# Voting with Partial Orders: The Plurality and Anti-Plurality Classes

Ulle Endriss, Federico Fioravanti

► To cite this version:

Ulle Endriss, Federico Fioravanti. Voting with Partial Orders: The Plurality and Anti-Plurality Classes. 2026. hal-05493098

HAL Id: hal-05493098

https://hal.science/hal-05493098v1

Preprint submitted on 4 Feb 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# Voting with Partial Orders: The Plurality and Anti-Plurality Classes

Ulle Endriss¹ and Federico Fioravanti* ²

¹Institute for Logic, Language and Computation (ILLC), University of Amsterdam, The Netherlands

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

## Abstract

In the theory of voting, the Plurality rule for preferences that come in the form of linear orders selects the alternatives most frequently appearing in the first position of those orders, while the Anti-Plurality rule selects the alternatives least often occurring in the final position. We explore extensions of these rules to preferences that are partial orders, offering axiomatic characterisations for them.

**Keywords:** Plurality, Anti-Plurality, Strict Partial Orders, Voting.

## 1 Introduction

We consider the problem of a group choosing alternatives from a fixed set of finitely many alternatives. In a deviation from the model most commonly studied in social choice theory (Arrow et al., 2002; Zwicker, 2016), we focus on situations in which each individual's preference is a *strict partial order*, allowing any given individual, for each pair of alternatives, to report either that she strictly prefers one over the other or that she does not wish to compare the two. The study of collective decision making on the basis of partial-order preferences deserves attention, as requiring voters to fully rank a (possibly) large number of alternatives may be overly demanding in many circumstances. Partial orders offer a less restrictive setting, giving voters

<footnote>*federico.fioravanti@univ-st-etienne.fr (corresponding author)</footnote>
more flexibility for expressing their true beliefs, while still allowing for the
option to express preferences as linear orders for those who wish to do so.

**Example 1.** Consider the following situation, inspired by an example first discussed by Terzopoulou and Endriss (2019). Suppose we want to run a competition to look for the best mobile app. We ask users to rank different mobile apps, namely *Instagram*, *Facebook*, *TikTok*, *Uber*, *Gmail*, and *Yahoo*. A particular user might be sure that she prefers *Instagram* to *Facebook*, and *Facebook* to *TikTok*, as she uses the three of them for posting videos; she also is sure that she prefers *Gmail* to *Yahoo*; but she might not be able to compare *Uber* to the rest of the apps, or *Gmail* to *Instagram*.

<figure><img src="image_2.png" /></figure>

So it is clear that *Instagram* and *Gmail* are among the most preferred apps,
and that *TikTok* and *Yahoo* are the least preferred ones. But it is less clear
whether *Uber* should be considered a good or a bad app, as it is not preferred
to other alternatives, but there also are no alternatives that are preferred
to it. Overall, it is not obvious which app should be considered *the* best for
this user, as the set of most preferred alternatives has a particular structure
(e.g., *Instagram* beats two other apps, while *Uber* has not been compared to
any other app). What is even less clear is how the preferences of multiple
users should be aggregated in a situation like this.

Δ

$\Delta$

For the standard model of social choice, in which voters report strict linear orders, the most widely used rule in practice is the Plurality rule, which selects the alternative(s) most frequently ranked at the top of an individual preference order. While the Plurality rule has been rightfully criticised in the literature for its shortcomings (Laslier, 2011), it does have the significant advantage of being particularly simple: every voter simply awards a point to her most preferred alternative. Can we preserve some of this simplicity when moving from linear to partial orders? The answer is not obvious, as an individual who only reports a partial order might not have a uniquely identified most preferred alternative.

If we stick to the idea that an individual should contribute more to her
most preferred alternatives, we arrive at what we shall call the *Plurality Class* of voting rules for partial orders. For instance, in the mobile apps
example, *Instagram*, *Gmail*, and *Uber* are all top alternatives (in the sense that there are no alternatives more preferred to them), but other preferences might have a different structure. One natural rule might assign 1 point to each of them (and 0 to all others), while another might assign each top alternative as many points as the number of alternatives it dominates in that preference (and still 0 to all non-top alternatives). Both of these rules belong to the Plurality Class, and both reduce to the standard Plurality rule for the special case of linear orders.

The core contribution of this paper is the identification and axiomatic characterisation of the Plurality Class and the closely related Anti-Plurality Class, which extends the well-known Anti-Plurality rule (also referred to as the Veto rule) for linear orders. In the standard framework, the Anti-Plurality rule selects the alternative(s) that are least frequently ranked last. We further provide characterisations of several important subclasses, including, in particular, the most natural interpretations of the Plurality and Anti-Plurality rules within this setting.

To characterise these two classes, we adapt many previously introduced axioms to our setting, while also proposing new ones. The standard Plurality and Anti-Plurality rules belong to the family of positional scoring rules, which are rules that assign scores to alternatives based on their positions in the linear orders, and then choose the alternative(s) that maximise the sum of the scores across all voters. Since Young (1975) characterised the scoring rules as the only ones satisfying Anonymity, Neutrality, Reinforcement, and Continuity, many different characterisations of the Plurality rule and the Anti-Plurality rule for linear orders have been obtained. Let us briefly mention some of the characterisations of the Plurality rule that can be found in the literature:<fnref n="1" /> Anonymity, Neutrality, Reinforcement, Continuity, and Reduction (Richelson, 1978);<fnref n="2" /> Anonymity, Neutrality, Reinforcement, Monotonicity, and Bottom-Invariance (Merlin and Naeve, 2000); and Anonymity, Neutrality, Reinforcement, Faithfulness, and Tops-Only (Sekiguchi, 2012). Moreover, Llamazares and Peña (2015) show that within the scoring rules that award one point to the first *k* alternatives, Plurality is the only one

<footnote n="1">¹Definitions of these axioms can be found in Section 2 and in Section 3, except for Reduction, which says that Pareto-dominated alternatives can be deleted; Monotonicity, which requires that if a voter improves the position of an already winning alternative, keeping everything else fixed, then that alternative becomes the only winner; Bottom- and Top-Invariance, which imply that reshuffling losing alternatives below or above, respectively, the winning candidate, does not change the outcome; and Minimal Veto, which says that every minority group should be assigned veto power.</footnote>

<footnote n="2">²Ching (1996) later showed that Continuity, in fact, is not necessary for this characterisation.</footnote>
that respects the Pareto principles. For the case of the Anti-Plurality rule, we can mention the following characterisations for linear orders: Anonymity, Neutrality, Reinforcement, Monotonicity, and Top-Invariance (Barberà and Dutta, 1982); Anonymity, Neutrality, Reinforcement, Continuity, and Minimal Veto (Baharad and Nitzan, 2005); and Anonymity, Neutrality, Reinforcement, Averseness, and Bottoms-Only (Kurihara, 2018).

While we are not aware of any attempts to characterise Plurality-like rules for partial orders, there is a growing literature that recognises the practical significance of partial orders in the context of social choice (Konczak and Lang, 2005; Terzopoulou, 2021). For instance, Pini et al. (2005) generalise both Arrow's Theorem (1951) and the Muller-Satterthwaite Theorem (1977) to the case of partial orders. Terzopoulou and Endriss (2019) consider the problem of aggregating incomplete pairwise preferences where the voters have a weight that depends on the number of alternatives they rank. Some work has focused on specific classes of partial orders, notably the top-truncated orders, where an individual strictly ranks her most preferred alternatives and it is assumed that she is indifferent between the remaining alternatives. In this setting, Baumeister et al. (2012) provide complexity results on manipulation and bribery, while Terzopoulou and Endriss (2021) characterise different variants of the Borda rule.

A related framework from which we draw some inspiration, and to which some of our results can be applied, is the setting of approval voting (Brams and Fishburn, 1978), where ballots merely indicate which alternatives a voter approves of. This framework has been the subject of extensive axiomatic analysis, with numerous characterisations of various approval voting rules available in the literature. For instance, Brandl and Peters (2022) provide eight distinct characterisations of the standard approval voting rule, while Alcalde-Unzu and Vorsatz (2009) and Bardal and Endriss (2025) focus one approval-based scoring rules, where a voter's weight depends on the size of the approval set they report. Aragones et al. (2011) show that, in certain environments, approval voting provides stronger incentives for participation than plurality voting.

The remainder of this paper is organised as follows. Section 2 presents the model of voting we study, introduces a number of voting rules for this setting, and defines and motivates the axioms we shall use. Section 3 presents our characterisation results for the two families of voting rules we focus on, inspired by the standard Plurality and Anti-Plurality rule. It also comments on a few applications of our results to approval voting. Finally, Section 4 contains some concluding remarks.
## 2 The Model

In this section we present the model of voting with partial orders that we study. It is a model of variable electorates, meaning that any finite subset of an infinite universe of voters might cast a ballot in any given election. Our model is one of the standard models of voting commonly studied in the literature (Arrow et al., 2002), except that individual preferences are assumed to be strict partial orders rather than strict linear orders.

We also recall the definition of the family of positional scoring rules for this model and then introduce the Plurality Class and the Anti-Plurality Class of voting rules belonging to this family. Finally, we propose a number of axioms for voting with partial orders.

