📈 economics

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

This paper explores and provides axiomatic characterizations for extensions of the Plurality and Anti-Plurality voting rules to settings where voter preferences are expressed as partial orders rather than linear orders.

Original authors: Ulle Endriss, Federico Fioravanti

Published 2026-05-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ulle Endriss, Federico Fioravanti

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are organizing a competition to find the best mobile app. Usually, in voting theory, we ask people to rank every single option from "best" to "worst" in a strict line, like a ladder. But in real life, asking someone to rank 50 apps from 1 to 50 is exhausting and often impossible. You might love Instagram and Facebook, but have no idea how Uber compares to Gmail. You just can't compare them.

This paper tackles the problem of how to vote when people's preferences are messy, incomplete, or "partial" rather than perfect lines. The authors, Ulle Endriss and Federico Fioravanti, ask: How do we adapt our most famous voting rules to handle this messiness without losing their simplicity?

Here is the breakdown of their work using everyday analogies.

The Two Main Rules: The "Top" and The "Bottom"

In standard voting, there are two famous ways to pick a winner:

Plurality: You give a point to your absolute favorite. The one with the most points wins.
Anti-Plurality (or Veto): You give a "negative point" (or a veto) to your absolute worst choice. The one with the fewest vetoes wins.

The authors ask: What happens when a voter doesn't have just one favorite or one worst choice, but a whole group of them?

The "Plurality Class" (The Top Group)

When you can't rank everything, you might have a "Top Set" of apps you like equally (e.g., Instagram, Gmail, and Uber are all "good," but you can't say which is #1).

The Old Way: In a perfect line, you pick one #1.
The New Way: The authors define a whole family of rules called the Plurality Class.
- Simple Plurality: Give 1 point to every app in your "Top Set" and 0 to the rest.
- Smart Plurality: Maybe Instagram beats 3 other apps in your mind, but Uber beats none. The authors say it's fair to give Instagram more points than Uber because Instagram is "more dominant" in your preference, even if both are in the top group.
- Uniform Plurality: The simplest version. Just give 1 point to everyone in the Top Set, no matter how they relate to each other.

The "Anti-Plurality Class" (The Bottom Group)

Similarly, you might have a "Bottom Set" of apps you hate equally (e.g., you think both Yahoo and a specific obscure app are terrible).

The New Way: The Anti-Plurality Class extends the "Veto" rule.
- Instead of just vetoing one worst choice, you might veto a whole group of worst choices.
- Just like the Plurality side, you can have rules that treat all bottom choices equally, or rules that give "extra negative points" to the ones that are clearly worse than others in the bottom group.

The "Rules of the Game" (Axioms)

To prove these new rules make sense, the authors use a set of logical "rules of the game" (called axioms). Think of these as the constitution for a fair election.

Anonymity & Neutrality: It doesn't matter who votes or what the apps are named. If everyone swaps their votes, the winner should just swap names too.
Reinforcement: If Group A votes and picks App X, and Group B votes and also picks App X, then if you combine the groups, App X should still win.
Continuity: A tiny group of voters shouldn't be able to completely override a massive majority. (Though they can break a tie).
Faithfulness (The "Top" Rule): If there is only one voter, the winner must be one of their top choices. You shouldn't pick an app they hate just because of the math.
Averseness (The "Bottom" Rule): If there is only one voter, the winner should not be one of their bottom choices (unless they hate everything equally).
Congruity (The "Agreement" Rule): If you add new voters who agree with the current winner (e.g., they don't rank the winner at the bottom), the winner shouldn't suddenly lose.
Contraction/Expansion: If a voter decides to narrow their list of favorites (removing some from the top), the people who were already winning should stay winning.

The Big Discovery

The authors proved that if a voting rule follows these specific "fairness" rules, it must belong to one of these families (Plurality or Anti-Plurality).

The Main Result: If you want a rule that respects the "Top Set" logic and follows the fairness rules, you are forced to use a rule from the Plurality Class.
The Specific Result: If you want the simplest version (where everyone in the top gets the exact same points, regardless of how they compare to each other), you must use the Uniform Plurality Rule.

