# Expert classification aggregation

Federico Fioravanti

► To cite this version:

Federico Fioravanti. Expert classification aggregation. Mathematical Social Sciences, 2025, 137, pp.102406. 10.1016/j.mathsocsci.2025.102406. hal-05444218

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# Expert Classification Aggregation

Federico Fioravanti*¹,²

¹Université Jean Monnet Saint-Étienne, CNRS, Université Lyon 2, emlyon business school, GATE, 42023, Saint-Étienne, France.

²Institute for Logic, Language and Computation, University of Amsterdam, Amsterdam, The Netherlands

## Abstract

We consider the problem where a set of individuals has to classify *m* objects into *p* categories by aggregating the individual classifications, and no category can be left empty. An aggregator satisfies *Expertise* if individuals are decisive either over the classification of a given object, or the classification into a given category. We show that requiring an aggregator to satisfy *Expertise* (or variants of it) and be either unanimous or independent leads to numerous impossibility results.

**Keywords:** Classification Aggregation; Expertise.

**JEL Classification:** D71.

## 1 Introduction

The problem where a set of *n* individuals has to classify a set of *m* objects into *p* different categories, and no categories should be left empty, can be seen as a relevant one in many scenarios. For example, consider a set of

<footnote>*federico.fioravanti@univ-st-etienne.fr</footnote>

<footnote>I am grateful to Fernando Tohmé, Ulle Endriss, Bernardo Moreno, Agustín Bonifacio, participants of the COMSOC seminar at the ILLC, and two anonymous reviewers for comments and suggestions that led to an improvement of the paper.</footnote>
managers that has to assign a set of workers to different tasks. One of the
natural considerations at hand could be that no task should be left unas-
signed. Another example is training neural networks to classify images of
cats, dogs, and rabbits, using a large database with an equal proportion of
images of each animal. Requiring each neural network to classify at least one
image of each animal serves as a sanity check for the training process.

Maniquet and Mongin (2016) propose a formal setting to consider this
problem, extending Kasher and Rubinstein’s (1997) analysis of the Group
Identification Problem, the situation where a set of individuals has to clas-
sify a subset of them into two categories.<fnref n="1" /> Inspired by Arrow’s (1951) axioms,
Pareto and Independence of Irrelevant Alternatives, they show that an aggre-
gator that satisfies Unanimity, which indicates that unanimous classifications
should be respected, and Independence, which requires considering each ob-
ject separately, must be dictatorial. Later on, Cailloux et al. (2024) weaken
the Unanimity axiom, to obtain a weakening of the impossibility result that
holds for $m > p \ge 2$, with the existence of an essential dictator.<fnref n="2" />

The intention of this paper is to follow the line of Maniquet and Mongin
(2016) and Cailloux et al. (2024), who study counterparts for the classifica-
tion aggregation problem of classical results in standard preference aggrega-
tion (with Arrow’s (1951) and Wilson’s (1972) theorems, respectively). In
this work, we draw inspiration from Sen’s (1970) impossibility result of a
Paretian liberal and adapt various versions of the liberal axiom to our con-
text. Sen’s (1970) shows that even a minimal level of individual rights (where
people have control over personal choices) can conflict with Pareto efficiency
(the idea that if everyone agrees something is better, we should choose that
outcome). In our setting, objects do not necessarily belong to individuals’
private spheres; hence, instead of referring to it as the liberal axiom, we
term it the expertise axiom. This property requires that a collective decision
process should ensure that each individual with expertise can unilaterally
determine the classification of a specific object or the objects assigned into a
particular category. The results obtained here are significant in cases where
individuals have a direct connection to some of the objects they need to
classify or the categories involved in the classification process. Consider, for

<footnote n="1">¹A similar scenario occurs when a group of experts must select one among themselves to receive an award. See, for example, Holzman and Moulin (2013); Tamura and Ohseto (2014), and Tamura (2016).</footnote>