### 2.1 Voting with Partial Orders

Let $A$ be a finite set of alternatives, with $|A| = m \ge 3$. A preference is a strict partial order $>_{\prec}$ over the set of alternatives, i.e., an irreflexive, antisymmetric, and transitive (but not necessarily complete) binary relation on $A$. We use $\mathcal{D}$ to denote the domain of all such preferences. For $a, b \in A$, we say that $a$ is (strictly) preferred to $b$ in case $a >_{\prec} b$. If neither $a >_{\prec} b$ nor $b >_{\prec} a$ holds, we say that $a$ is incomparable to $b$. We use $>_{\prec}^{ab}$ to denote the preference $>_{\prec}$ where alternative $a$ is re-labelled as $b$ and alternative $b$ is re-labelled as $a$, while everything else remains the same. The top of a preference $>_{\prec}$ is the set of alternatives that are not dispreferred to any other alternative:

$$T(>_{\prec}) = \{a \in A \mid x \not\prec a \text{ for all } x \in A\}.$$

The non-top of a preference $>_{\prec}$ is the set $NT(>_{\prec}) = A \setminus T(>_{\prec})$. Analogously, the bottom of $>_{\prec}$ is the set of alternatives that are not preferred to any other alternatives:

$$B(>_{\prec}) = \{b \in A \mid b \not\prec x \text{ for all } x \in A\}.$$

The non-bottom of $>_{\prec}$ is the set $NB(>_{\prec}) = A \setminus B(>_{\prec})$. Note that it is not necessarily the case that $T(>_{\prec}) \cap B(>_{\prec}) = \emptyset$. For instance, in Example 1, alternative Uber belongs to both the top and the bottom.<fnref n="3" />

We consider elections in which a finite electorate $N \subset \mathbb{N}$ of agents (or voters) participate. That is, we are working in a model of voting with variable electorates. Each agent $i \in N$ casts her vote by reporting a preference

<footnote n="3">³Although it is possible for an alternative to belong to the intersection of $T(>_{\prec})$ and $B(>_{\prec})$, each voting rule considered in this paper will focus on either the top alternatives or the bottom alternatives, but never both.</footnote>
>$i \in \mathcal{D}$, giving rise to a *profile* of individual preferences. Formally, a profile for the electorate *N* is a function from *N* to *D*, but for notational convenience we will denote profiles as vectors $> = (>_{i})_{i \in N}$. We write $\mathcal{D}^N$ for the set of all such profiles. We use $>_{-i}$ to denote the profile where the preference of agent *i* in profile $>$ has been omitted; and we use $(>_{i'}, >_{-i})$ to denote the profile we obtain when we replace the preference of agent *i* in $>$ with $>_{i'}$. Given two profiles $> \in \mathcal{D}^N$ and $>_{'} \in \mathcal{D}^{N'}$, with $N \cap N' = \emptyset$, we write $(> , >') \in \mathcal{D}^{N \cup N'}$ for their concatenation, i.e., the profile in which both groups report their preferences. Given a preference $> \in \mathcal{D}$, we use $\succeq$ to denote the profile consisting of *m*! preferences with the same graph structure as $>$, where each alternative appears in every position exactly once.<fnref n="4" />

Finally, a *voting rule* is a function $F : \bigcup_{N \subset \mathbb{N}} \mathcal{D}^N \to 2^A \setminus \{\emptyset\}$, mapping any given profile of preferences (for any finite electorate $N \subset \mathbb{N}$) to a non-empty set of alternatives, the *winners* of the election in question. That is, we are working with voting rules that might return several alternatives that are tied for winning the election. Below, we introduce some concrete examples for natural definitions of voting rules in this setting.

## 2.2 Specific Classes of Voting Rules

For the standard model of voting with strict linear orders, the class of *positional scoring rules* (Young, 1975; Zwicker, 2016) includes some of the best-known and most widely used rules, such as the Borda rule and the Plurality rule. Following Kruger and Terzopoulou (2020), we now generalise the definition of this class to our model of voting with partial orders.

A *scoring function* $s: A \times \mathcal{D} \to \mathbb{R}$ maps any given alternative $a \in A$ in a given preference $> \in \mathcal{D}$ to a real number $s_>(a)$, which we call the *score* of $a$. We are specifically interested in scoring functions that are positional, meaning that the score of an alternative only depends on its position in the given preference. Formally, a scoring function $s$ is *positional* if, for all permutations $\sigma: A \to A$, all preferences $> \in \mathcal{D}$, and all alternatives $a \in A$, it is the case that $s_>(a) = s_{\sigma(>)}(\sigma(a))$, where $\sigma(>) = \{(\sigma(a), \sigma(b)) | a > b\}$.

Intuitively, a positional scoring function assigns scores to positions in a graph. Three well-known positional scoring functions for linear orders are the Borda scoring function, giving $m-1$ points to the first alternative, $m-2$ points to the second alternative, and so forth; the Plurality scoring function, assigning one point to the first alternative and zero to the rest; and the Anti-Plurality scoring function, assigning zero points to all the alternatives except

<footnote n="4">⁴As we are going to be interested in rules where the names of the voters are not relevant, this profile is well defined.</footnote>
for the last-ranked one, which gets − 1 points.

For every positional scoring function s there is an associated positional
scoring rule F_s, which is the voting rule defined as follows:

$$
F_s(\succ) = \operatorname*{argmax}_{a \in A} \sum_{i \in N} s_{\succ_i}(a) \quad \text{for any profile } \succ \in \mathcal{D}^N.
$$

Note that applying an affine transformation to a scoring function does not
change the voting rule that is being induced.<fnref n="5" /> Indeed, while an affine trans-
formation will change the scores each alternative receives, it will not change
whether or not a given alternative receives the highest score and thus wins
the election. So if s' is an affine transformation of s, then F_s = F_{s'}, meaning
that for every given positional scoring rule F_s there are infinitely many ways
of representing that one rule.

**Example 2.** Consider the following profile, in which two agents report preferences regarding three alternatives:

<figure><img src="image_3.png" /></figure>

A natural way in which one might generalise the Borda scoring function to
the case of partial orders would be to define $s_>(a)$ as the cardinality of the
set $\{x \in A \mid a \succ x\}$, i.e., by equating the Borda score of an alternative $a$
with the number of other alternatives dominated by $a$. Under this rule, for
the above profile, alternative $a$ wins the election with a score of 2, while $b$
and $c$ each only receive a score of 1.

A natural way of generalising the Plurality scoring function to our setting
would be to set $s_>(a)$ to 1 if $\{x \in A \mid x \succ a\}$ is empty, and to set it to 0
otherwise. Under the corresponding rule, we obtain a three-way tie.

Finally, if we consider, say, the affine transformation $x \mapsto 4x - 1$, in the
first case $a$ gets 6 points, while $b$ and $c$ get 2 points each; and in the second
case $a$, $b$, and $c$ get 2 points each. So the winners indeed do not change. $\Delta$

But this particular way of generalising the Plurality rule is just one of many
options. We now propose a class of positional scoring rules for strict partial
orders that, we believe, naturally captures one of the options available for
generalising the Plurality rule for linear orders.

<footnote n="5">⁵Applying an affine transformation $x \mapsto \alpha x + \beta$ with $\alpha \in \mathbb{R}_{>0}$ and $\beta \in \mathbb{R}$ to a scoring function $s$ means replacing, for every preference $\succ \in \mathcal{D}$ and every alternative $a \in A$, the original score $s_>(a)$ with the new score $\alpha \cdot s_>(a) + \beta$.</footnote>
**Definition 1.** A positional scoring rule $F_s$ belongs to the **Plurality Class** if the associated scoring function $s$ satisfies the following property: Given a preference $\succ \in \mathcal{D}$, there exists a $k' \in \mathbb{R}$ such that for all $a \in T(\succ)$ and all $b \in NT(\succ)$, it is the case that $s_>(a) \ge s_>(b) = k'$, with at least one of these inequalities being strict.

Thus, we require that all the alternatives not in the top receive the same score $k'$, and that score must not be greater than the scores awarded to the alternatives at the top. But we are free to award different scores to different top alternatives. This way, we are able to exploit the preference's top structure and, for example, award points to a top alternative according to the number of alternatives it dominates. For simplicity, and to highlight the clear analogy to the Plurality rule for linear orders, we are going to assume that $k' = 0$ for any preference $\succ \in \mathcal{D}$.<fnref n="6" />

We stress once more that for two partial orders with different graph structures, the scoring function can assign different scores to the alternatives. The following example illustrates this point.

**Example 3.** Let $s$ be the scoring function that assigns to any alternative in the top a score that is equal to the number of alternatives it dominates, while to every alternative not in the top it assigns a score of 0. Here are the scores assigned to the alternatives under consideration for two possible preferences:

<figure><img src="image_4.png" /></figure>

The positional scoring rule $F_s$ induced by the scoring function $s$ belongs to the Plurality Class. $\Delta$

This example can be viewed as a hybrid of the Borda and Plurality scoring rules: the Borda rule, where alternatives receive points based on the number of dominated options, and the Plurality rule, where a voter contributes only to her most preferred alternatives. Scoring rules in this class may be particularly relevant when the goal is to avoid selecting losing alternatives, while still differentiating among the non-losing ones. For instance, consider

<footnote n="6">⁶This can be done without loss of generality, as an additive shift can be applied to every preference relation individually.</footnote>
partial orders that represent outcomes of various sports competitions in
which teams may play an unequal number of games. In such cases, we may
want to identify teams that have not lost any games in specific competitions,
while also taking into account the number of games they have won.

