A House Monotone, Coherent, and Droop Proportional Ranked Candidate Voting Method

Ross Hyman

arxiv: 2506.12318 · v3 · submitted 2025-06-14 · 💰 econ.TH

A House Monotone, Coherent, and Droop Proportional Ranked Candidate Voting Method

Ross Hyman This is my paper

Pith reviewed 2026-05-19 09:52 UTC · model grok-4.3

classification 💰 econ.TH

keywords ranked candidate votingPhragmen procedureDroop proportionalityhouse monotonicitycoherenceInstant Runoff Votingproportional representation

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The pith

A Phragmén-based ranked voting method produces proportional candidate lists that satisfy Droop proportionality along with house monotonicity and coherence.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper describes a ranked candidate voting method built from Phragmen's procedure that generates a top-down list of candidates. The list meets the Droop proportionality criterion used by Single Transferable Vote. It further satisfies house monotonicity and coherence, the ranked analogs of divisor-method properties that avoid the Alabama and New State paradoxes. The highest-ranked candidate is always the Instant Runoff winner and belongs to at least one Droop-proportional set of any size N. A reader would care because the construction tries to combine proportionality with monotonicity guarantees that many existing ranked methods lack.

Core claim

A ranked candidate voting method based on Phragmen's procedure can be used to produce a top-down proportional candidate list. The method complies with the Droop proportionality criterion satisfied by Single Transferable Vote. It also complies with house monotonicity and coherence, which are the ranked-candidate analogs of the divisor methods properties of always avoiding the Alabama and New State paradoxes. The highest ranked candidate in the list is the Instant Runoff winner, which is in at least one Droop proportional set of N winners for all N.

What carries the argument

The Phragmén-based procedure that iteratively builds the ranked candidate list while preserving proportionality conditions at each prefix.

Load-bearing premise

That the specific algorithmic definition of the Phragmén-based procedure actually delivers all stated properties simultaneously without counterexamples in edge cases or implementation details.

What would settle it

A specific profile of ranked ballots in which the generated list either fails to contain a Droop-proportional set of winners or changes its prefix when the number of seats increases, violating house monotonicity.

read the original abstract

A Ranked candidate voting method based on Phragmen's procedure is described that can be used to produce a top-down proportional candidate list. The method complies with the Droop proportionality criterion satisfied by Single Transferable Vote. It also complies with house monotonicity and coherence, which are the ranked-candidate analogs of the divisor methods properties of always avoiding the Alabama and New State paradoxes. The highest ranked candidate in the list is the Instant Runoff winner, which is in at least one Droop proportional set of N winners for all N.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a Phragmén variant for ranked candidate lists that claims Droop proportionality at every prefix, house monotonicity, and coherence, with the top pick matching the IRV winner.

read the letter

This paper defines a ranked candidate voting method based on Phragmén's procedure. It produces a top-down list that meets the Droop proportionality criterion for any number of seats, stays house monotone, and is coherent in the sense that it avoids the ranked versions of the Alabama and New State paradoxes. The first candidate in the list is the Instant Runoff winner. The work does a solid job of defining the procedure in enough detail to follow the steps and see how it differs from standard Phragmén or STV. It shows the connection to existing criteria and gives examples that illustrate the properties. That makes the proposal concrete rather than purely abstract. The soft spot is the strength of the evidence for the claims. The paper appears to establish the properties through the construction and some test cases rather than a general proof that covers all preference profiles. The stress test raises a fair point about possible issues with tie-breaking rules or how the sequential updates handle quotas. If those details have any ambiguity, one of the three main properties could fail on a particular profile. A reader would want to see either a formal argument or more systematic checking to be confident the combination holds up. This kind of paper is for researchers in social choice theory who are looking at ways to combine proportionality with monotonicity in ranked systems. It could be useful for someone designing or evaluating voting methods for proportional representation, especially if they care about avoiding certain paradoxes. I would recommend sending it to peer review. The topic is relevant and the method is specified clearly enough that referees can examine whether the properties actually follow from the definition.

Referee Report

2 major / 1 minor

Summary. The manuscript describes a ranked candidate voting method derived from Phragmén's procedure that generates a top-down proportional candidate list. It claims to satisfy the Droop proportionality criterion (as in STV), house monotonicity, and coherence (the ranked analogs of avoiding Alabama and New State paradoxes), while ensuring the top-ranked candidate is the Instant Runoff winner and belongs to at least one Droop-proportional winning set for every prefix length N.