<footnote n="2">²If for every classification problem, objects are always classified according to a permutation of the classification of a given individual, it is essentially a dictatorship.</footnote>
instance, a group of editors responsible for overseeing the publication of a
'Handbook in Economics', where each major area of the discipline must be
represented by a dedicated chapter. Some editors may submit papers for
specific chapters themselves, while others may have specialised expertise in
particular fields of economics. In this context, two reasonable approaches
could be proposed: allowing editors who are also authors to decide the ap-
propriate chapter for their own submissions or assigning the task to editors
with expertise in the corresponding chapters. These ideas can be captured
by an expertise axiom, indicating different ways in which an individual can
be decisive over the classification of an object or into a category. We show
that in general, the expertise axiom is incompatible with natural properties
such as Unanimity or Independence. Furthermore, some versions of expertise
alone are not even satisfiable in this setting, where no category can be left
empty. This represents a departure from results in the Group Identification
Problem, where there are aggregators where the individuals are decisive for
certain classifications (Miller, 2008; Fioravanti and Tohmé, 2021). The key
distinction between these two settings is that in the Group Identification
Problem, it is not necessary that all the categories get an object classified
into.

The implications of the impossibility results presented in this paper ex-
tend to several domains where classification tasks are fundamental. For in-
stance, in organisational decision-making, where tasks or resources must be
allocated to individuals or groups, the expertise axiom reflects the need to
respect individual preferences based on their expertise. The results here
imply that in striving to ensure that no single individual’s preferences dom-
inate across all tasks, we may encounter situations where it is impossible to
maintain other desirable properties such as Unanimity or Independence. This
finding could be particularly relevant in settings where collaborative decision-
making processes are used, such as in project management or committee-
based allocations. Similarly, in the field of multi-agent systems, where mul-
tiple autonomous agents must classify resources or tasks into categories such
as ‘high priority’, ‘medium priority’, and ‘low priority’, the expertise axiom
might reflect the need for each agent to have control over at least one task or
resource. However, the impossibility results indicate that achieving a fair and
efficient aggregation of these classifications might be inherently challenging
when trying to balance ‘control’, Unanimity, and Independence. This has
practical implications for the design of decentralised decision-making algo-
rithms, used, for example, in distributed computing or collaborative robotics.
The plan of the paper is as follows. Section 2 presents the basic notions and axioms that we use, while we present the results in Section 3. Finally, Section 4 contains some concluding remarks.

# 2 Basic Notions and Axioms

Let $N = \{1, \dots, n\}$, with $n \ge 2$, be a finite set of individuals and let $X = \{x_1, \dots, x_m\}$ be a set of $m$ objects that need to be classified into the $p$ categories of a set $P$, with $p \ge 2$. In this setting, introduced by Maniquet and Mongin (2016), classifications are surjective mappings $c : X \to P$, that is, every category must have at least one object classified into. Thus we have that $m \ge p \ge 2$. We use $\mathcal{C}$ to denote the set of classifications, and every $\mathbf{c} = (c_1, \dots, c_n) \in \mathcal{C}^N$ is a classification aggregation problem, with $c_i$ indicating the classification given by individual $i$. A classification aggregation function (CAF) is a mapping $\alpha : \mathcal{C}^N \to \mathcal{C}$ such that $\alpha(\mathbf{c})(x)$ indicates the category where object $x$ is classified into. We call the outcome of $\alpha$ the social classification. Given a category $t \in P$, we denote the inverse image of $t$ as $\alpha(\mathbf{c})^{-1}(t)$. Formally, $\alpha(\mathbf{c})^{-1}(t) = \{x \in X \mid \alpha(\mathbf{c})(x) = t\}$.

In the following, we introduce a number of axioms, i.e., fundamental normative requirements that, in this specific classification process, we consider a reasonable CAF should satisfy. The first property states that if there is an object that is unanimously classified by the individuals, then the CAF has to classify that object accordingly. This property is the unary interpretation of Arrow's (1951) and Sen's (1970) Pareto principle, introduced by Maniquet and Mongin (2016).