But we can also take the stance that all alternatives that are part of the
same top set in a voter's preference should receive the same score. Thus, we
consider the following class.

**Definition 2.** A positional scoring rule $F_s$ in the Plurality Class belongs to the *Simple Plurality Class* if the associated scoring function $s$ satisfies the following property: For any given preference $\succ \in \mathcal{D}$, there exists a $k \in \mathbb{R}$ such that $s_{\succ}(a) = k$ for all $a \in T(\succ)$.

Observe that for rules in this class, we still can make $k$, the score awarded to each alternative in a given top set, dependent on the preference $\succ$ at hand. Now, we might feel that an alternative that is alone in the top should be considered at least as important as alternatives that are not alone in the top. We propose the following subclass of the simple rules.

**Definition 3.** A positional scoring rule $F_s$ in the Simple Plurality Class belongs to the *Monotonic Simple Plurality Class* if, for any two preferences $\succ, \succ' \in \mathcal{D}$ and any two alternatives $a \in T(\succ)$ and $a' \in T(\succ')$, it is the case that $|T(\succ)| \le |T(\succ')|$ entails $s_{\succ}(a) \ge s_{\succ'}(a')$.

Restricting the scoring function even further, let us now focus on the posi-
tional scoring rules in the Plurality Class that assign the same score to all
the top alternatives. Such a Plurality rule is *uniform* in the sense that the
score assigned to a top alternative does not depend on the structure of the
preference but is the same across all possible preferences, and the same is
true for the alternatives that are not in the top. Note that, in view of what
we said earlier about positional scoring rules being invariant under affine
transformations, there is in fact only a single rule in the Plurality Class that
is uniform—but it can be represented by infinitely many different scoring
functions. We call this unique rule the *Uniform Plurality* rule. The canon-
ical way of defining it would be as the rule that assigns 1 point to every
alternative in the top.

**Definition 4.** The *Uniform Plurality* rule $F_s$ is the rule belonging to the *Monotonic Simple Plurality Class* that is induced by the scoring function $s$ with $s_{\succ}(a) = 1$ for every preference $\succ \in \mathcal{D}$ and every alternative $a \in T(\succ)$.

This rule might be regarded as the most natural extension of the Plurality
rule for linear orders to our model of voting with partial orders, and it is
clearly appealing due to its simplicity. It captures the most basic intuition
about the Plurality rule for linear orders, selecting the alternatives that
most frequently appear in the top positions of the ballots. Having said this,
focusing *only* on the Uniform Plurality Rule would not do justice to the rich
variety in which voters can express themselves in our model of voting with
partial orders; so certain other rules belonging to the Plurality Rule also
have their place.

We now move on to defining corresponding classes of positional scor-
ing rules for partial orders that generalise the Anti-Plurality rule for linear
orders.

**Definition 5.** A positional scoring rule $F_s$ belongs to the **Anti-Plurality Class** if the associated scoring function $s$ satisfies the following property: Given a preference $> \in \mathcal{D}$, there exists a $k'$ such that for all $a \in NB(>)$ and all $b \in B(>)$, it is the case that $k' = s_>(a) \ge s_>(b)$, with at least one of these inequalities being strict.

Again, for simplicity, and to highlight the clear analogy to the Anti-Plurality
rule for linear orders, we are going to assume that $k' = 0$.

**Definition 6.** A positional scoring rule $F_s$ in the *Anti-Plurality Class belongs to the Simple Anti-Plurality Class* if the associated scoring function $s$ satisfies the following property: For any given preference $> \in \mathcal{D}$, there exists a $k \in \mathbb{R}$ such that $s_>(a) = k$ for all $a \in B(>)$.

**Definition 7.** A positional scoring rule $F_s$ in the *Simple Anti-Plurality Class* belongs to the *Monotonic Simple Anti-Plurality Class* if, for any two preferences $>$, $>'$ $\in \mathcal{D}$ and any two alternatives $b \in B(>)$, and $b' \in B(>')$, it is the case that $|B(>)| \le |B(>')|$ entails $s_>(b) \le s_{>'}(b')$.

Thus, here the alternatives that are part of the smaller bottom sets receive
the lower scores, while for the Monotonic Simple Plurality Class the alter-
natives in the smaller top sets receive the higher scores. We again obtain a
single most canonical rule for this class.

**Definition 8.** The *Uniform Anti-Plurality rule* $F_s$ is the rule belonging to the *Monotonic Simple Anti-Plurality Class* that is induced by the scoring function $s$ with $s_>(b) = -1$ for every preference $> \in \mathcal{D}$ and every alternative $b \in B(>)$.

Again, this rule captures the most basic intuition about the Anti-Plurality
rule for linear orders, selecting those alternatives that least often appear in
the bottom of a preference.
## 2.3 Axioms

Next, we introduce a number of axioms that each encode a normative requirement that, one might argue, any reasonable voting rule should satisfy. These axioms will later turn out to be useful for our characterisation results.

The first four axioms, Anonymity, Neutrality, Reinforcement, and Continuity, are very common in the literature on scoring rules and are used by Young (1975) to characterise them. The first two are symmetry requirements that essentially state that the names of the agents and alternatives should not be considered important for choosing a winner. We formulate Anonymity in terms of a bijection $\mu: N \to N'$ for any two electorates $N, N' \subset \mathbb{N}$ with the same number of agents. Similarly, we formulate Neutrality in terms of a permutation $\sigma: A \to A$ on the set of alternatives, which extends to both sets $S \subseteq A$ of alternatives and preferences $\succ \in \mathcal{D}$ in a natural manner: $\sigma(S) = \{\sigma(x) | x \in S\}$ and $\sigma(\succ) = \{\sigma(x), \sigma(y) | x \succ y\}$.

**Axiom 1 (Anonymity).** For any two electorates $N, N' \subset \mathbb{N}$ with the same number of agents and any bijection $\mu: N \to N'$, it should be the case that $F((\succ_i)_{i \in N}) = F((\succ_{\mu(i)})_{i \in N})$.

**Axiom 2 (Neutrality).** For any profile $\succ = (\succ_i)_{i \in N}$ and any permutation $\sigma: A \to A$, it should be the case that $\sigma(F((\succ_i)_{i \in N})) = F((\sigma(\succ_i))_{i \in N})$.

Thus, Anonymity says that changing the names of the voters should not change the winners, while Neutrality says that any change in the names of the alternatives in the profile should then also be applied to the original winners to obtain the new winners.

The next axiom, which is also known as Consistency, states that if two disjoint electorates agree on some winning alternatives, then those alternatives must be selected when we consider a profile where the members of both electorates cast a vote.

**Axiom 3 (Reinforcement).** For any two profiles $\succ \in \mathcal{D}^N$ and $\succ' \in \mathcal{D}^{N'}$ with $N \cap N' = \emptyset$ such that $F(\succ) \cap F(\succ') \neq \emptyset$, it should be the case that $F(\succ, \succ') = F(\succ) \cap F(\succ')$.

The next axiom, called the Archimedean property by Smith (1973) and Young (1975), and Overwhelming Majority by Myerson (1995), stipulates that, although a small number of agents cannot completely over-ride the decision of a large majority, they might still break ties in favour of some of the alternatives selected by the majority.
**Axiom 4 (Continuity).** For any two profiles $\succ \in \mathcal{D}^N$ and $\succ' \in \mathcal{D}^{N'}$ such that $N \cap N' = \emptyset$, there should exist a bound $K$ such that, for every natural number $k > K$, the following inclusion holds:

$$F(\succ, \dots, \succ, \succ') \subseteq F(\succ),$$

where, for assigning identities to the voters of the duplicated profiles, we use the minimum number in the set $N \setminus \{N \cup N'\}$ that has not been used, following the order of the original profile.

Now we present an example to clarify the previously introduced notation.

**Example 4.** Let $\succ \in \mathcal{D}^N$ and $\succ' \in \mathcal{D}^{N'}$, with $N = \{2, 4\}$ and $N' = \{1, 7, 9\}$. Then $(\succ, \succ, \succ, \succ') = (\succ_2, \succ_4, \succ_3, \succ_5, \succ_6, \succ_8, \succ'_1, \succ'_7, \succ'_9)$, with $\succ_2 = \succ_3 = \succ_6$ and $\succ_4 = \succ_5 = \succ_8$. $\Delta$

In the sequel, we are going to write $k\succ$ as a shorthand for the profile $(\succ, \dots, \succ)$ consisting of a concatenation of $k$ copies of the basic profile $\succ$.

The next two axioms are minimal responsiveness and efficiency requirements for a voting rule. The first one is a version of the Faithfulness axiom due to Young (1974), and the second one is a version of Averseness introduced by Kurihara (2018). The former states that, for the special case of single-voter profiles, the voting rule must select alternatives that are in the top of that agent. Thus, dominated alternatives are not selected. The latter states that for such profiles, it should not select all of the bottom alternatives. So, not all dominated alternatives are selected.

