and suppose that
$C_{k_1,k_2}$ in Theorem~\ref{th:linearJ3} has rank one for every
pair $k_1\neq k_2$. For each pair $(k_1,k_2)$, let $k_3$ denote the remaining alternative. Define the ambient indifference line
{\small$$H_{k_1,k_2}
:=
\left\{
\mathbf p\in\operatorname*{\mathrm{relint}}(\Delta_2):
\bigl(\gamma_{\{1,2,3\}}+\gamma_{\{k_2,k_3\}}\bigr)
(p_{k_1}-\pi^*_{k_1})
=
\bigl(\gamma_{\{1,2,3\}}+\gamma_{\{k_1,k_3\}}\bigr)
(p_{k_2}-\pi^*_{k_2})
\right\}.$$}The three pairwise lines are distinct, intersect at
$\boldsymbol\pi^*$, and divide $\operatorname*{\mathrm{relint}}(\Delta_2)$
into six nonempty open cells. Moreover, on the attainable probability set,
$H_{k_1,k_2}
\cap
\operatorname*{\mathrm{relint}}
(\Delta^{B,\cup_{\boldsymbol\beta}}_2)
=
\mathcal I^{B,\cup_{\boldsymbol\beta}}_{k_1,k_2}
\cap
\operatorname*{\mathrm{relint}}
(\Delta^{B,\cup_{\boldsymbol\beta}}_2).$
For each strict complete ordering $j\succ h\succ \ell$, the corresponding attainable ranking region is the trace of the associated ambient cell on $\operatorname*{\mathrm{relint}}(\Delta^{B,\cup_{\boldsymbol\beta}}_2)$. These traces are pairwise disjoint across different strict rankings (and need not all be nonempty).
\end{corollary}
The corollary delivers a ``quantile'' multinomial choice model, in the sense defined above, at the \emph{Behavioral environment} level uniformly over $\boldsymbol{\beta} \in \mathcal{B}_0$. The partition of $\operatorname*{\mathrm{relint}}(\Delta^{B,\cup_{\boldsymbol{\beta}} }_{J-1})$ in this ``quantile'' model depends only on the attention probabilities. The uniformity over $\boldsymbol\beta\in\mathcal B_0$ concerns the ranking rule, not the set of probability vectors attained, with different values of $\boldsymbol\beta$ potentially generating different traces of the six ambient cells. For every pair $(k_1,k_2)$, the position of the corresponding probability vector relative to $H_{k_1,k_2}$ uniquely determines the sign of $x_{k_1}\beta_{k_1}-x_{k_2}\beta_{k_2}$, automatically giving ranking recovery.
\vskip 0.1in
\noindent \textbf{Case $J \geq 4$.} For $J \geq 4$, the pairwise separator logic extends as follows. For a given pair $(k_1,k_2)$, a two-coordinate rank-one condition analogous to the one in Theorem \ref{th:linearJ3} and Corollary \ref{cor:linearJ3allpairs} requires the probabilities of the consideration sets $S \cup \{k_1\}$ and $S \cup \{k_2\}$, with
$\varnothing\neq S\subseteq\mathcal J\setminus\{k_1,k_2\}$, to be proportional across $S$. Whenever there is an active consideration set containing both $k_1$ and $k_2$ together with outside alternatives, this proportionality factor has to be 1, yielding exact attention neutrality. This condition is sufficient for a common linear separator in the $(p_{k_1},p_{k_2})$-coordinates for ranking $k_1$ relative to $k_2$ on all attainable configurations. Under conditions analogous to the joint index difference variation used for the converse in Theorem~\ref{th:linearJ3}, it is also necessary for this distribution-robust representation. Importantly though, the failure of the corresponding two-coordinate rank-one condition does not rule out more general linear separators involving the choice probabilities of other alternatives.\footnote{These results highlight the difference between the restrictions on attention probabilities required for generalized linear separators in this paper and the symmetry restrictions imposed by the RCL model in \citet{BarseghyanMolinariThirkettle2019}.} These results are developed in Appendix~B.
\vskip 0.1in
\noindent \textbf{Graphical illustrations.} Figure \ref{fig:graphical_ill_simplex} illustrates two consideration structures satisfying Corollary \ref{cor:linearJ3allpairs}, showing how the shape of the ambient probability space $\operatorname*{\mathrm{relint}}(\Delta^{B,\cup_{\boldsymbol{\beta}} }_{J-1})$, $\pi^*$ and the slopes of the indifference lines depend on $\{\gamma_A\}$. In both cases, any two elements are connected in the sense of Definition \ref{def:cs_connected}. The figure illustrates the geometry described in Corollary \ref{cor:linearJ3allpairs}. Namely, in each panel, the three ambient indifference lines intersect at $\boldsymbol\pi^*$ and partition $\operatorname*{\mathrm{relint}}(\Delta^{B,\cup_{\boldsymbol{\beta}} }_{J-1})$ into six cone-shaped truncated regions with each such region corresponding to a unique complete strict ordering $j\succ h\succ \ell$ on the set of alternatives. The truncation arises from the consideration structure. The feasible probability set selects the attainable traces of these regions. Thus, we have a common linear ranking arrangement, and which ranking regions are actually populated depends on the attainable index configurations.