**Definition 1 (Unanimity).** A CAF is unanimous if for all $\mathbf{c} \in \mathcal{C}^N$, all $x \in X$, and all $t \in P$ such that $c_1(x) = \dots = c_n(x) = t$, it is the case that $\alpha(\mathbf{c})(x) = t$.

Next, we introduce different interpretations of how an individual can be decisive in a classification problem. Individuals can have decision power over objects, categories, or pairs of objects and categories. We say that individual $i$ is decisive over object $x$, if for all $\mathbf{c} \in \mathcal{C}^N$, and all $t \in P$, it is the case that $c_i(x) = t$ implies $\alpha(\mathbf{c})(x) = t$.<fnref n="3" /> An individual $i$ is categorically decisive over category $t$, if for all $\mathbf{c} \in \mathcal{C}^N$, and all $x \in X$, it is the case that $\alpha(\mathbf{c})(x) = t$

<footnote n="3">³Although this definition is expressed as an implication for presentation purposes, it is equivalent to using a bi-conditional, given that an individual must classify every object.</footnote>
implies $c_i(x) = t$. Finally, individual $i$ is minimally decisive over the object $x$ and category $t$, if for all $c \in C^N$, it is the case that $c_i(x) = t$ if, and only if, $\alpha(c)(x) = t$.<fnref n="4" /> Now we present three axioms requiring the existence of two decisive individuals, an adaptation to this setting of Sen's (1970) minimally liberal principle. The goal is to ensure a minimum number of decisive individuals, sufficient to prevent the emergence of a dictator while remaining practically demanding.<fnref n="5" />

**Definition 2 (Expertise).** A CAF is expert if there exist two individuals $i, j \in N$ and two different objects $x, y \in X$ such that for all $c \in C^N$, individuals $i$ and $j$ are decisive over $x$ and $y$, respectively.

**Definition 3 (Categorical Expertise).** A CAF is categorically expert if there exist two individuals $i, j \in N$ and two categories $t, t' \in P$ such that for all $c \in C^N$, individuals $i$ and $j$ are categorically decisive over $t$ and $t'$, respectively.

**Definition 4 (Minimal Expertise).** A CAF is minimally expert if there exist two individuals $i, j \in N$ and two pairs $(x, t), (y, t') \in X \times P$, with $x \neq y$, such that for all $c \in C^N$, individuals $i$ and $j$ are minimally decisive over $(x, t)$ and $(y, t')$, respectively.

It is easy to see that Minimal Expertise is weaker than Expertise, with the former being implied by the latter. Finally, we present a unary interpretation of Arrow's (1951) Independence of Irrelevant Alternatives, introduced by Maniquet and Mongin (2016), which states that the social classification of an object in two different profiles is the same if the individual's classifications of that object are the same in both profiles.

**Definition 5 (Independence).** A CAF is independent if given $c, c' \in C^N$ and $x \in X$ such that $c_i(x) = c'_i(x)$ for all $i \in N$, it is the case that $\alpha(c)(x) = \alpha(c')(x)$.

<footnote n="4">⁴These formal definitions provide different interpretations of how an individual can be decisive in this setting, which are useful for identifying impossibility results. While other definitions, such as those presented in Remark 1, are of interest, they fall outside the scope of this work, which focuses on illustrating the complexity of aggregation processes with decisive individuals.</footnote>

<footnote n="5">⁵The objects of Definition 2 and Definition 4 must be different, otherwise an object might end up being classified into two different categories (a contradiction to the definition of a classification function).</footnote>
# 3 Results

We start by looking at the implications of Minimal Expertise, and its con-
junction with Unanimity. Our first result resembles Sen's (1970) impossibility
of a Paretian liberal, highlighting the conflict that arises between the Pareto
principle and individual expertise (individuals being decisive over objects or
categories in which they are experts).