**Axiom 5 (Partial Faithfulness).** For any single-voter profile $\succ \in \mathcal{D}$, it should be the case that $F(\succ) \subseteq T(\succ)$.

**Axiom 6 (Partial Averseness).** For any single-voter profile $\succ \in \mathcal{D}$ with $B(\succ) \neq A$, it should be the case that $B(\succ) \not\subseteq F(\succ)$.

Note that Partial Averseness is a weaker requirement than Partial Faithfulness, in the formal sense of being implied by the latter. Now we present stronger versions of these axioms.

**Axiom 7 (Faithfulness).** For any single-voter profile $\succ \in \mathcal{D}$, it should be the case that $F(\succ) = T(\succ)$.

**Axiom 8 (Averseness).** For any single-voter profile $\succ \in \mathcal{D}$ with $B(\succ) \neq A$, it should be the case that $B(\succ) \cap F(\succ) = \emptyset$.
It is easy to see that in the case where preferences are linear orders, the
partial and non-partial versions of these axioms coincide.

The following axioms are inspired by the approval voting framework
introduced by Brams and Fishburn (1978). We adapt axioms from Alcalde-
Unzu and Vorsatz (2009) and Bardal and Endriss (2025), and introduce new
ones tailored to situations where one might want to use an Anti-Plurality
rule. The next two axioms can be interpreted as agreement properties: if
voters who support the same alternatives as those collectively chosen by a
given electorate are added to that electorate, then the same outcome should
continue to be supported in the expanded electorate.

**Axiom 9** (T-Congruity). Let $x \in A$, $\succcurlyeq \in \mathcal{D}^N$ and $\succcurlyeq' \in \mathcal{D}^{N'}$ be such that $N \cap N' = \emptyset$, $x \notin F(\succ)$, and $x \notin T(\succ_i')$ for all $i \in N'$. Then $x \notin F(\succ, \succ')$.

**Axiom 10** (B-Congruity). Let $x \in A$, $\succ \in \mathcal{D}^N$ and $\succ' \in \mathcal{D}^{N'}$ be such that $N \cap N' = \emptyset$, $x \in F(\succ)$, and $x \notin B(\succ_i')$ for all $i \in N'$. Then $x \in F(\succ, \succ')$.

The final set of axioms dictates that selected alternatives should not be
harmed when a voter changes their support for non-selected alternatives.
We make a distinction according to where these selected and removed alter-
natives are in the voter's preference.

**Axiom 11** (Contraction). Let $\succ \in \mathcal{D}^N$ and $\succ'_i \in \mathcal{D}$ be such that $T(\succ'_i) \subseteq T(\succ_i)$ and $F(\succ) \cap T(\succ'_i) \neq \emptyset$. Then $F(\succ) \cap T(\succ'_i) \subseteq F(\succ'_i, \succ_{-i})$.

**Axiom 12** (Strong Contraction). Let $\succ \in \mathcal{D}^N$ and $\succ'_i \in \mathcal{D}$ be such that $T(\succ'_i) \subseteq T(\succ_i)$. Then $F(\succ) \setminus (T(\succ_i) \setminus T(\succ'_i)) \subseteq F(\succ'_i, \succ_{-i})$.

**Axiom 13** (Expansion). Let $\succ \in \mathcal{D}^N$ and $\succ'_i \in \mathcal{D}$ be such that $B(\succ_i) \subseteq B(\succ'_i)$ and $F(\succ) \cap B(\succ_i) \neq \emptyset$. Then $F(\succ) \cap B(\succ_i) \subseteq F(\succ'_i, \succ_{-i})$.

**Axiom 14** (Strong Expansion). Let $\succ \in \mathcal{D}^N$ and $\succ'_i \in \mathcal{D}$ be such that $B(\succ_i) \subseteq B(\succ'_i)$. Then $F(\succ) \setminus (B(\succ'_i) \setminus B(\succ_i)) \subseteq F(\succ'_i, \succ_{-i})$.

# 3 Results

In this section, we provide axiomatic characterisations of the classes of voting
rules defined in Section 2.2 in terms of the axioms put forward in Section 2.3.

For the class of positional scoring rules, it is well-understood that they
can be characterised in terms of Anonymity, Neutrality, Reinforcement, and
Continuity, and results of this kind have been obtained for a number of dif-
ferent models. Originally, Smith (1973) and Young (1975) proved this to be
the case for the classical model of voting with linear orders. Our model is closest to that of Kruger and Terzopoulou (2020), who obtained the same kind of characterisation for the case of voting with incomplete preferences that are merely assumed to be acyclic (but not necessarily transitive). Although their result does not directly imply the corresponding result for our domain (which is a subdomain of the domain considered by Kruger and Terzopoulou),<fnref n="7" /> a careful examination of their proof shows that it holds for our model as well (so we omit the proof). We note that Myerson (1995) and Pivato (2013) proved similar results for a model without an ordering assumption, and in principle it would also be possible to derive our Theorem 1 below as a corollary to their results.

**Theorem 1.** *A voting rule satisfies Anonymity, Neutrality, Reinforcement, and Continuity if, and only if, it is a positional scoring rule.*

This result is very useful, as it allows us to directly focus on voting rules that are positional scoring rules, which have a very well-defined structure.

## 3.1 The Plurality Class

We are now ready to provide axiomatic characterisations of the different Plurality Classes we defined earlier.

**Theorem 2.** *A voting rule satisfies Anonymity, Neutrality, Reinforcement, Continuity, Partial Faithfulness, and T-Congruity if, and only if, it belongs to the Plurality Class.*

*Proof.* First, we show that for the particular case where the profile of preferences consists of only linear orders, $F$ is the standard Plurality rule. By Theorem 1, we know that $F$ is a positional scoring rule. Given a linear order $L$, let $s_i$ be the score assigned to the alternative in position $i$. We need to show that $s_1 > s_k$ and $s_k = s_{k'}$ for all $k, k' \neq 1$. By Partial Faithfulness, it is the case that $s_1 > s_k$ for all $k \neq 1$. Now suppose there are at least two positions $k', k'' \neq 1$ such that $s_{k'} \neq s_{k''}$. W.l.o.g., we can assume that $s_{k'}$ and $s_{k''}$ are the second and third highest scores, respectively, and $s_{k''} = 0$.

<footnote n="7">⁷It might, at first, seem counterintuitive that a characterisation result established for one domain does not immediately transfer to its subdomains, but there are numerous counterexamples for such transfers in the literature. One example is the Gibbard-Satterthwaite Theorem (Gibbard, 1973; Satterthwaite, 1975). On the full domain of linear preferences, it establishes a characterisation of the family of dictatorships in terms of strategy-proofness and surjectivity. But on the single-peaked domain, these axioms admit also many other rules, such as Black's median-voter rule (Black, 1948).</footnote>
Let $L_1$ be a linear order with $a$ in the first position, $b$ in position $k'$, and $c$ in position $k''$; and let $L_2$ be a linear order with $b$ in the first position, $a$ in position $k'$, and $c$ in position $k''$. Now consider a profile $> = (\ell L_1, \ell' L_2)$ such that:

$$ \ell = \ell' + 1 \qquad (1) $$

and

$$ \left[ \frac{s_1 - s_{k'}}{s_{k'}} \right] s_1 \le \ell s_{k'} + \ell' s_1 + \left[ \frac{s_1 - s_{k'}}{s_{k'}} \right] s_{k'}. \qquad (2) $$

We have $F(>’) = \{a\}$ and the aggregated score for $a$ is $\ell s_1 + \ell' s_{k'}$, for $b$ is $\ell s_{k'} + \ell' s_1$, for $c$ is 0, and for any other alternative is at most $(\ell + \ell') s_{k'}$. Now let $L_3$ be a linear order with $c$ in the first position, $b$ in the position $k'$, and $a$ in the position $k''$. Consider the profile $>'_c = (\lceil \frac{s_1-s_{k'}}{s_{k'}} \rceil L_3)$. The aggregated scores in $F(>’, >’)$ are: $\ell s_1 + \ell' s_{k'}$ for $a$; $\ell s_{k'} + \ell' s_1 + \lceil \frac{s_1-s_{k'}}{s_{k'}} \rceil s_{k'}$ for $b$; $\lceil \frac{s_1-s_{k'}}{s_{k'}} \rceil s_1$ for $c$; and for the other alternatives at most $(\ell + \ell' + \lceil \frac{s_1-s_{k'}}{s_{k'}} \rceil) s_{k'}$.

Given Equations (1) and (2), we have that $b \in F(>’, >’)$, violating T-Congruity. So we can conclude that $s_{k'} = s_{k''} = 0$ for all $k’, k'' \ne 1$, and that $F$ acts as the standard Plurality rule when preferences are linear.