\begin{figure}[ht]
\centering
\begin{minipage}{0.45\textwidth}
\centering
\includegraphics[width=\linewidth]{Simplex2.png}
\end{minipage}
\hfill
\begin{minipage}{0.54\textwidth}
\centering
\includegraphics[width=\linewidth]{Simplex3.png}
\end{minipage}
\caption{LEFT: case $\gamma_{12}=1/2$, $\gamma_{23}=1/3$, $\gamma_{13}=1/6$. Here $\Delta^{B, \cup_{\boldsymbol{\beta}}}_{2}$ is bounded by $p_1\le \gamma_{12}+\gamma_{13}$, $p_2\le \gamma_{12}+\gamma_{23}$, $p_3\le \gamma_{13}+\gamma_{23}$. RIGHT: case $\gamma_{23}=2/3$, $\gamma_{12}=1/3$. $\Delta^{B, \cup_{\boldsymbol{\beta}}}_{2}$ is bounded by $p_1\le \gamma_{12}$, $p_3\le \gamma_{23}$.}
\label{fig:graphical_ill_simplex}
\end{figure}
\vskip 0.1in
\noindent \textbf{Lack of connectivity.} Let's briefly analyze what happens when the consideration set structure is not fully connected as for $J=3$ the implications are particularly transparent.
\textit{Case 1: Two elements connected, one isolated} (e.g., 2 and 3 connected, 1 isolated).
Then $P(Y=1|\mathbf{x})=\gamma_{\{1\}}$ is constant, and the feasible set collapses to the line segment $\Delta^{B, \cup_{\boldsymbol{\beta}}}_{2} \subseteq \{\mathbf{p}\in\Delta_2 : p_1 = \gamma_{\{1\}}\}$.
Comparisons for alternatives 1 and 2 ($x_1\beta_1 \geq (\leq) x_2\beta_2$) provide no information about the split between $p_2$ and $p_3$ within this segment, and $\mathcal{I}^{B, \cup_{\boldsymbol{\beta}}}_{1,2} = \mathcal{P}^{+,B, \cup_{\boldsymbol{\beta}}}_{1,2} = \mathcal{P}^{+,B, \cup_{\boldsymbol{\beta}}}_{2,1}$.
In contrast, comparisons between 2 and 3 remain informative in $\operatorname*{\mathrm{relint}}(\Delta^{B, \cup_{\boldsymbol{\beta}}}_2)$: $\mathcal{I}^{B, \cup_{\boldsymbol{\beta}}}_{2,3}$ reduces to a single point $(\pi_1^*,\pi_2^*,\pi_3^*)$, while $\mathcal{P}^{+,B, \cup_{\boldsymbol{\beta}}}_{2,3} \cap \operatorname*{\mathrm{relint}}(\Delta^{B, \cup_{\boldsymbol{\beta}}}_2) \subseteq \{p_2 - \pi_2^* > p_3 - \pi_3^*\}$ and $ \mathcal{P}^{+}_{3,2} \cap \operatorname*{\mathrm{relint}}(\Delta^{B, \cup_{\boldsymbol{\beta}}}_2) \subseteq \{p_2 - \pi_2^* < p_3 - \pi_3^*\}$.
\textit{Case 2: All three elements mutually disconnected.}
Here $P(Y=j|\mathbf{x})=\gamma_{\{j\}}$ is constant for each $j=1,2,3$, so $\Delta^{B, \cup_{\boldsymbol{\beta}}}_{2} = \{\mathbf{p}\in\Delta_2 : p_1=\gamma_{\{1\}}, \, p_2=\gamma_{\{2\}}, \, p_3=\gamma_{\{3\}}\}$ is a single point (dimension 0). Consequently, for every pair $k_1\neq k_2$, $\mathcal{I}^{B, \cup_{\boldsymbol{\beta}}}_{k_1,k_2} = \mathcal{P}^{+,B, \cup_{\boldsymbol{\beta}}}_{k_1,k_2} = \Delta^{B, \cup_{\boldsymbol{\beta}}}_{2}$.
In both cases, missing connections severely limit (or completely eliminate) the identifying power of $\mathbf{x}$ for relative probabilities among disconnected alternatives. For general $J$, once the partitioning $\{B_m\}$ of structure implied by connectivity is identified (see Proposition \ref{prop:notconnected} and the subsequent discussion), the analysis reduces to studying each component $B_m$ separately. Within each $B_m$ the feasible set is effectively restricted to a lower-dimensional (namely, $(|B_m|-1)$-dimensional) subset in $\Delta_{J-1}$, and only comparisons among alternatives inside $B_m$ remain informative. The analysis within each such component then proceeds exactly as in the fully connected case.