**Theorem 1.** For $m \ge p \ge 2$, there is no CAF that satisfies Unanimity and Minimal Expertise.

*Proof.* Let $\alpha$ be a CAF that satisfies Unanimity and Minimal Expertise, and assume, without loss of generality, that the individual 1 is decisive over the pair $(x, t')$ and the individual 2 is decisive over the pair $(y, t'')$. Now suppose that $t' \neq t'', t' = t_1, t'' = t_2$, and consider the following classification aggregation problem $\mathbf{c}:^6$

<table><thead><tr><th><b>c</b></th><th>1</th><th>2</th><th>N \ {1, 2}</th></tr></thead><tbody><tr><th scope="row">t<sub>1</sub></th><td>y</td><td>x</td><td>x</td></tr><tr><th scope="row">t<sub>2</sub></th><td>{x} &cup; A<sub>2</sub></td><td>{y} &cup; A<sub>2</sub></td><td>{y} &cup; A<sub>2</sub></td></tr><tr><th scope="row">t<sub>3</sub></th><td>A<sub>3</sub></td><td>A<sub>3</sub></td><td>A<sub>3</sub></td></tr><tr><th scope="row">...</th><td>...</td><td>...</td><td>...</td></tr><tr><th scope="row">t<sub>p</sub></th><td>A<sub>p</sub></td><td>A<sub>p</sub></td><td>A<sub>p</sub></td></tr></tbody></table>

where $A_i \subset X$, $A_i \neq \emptyset$ for $i = \{3, \dots, p\}$, for all $i$ and $j$ we have that $A_i \cap A_j = \emptyset$, and that $\cup_{i=2}^p A_i = X \setminus \{x, y\}$. So, the classification given by all the individuals is the same for every object except for $x$ and $y$. Thus, by Unanimity, $A_i \subseteq \alpha(\mathbf{c})^{-1}(t_i)$ for $i \in \{3, \dots, p\}$. In particular, by Unanimity and Minimal Expertise, we have that $\{y\} \cup A_2 \subseteq \alpha(\mathbf{c})^{-1}(t_2)$ and $\alpha(\mathbf{c})(x) \neq t_1$. Thus $\alpha(\mathbf{c})^{-1}(t_1) = \emptyset$, concluding that $\alpha(\mathbf{c})$ is not a classification function. The proof is similar for the case where $t' = t''$. If we assume $t' = t'' = t_1$, then using the same classification aggregation problem $\mathbf{c}$, we have that $x, y \notin \alpha(\mathbf{c})^{-1}(t_1)$, and thus $\alpha(\mathbf{c})^{-1}(t_1) = \emptyset$. $\square$

**Remark 1.** If we consider that an individual $i$ can be *minimally semi-decisive* for a pair $(x, t)$, that is, if $c_i(x) = t$ implies $\alpha(\mathbf{c})(x) = t$, then there are CAF's that are unanimous and can have two or more minimally

<footnote n="6">⁶The rows in the table indicate the objects that the individuals classify into a given category, while the columns are the individual's classifications.</footnote>
semi-decisive individuals.<fnref n="7" /> One example is the following CAF. Let $x_1 \succ x_2 \succ \dots \succ x_m$ and $t_1 \succ t_2 \succ \dots \succ t_p$ be given orders for the objects and the categories, respectively, and assume 1 and 2 are minimally semi-decisive over the pairs $(x_1, t_1)$ and $(x_2, t_2)$, respectively. For the cases where $c_1(x_1) = t_1$ or $c_2(x_2) = t_2$, the CAF classifies those objects accordingly. Then, classifies all the unanimous classifications accordingly. And finally, it classifies the first unassigned object to the first empty category, the second unassigned object to the second empty category, and so on, following the given orders, until all the objects are classified. For the case where $m > p$, when all the categories have one object, unassigned objects are classified into the categories following the given order. If neither $c_1(x_1) = t_1$ nor $c_2(x_2) = t_2$, the CAF skips the first step. This rule also works for the case where 1 and 2 are minimally semi-decisive over $(x_1, t)$ and $(x_2, t)$, when $m > p$. For $m = p$, there is no CAF for this case, as one category might remain without an object being classified into.