Now, let $> \in \mathcal{D}$ be any given preference. We need to show that for all $c \in NT(>)$ we get $s_>(c) = k$; that there is an $a \in T(>)$ such that $s_>(a) > k$; and that for all $b \in T(>)$ we get $s_>(b) \ge k$. W.l.o.g., we can assume that there is a $c \in NT(>)$, in a position such that $s_>(c) = k = 0 \ge s_>(d)$ for all $d \in NT(>)$ (so $c$ gets the highest possible score for a non-top alternative). By Partial Faithfulness, there is an alternative $a \in T(>)$ in a position such that $s_>(a) > 0$. Let $a \in F(>)$ (so $a$ gets the highest score possible). Now, suppose there is an alternative $b \in T(>)$ in a position such that $s_>(b) < 0$. Let $L_1$ and $L_2$ be two linear orders such that $b$ and $c$ are in the first position, respectively. Now consider the profile $> = (\ell L_1, \ell' L_2)$ such that:

$$ \ell = \ell' + 1 \qquad (3) $$

and

$$ \left[ \frac{1}{-s_>(b)} \right] s_>(a) \le \ell'. \qquad (4) $$

Let $> = (\ell L_1, \ell' L_2)$, so we have $F(>’) = \{b\}$. Now consider the profile $>'_c = (>’, \lceil \frac{1}{-s_>(b)} \rceil >’)$. The aggregated score for $b$ is $\ell’$; the one for $b$ is $\ell + \lceil \frac{1}{-s_>(b)} \rceil s_>(b)$; and any other alternative gets at most $\lceil \frac{1}{-s_>(b)} \rceil s_>(a)$.
Then, by Equations (3) and (4), $c \in F(\succ')$ and thus T-Congruity is violated. We can conclude that when $b \in T(\succ)$, it is the case that $s_>(b) \ge 0$.

Using a similar argument, we can show that there cannot be a position for a non-top alternative where it gets a negative score. Thus, we can conclude that $s_>(c) = 0$ for all $c \in NT(\succ)$. So $F$ indeed belongs to the Plurality Class. $\square$

If $F$ is a positional scoring rule, Faithfulness implies that all the alternatives in the top of a preference receive the same score.

**Corollary 3.** *A voting rule satisfies Anonymity, Neutrality, Reinforcement, Continuity, Faithfulness, and T-Congruity if, and only if, it belongs to the Simple Plurality Class.*

Contraction requires the scoring rule to assign more points to top alternatives when they are less accompanied by other top alternatives. This axiom plays a crucial role in our characterisation of the Monotonic Simple Plurality Class. Note that relative to Corollary 3 characterising the next larger class, we can weaken Faithfulness to Partial Faithfulness, and we can drop T-Congruity entirely.

**Theorem 4.** *A voting rule satisfies Anonymity, Neutrality, Reinforcement, Continuity, Partial Faithfulness, and Contraction if, and only if, it belongs to the Monotonic Simple Plurality Class.*

*Proof.* Let $\succ \in \mathcal{D}$ and assume that $|T(\succ)| \ge 2$.<fnref n="8" /> By Theorem 1, we know that $F$ is a positional scoring rule. By Partial Faithfulness, there is an alternative $a$ in a position such that $s_>(a) > s_>(c)$ for all $c \in NT(\succ)$, and $s_>(a) \ge s_>(b)$ for all $b \in T(\succ)$. Let $b \in T(\succ)$ and suppose that $s_>(a) > s_>(b)$. Thus, $a \in F(\succ)$ and $b \notin F(\succ)$. Now consider $\succ^{ab}$, where the $a$- and $b$-labels are switched. By Contraction, $a \in T(\succ^{ab}) \cap F(\succ) \subseteq F(\succ^{ab}) \subseteq F(\succ)$. But by Neutrality, $b \in F(\succ^{ab})$ and $a \notin F(\succ^{ab})$, obtaining a contradiction. Thus, we can conclude that there exists a value $k$ such that $s_>(a) = k$ for all $a \in T(\succ)$, and $k > s_>(c)$ for all $c \in NT(\succ)$.

Now assume that $|NT(\succ)| \ge 2$ and that there are two positions where alternatives $c, d \in NT(\succ)$ get scores $s_>(c) > s_>(d)$. Consider the profile $\succ^c$.<fnref n="9" /> By Neutrality and Anonymity, $F(\succ^c) = A$. Now consider $\succ_i^{cd}$, that is, the preference $\succ$ with the labels of $c$ and $d$ switched and submitted by

<footnote n="8">⁸If $|T(\succ)| = 1$, by Partial Faithfulness we trivially reach the same conclusion.</footnote>

<footnote n="9">⁹We remind the reader that $\succ$ is the profile where each individual preference has the same graph structure as $\succ$ and every alternative appears exactly once in every position.</footnote>
individual *i*. Thus $T(>) = T(\succ_i^{cd}) \cap F(>)$ and by Contraction, we have that $T(>) \subseteq F(\succ_i^{cd}, \succ_{-i})$. But $c \notin F(\succ_i^{cd}, \succ_{-i})$, as it gets a lower score in this profile, obtaining a contradiction. Thus there is a *k* such that $s_{\succ}(c) = k$ for all $c \in NT(>)$.

As Contraction only considers the alternatives in the top of a preference and not the graph structure of the top, and by Neutrality, we have that for any two preferences $>}$ and $>'$ such that $|T(>)| = |T(>')|$, it is the case that $s_{>}(a) = s_{>'}(b)$ for any $a \in T(>)$ and $b \in T(>')$, or any $a \in NT(>)$ and $b \in NT(>')$.

We can normalise the scores such that for all $> \in \mathcal{D}$ we get $s_{>} (c) = 0$ for all $c \in NT(>)$. Now consider two preferences $>_{i}, >'_{i} \in \mathcal{D}$ where $|T(>_{i})| < |T(>_{i}')|$, and $k$ and $k'$ are the scores for the top alternatives of $>}$ and $>'$, respectively. Suppose $k < k'$. Consider the profile $>^{'}$. By Neutrality and Anonymity, $F(>^{'}) = A$. We have that $T(>_{i}) \cap F(>') = T(>_{i})$ and thus; by Contraction, $T(>_{i}) \subseteq F(>_{i}, >'_{-i})$. But in $F(>_{i}, >'_{-i})$, all the alternatives in $T(>_{i}')$ get their scores reduced, while those not in the top of $T(>')$ remain with the same score, obtaining $F(>_{i}, >'_{-i}) = A \setminus T(>_{i}')$. With this contradiction, we can conclude that $k' \le k$, and thus, that $F$ belongs to the Monotonic Simple Plurality Class. $\square$

We now introduce a further condition known as Tops-Only. Importantly, we think of Tops-Only as a purely technical property, with no immediate normative appeal in and of its own. However, as we are going to see soon, it will turn out to be implied by another property that does have such appeal. This condition requires that only the top-ranked alternative(s) in each voter's preference matter to the outcome. Formally, a voting rule satisfies Tops-Only if, for any two profiles $>,$ $>'' \in \mathcal{D}^N$ with $T(>_{i}) = T(>''_{i})$ for all $i \in N$, it is the case that $F(>)=F(>'').^{10}$ We show that any rule satisfying Strong Contraction necessarily depends only on the voters' top-ranked alternatives.

**Proposition 5.** *Strong Contraction implies Tops-Only.*

*Proof.* Let $>,$ $>'' \in \mathcal{D}^N$ be two profiles such that, for all $i \in N$, it is the case that $T(>_{i}) = T(>''_{i})$. For any $i \in N$, we have that $T(>_{i}) \setminus T(>''_{i}) = \emptyset$, and thus, by Strong Contraction, $F(>_{i}) \subseteq F(>''_{i}, >_{-i})$. By iterating this process for all $i \in N$, we obtain that $F(>_{i}) \subseteq F(>''_{i})$. With the same reasoning, we obtain that $F(>''_{i}) \subseteq F(>_{i})$, concluding that $F(>_{i}) = F(>''_{i})$. $\square$

<footnote n="10">¹⁰For further motivation regarding this property, and its use for a characterisation result, we refer to Sekiguchi (2012).</footnote>
To characterise the Uniform Plurality rule, we build on a characterisation of the Plurality rule for linear orders by Sekiguchi (2012) in terms of Anonymity, Neutrality, Reinforcement, Partial Faithfulness, and Tops-Only.

**Theorem 6.** *A voting rule satisfies Anonymity, Neutrality, Reinforcement, Partial Faithfulness, and Strong Contraction if, and only if, it is the Uniform Plurality rule.*

*Proof.* By Proposition 5, we have that any rule satisfying our axioms also satisfies Tops-Only. By a result due to Sekiguchi (2012), we know that when the profile consists of linear orders, a voting rule satisfies all the axioms if, and only if, it is the Plurality rule. We normalise the rule such that the score for the non-top alternatives of a linear order is equal to 0, and let $s_L > 0$ be the score for the top alternative.

By the aforementioned result due to Pivato (2013), we know that if a rule satisfies Anonymity, Neutrality and Reinforcement, it is a composite scoring rule, i.e., a rule that first applies a scoring rule and then uses a second scoring rule to break any possible ties.

Let $\succ \in \mathcal{D}$ be a generic preference. By Partial Faithfulness and Tops-Only, we have that $F(\succ) = T(\succ)$ (to see this, consider the top with all symmetric alternatives). This implies that for all $a \in T(\succ)$, $s_\succ(a) = k$ and that for all $b \in NT(\succ)$, $k > s_\succ(b)$.