\subsubsection{When partitioning of $\Delta_{J-1}$ by linear hyperplanes fails (Behavioral environment level)}
\label{sec:behavioralmultistructuralmodel}
When $\mathrm{rank}(C_{k_1,k_2})=2$ for $J=3$, a direct calculation
shows what goes wrong. On the partial indifference locus \eqref{partialindiff},
{\small$$P(Y=k_1\mid\mathbf{x})-\pi_{k_1}^*
=
P(Y=k_2\mid\mathbf{x})-\pi_{k_2}^*
+
\bigl(\gamma_{\{k_1,k_3\}}-\gamma_{\{k_2,k_3\}}\bigr)
\bigl(G_{1,\mathbf{x}}(-x_{k_2}\beta_{k_2}+x_{k_3}\beta_{k_3})-1/2\bigr),$$}with $G_{1,\mathbf{x}}$ denoting the survival function of the difference between two errors conditional on $\mathbf{x}$ (Assumption~\ref{assn:multi_distributionMED} ensures that this is the same for all index pairs). Since
$\gamma_{\{k_1,k_3\}}-\gamma_{\{k_2,k_3\}}\neq0$, as implied by
$\mathrm{rank}(C_{k_1,k_2})=2$, the additional term
$\bigl(\gamma_{\{k_1,k_3\}}-\gamma_{\{k_2,k_3\}}\bigr)
\bigl(G_{1,\mathbf{x}}(-x_{k_2}\beta_{k_2}+x_{k_3}\beta_{k_3})-1/2\bigr)$ need not vanish and generally depends on the conditional distribution
$\boldsymbol{\varepsilon}|\mathbf{x}$ and on $\mathbf{x}$ in a nonlinear way. Varying $\mathbf{x}$ while preserving partial indifference
\eqref{partialindiff}, together with variation in
$\mathbb{P}_{\boldsymbol{\varepsilon}\mid\mathbf{x},\,\mathbf{x}\in\mathbb{R}^{M_e}}\in\mathcal{P}$, will generally result in
$\mathcal{I}^{B,\cup_{\boldsymbol{\beta}}}_{k_1,k_2}$ that retains the full affine and topological dimension of
$\Delta^{B,\cup_{\boldsymbol{\beta}}}_{2}$. Moreover, as illustrated in more detail below, the overlap of
$\mathcal{P}^{+,B,\cup_{\boldsymbol{\beta}}}_{k_1,k_2}\cap
\operatorname*{\mathrm{relint}}(\Delta^{B,\cup_{\boldsymbol{\beta}}}_{2})$
and
$\mathcal{P}^{+,B,\cup_{\boldsymbol{\beta}}}_{k_2,k_1}\cap
\operatorname*{\mathrm{relint}}(\Delta^{B,\cup_{\boldsymbol{\beta}}}_{2})$
may likewise have the full affine and topological dimension of
$\Delta^{B,\cup_{\boldsymbol{\beta}}}_{2}$.
Thus, once the rank condition fails, the ``quantile'' multinomial choice model described above is generally no longer available at the \emph{Behavioral environment} level uniformly over
$\boldsymbol{\beta}\in\mathcal{B}_0$.\footnote{The partial inferential value of choice probabilities at this level still applies and is discussed in Appendix~D.} In particular, ranking recovery at that level may fail even though choice probabilities can continue to contain ranking information on more restricted sets.\footnote{On a restricted attainable set, a violation of the rank condition need not be exposed by the available index configurations.} Such a failure, however, does not rule out ranking recovery under a more specific structural model as one can move down to the \emph{Structural environment} level and fix
$\mathbb{P}_{\boldsymbol{\varepsilon}\mid\mathbf{x}}$, thus fixing function $G_{1,\mathbf{x}}$. The relevant question becomes whether the resulting model-specific mapping from indices into choice probabilities still separates the rankings.
\subsubsection{A more modest perspective: the \textit{Structural environment} level?}
\label{sec:cs_structural}
A \emph{Structural environment} ``quantile'' property requires a common ranking partition of its probability domain $\Delta^{M,\cup_{\boldsymbol{\beta}}}_{J-1}$, with this domain jointly attainable as $\boldsymbol{\beta}$ varies over $\mathcal B_0$. Even when it exists, a particular $\boldsymbol{\beta}$ may attain only a trace of the partition. When it fails to exist, ranking recovery within the \emph{Structural environment} may still be possible for a fixed $\boldsymbol{\beta}$.
Even for the fixed-$\boldsymbol{\beta}$ structural model level, ranking recovery is not automatic.
The fundamental difficulty is that Assumption~\ref{assn:multi_distributionMED}
imposes no restriction on how the joint distribution of $\boldsymbol{\varepsilon}|\mathbf{x}$
changes with $\mathbf{x}$.
If those changes are discontinuous in $\mathbf{x}$, the indifference set
$\mathcal{I}^{M,\boldsymbol{\beta}}_{k_1,k_2}$ need not be in a $(J-2)$-dimensional manifold, and even if it is, it may have
self-intersections or other singularities that prevent separation of the ordering regions.
\vskip 0.1in
\noindent \textit{Stochastic independence of $\boldsymbol{\varepsilon}$ and $\mathbf{x}$.}
Consider $J=3$ with $\boldsymbol{\varepsilon}$ independent of $\mathbf{x}$. This case provides an example of $\mathcal{I}^{M, \cup \boldsymbol{\beta}}_{k_1,k_2}$ being a well-behaved $(J-2)$-manifold with desirable separability properties and then highlights how this separability may fail once independence is relaxed.
Suppose $\gamma_{123},\gamma_{12},\gamma_{13},\gamma_{23}>0$ with
$\gamma_{123}+\gamma_{12}+\gamma_{13}+\gamma_{23}=1$, and suppose
$\gamma_{12},\gamma_{13},\gamma_{23}$ are distinct. Let $C_2$ denote the
bivariate copula of $(\varepsilon_{k_2}-\varepsilon_{k_1}, \varepsilon_{k_3}-\varepsilon_{k_1})$, and let $F$ denote the common marginal c.d.f.\ of
$\varepsilon_{k_2}-\varepsilon_{k_1}$ for every permutation
$(k_1,k_2,k_3)$ of $\mathcal{J}$. In line with Assumption~\ref{assn:multi_distributionMED}, $C_2$ and $F$
are absolutely continuous. For $k\neq j$, define
$a_{k,j}(\mathbf{x})=x_k\beta_k-x_j\beta_j$. Then, for example,
{\small$$
P(Y=k_1|\mathbf{x})
= \gamma_{123}
C_2\!\left(
F(a_{k_1,k_2}(\mathbf{x})),
F(a_{k_1,k_3}(\mathbf{x}))
\right)
+ \gamma_{k_1k_2}F(a_{k_1,k_2}(\mathbf{x}))
+ \gamma_{k_1k_3}F(a_{k_1,k_3}(\mathbf{x})),
$$}with analogous expressions for the other alternatives.