For the particular case of 2 objects and 2 categories, even the sole requirement of the existence of minimally decisive individuals is excessively demanding.

**Proposition 2.** For $m = p = 2$, there is no CAF that satisfies Minimal Expertise.

*Proof.* Let $\alpha$ be a CAF that satisfies Minimal Expertise, and assume, without loss of generality, that the individual 1 is decisive over the pair $(x, t')$ and the individual 2 is decisive over the pair $(y, t'')$. Now suppose that $t' \neq t''$, $t' = t_1$, $t'' = t_2$ and consider the following classification aggregation problem $\mathbf{c}$:

<table><thead><tr><th><b>c</b></th><th>1</th><th>2</th><th>N \ {1, 2}</th></tr></thead><tbody><tr><th scope="row">t<sub>1</sub></th><td>y</td><td>x</td><td>x</td></tr><tr><th scope="row">t<sub>2</sub></th><td>x</td><td>y</td><td>y</td></tr></tbody></table>

By Minimal Expertise, we have that $y \in \alpha(\mathbf{c})^{-1}(t_2)$ and $\alpha(\mathbf{c})(x) \neq t_1$. Thus $\alpha(\mathbf{c})^{-1}(t_1) = \emptyset$, concluding that $\alpha(\mathbf{c})$ is not a classification function. The proof is similar for the case where $t' = t''$ (we can use the previous classification aggregation problem $\mathbf{c}$). □

<footnote n="7">⁷Note that in Sen (1970), requiring individuals to be semi-decisive still leads to an impossibility result.</footnote>
**Remark 2.** There exist minimally expert CAFs for $m \ge p > 2$. Let individuals 1 and 2 be minimally decisive over $(x, t')$ and $(y, t'')$, respectively, with $t' \neq t''$. A CAF that for the cases where $c_1(x_1) = t'$ or $c_2(x_2) = t''$, classifies those objects accordingly and assigns the remaining objects so no category is left empty, and otherwise, classifies the objects according to a pre-given classification, satisfies Minimal Expertise. When $t' = t''$, we need that $m > p > 2$ (otherwise there might be categories that are left empty).

Next, we demonstrate that requiring a CAF to be both independent and minimally expert is quite demanding, though not as stringent as requiring the aggregator to be unanimous.

**Proposition 3.** Let $p+1 \ge m \ge p$. There is no CAF that satisfies Minimal Expertise and Independence.

*Proof.* Let $\alpha$ be a CAF that satisfies Independence and Minimal Expertise, and assume, without loss of generality, that individual 1 is minimally decisive over the pair $(x, t')$ and individual 2 is minimally decisive over the pair $(y, t'')$. If $m = p$, every classification is such that there is exactly one object in every category, and if $m = p + 1$, every classification is such that there is exactly one object in every category but one, that has two objects classified into. Let $t' \neq t''$, $t' = t_1$, $t' = t_2$, and consider the following classification aggregation problem $c$:

<table><thead><tr><th><b><i>c</i></b></th><th>1</th><th>2</th><th><i>N</i> \ {1, 2}</th></tr></thead><tbody><tr><td><i>t</i><sub>1</sub></td><td><i>x</i></td><td><i>y</i></td><td><i>x</i></td></tr><tr><td><i>t</i><sub>2</sub></td><td><i>y</i></td><td><i>x</i></td><td><i>y</i></td></tr><tr><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td></tr></tbody></table>

such that the rest of the table is completed to be a classification aggregation problem. Thus, by Minimal Expertise, we have that $x \in \alpha(c)^{-1}(t_1)$ and $y \notin \alpha(c)^{-1}(t_2)$. Assume that $z \in \alpha(c)^{-1}(t_2)$, with $z \neq y$. Now consider the following classification aggregation problem $c'$:

<table><thead><tr><th><b><i>c</i>'</b></th><th>1</th><th>2</th><th><i>N</i> \ {1, 2}</th></tr></thead><tbody><tr><td><i>t</i><sub>1</sub></td><td><i>x</i></td><td><i>x</i></td><td><i>x</i></td></tr><tr><td><i>t</i><sub>2</sub></td><td><i>y</i></td><td><i>y</i></td><td><i>y</i></td></tr><tr><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td></tr></tbody></table>
such that the rest of the table is completed to be a classification aggregation problem, and the rows from $t_3$ to $t_p$ are the same as in **c**. Then, by Independence and Minimal Expertise, we have that $\{z, y\} = \alpha(c')^{-1}(t_2)$, and thus, $\{x\} = \alpha(c')^{-1}(t_1)$. Otherwise, there are two categories with two objects classified into. For the case where $m = p$, this already leads to a contradiction. Finally, consider the classification aggregation problem **c''**:

<table><thead><tr><th><b><i>c</i>''</b></th><th>1</th><th>2</th><th><i>N</i> \ {1, 2}</th></tr></thead><tbody><tr><td><i>t</i><sub>1</sub></td><td><i>y</i></td><td><i>x</i></td><td><i>x</i></td></tr><tr><td><i>t</i><sub>2</sub></td><td><i>x</i></td><td><i>y</i></td><td><i>y</i></td></tr><tr><td>&#8942;</td><td>&#8942;</td><td>&#8942;</td><td>&#8942;</td></tr></tbody></table>

such that the rest of the table is completed to be a classification aggregation problem, and the rows from $t_3$ to $t_p$ are the same as in **c'**. By Independence and Minimal Expertise, we have that $\{z, y\} = \alpha(c'')^{-1}(t_2)$, $x \notin \alpha(c'')^{-1}(t_1)$ and $\alpha(c'')(w) = \alpha(c')(w) \neq t_1$ for all $w \in X \setminus \{x, y, z\}$. Hence the category $t_1$ is left empty and $\alpha(c'')$ is not a classification function.

For $t' = t''$, we can assume that $t' = t_1$. Thus, we need that $m > p$ and we are in the case $m = p + 1$. Consider the classification aggregation problems **c** and **c''** previously introduced. By Minimal expertise, $x, y \in \alpha(c)^{-1}(t_1)$. Then, it must be that there is exactly one object in each category $t_i \neq t_1$. Let $z_i \in X \setminus \{x, y\}$ and $\{z_i\} = \alpha(c)^{-1}(t_i)$, for $i \in \{2, \dots, p\}$. By Independence, we have that $z_i \in \alpha(c'')^{-1}(t_i)$, for $i \in \{2, \dots, p\}$, and by Minimally Expertise we have that $x, y \notin \alpha(c'')^{-1}(t_1)$. So $\alpha(c'')^{-1}(t_1) = \emptyset$, and thus $\alpha(c'')$ is not a classification function. □

In light of the Group Identification Problem, this result might be seen as surprising, as there are independent and expert rules. An example is the Liberal aggregator, where every individual is decisive over their own classification (see, for more examples, Fioravanti and Tohmé, 2021).

**Remark 3.** There are independent and expert (thus minimally expert) CAF's for $m \ge p + 2$. For example, assume 1 is decisive over $x_1$ and 2 is decisive over $x_2$, and consider the CAF that fixes all the objects in $X \setminus \{x_1, x_2\}$ to $p$ different categories, and then assigns $x_1$ and $x_2$ according to individuals 1 and 2 classifications. Moreover, if $m \ge p + n$, where $n$ is the number of individuals, we can have a rule such that all the individuals are decisive over an object, and under the same reasoning of the previous example, obtain an independent and expert CAF.
Now we turn our attention to the analysis of Expertise alone. We have already shown that the conjunction of either Unanimity or Independence and Minimal Expertise is rather demanding, so it is expected that at least for some configurations of the number of objects and categories, just Expertise alone is prohibitive. That is what the next result shows.