Now suppose there are two positions not at the top such that alternatives $a$ and $b$ get scores $x = s_\succ(a) > s_\succ(b) = y$. Let $\mathcal{L} = \{\succ_1, \succ_2\}$ with $\succ_1$ and $\succ_2$ being linear orders such thats $a \succ_1 b \succ_1 \dots$ and $b \succ_2 a \succ_2 \dots$, and all the alternatives different from $a$ and $b$ being ordered the same. As $F$ is the plurality rule when considering linear orders, we have that $F(\mathcal{L}) = \{a, b\}$. Let $n$ be such that $ns_L + x > k$. Thus $F(n\mathcal{L}, \succ) = \{a\}$. Now consider $\succ^{ab}$, to obtain $F(n\mathcal{L}, \succ^{ab}) = \{b\}$ (as now $b$ gets $ns_L + x$ points). But this violates Tops-Only (and thus Strong Contraction). So we conclude that for all $b \in NT(\succ)$ it is the case that $s_\succ(b) = k' < k$. We normalise the scoring rule such that $k' = 0$.

By Neutrality and Tops-Only, we only care about the number of alternatives that are at the top. Suppose there are two different preferences $\succ, \succ^*$ such that $k = s_\succ(a) \ge s_{\succ^*}(b) = k^*$ for $a \in T(\succ)$ and $b \in T(\succ^*)$, and $|T(\succ)| > |T(\succ^*)|$. Let $b \in NT(\succ)$, $b \in T(\succ^*)$, and $a \in NT(\succ^*)$, $a \in T(\succ)$ (there is no loss of generality because of Neutrality and Tops-Only). Let $\mathcal{L} = \{\succ_1, \succ_2\}$ be a profile consisting of two linear orders with $a \succ_1 b \succ_1 \dots$ and $a \succ_2 b \succ_2 \dots$, and all the alternatives different from $a$ and $b$ are ordered the same. For a sufficiently large $n$ we have by Anonymity and Neutrality that $F(n\mathcal{L}, \succ, \succ^*, \succ^{ab}, \succ^{*(ab)}) = \{a, b\}$. Now remove alternatives
different from *a* from the top of ≻, until we get a preference ≻' such that
|T(≫')| = |T(≫*)|. As now *a* gets a lower score than *b* (as the contribution
from ≻' is less) we have F(nL, ≻, ≻', ≻<sup>ab</sup>, ≻<sup>*(*ab*)</sup>) = {*b*}, violating Strong
Contraction. Thus *k* ≤ *k*^*.

Following a similar line of reasoning, if we assume that |T(≫)| < |T(≫*)|,
we obtain that k ≥ k*. We conclude that k = k*. By setting k = 1, we
obtain that F is the Uniform Plurality rule. □

An interesting and immediate question that arises in view of our previous
results is whether all of the axioms mentioned are in fact necessary for the
characterisation, that is, whether those axioms are independent. We now
show that this is indeed the case, by providing examples of voting rules that
satisfy all but one of the axioms featuring in the theorems.

The most interesting of these is the example of a voting rule that satisfies all of the axioms but Continuity, as that axiom does not feature in the characterisation of the Plurality rule in the standard model of voting with linear preferences due to Sekiguchi (2012). Also, recall that Ching (1996) showed that Continuity can be dropped from Richelson’s (1978) original characterisation. So the need for Continuity is a crucial difference between our Theorems 2 and 4, and related results for the standard model. Loosely speaking, the reason for this difference is that, for our model of voting with partial orders, the different structures of the top sets permit us to vary the number of points awarded to the most preferred alternatives, making T-Congruity and Contraction less demanding.

**Proposition 7.** *The axioms of Anonymity, Neutrality, Reinforcement, Continuity, Partial Faithfulness, and T-Congruity/Contraction are logically independent.*

*Proof.* So here we are claiming the independence of two sets of six axioms each. To prove the claim, for each of the six axioms featuring in Theorem 2 and 4, we provide an example of a voting rule that satisfies all of the axioms but one, and that does not belong to the Plurality Class. In most cases the fact that the rules provided have the required properties is straightforward to verify; where this is not the case, we provide additional details.

*Anonymity, Neutrality, Reinforcement, Continuity, and Partial Faithfulness but neither T-Congruity nor Contraction:* the Bordà-style rule that assigns to each alternative *a* in a partial order a score equal to the number of alternatives that *a* is preferred to.

*Anonymity, Neutrality, Reinforcement, Continuity, T-Congruity, and Contraction but not Partial Faithfulness:* the rule that always selects the full set
of alternatives.

*Anonymity, Neutrality, Reinforcement, Partial Faithfulness, T-Congruity, and Contraction but not Continuity:* the rule defined by means of the following two-step procedure. In the first step, select all the winning alternatives returned by the Uniform Plurality rule. In the second step, select as winning alternatives from this set those alternatives that appear as a unique top element in at least one of the agents’ preferences—in case there are such alternatives; otherwise, pick all alternatives selected in the first step.<fnref n="11" /> See also Example 5.

*Anonymity, Neutrality, Continuity, Partial Faithfulness, T-Congruity, and Contraction but not Reinforcement:* the rule such that for single-voter profiles selects the top of the sole voter and for other profiles selects the Uniform Plurality winners and all the alternatives that get exactly one point less than the Uniform Plurality winners. The following example shows that this rule indeed violates Reinforcement. If in profile ≻ the alternatives a and b appear 10 and 9 times in the top, respectively, and in ≻' they appear 7 and 6 times, respectively, then {a} = F(≫, ≻') ≠ F(≫) ∩ F(≫') = {a, b}.

*Anonymity, Reinforcement, Continuity, Partial Faithfulness, T-Congruity, and Contraction but not Neutrality:* the rule assigning points to alternatives just like the Uniform Plurality rule but that doubles the score of one fixed alternative a ∈ A before determining the winners.

*Neutrality, Reinforcement, Continuity, Partial Faithfulness, T-Congruity, and Contraction but not Anonymity:* the rule assigning points to alternatives just like the Uniform Plurality rule except for voter 1's top alternatives, which each receive 2 points. □

**Example 5.** To exemplify the operation of the rule demonstrating that Continuity is a necessary axiom for our characterisation of the Plurality Class, consider the following two preferences, ≻₁ and ≻₂:

<figure><img src="image_5.png" /></figure>

In the two-agent profile (≫₁, ≻₂), in the first step we select {a, b, c}, but as only a appears as a unique top, in the second step we obtain {a}.

<footnote n="11">¹¹The inspiration for the definition of this voting rule comes from an example used by Myerson (1995).</footnote>
The failure of Continuity can now be observed by noting that the rule instead returns {$b, c$} for any profile that is composed of $k > 1$ copies of ($>_{1}, >_{2}$) and one further copy of ($>_{2}$). Indeed, for any such profile, we select {$b, c$} in the first step, after which no further selection will occur in the second step. $\Delta$

The next statement, regarding the independence of the axioms of Theorem 6, is presented without proof, as the same examples as in the proof of Proposition 7 can be used (except that certain rules also satisfying Continuity are not relevant here).

**Proposition 8.** *The axioms of Anonymity, Neutrality, Reinforcement, Partial Faithfulness, and Strong Contraction are logically independent.*

## 3.2 The Anti-Plurality Class

In this section, we present the characterisation of the voting rules that make up the Anti-Plurality Class. Some of the proofs are similar to those for the corresponding results regarding the Plurality Class.<fnref n="12" />

**Theorem 9.** *A voting rule satisfies Anonymity, Neutrality, Reinforcement, Continuity, Partial Averseness, and B-Congruity if, and only if, it belongs to the Anti-Plurality class.*

*Proof.* By Theorem 1, we know that $F$ is a positional scoring rule. Let $> \in \mathcal{D}$ and suppose there are alternatives $a \in NB(>)$ and $c \in B(>)$ such that $s_>(a) < s_>(c)$. We consider the profile $\bar{>}$, and by Anonymity and Neutrality, we have that $F(\bar{>}) = A$. Thus $a \notin F(\bar{>}, >)$, contradicting B-Congruity. Then $s_>(a) \ge s_>(c)$ for all $a \in NB(>)$ and $c \in B(>)$.

Using a similar argument we can show that $s_>(a) = s_>(d) = k$ for all $a, d \in NB(>)$.

By Partial Averseness, there is at least one alternative $b \in B(>)$ that gets a score $s_>(b) < k$, concluding that $F$ belongs to the Anti-Plurality class. $\square$

Averseness forces the scoring rule to give, within a preference, the same number of points to all the bottom alternatives.

**Corollary 10.** *A voting rule satisfies Anonymity, Neutrality, Reinforcement, Continuity, Averseness, and B-Congruity if, and only if, it belongs to the Simple Anti-Plurality class.*

<footnote n="12">¹²But it also is not the case that the results for the Anti-Plurality Class and the Plurality Class are ‘duals’ in any immediate sense.</footnote>
Recall that Expansion requires the scoring rule to assign more points to bottom alternatives when they occur together with other bottom alternatives.