They did the exact same thing for the "Anti-Plurality" (veto) side, proving that the only fair ways to handle "bottom sets" are the rules in the Anti-Plurality Class.

The "Approval Voting" Connection

The paper also shows a cool trick: Approval Voting (where you just tick the boxes of apps you like) is actually a special case of this messy "partial order" voting.

In Approval Voting, your "Top Set" is the apps you tick, and your "Bottom Set" is the apps you didn't tick.
The authors show that their new math proves the standard Approval Voting rule is actually just the Uniform Plurality rule (or the Uniform Anti-Plurality rule) applied to this specific type of ballot.

Summary

This paper is like a bridge. It takes the simple, well-known rules of voting (Plurality and Veto) and builds a sturdy bridge to the complex, messy reality of human preference (where we often can't compare everything). They proved that there is a specific, logical family of rules that fits this bridge perfectly, ensuring that even when voters are uncertain or have incomplete lists, the election remains fair and predictable.

Technical Summary: Voting with Partial Orders – The Plurality and Anti-Plurality Classes

1. Problem Statement

The paper addresses the problem of collective decision-making when individual preferences are represented as strict partial orders rather than the standard linear orders (complete rankings) used in classical social choice theory. In many practical scenarios, requiring voters to fully rank a large set of alternatives is overly demanding. Partial orders allow voters to express strict preferences for some pairs while remaining indifferent or unable to compare others.

The central challenge is to extend the well-known Plurality rule (selecting the most frequently top-ranked alternative) and the Anti-Plurality rule (selecting the least frequently bottom-ranked alternative, also known as Veto) to this domain. A key difficulty arises because, in a partial order, a voter may have multiple "top" alternatives (none preferred to them) or multiple "bottom" alternatives (none preferred to them), and the structure of these sets can vary significantly between voters. The paper seeks to identify and axiomatically characterise the natural classes of voting rules that generalize these standard rules to partial orders.

2. Methodology

The authors employ an axiomatic approach within a model of variable electorates (where any finite subset of an infinite universe of voters can participate).

The Model

Alternatives: A finite set $A$ with $|A| = m \geq 3$ .
Preferences: A strict partial order $\succ$ (irreflexive, antisymmetric, transitive).
Key Sets:
- $T(\succ)$ : The set of top alternatives (maximal elements).
- $B(\succ)$ : The set of bottom alternatives (minimal elements).
- $NT(\succ)$ and $NB(\succ)$ : The non-top and non-bottom sets, respectively.
Voting Rules: Functions mapping profiles of partial orders to a non-empty set of winning alternatives.

Scoring Rules

The paper focuses on positional scoring rules, where an alternative's score depends only on its position in the preference graph. The authors define several classes of rules based on how scores are assigned to top and bottom sets:

Plurality Class: Non-top alternatives receive a constant score $k'$ , while top alternatives receive scores $\geq k'$ , with at least one top alternative receiving strictly more.
Simple Plurality Class: All top alternatives receive the same score $k$ .
Monotonic Simple Plurality Class: The score difference between top and non-top alternatives increases as the size of the top set decreases (rewarding "more decisive" top rankings).
Uniform Plurality Rule: A specific rule where every top alternative gets 1 point and every non-top gets 0.
Anti-Plurality Classes: Defined symmetrically, focusing on bottom sets (non-bottoms get score $k$ , bottoms get $\leq k$ ).

Axioms

The characterisations rely on a set of axioms, including standard ones adapted for partial orders and new ones inspired by approval voting:

Standard: Anonymity, Neutrality, Reinforcement (Consistency), and Continuity.
Responsiveness: Partial Faithfulness (single voter selects top), Faithfulness (single voter selects exactly top), Partial Averseness (single voter does not select all bottom), Averseness (single voter selects no bottom).
Congruity: T-Congruity (adding voters who don't rank $x$ top shouldn't make $x$ win) and B-Congruity (adding voters who don't rank $x$ bottom shouldn't make $x$ lose).
Contraction/Expansion: Constraints on how the outcome changes when a voter shrinks their top set or expands their bottom set.
Technical Properties: Tops-Only and Bottoms-Only (outcomes depend only on top/bottom sets).