On $\mathcal{I}^{M,\cup\boldsymbol{\beta}}_{k_1,k_2}$,
$a_{k_1,k_2}(\mathbf{x})=0$ and
$a_{k_1,k_3}(\mathbf{x})=a_{k_2,k_3}(\mathbf{x})$. Write this common
difference as $a$ and set $u=F(a)$, $h(u)=C_2(1/2,u)$. The partial indifference locus can then be equivalently parametrized by $a$ or $u\in(0,1)$:
\begin{equation}
\label{k1choiceoncurve}
P(Y=j|\mathbf{x})
=\gamma_{123}h(u)
+\frac{\gamma_{k_1k_2}}{2}
+\gamma_{j k_3}u, \quad j \in \{k_1,k_2\},
\end{equation}
and $P(Y=k_3|\mathbf{x})=1-P(Y=k_1|\mathbf{x})-P(Y=k_2|\mathbf{x})$. Since $\gamma_{k_1k_3},\gamma_{k_2k_3}>0$, both coordinates
are strictly increasing in $u$. Thus, under stochastic independence of
$\boldsymbol{\varepsilon}$ and $\mathbf{x}$, the image of pairwise index
indifference is a well defined one-dimensional curve in
$\operatorname*{\mathrm{relint}}(\Delta^{M,\cup \boldsymbol{\beta}}_2)$. Its shape, however, depends on the difference copula through $h$ and is generally nonlinear but it does not depend on a particular $\boldsymbol{\beta} \in \mathcal{B}_0$ as it is fully indexed by $u$.
The left panel of Figure \ref{fig:ind_assumption} illustrates this nonlinearity
for i.i.d. standard normal unobservables. It depicts the slope ratio $\frac{p_1'(a)}{p_2'(a)}$, where $p_k(a)$ is a special case of \eqref{k1choiceoncurve}: $p_k(a)
=\gamma_{123}
\Phi_2\!\left(0,\frac{a}{\sqrt{2}};\frac12\right)
+\frac{\gamma_{12}}{2}
+\gamma_{k3}\Phi\!\left(\frac{a}{\sqrt{2}}\right)$, $k=1,2$. For attention probabilities used in the figure,
the slope ratio varies smoothly from
$\approx1.20$
to $\approx 12$. Hence the partial indifference locus cannot be
a straight line.
The right panel illustrates that $\mathcal{I}_{k_1,k_2}^{M,\cup_{ \boldsymbol{\beta}}}$ also depends on the
distribution of the unobservables, even when every candidate
distribution is fully exchangeable, absolutely continuous, and
independent of $\mathbf{x}$. Three specifications are compared. In the
first, $\boldsymbol{\varepsilon}
=\boldsymbol{V}+0.25\boldsymbol{Z}$,
where $\boldsymbol{V}$ is uniformly distributed over the six coordinate
permutations of $(0,1,2)^{\top}$, independently of
$\boldsymbol{Z}\sim \mathcal{N}(\boldsymbol{0},I_3)$ (smooth permutation-Gaussian mixture). The second specification is
the i.i.d. normal benchmark
$\boldsymbol{\varepsilon}=\boldsymbol{Z}$. In the third, $\boldsymbol{\varepsilon}=S\boldsymbol{Z}$, where
$P(S=1)=P(S=10^4)=\frac12$,
with $S$ independent of $\boldsymbol{Z}$ (common scale normal). The display focuses on $u\in[0.08,0.30]$, where the
differences between curves are particularly visible.\footnote{This comparison is not intended to
hold the marginal difference distribution fixed and each specification is
parameterized by its own percentile $u=F(a)$.} Thus, at the \textit{Structural environment} level, $\mathcal{I}_{k_1,k_2}^{M,\cup_{\boldsymbol{\beta}}}$ is not determined by the attention probabilities alone but it does meaningfully separate $\operatorname*{\mathrm{relint}}(\Delta^{M,\cup_{\boldsymbol{\beta}}}_2)$
into disjoint ordering regions: for $x_{k_1}\beta_{k_1}>x_{k_2}\beta_{k_2}$
it holds that $P(Y=k_1|\mathbf{x})>p_1(a)$ and
$P(Y=k_2|\mathbf{x})<p_2(a)$, placing the choice probability vector strictly
above the indifference curve, while the inequalities reverse when
$x_{k_1}\beta_{k_1}<x_{k_2}\beta_{k_2}$.
\begin{figure}[ht]
\centering
\begin{minipage}{0.48\textwidth}
\centering
\includegraphics[width=0.95\linewidth]
{Nonlinearity1_illustration.png}
\end{minipage}
\hfill
\begin{minipage}{0.48\textwidth}
\centering
\includegraphics[width=0.95\linewidth] {Nonlinearity1_3curves.png}
\end{minipage}
\caption{
In both panels,
$\gamma_{123}=0.80$, $\gamma_{13}=0.18$,
$\gamma_{23}=0.015$, and $\gamma_{12}=0.005$.
\\
Left: the slope ratio $p_1'(a)/p_2'(a)$ under i.i.d.
standard normal unobservables.