**Theorem 11.** A voting rule satisfies Anonymity, Neutrality, Reinforcement, Continuity, Partial Averseness, and Expansion if, and only if, it belongs to the Monotonic Simple Anti-Plurality class.

*Proof.* By Theorem 1, we know that $F$ is a positional scoring rule. Let $\succ_i \in \mathcal{D}$ and assume $|B(\succ)| \ge 2$. Suppose there are two alternatives $b, c \in B(\succ)$ such that $s_{\succ}(b) < s_{\succ}(c)$. When we consider the profile $\bar{\succ}$, we know that $F(\bar{\succ}) = A$. Consider $\succ_i^{bc}$ where $b$ and $c$ are relabelled. Thus $F(\succ_i^{bc}, \bar{\succ}_{-i}) = \{c\}$. But $B(\succ) \subseteq B(\succ_i^{bc})$ and $B(\succ) \cap F(\bar{\succ}) \ne \emptyset$, so we have, by Expansion, that $B(\succ) \subseteq F(\succ_i^{bc}, \bar{\succ}_{-i})$, obtaining a contradiction. We conclude that there is a $k$ such that $s_{\succ}(b) = k$ for all $b \in B(\succ)$.

Now assume that $|NB(\succ)| \ge 2$ and that $s_{\succ}(a) \ne s_{\succ}(b)$ for some alternatives $a, b \in NB(\succ)$. Using a similar argument as before, we obtain that there exists a $k'$ such that $s_{\succ}(a) = k'$ for all $a \in NB(\succ)$. Finally, by Partial Averseness, we have that $k < k'$, concluding that $F$ belongs to the Simple Anti-Plurality class.

As Expansion only considers the alternatives in the bottom of a preference and not the structure, and by Neutrality, we have that for any two preferences $\succ$ and $\succ'$ such that $|B(\succ)| = |B(\succ')|$, it is the case that $s_{\succ}(a) = s_{\succ'}(b)$ for any $a \in B(\succ)$ and $b \in B(\succ')$, or any $a \in NB(\succ)$ and $b \in NB(\succ')$.

We normalise the scores such that for all $\succ \in \mathcal{D}$ we get $s_{\succ}(a) = 0$ for all $a \in NB(\succ)$. Now consider two preferences $\succ_i, \succ'_i \in \mathcal{D}$ where $|B(\succ_i)| < |B(\succ'_i)|$, and $k$ and $k'$ are the scores for the bottom alternatives of $\succ$ and $\succ'$, respectively. Suppose $k > k'$. By Neutrality, and w.l.o.g., we can assume that $B(\succ_i) \subset B(\succ'_i)$. Again, we consider the profile $\bar{\succ}$ and find that $F(\bar{\succ}) = A$. We have that $B(\succ_i) \cap F(\bar{\succ}) = B(\succ_i)$ and thus, by Expansion, $B(\succ_i) \subseteq F(\succ'_i, \bar{\succ}_{-i})$. But in $F(\succ'_i, \bar{\succ}_{-i})$ all the alternatives in $B(\succ'_i)$ get their scores reduced, while those in $NB(\succ'_i)$ remain with the same score, obtaining $F(\succ'_i, \bar{\succ}_{-i}) = A \setminus B(\succ'_i)$. With this contradiction, we can conclude that $k \le k'$. Thus, we obtain that $F$ belongs to the Monotonic Simple Anti-Plurality class. $\square$

We now introduce another technical property: Bottoms-Only. It requires that only the least-preferred alternatives of the voters influence the outcome. Formally, a rule satisfies Bottoms-Only if for any two profiles $\succ, \succ' \in \mathcal{D}^N$
with $B(\succ_i) = B(\succ'_i)$ for all $i \in N$, it is the case that $F(\succ) = F(\succ')$.<fnref n="13" /> It turns out that any rule satisfying Strong Expansion depends only on the voters' bottom-ranked alternatives. We omit the proof, which is analogous to that of Proposition 5.

**Proposition 12.** *Strong Expansion implies Bottoms-Only.*

Here again, to characterise the Uniform Anti-Plurality rule, we build on a characterisation of the Anti-Plurality rule for linear orders by Kurihara (2018) in terms of Anonymity, Neutrality, Reinforcement, Partial Averseness, and Bottoms-Only.

**Theorem 13.** *A voting rule satisfies Anonymity, Neutrality, Reinforcement, Partial Averseness, and Strong Expansion if, and only if, it is the Uniform Anti-Plurality rule.*

*Proof.* By Proposition 12, we have that any rule that satisfies our axioms also satisfies Bottoms-Only. By the result of Kurihara (2018), we know that when the profile consists of linear orders, a voting rule satisfies all the axioms if, and only if, it is the Anti-Plurality rule. Normalise the rule such that the score for the non-bottom alternatives of a linear order is 0, and let $s_L < 0$ be the score for the bottom alternative.

By the result due to Pivato (2013) mentioned earlier, we know that a rule satisfies Anonymity, Neutrality and Reinforcement if, and only if, it is a composite scoring rule.

Let $\succ \in \mathcal{D}$ be a generic preference. By Neutrality, Partial Averseness and Bottoms Only, we have that $F(\succ) = NB(\succ)$ (to see this, consider the preference where all the non-bottom and all the bottom alternatives are symmetric). So $s_>(c) = k$ for all $c \in NB(\succ)$ and $k > s_>(b)$ for all $b \in B(\succ)$. Now suppose there are positions at the bottom that get different scores, select the two highest scores, and assume that $a$ and $b$, with $a, b \in B(\succ)$, get those scores such that $s_>(a) > s_>(b)$. Let $\mathbf{L}^a = \{L_1^a, \dots, L_{m-2}^a\}$ be a profile of $m-2$ linear orders such that $a$ is always on top, $b$ is always second, and the rest of the alternatives appear exactly once at the bottom. Now let $\mathbf{L}^b = \{L_1^b, \dots, L_{m-2}^b\}$ be the same as $\mathbf{L}^a$ but with $a$ and $b$ swapped. Let $n$ be such that

$$s_>(a) > 2ns_L + k.$$

Thus $F(\succ, n(\mathbf{L}^a, \mathbf{L}^b)) = \{a\}$. Now consider the profile where $a$ and $b$ are relabelled. Then, by Neutrality, we have that $F(\succ^{ab}, n(\mathbf{L}^b, \mathbf{L}^a)) = \{b\}$.

<footnote n="13">¹³For a possible motivation regarding this property, and its use for a characterisation result, we refer to Kurihara (2018).</footnote>
But this violates Bottoms-Only (and thus Strong Expansion). Then we can conclude that $s_{>}(a) = s_{>}(b)$ for all $a, b \in B(>)$). Now normalise the score such that for every preference $> $ we get $s_{>}(a) = 0$ for all $a \in NB(>)$.

The remainder of the proof now proceeds in an analogous manner to the proof of Theorem 6. □

Next, as we previously did for the results of Section 3.1, we show that the axioms used for the results in Section 3.2 are logically independent. Again, it is interesting to observe that Continuity is required for Theorems 9 and 11, in contrast to the result by Kurihara (2018) for the case of linear orders. This is so because of the different structures that the bottom set can have, making B-Congruity and Expansion less demanding.

**Proposition 14.** *The axioms of Anonymity, Neutrality, Reinforcement, Continuity, Partial Averseness, and B-Congruity/Expansion are logically independent.*

*Proof.* We again provide a counterexample for each proper subset of the relevant sets of axioms.

*Anonymity, Neutrality, Reinforcement, Continuity, and Partial Averseness but neither B-Congruity nor Expansion:* the Borda-style rule also used in the proof of Proposition 7.

*Anonymity, Neutrality, Reinforcement, Continuity, B-Congruity, and Expansion but not Partial Averseness:* the rule that always selects the full set of alternatives.

*Anonymity, Neutrality, Reinforcement, Partial Averseness, B-Congruity, and Expansion but not Continuity:* the rule defined by means of the following two-step procedure. In the first step, select all the winning alternatives returned by the Uniform Anti-Plurality rule. In the second step, select as winning alternatives from this set those alternatives that do not appear as a unique bottom element in at least one of the agents’ preferences—in case there are such alternatives; otherwise, pick all alternatives selected in the first step.

*Anonymity, Neutrality, Continuity, Partial Averseness, B-Congruity, and Expansion but not Reinforcement:* the rule that for a single-voter profile selects all the non-bottom alternatives and for other profiles selects the Uniform Anti-Plurality winners and all the alternatives that get exactly one point less than the Uniform Anti-Plurality winners.

*Anonymity, Continuity, Reinforcement, Partial Averseness, B-Congruity, and Expansion but not Neutrality:* the rule that assigns points to alternatives
just like the Uniform Anti-Plurality rule but that doubles the score of one
fixed alternative $a \in A$ before determining the winners.

*Neutrality, Reinforcement, Continuity, Partial Averseness, B-Congruity, and Expansion but not Anonymity:* the rule that assigns points to alternatives just like the Uniform Anti-Plurality rule, except for voter 1's bottom alternatives, which receive -2 points. □

The next statement, regarding the independence of the axioms featuring in
Theorem 13, is presented without proof, as the same examples of Proposition 14 can be used.