**Proposition 4.** Let $m = p$. There is no CAF that satisfies Expertise.

*Proof.* If $m = p$, then there is exactly one object in each category. Let $\alpha$ be a CAF that satisfies Expertise, and assume, without loss of generality, that individual 1 is decisive over object $x$ and individual 2 is decisive over object $y$. Consider the following classification aggregation problem $c$:

<table><thead><tr><th>c</th><th>1</th><th>2</th><th>N \ {1, 2}</th></tr></thead><tbody><tr><td>t<sub>1</sub></td><td>y</td><td>x</td><td>x</td></tr><tr><td>t<sub>2</sub></td><td>x</td><td>y</td><td>y</td></tr><tr><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td></tr></tbody></table>

such that the rest of the table is completed to be a classification aggregation problem. Then we have that $x$ and $y$ are both classified into $t_2$, so $\alpha(c)$ is not a classification function. □

**Remark 4.** There are expert CAF's for $m = p + 1$. The CAF that classifies $x$ and $y$ according to the classifications given by the individuals that are decisive over them, and then classifies the rest of the objects following a given order such that no category is left empty, is an expert CAF.

**Remark 5.** There is a CAF that satisfies Unanimity and Independence, but not Minimal Expertise. It is the dictatorial CAF, such that for all $c \in C^N$, it is the case that $\alpha(c) = c_i$, for a fixed $i \in N$. Moreover, it is the unique CAF that satisfies those two properties (Maniquet and Mongin, 2016).

Our final result shows that requiring individuals to have decisive power over the categories turns out to be prohibitive, even if we do not impose extra normative requirements.

**Proposition 5.** There is no CAF that satisfies Categorical Expertise.

*Proof.* Let $\alpha$ be a CAF that satisfies Categorical Expertise, and assume, without loss of generality, that the individual 1 is decisive over the category $t'$ and the individual 2 is decisive over the category $t''$. Let $t' \neq t''$, $t' = t_1$, $t' = t_2$, and consider the following classification aggregation problem $c$:
<table><thead><tr><th><b>c</b></th><th>1</th><th>2</th><th>N \ {1, 2}</th></tr></thead><tbody><tr><td>t<sub>1</sub></td><td>x</td><td>y</td><td>x</td></tr><tr><td>t<sub>2</sub></td><td>y</td><td>x</td><td>y</td></tr><tr><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td></tr></tbody></table>

such that the rest of the table is completed to be a classification aggregation problem. Then, by Categorical Expertise, we have that $\emptyset \neq \alpha(c)^{-1}(t_1) \subseteq \{x\}$ and $\emptyset \neq \alpha(c)^{-1}(t_2) \subseteq \{x\}$, leading to a contradiction.

For the case $t' = t''$, we can assume that $t' = t_1$ and consider the following classification aggregation problem $c'$:

<table><thead><tr><th><b>c'</b></th><th>1</th><th>2</th><th>N \ {1, 2}</th></tr></thead><tbody><tr><td>t<sub>1</sub></td><td>x</td><td>y</td><td>y</td></tr><tr><td>t<sub>2</sub></td><td>y</td><td>x</td><td>x</td></tr><tr><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td></tr></tbody></table>

such that the rest of the table is completed to be a classification aggregation problem. Then, by Categorical Expertise, we have that $\emptyset \neq \alpha(c)^{-1}(t_1) \subseteq \{x\}$ and $\emptyset \neq \alpha(c)^{-1}(t_2) \subseteq \{y\}$, leading to a contradiction. □

If we consider this result in light of the Group Identification Problem, it strikes us as surprising. When the requirement of not leaving a category empty is not imposed, it is easy to think of a rule that satisfies Categorical Expertise. For example, for the case of only two categories, where objects are the same individuals (the ‘Who is a J?’ original setting, Kasher and Rubinstein, 1997), the aggregator that classifies an individual into category J if, and only if, individuals 1 and 2 classify them as J, and not in J otherwise, is categorically expert.