\\
Right: a zoomed plot of
$\mathcal{I}^{M, \cup_{\boldsymbol{\beta}}}_{1,2}$ in the original $(p_1,p_2)$ coordinates for
the smooth permutation-Gaussian mixture (solid), the i.i.d. normal model (dashed),
and the two-scale Gaussian mixture (dotted) described in the text. The
displayed range is $u\in[0.08,0.30]$.}
\label{fig:ind_assumption}
\end{figure}
\vskip 0.05in
\noindent \textit{Distribution of $\boldsymbol{\varepsilon}|\mathbf{x}$ depends on $\mathbf{x}$.} When the distribution of $\boldsymbol{\varepsilon}|\mathbf{x}$ {depends} on $\mathbf{x}$, the findings change. Continuing with $J=3$ and now even fixing $\boldsymbol{\beta}$, the set $\mathcal{I}^{M, \boldsymbol{\beta}}_{k_1,k_2}$ is described as a collection of choice probabilities with
$$P(Y=j|\mathbf{x})
= \gamma_{123}\, C_{2,\mathbf{x}}(0.5,\,F_{\mathbf{x}}(a_{k_1,k_3}(\mathbf{x})))
+ \gamma_{\{k_1,k_2\}}/2
+ \gamma_{\{j,k_3\}}\, F_{\mathbf{x}}(a_{k_1,k_3}(\mathbf{x})), \quad j \in \{k_1,k_2\},$$
where
$C_{2,\mathbf{x}}$ denotes the bivariate copula of $(\varepsilon_{k_2}-\varepsilon_{k_1},\varepsilon_{k_3}-\varepsilon_{k_1}) | \mathbf{x}$ and $F_{\mathbf{x}}$ is the c.d.f. of $\varepsilon_{k_1}-\varepsilon_{k_2}|\mathbf{x}$. Thus, the choice probabilities depend on what can be taken to be a parameter $F_\mathbf{x}(a_{k_1,k_3}(\mathbf{x}))$ but also depend on $C_{2,\mathbf{x}}$.
First, $\mathcal{I}^{M,\boldsymbol{\beta}}_{k_1,k_2}$ in
$\operatorname*{\mathrm{relint}}(\Delta^{M,\boldsymbol{\beta}}_2)$ need not have topological
dimension one less than that of $\operatorname*{\mathrm{relint}}(\Delta^{M,\boldsymbol{\beta}}_2)$. Even when it does, it need not be contained in a
codimension-one embedded manifold of
$\operatorname*{\mathrm{relint}}(\Delta^{M,\boldsymbol{\beta}}_2)$. This is once again because
Assumption~\ref{assn:multi_distributionMED} imposes no restriction on how
the joint distribution of $\boldsymbol{\varepsilon}|\mathbf{x}$ varies
with $\mathbf{x}$, allowing it potentially to vary discontinuously. Even if, in a particular structural model,
$\mathcal{I}^{M,\boldsymbol{\beta}}_{k_1,k_2}$ is contained in a
codimension-one embedded manifold, this alone does not guarantee the required
separation property. The way the ordering images $\mathcal{P}^{+,M,\boldsymbol{\beta}}_{k_1,k_2}$ and
$\mathcal{P}^{+,M,\boldsymbol{\beta}}_{k_2,k_1}$ are generated may itself
vary discontinuously with $\mathbf{x}$, as illustrated below, so that
probability vectors associated with opposite rankings can lie on the same
side of, or overlap across, the partial indifference locus. Consequently,
$\mathcal{I}^{M,\boldsymbol{\beta}}_{k_1,k_2}$ may fail to separate
$\operatorname*{\mathrm{relint}}(\Delta^{M,\boldsymbol{\beta}}_2)$ into the two disjoint ordering
regions
$\mathcal{P}^{+,M,\boldsymbol{\beta}}_{k_1,k_2}
\cap \operatorname*{\mathrm{relint}}(\Delta^{M,\boldsymbol{\beta}}_2)$
and
$\mathcal{P}^{+,M,\boldsymbol{\beta}}_{k_2,k_1}
\cap \operatorname*{\mathrm{relint}}(\Delta^{M,\boldsymbol{\beta}}_2)$. Imposing suitable continuity in $\mathbf{x}$ of the
distribution of $\boldsymbol{\varepsilon}|\mathbf{x}$ would not by
itself resolve the problem. Such continuity may produce a
codimension-one indifference image, but it need not have separation properties as self-intersections or other singularities
may remain. Thus, additional structure is required for
$\mathcal{I}^{M,\boldsymbol{\beta}}_{k_1,k_2}$ to deliver the desired
separation property.
The right panel of Figure \ref{fig:ind_assumption} can help illustrate what can happen under such dependence. Suppose all $x_k$ are one-dimensional with $\beta_k=1$. Let the
conditional unobservable distribution be the smooth permutation-Gaussian model
when $x_1>x_2$, the two-scale Gaussian mixture when $x_2>x_1$, and the
i.i.d. normal model when $x_1=x_2$. The pairwise indifference locus then
jumps between the upper and lower curves in the right panel as
$\mathbf{x}$ crosses $x_1=x_2$. Since these loci are distinct and bound
a nondegenerate strip of the simplex over the displayed range of $u$,
the regions $\mathcal{P}^{+,M, \boldsymbol{\beta}}_{1,2}$ and
$\mathcal{P}^{+,M, \boldsymbol{\beta}}_{2,1}$ overlap inside
$\operatorname*{\mathrm{relint}}(\Delta^{M,\boldsymbol{\beta}}_2)$. Thus, even though each fixed
conditional distribution produces a smooth indifference curve, allowing
the distribution to vary discontinuously with $\mathbf{x}$ destroys
ranking recovery even for the fixed $\boldsymbol{\beta}$.
Even imposing an $\mathbf{x}$-invariant trivariate copula, including the independence one of \citet{Manski1975}, does not generally restore the separation property of $\mathcal{I}^{M,\boldsymbol{\beta}}_{k_1,k_2}$. The induced bivariate copula of $(\varepsilon_{k_1}-\varepsilon_{k_2},\varepsilon_{k_1}-\varepsilon_{k_3})|\mathbf{x}$ may still vary with $\mathbf{x}$ through $F_{\mathbf{x}}$. Thus, outside the settings covered by Theorem~\ref{th:linearJ3}, we need additional assumptions on how $\boldsymbol{\varepsilon}|\mathbf{x}$ varies with $\mathbf{x}$ for $\mathcal{I}^{M, \boldsymbol{\beta}}_{k_1,k_2}$ to yield a meaningful partition and guarantee ranking recovery.