**Proposition 15.** *The axioms of Anonymity, Neutrality, Reinforcement, Partial Averseness, and Strong Expansion are logically independent.*

**3.3 Applications to Approval Voting**

In the approval voting model, voters submit a ballot indicating the candidates they approve of, and it is typically assumed that any candidates not included in the ballot are disapproved by the voter (Brams and Fishburn, 1978). This type of ballot can be simulated in our setting as a ballot where an alternative is either in the top or in the bottom of the preference. Formally, in our model, a preference $\succ \in \mathcal{D}$ is an approval ballot if $T(\succ) \cap B(\succ) = \emptyset$ and $T(\succ) \cup B(\succ) = A$.

It is not hard to see that for approval ballots, Averseness becomes equivalent to Partial Faithfulness, and Strong Expansion becomes equivalent to Strong Contraction. In our setting, the size approval voting rules characterised by Alcalde-Unzu and Vorsatz (2009) and Bardal and Endriss (2025) are equivalent to the rules of the Monotonic Simple Plurality Class; while the standard approval voting rule (see Brandl and Peters, 2022, for several characterisations), is equivalent, for approval ballots, to both the Uniform Plurality and Uniform Anti-Plurality rules. These observations lead to the following new characterisations.

**Corollary 16.** *A voting rule for approval ballots satisfies Anonymity, Neutrality, Reinforcement, Continuity, Partial Faithfulness, and Contraction if, and only if, it is a size approval voting rule.*

**Corollary 17.** *A voting rule for approval ballots satisfies Anonymity, Neutrality, Reinforcement, Partial Faithfulness, and Strong Contraction if, and only if, it is the standard approval voting rule.*
**Corollary 18.** A voting rule for approval ballots satisfies Anonymity, Neutrality, Reinforcement, Partial Averseness, and Strong Contraction if, and only if, it is the standard approval voting rule.

We can also consider an approval voting rule that works in the ‘opposite’ direction of size approval voting. Under this rule, an approved alternative receives more points the more alternatives it is approved alongside (while the opposite is true for size approval voting rules). Such rules favour voters who are willing to compromise and approve more alternatives (see also Example 6 in the work of Bardal and Endriss, 2025).

Let us call this family of voting rules the *anti-size approval rules*.

**Corollary 19.** A voting rule for approval ballots satisfies Anonymity, Neutrality, Reinforcement, Continuity, Partial Averseness, and Expansion if, and only if, it is an anti-size approval voting rule.

We emphasise that all these characterisation results are derived without directly relying on the technical assumption of Tops-Only, which is implicit in the approval voting model.

# 4 Conclusion

We have provided axiomatic characterisations of generalised versions of the Plurality and Anti-Plurality rules for settings in which preferences are represented by partial orders. In the classical setting, where preferences are linear orders, the Plurality (Anti-Plurality) rule assigns one point (one negative point) to the most (least) preferred alternative and zero points to all others. We have extended this defining feature to partial orders in a way that respects the structural richness of such preferences, for example, recognising that two top alternatives may differ in the number of alternatives they dominate, and thus may not be equally ‘top-ranked.’

When considering linear orders, the characterisations we have presented all converge to familiar standards established in the existing literature.

Moreover, we demonstrate that several of our findings carry over to the approval voting setting, enabling new characterisations of some well-known and new approval voting rules through our axiomatic lens.

An intriguing path for future investigation lies in exploring similar generalisations of other scoring rules to the domain of partial orders, including but not limited to the Borda, Cumulative, and Stepwise scoring rules.<fnref n="14" />

<footnote n="14">¹⁴See Kruger and Terzopoulou (2020) for definitions of the Cumulative and Stepwise scoring rules.</footnote>
**Acknowledgements.** We are grateful to Fernando Tohmé, Jordi Massó, Tuva Bardal, several attendees of the COMSOC Seminar at the University of Amsterdam, as well as the Conference on Mechanism and Institution Design held in Budapest in 2024, and an anonymous reviewer for comments and suggestions that led to improvements of the paper.

**Statements and Declarations.** The authors have no competing interests to declare that are relevant to the content of this article.

## References

Alcalde-Unzu, J. and Vorsatz, M. (2009). Size approval voting. *Journal of Economic Theory*, 144(3):1187–1210.

Aragones, E., Gilboa, I., and Weiss, A. (2011). Making statements and approval voting. *Theory and Decision*, 71(4):461–472.

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

Arrow, K. J., Sen, A. K., and Suzumura, K., editors (2002). *Handbook of Social Choice and Welfare*, volume 1. Elsevier.

Baharad, E. and Nitzan, S. (2005). The inverse plurality rule—an axiomatication. *Social Choice and Welfare*, 25(1):173–178.

Barberà, S. and Dutta, B. (1982). Implementability via protective equilibria. *Journal of Mathematical Economics*, 10(1):49–65.

Bardal, T. and Endriss, U. (2025). Axiomatic analysis of approval-based scoring rules. *Games and Economic Behavior*, 153:345–358.

Baumeister, D., Faliszewski, P., Lang, J., and Rothe, J. (2012). Campaigns for lazy voters: Truncated ballots. In *Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems (AAMAS)*.

Black, D. (1948). On the rationale of group decision-making. *Journal of Political Economy*, 56(1):23–34.

Brams, S. J. and Fishburn, P. C. (1978). Approval voting. *The American Political Science Review*, 72(3):831–847.

Brandl, F. and Peters, D. (2022). Approval voting under dichotomous preferences: A catalogue of characterizations. *Journal of Economic Theory*, 205:105532.
Ching, S. (1996). A simple characterization of plurality rule. *Journal of Economic Theory*, 71(1):298-302.

Gibbard, A. (1973). Manipulation of voting schemes: A general result. *Econometrica*, 41(4):587-601.

Konczak, K. and Lang, J. (2005). Voting procedures with incomplete preferences. In *Proceedings of the 1st Multidisciplinary Workshop on Advances in Preference Handling (MPREF)*.

Kruger, J. and Terzopoulou, Z. (2020). Strategic manipulation with incomplete preferences: Possibilities and impossibilities for positional scoring rules. In *Proceedings of the 19th International Conference on Autonomous Agents and Multiagent Systems (AAMAS)*.

Kurihara, T. (2018). A simple characterization of the anti-plurality rule. *Economics Letters*, 168:110-111.

Laslier, J.-F. (2011). And the loser is ... plurality voting. In *Electoral Systems: Paradoxes, Assumptions, and Procedures*, pages 327-351. Springer.

Llamazares, B. and Peña, T. (2015). Scoring rules and social choice properties: Some characterizations. *Theory and Decision*, 78(3):429-450.

Merlin, V. and Naeve, J. (2000). Implementation of social choice functions via demanding equilibria. Diskussionspapiere aus dem Institut für Volkswirtschaftslehre der Universität Hohenheim 191/2000, Department of Economics, University of Hohenheim, Germany.

Muller, E. and Satterthwaite, M. A. (1977). The equivalence of strong positive association and strategy-proofness. *Journal of Economic Theory*, 14(2):412-418.

Myerson, R. B. (1995). Axiomatic derivation of scoring rules without the ordering assumption. *Social Choice and Welfare*, 12(1):59-74.

Pini, M. S., Rossi, F., Venable, K. B., and Walsh, T. (2005). Aggregating partially ordered preferences. In *Proceedings of the 10th Conference on Theoretical Aspects of Rationality and Knowledge (TARK)*. National University of Singapore.

Pivato, M. (2013). Variable-population voting rules. *Journal of Mathematical Economics*, 49(3):210-221.
Richelson, J. T. (1978). A characterization result for the plurality rule. *Journal of Economic Theory*, 19(2):548–550.

Satterthwaite, M. A. (1975). Strategy-proofness and Arrow's conditions: Existence and correspondence theorems for voting procedures and social welfare functions. *Journal of Economic Theory*, 10(2):187–217.

Sekiguchi, Y. (2012). A characterization of the plurality rule. *Economics Letters*, 116(3):330–332.

Smith, J. H. (1973). Aggregation of preferences with variable electorate. *Econometrica*, 41(6):1027–1041.

Terzopoulou, Z. (2021). *Collective Decisions with Incomplete Individual Opinions*. PhD thesis, University of Amsterdam.

Terzopoulou, Z. and Endriss, U. (2019). Aggregating incomplete pairwise preferences by weight. In *Proceedings of the 28th International Joint Conference on Artificial Intelligence (IJCAI)*.

Terzopoulou, Z. and Endriss, U. (2021). The Borda class: An axiomatic study of the Borda rule on top-truncated preferences. *Journal of Mathematical Economics*, 92:31–40.

Young, H. (1974). An axiomatization of Borda's rule. *Journal of Economic Theory*, 9(1):43–52.

Young, H. P. (1975). Social choice scoring functions. *SIAM Journal on Applied Mathematics*, 28(4):824–838.

Zwicker, W. S. (2016). Introduction to the theory of voting. In Brandt, F., Conitzer, V., Endriss, U., Lang, J., and Procaccia, A. D., editors, *Handbook of Computational Social Choice*, chapter 2, pages 23–56. Cambridge University Press.