3. Key Contributions and Results

Characterisation of Scoring Rules

Theorem 1 establishes that a voting rule satisfies Anonymity, Neutrality, Reinforcement, and Continuity if and only if it is a positional scoring rule. This adapts Young's (1975) classic result to the domain of partial orders.

The Plurality Family

The paper provides axiomatic characterisations for the Plurality classes:

Theorem 2: A rule belongs to the Plurality Class iff it satisfies Anonymity, Neutrality, Reinforcement, Continuity, Partial Faithfulness, and T-Congruity.
Corollary 3: The Simple Plurality Class is characterised by replacing Partial Faithfulness with Faithfulness.
Theorem 4: The Monotonic Simple Plurality Class is characterised by adding the Contraction axiom to the axioms of the Simple Plurality Class.
Theorem 6: The Uniform Plurality Rule is characterised by Anonymity, Neutrality, Reinforcement, Partial Faithfulness, and Strong Contraction. The proof relies on showing that Strong Contraction implies Tops-Only, reducing the problem to the linear order case where Sekiguchi's (2012) characterisation applies.

The Anti-Plurality Family

Symmetric results are provided for the Anti-Plurality classes:

Theorem 9: A rule belongs to the Anti-Plurality Class iff it satisfies Anonymity, Neutrality, Reinforcement, Continuity, Partial Averseness, and B-Congruity.
Corollary 10: The Simple Anti-Plurality Class is characterised by replacing Partial Averseness with Averseness.
Theorem 11: The Monotonic Simple Anti-Plurality Class is characterised by adding the Expansion axiom.
Theorem 13: The Uniform Anti-Plurality Rule is characterised by Anonymity, Neutrality, Reinforcement, Partial Averseness, and Strong Expansion.

Independence of Axioms

The authors demonstrate the logical independence of the axioms used in these characterisations (Propositions 7, 8, 14, 15) by constructing counter-example rules that satisfy all but one axiom. Notably, they highlight that Continuity is necessary for their characterisations of the Plurality and Anti-Plurality classes in the partial order setting, whereas it is often dispensable for the standard linear order Plurality rule (e.g., Ching, 1996). This necessity arises because the varying structures of top/bottom sets in partial orders make axioms like T-Congruity and Contraction less restrictive without Continuity.

Applications to Approval Voting

The paper shows that their results apply to Approval Voting, where a ballot is a partial order with disjoint top and bottom sets covering all alternatives ( $T(\succ) \cap B(\succ) = \emptyset$ and $T(\succ) \cup B(\succ) = A$ ).

Corollary 16: Size approval voting rules (where score depends on the size of the approval set) are characterised by the axioms of the Monotonic Simple Plurality Class (with Partial Faithfulness).
Corollaries 17 & 18: The standard approval voting rule is characterised by the axioms of the Uniform Plurality and Uniform Anti-Plurality rules, respectively.
Corollary 19: "Anti-size" approval rules (favouring larger approval sets) are characterised by the Monotonic Simple Anti-Plurality axioms.

4. Significance and Claims

The paper claims to provide the first axiomatic characterisations of Plurality-like and Anti-Plurality-like rules for the domain of partial orders. Its significance lies in:

Generalisation: It successfully extends the intuitive logic of Plurality and Anti-Plurality (focusing on top and bottom positions) to a setting where "top" and "bottom" are sets rather than single elements, respecting the structural richness of partial orders.
Unification: It unifies the understanding of these rules across linear orders, partial orders, and approval voting. The authors note that their characterisations for linear orders converge to established literature results, validating the generalisation.
New Insights for Approval Voting: By deriving these results from the partial order framework, the paper offers new, independent axiomatic characterisations for standard and size-based approval voting rules, avoiding the direct reliance on the "Tops-Only" assumption often implicit in approval voting literature.
Subtlety of the Domain: The results illustrate the specific subtleties of voting with partial orders, particularly the necessity of the Continuity axiom in this broader context, which distinguishes it from the linear order case.

The authors conclude that while they have focused on Plurality and Anti-Plurality, their framework opens the door for similar generalisations of other scoring rules (e.g., Borda, Cumulative, Stepwise) to the domain of partial orders.