# 4 Final Remarks

In this paper, we address the problem of classifying *m* objects into *p* different categories, ensuring that no category remains empty and recognizing that certain individuals, deemed experts, have decisive power over objects or categories. Even with the minimal requirement that only two individuals possess decisiveness, the potential for aggregators emerges only under specific configurations of the number of objects and categories. We derive a result analogous to Sen's (1970) impossibility theorem of a Paretian liberal,
<table><caption>Table 1: Summary of our results.</caption><thead><tr><th></th><th>Without additional axioms</th><th>Unanimity</th><th>Independence</th></tr></thead><tbody><tr><th>M. Expertise</th><td>No, for m = p = 2 (Prop. 2)<br>Yes, for m ≥ p > 2 (Rem. 2)</td><td>No (Thm. 1)</td><td>No, for p + 1 ≥ m ≥ p (Prop. 3)<br>Yes, for m ≥ p + 2 (Rem. 3)</td></tr><tr><th>Expertise</th><td>No, for m = p (Prop. 4)<br>Yes, for m ≥ p + 1 (Rem. 3 and 4)</td><td>No (Thm. 1)</td><td>No, for p + 1 ≥ m ≥ p (Prop. 3)<br>Yes, for m ≥ p + 2 (Rem. 3)</td></tr><tr><th>C. Expertise</th><td>No (Prop. 5)</td><td>No (Prop. 5)</td><td>No (Prop. 5)</td></tr></tbody></table>

demonstrating that there is no Classification Aggregation Function that satisfies both Unanimity and Minimal Expertise, that is when individuals have decisive power over pairs of objects and categories. This finding illustrates how the Pareto principle remains in conflict with individual expertise within this context. The best we can do is to show the existence of independent and expert CAFs, for the cases where there are at least two more objects than categories, although for particular values ($m = p$) even Expertise alone cannot be satisfied. We also show that it is not possible in this setting to require individuals to be decisive over categories, even if we do not impose additional normative requirements. A summary of our results can be found in Table 1.

In essence, the expertise axiom is a powerful tool for ensuring that decision-making processes are not dominated by a single perspective. However, its application necessitates careful consideration of the trade-offs involved, particularly when other principles like fairness, consistency, and independence are also valued. The results in this paper highlight the complexities that arise when attempting to create classification systems that are both fair and responsive to a diversity of decision-makers, offering critical insights into the design of such systems across various domains.

## References

* Arrow, K. J. (1951). *Social Choice and Individual Values*. Wiley: New York.

* Cailloux, O., Hervouin, M., Ozkes, A. I., and Sanver, M. R. (2024). Classification aggregation without unanimity. *Mathematical Social Sciences*, 128:6–9.
Fioravanti, F. and Tohmé, F. (2021). Alternative axioms in group identification problems. *Journal of Classification*, 38(2):353–362.

Holzman, R. and Moulin, H. (2013). Impartial nominations for a prize. *Econometrica*, 81(1):173–196.

Kasher, A. and Rubinstein, A. (1997). On the question “Who is a J?”: A social choice approach. *Logique et Analyse*, 40(160):385–395.

Maniquet, F. and Mongin, P. (2016). A theorem on aggregating classifications. *Mathematical Social Sciences*, 79:6–10.

Miller, A. D. (2008). Group identification. *Games and Economic Behavior*, 63(1):188–202.

Sen, A. (1970). The impossibility of a Paretian liberal. *Journal of Political Economy*, 78(1):152–157.

Tamura, S. (2016). Characterizing minimal impartial rules for awarding prizes. *Games and Economic Behavior*, 95:41–46.

Tamura, S. and Ohseto, S. (2014). Impartial nomination correspondences. *Social Choice and Welfare*, 43:47–54.

Wilson, R. (1972). Social choice theory without the Pareto Principle. *Journal of Economic Theory*, 5(3):478–486.