Significance. If the simultaneous satisfaction of these properties is established, the result would be a meaningful contribution to social choice theory by combining proportionality, monotonicity, and coherence in a single ranked-ballot method, properties that are frequently in tension. The explicit link to the IRV winner and construction from an established procedure like Phragmén provide a concrete algorithmic foundation that could support further theoretical and computational work.

major comments (2)

The central claims of simultaneous Droop proportionality for every N, house monotonicity, and coherence rest on the specific sequential update rules of the Phragmén-based procedure. Without a general proof or exhaustive verification that these hold for all preference profiles (including edge cases in quota handling and tie resolution), the construction-plus-examples approach leaves the load-bearing assertions unconfirmed.
The assertion that the highest-ranked candidate is always the IRV winner and lies in some Droop-proportional set of N winners for all N requires a formal argument showing invariance under the method's sequential steps; illustrative examples alone do not rule out counterexamples on profiles where IRV and Droop sets diverge.

minor comments (1)

Clarify the precise tie-breaking rule and quota-update formula in the method definition to allow independent verification of the claimed properties.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful and constructive review. The comments highlight important points regarding the rigor of our proofs, and we address each one below while committing to strengthen the manuscript accordingly.

read point-by-point responses

Referee: The central claims of simultaneous Droop proportionality for every N, house monotonicity, and coherence rest on the specific sequential update rules of the Phragmén-based procedure. Without a general proof or exhaustive verification that these hold for all preference profiles (including edge cases in quota handling and tie resolution), the construction-plus-examples approach leaves the load-bearing assertions unconfirmed.

Authors: We agree that the current presentation relies on the construction and illustrative examples, which may leave some readers seeking a more explicit general argument. The sequential Phragmén update rules are defined to preserve Droop quota satisfaction at every prefix length by construction, with house monotonicity following directly from never removing previously selected candidates and coherence from the uniform treatment of preference profiles when the electorate changes. We will add a new section containing a formal inductive proof that these properties hold for arbitrary profiles, with explicit handling of quota thresholds and tie-breaking rules. revision: yes
Referee: The assertion that the highest-ranked candidate is always the IRV winner and lies in some Droop-proportional set of N winners for all N requires a formal argument showing invariance under the method's sequential steps; illustrative examples alone do not rule out counterexamples on profiles where IRV and Droop sets diverge.

Authors: The first selection step of the procedure is equivalent to the initial support calculation in IRV, guaranteeing that the top candidate is the IRV winner. Because each subsequent addition maintains the invariant that the current prefix belongs to a Droop-proportional set, the top candidate remains in at least one such set for every larger N. We will insert a dedicated lemma with a short inductive argument establishing this invariance and will explicitly check the behavior on profiles where IRV and minimal Droop sets are known to differ. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from established Phragmén procedure

full rationale

The manuscript defines a ranked voting method by adapting the known Phragmén procedure to produce a top-down list. It asserts compliance with Droop proportionality (as in STV), house monotonicity, and coherence (ranked analogs of divisor-method properties), plus the property that the top candidate is the IRV winner and lies in a Droop-proportional set for every prefix length. These claims rest on the algorithmic construction and reference to standard external criteria rather than any self-definitional loop, fitted-input prediction, or load-bearing self-citation chain. No equation or step in the abstract reduces a claimed property to its own inputs by construction; the derivation therefore remains independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the method is presented as an adaptation of the known Phragmén procedure.

pith-pipeline@v0.9.0 · 5604 in / 1208 out tokens · 34970 ms · 2026-05-19T09:52:10.186255+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

The method complies with the Droop proportionality criterion satisfied by Single Transferable V ote

A House Monotone, Coherent, and Droop Proportional Ranked Candidate Voting Method Ross Hyman Research Computing Center, University of Chicago rhyman@uchicago.edu Abstract A Ranked candidate voting method based on Phragmén’s procedure is described that can be used to produce a top-down proportional candidate list. The method complies with the Droop proport...

work page 1900
[2]

and most countries that elect parlements in proportional representation elections also use divisor methods to determine how many seats should be apportioned proportionally to each party’s vote (Pukelsheim, F. 2017). However, Single Transferable V ote (STV) (Tideman, Nicolaus 2006), a proportional representation election method used in Australia, Ireland, ...

work page 2017
[3]

is satisfied if at least K of those L candidates are elected. Droop proportionality guarantees that if there are an odd number of seats, and there is a solid coalition of more than half the voters preferring a sufficient number of the candidates above all others, then more than half the seats will be filled from the set of those preferred candidates. This...

work page 1999
[4]

This is because if more than KQN ballots prefer L candidates above all others, then more than KQN+1 ballots prefer at least the same L candidates above all others

have shown that every bottom-up method based on a Droop proportional method is house monotonic and Droop proportional. This is because if more than KQN ballots prefer L candidates above all others, then more than KQN+1 ballots prefer at least the same L candidates above all others. This is because KQN > KQN+1 and every subset of a solid coalition is itsel...

work page 1998
[5]

To determine priorities, their methods assign support from a ballot to its highest ranked candidate that is not elected or excluded

developed ranked-candidate election methods based on Phragmén’s method. To determine priorities, their methods assign support from a ballot to its highest ranked candidate that is not elected or excluded. They borrow, from STV, the idea of using a quota to determine if an election or exclusion of a candidate should occur, electing the highest priority can...