One such assumption is a common location-scale structure for the
unobservables: $\varepsilon_j=\mu(\mathbf{x})+s(\mathbf{x}) \widetilde{\varepsilon}_{j}$, $s(\mathbf{x})>0,$
where $\widetilde{\boldsymbol\varepsilon}
=(\widetilde\varepsilon_1,\ldots,\widetilde\varepsilon_J)^{\top}$ is independent of $\mathbf x$.
Fix alternative 1 as a reference and define $d^{\boldsymbol\beta}(\mathbf x)
=
\left(0,
\frac{x_2\beta_2-x_1\beta_1}{s(\mathbf x)},\ldots,
\frac{x_J\beta_J-x_1\beta_1}{s(\mathbf x)}
\right)^{\top}$. Let $\mathcal E^o_{\mathrm{diff}}$ be the relative interior of the support of
$(0, \widetilde\varepsilon_1-\widetilde\varepsilon_2,\ldots,
\widetilde\varepsilon_1-\widetilde\varepsilon_J)^{\top}$ in its $(J-1)$-dimensional affine hull, and let $\mathcal D_{\boldsymbol\beta}
=
\{d^{\boldsymbol\beta}(\mathbf x):\mathbf x\in\mathbb R^{M_e}\}
\cap\mathcal E^o_{\mathrm{diff}}$ denote the attainable normalized index set restricted to
$\mathcal E^o_{\mathrm{diff}}$. Note that $(d^{\boldsymbol\beta})_j-(d^{\boldsymbol\beta})_k$ and $x_j\beta_j-x_k\beta_k$ have the same sign. Theorem \ref{th:cs_heteroskedasticity} proves a common structural partitioning at the \textit{Structural environment} uniformly over $\boldsymbol{\beta}$ and, in particular, implies the ranking recovery property.
\begin{theorem}[Common structural partition and ranking recovery]
\label{th:cs_heteroskedasticity}
Suppose Assumptions~\ref{assn:indCS}--\ref{assn:rationalCS} hold, every two alternatives are connected in the sense of Definition~\ref{def:cs_connected}, and $\varepsilon_j=\mu(\mathbf x)+s(\mathbf x)\widetilde\varepsilon_j$,
$s(\mathbf x)>0$, where $\widetilde{\boldsymbol\varepsilon}$ is independent of $\mathbf x$ and has an absolutely continuous exchangeable distribution whose support is convex and has nonempty interior. Then
\vskip 0.1in
A. (Ranking recovery) The map $\Psi:\mathcal E^o_{\mathrm{diff}}
\longrightarrow \operatorname*{\mathrm{relint}}(\Delta_{J-1})$
under which the choice probability vector at $(\mathbf x,\boldsymbol\beta)$ is $\Psi(d^{\boldsymbol\beta}(\mathbf x))$ whenever $d^{\boldsymbol\beta}(\mathbf x)\in\mathcal E^o_{\mathrm{diff}}$ is a topological embedding. Hence, for every $\boldsymbol\beta \in \mathcal{B}_0$, each probability vector in $\Psi(\mathcal D_{\boldsymbol\beta})$ uniquely determines the normalized index vector and therefore all pairwise rankings of the deterministic utility indices.
\vskip 0.1in
B. (Common pairwise separation) $\Psi\!\left(\mathcal D_{\boldsymbol\beta}\cap\{d:d_j=d_k\}\right)
=
\Psi(\mathcal D_{\boldsymbol\beta})
\cap
\Psi\!\left(\mathcal E^o_{\mathrm{diff}}\cap\{d:d_j=d_k\}\right)$ for every $\boldsymbol\beta\in\mathcal B_0$ and $j,k \in \mathcal{J}$, $j\neq k$. The same identity holds with $=$ replaced by $>$ or by $<$. The second set on the right-hand side of each identity is common to all
$\boldsymbol\beta$, and $\boldsymbol\beta$ changes only its attained portion.
\vskip 0.1in
C. (Common structural partitioning) For every $j,k \in \mathcal{J}$, $j\neq k$, $\Psi\!\left(
\mathcal E^o_{\mathrm{diff}}\cap\{d:d_j=d_k\}
\right)$ is an embedded $(J-2)$-dimensional topological manifold in
$\Psi(\mathcal E^o_{\mathrm{diff}})$. Its complement in
$\Psi(\mathcal E^o_{\mathrm{diff}})$ has exactly two connected components, $\Psi\!\left(
\mathcal E^o_{\mathrm{diff}}\cap\{d:d_j>d_k\}
\right)$ and
$\Psi\!\left(
\mathcal E^o_{\mathrm{diff}}\cap\{d:d_j<d_k\}
\right)$. Moreover, $0\in\mathcal E^o_{\mathrm{diff}}$, and the $J!$ strict order
regions are all nonempty and connected, with $\Psi(0)$ in the closure
of each.
\end{theorem}
Thus, Theorem \ref{th:cs_heteroskedasticity} establishes that the \emph{Structural environment} induces a common ranking partition of $\Psi(\mathcal E^o_{\mathrm{diff}})$, with potentially only a trace of it attained for a given $\boldsymbol\beta$. It is more than fixed-$\boldsymbol{\beta}$ ranking recovery as a structural ranking rule is common to all $\boldsymbol{\beta}$. If, in addition,
$\mathcal E^o_{\mathrm{diff}}\subseteq \cup_{\boldsymbol{\beta} \in \mathcal{B}_0} \mathcal D_{\boldsymbol\beta}$,
the common structural partition is fully attained as $\boldsymbol{\beta}$ varies. The \emph{Structural environment} is then ``quantile'' in the sense defined above.
Finally, since $\Psi|_{\mathcal D_{\boldsymbol\beta}}$ is a homeomorphism onto its image, the intrinsic dimension and separation properties of any attainable partition are preserved. E.g., if $\mathcal D_{\boldsymbol\beta}$ is an $r$-dimensional manifold and $d_j=d_k$ cuts it in a separating $(r-1)$-dimensional manifold, the corresponding probability space trace has the same dimension and separation property.
\subsection{Distributional route: ``quantile'' version?}
\label{sec:distributionalmulti}
Let's now maintain the standard random utility paradigm and ask when restrictions on the distribution of $\boldsymbol\varepsilon|\mathbf{x}$ generate a ``quantile'' ranking partition or deliver only ranking recovery on the attainable probability set. Suppose generally that $\boldsymbol\varepsilon|\mathbf x$ has a convex support with nonempty interior and an absolutely continuous distribution, with copula $C_{\mathbf x}$ and marginal c.d.f.s $F_{j,\mathbf x}$.
Under complete index indifference \eqref{totalindiff}, the image in the probability simplex is $\pi_k^*(\mathbf x)
=
P\!\left(
\varepsilon_k\geq\varepsilon_j\ \text{for all }j
\mid\mathbf x
\right)$,
$ k=1,\ldots,J$. If the conditional distribution of $\boldsymbol\varepsilon$ varies freely
with $\mathbf x$, then $\boldsymbol\pi^*(\mathbf x)$ can also vary freely and
need not vary continuously. Thus even total indifference need not have a
single image at the \textit{Structural environment} level.\footnote{It need not have a
single image even at the \textit{empirical specification} level, but it is not analyzed in the paper.} A fixed copula $C_{\mathbf x}=C$ and common marginals $F_{j,\mathbf x}=F_{\mathbf x}$ for every $j$ stabilize this point: $\pi_k^{*,M}
=
\int_0^1
\frac{\partial C}{\partial u_k}(v,\ldots,v)\,dv$, $k=1,\ldots,J$, does not depend on $\mathbf x$.\footnote{Unless $C$ is exchangeable, this point need
not be the barycentre.} Constancy of the total indifference image point is not,
however, enough for ranking recovery. As the shape of $F_{\mathbf x}$ changes
with $\mathbf x$, the map from index differences to probabilities also changes
with $\mathbf x$, so probability vectors generated by opposite index rankings
may overlap. Appendix~E gives an explicit overlap construction, and
Proposition~\ref{prop:distributional_no_uniform_recovery} there formalizes
the resulting failure of distribution-robust ranking recovery even
for a fixed $\boldsymbol{\beta}$.
Just like in the limited attention models in Section \ref{sec:cs_structural}, a common location-scale structure removes this source of ambiguity by producing one fixed probability map and letting its indifference boundaries to be nonlinear. It is generally asymmetric due to copula non-exchangeability (in Section \ref{sec:cs_structural} asymmetry of such a map was driven by non-symmetric attention probabilities).
\begin{theorem}[Common structural partition and ranking recovery]
\label{th:nonexch_specialcase}
Suppose the standard random utility paradigm holds, and
$\varepsilon_j=\mu(\mathbf x)+s(\mathbf x)\widetilde\varepsilon_j$, $s(\mathbf x)>0$,
where $\widetilde{\boldsymbol\varepsilon}$ is independent of $\mathbf x$ with its distribution having convex support and nonempty interior. Suppose also that its marginals have the same absolutely continuous c.d.f. $\widetilde F$, and its absolutely continuous copula $\widetilde C$ is generally non-exchangeable. Let $\mathcal E^o_{\mathrm{diff}}$, $d^{\boldsymbol\beta}(\mathbf x)$, and $\mathcal D_{\boldsymbol\beta}$ be defined as in Section \ref{sec:cs_structural} for this vector $\widetilde{\boldsymbol\varepsilon}$.
\vskip 0.1in
A.(Ranking recovery) The map $\Psi:\mathcal E^o_{\mathrm{diff}}
\longrightarrow\operatorname*{\mathrm{relint}}(\Delta_{J-1})$
such that the choice probability vector at $(\mathbf x,\boldsymbol\beta)$ is $\Psi(d^{\boldsymbol\beta}(\mathbf x))$ whenever $d^{\boldsymbol\beta}(\mathbf x)\in\mathcal E^o_{\mathrm{diff}}$ is a topological embedding. For every $\boldsymbol\beta \in \mathcal{B}_0$, each probability vector in $\Psi(\mathcal D_{\boldsymbol\beta})$, thus, uniquely determines the normalized index vector and all pairwise index rankings.
\vskip 0.05in
If $0\in\mathcal E^o_{\mathrm{diff}}$ (complete indifference is feasible), then $\Psi(0)$ is the vector with coordinates $\pi_j^{*,M}
=
\int_0^1
\frac{\partial\widetilde C}{\partial u_j}(v,\ldots,v)\,dv$,
$j=1,\ldots,J$.
\vskip 0.1in
B.(Common pairwise separation) $\Psi\!\left(\mathcal D_{\boldsymbol\beta}\cap\{d:d_j=d_k\}\right)
=
\Psi(\mathcal D_{\boldsymbol\beta})
\cap
\Psi\!\left(\mathcal E^o_{\mathrm{diff}}\cap\{d:d_j=d_k\}\right)$ for every $\boldsymbol\beta\in\mathcal B_0$ and $j,k \in \mathcal{J}$, $j\neq k$, with the analogous identities for $>$ and $<$. The second set on the right-
hand side of each identity (for each $=$, $>$, $<$) is common to all $\boldsymbol\beta$ but its attainable trace may depend on $\boldsymbol\beta$.
\vskip 0.1in
C. (Common structural partitioning) For every $j\neq k$, it holds that $\Psi\!\left( \mathcal E^o_{\mathrm{diff}}\cap\{d:d_j=d_k\} \right)$ is an embedded $(J-2)$-dimensional topological manifold in $\Psi(\mathcal E^o_{\mathrm{diff}})$. Its complement in $\Psi(\mathcal E^o_{\mathrm{diff}})$ has exactly two connected components, $\Psi\!\left( \mathcal E^o_{\mathrm{diff}}\cap\{d:d_j>d_k\} \right)$ \quad\text{and}\quad $\Psi\!\left( \mathcal E^o_{\mathrm{diff}}\cap\{d:d_j<d_k\} \right)$. If $0\in\mathcal E^o_{\mathrm{diff}}$, the $J!$ strict order regions are all nonempty and connected, and all have $\boldsymbol\pi^{*,M}=\Psi(0)$ in their closure.
\end{theorem}
The same discussion as after Theorem \ref{th:cs_heteroskedasticity} applies here. The common map $\Psi$ guarantees more than fixed-$\boldsymbol{\beta}$ recovery and gives the common structural ranking
partition of $\Psi(\mathcal E^o_{\mathrm{diff}})$, with a given $\boldsymbol{\beta}$ possibly tracing only part of it. If, in
addition, $\mathcal E^o_{\mathrm{diff}}
\subseteq
\bigcup_{\boldsymbol\beta\in\mathcal B_0}
\mathcal D_{\boldsymbol\beta}$,
the common structural partition is jointly attained as
$\boldsymbol\beta$ varies and the \emph{Structural environment} is ``quantile''.
Figure~\ref{fig:nonexch_gaussian} illustrates both the structural
indifference curve and its dependence on the unobservables distribution, for simplicity taking $s(\mathbf{x})=1$. In each
case, $\widetilde{\boldsymbol\varepsilon}$ is a mean-zero trivariate normal
vector with unit variances. Let $\rho_{kj}$ denote the pairwise correlation between $\widetilde{\varepsilon}_k$ and $\widetilde{\varepsilon}_j$. We consider the following three cases of $(\rho_{12},\rho_{13},\rho_{23})$: (a) case 1 is the case of all positive correlations
$(0.15,\,0.35,\,0.55)$; (b) case 2 is the case of one negative and two positive correlations $ (-0.20,\,0.25,\,0.60)$; (c) case 3 is the case of two negative and one positive
$(-0.3,\,-0.1,\,0.7)$. For these three cases, Figure \ref{fig:nonexch_gaussian} plots $(p_1(a),p_2(a))$ across $a$, where
\begin{equation*}
p_k(a) = \int_{-\infty}^{+\infty} \frac{\partial C}{\partial u_k}( F(e_1), F(e_1), F(e_1 -a)) dF(e_1), \quad k=1,2,
\end{equation*}
which fully characterize partial indifference cases $d_2=0$ with $a$ standing for $d_3$. Figure \ref{fig:nonexch_gaussian} confirms that different correlation structures produce different curvatures. The map $\Psi$ is common across values of
$\mathbf x$ only after the distribution of
$\widetilde{\boldsymbol\varepsilon}$ has been fixed. Within any one of these
structural environments, a particular $\boldsymbol\beta$ may still trace only
a segment or another proper subset of the corresponding curve.
\begin{figure}[ht]
\centering
\includegraphics[width=0.7\linewidth]{Nonexchahg_Indep2.png}
\caption{\label{fig:nonexch_gaussian}
Indifference between alternatives $1$ and $2$ under three
non-exchangeable trivariate normal unobservables distributions. Each curve plots
$(p_1,p_2)$ along $d_2=0$ as $d_3$ varies. The probability of alternative
$3$ is $p_3=1-p_1-p_2$.}
\end{figure}
If the conditional distribution of $\boldsymbol\varepsilon$ varies with
$\mathbf x$ in a way that cannot be absorbed by a common location and scale, the challenges are analogous to those discussed in Section \ref{sec:cs_structural}. Namely, there need not be a common index representation of choice probabilities across covariate values or a single map $\Psi$ that applies at every covariate value. Different
values of $\mathbf x$ may then be associated with different structural maps,
such as those generated by the three Gaussian specifications in
Figure~\ref{fig:nonexch_gaussian}. Even if each map separately generates a
smooth partial indifference codimension-one manifold, these manifolds may intersect or overlap when combined
across covariate values, and probability vectors associated with opposite
index rankings may coincide, producing ranking ambiguity.
\section{Conclusion}
\label{sec:conclusion}
A central insight of \citet{Manski1975} is that choice probabilities can
reveal ordinal information about deterministic utilities without any
parametric model of the utility unobservables. That insight relies on a particular environment of utility maximization with conditionally i.i.d. unobservables, which can be considered the natural multinomial counterpart of the median benchmark in binary choice. This paper takes \citet{Manski1975} as a starting point and asks how far the ordinal interpretation of choice probabilities extends once that benchmark is left behind, whether through behavioral departures from utility maximization, such as exogenous limited attention, or through asymmetries in the distribution of utility unobservables. The extension is far from automatic. With several alternatives, interactions across multiple comparison margins can lead to ranking ambiguity with no counterpart in binary choice.
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