work page 2003
[6]

Claim 5: For every N, the IRV winner is in at least one Droop compliant set of N winners. Proof: If it were the case that the IRV winner was not in any Droop compliant sets for N winners, then there would be a way to separate the ballots in up to N+1 solid coalitions, up to N of which exceed QN, such that the IRV winner is not in the preferred candidate s...

work page 1994
[7]

If it were the case that the new M-winner was not in any Droop compliant set for N winners that includes the M-1 previous winners, then there would be a way to divide the ballots into up to N+1 solid coalitions, up to N of which exceed QN, such that all M-1 previously elected winners are in the preferred candidate sets for the up to N solid coalitions tha...

work page 2025
[8]

Committee Monotonicity and Proportional Representation for Ranked Preferences

“Committee Monotonicity and Proportional Representation for Ranked Preferences.” arXiv. https://doi.org/10.48550/arXiv.2406.19689v3. Balinski, M and Young, H

work page doi:10.48550/arxiv.2406.19689v3
[9]

What Is Just?

“What Is Just?” The American Mathematical Monthly 112 (6): 502–11. https://doi.org/10.1080/00029890.2005.11920221. Brill, Markus, Rupert Freeman, Svante Janson, and Martin Lackner

work page doi:10.1080/00029890.2005.11920221 2005
[10]

Phragmén’s V oting Methods and Justified Representation

“Phragmén’s V oting Methods and Justified Representation.” Mathematical Programming 203 (1): 47–76. https://doi.org/10.1007/s10107-023-01926-8. Hyman, Ross, Deb Otis, Seamus Allen, and Greg Dennis

work page doi:10.1007/s10107-023-01926-8
[11]

A Majority Rule Philosophy for Instant Runoff V oting

“A Majority Rule Philosophy for Instant Runoff V oting.” Constitutional Political Economy 35 (3): 425–36. https://doi.org/10.1007/s10602-024-09442-3. Otten, J

work page doi:10.1007/s10602-024-09442-3
[12]

Tideman, Nicolaus

http://lists.electorama.com/pipermail/election-methods-electorama.com//2002-November/074151.html. Tideman, Nicolaus

work page 2002

[1] [1]

The method complies with the Droop proportionality criterion satisfied by Single Transferable V ote

A House Monotone, Coherent, and Droop Proportional Ranked Candidate Voting Method Ross Hyman Research Computing Center, University of Chicago rhyman@uchicago.edu Abstract A Ranked candidate voting method based on Phragmén’s procedure is described that can be used to produce a top-down proportional candidate list. The method complies with the Droop proport...

work page 1900

[2] [2]

and most countries that elect parlements in proportional representation elections also use divisor methods to determine how many seats should be apportioned proportionally to each party’s vote (Pukelsheim, F. 2017). However, Single Transferable V ote (STV) (Tideman, Nicolaus 2006), a proportional representation election method used in Australia, Ireland, ...

work page 2017

[3] [3]

is satisfied if at least K of those L candidates are elected. Droop proportionality guarantees that if there are an odd number of seats, and there is a solid coalition of more than half the voters preferring a sufficient number of the candidates above all others, then more than half the seats will be filled from the set of those preferred candidates. This...

work page 1999

[4] [4]

This is because if more than KQN ballots prefer L candidates above all others, then more than KQN+1 ballots prefer at least the same L candidates above all others

have shown that every bottom-up method based on a Droop proportional method is house monotonic and Droop proportional. This is because if more than KQN ballots prefer L candidates above all others, then more than KQN+1 ballots prefer at least the same L candidates above all others. This is because KQN > KQN+1 and every subset of a solid coalition is itsel...

work page 1998

[5] [5]

To determine priorities, their methods assign support from a ballot to its highest ranked candidate that is not elected or excluded

developed ranked-candidate election methods based on Phragmén’s method. To determine priorities, their methods assign support from a ballot to its highest ranked candidate that is not elected or excluded. They borrow, from STV, the idea of using a quota to determine if an election or exclusion of a candidate should occur, electing the highest priority can...

work page 2003

[6] [6]

Claim 5: For every N, the IRV winner is in at least one Droop compliant set of N winners. Proof: If it were the case that the IRV winner was not in any Droop compliant sets for N winners, then there would be a way to separate the ballots in up to N+1 solid coalitions, up to N of which exceed QN, such that the IRV winner is not in the preferred candidate s...

work page 1994

[7] [7]

If it were the case that the new M-winner was not in any Droop compliant set for N winners that includes the M-1 previous winners, then there would be a way to divide the ballots into up to N+1 solid coalitions, up to N of which exceed QN, such that all M-1 previously elected winners are in the preferred candidate sets for the up to N solid coalitions tha...

work page 2025

[8] [8]

Committee Monotonicity and Proportional Representation for Ranked Preferences

“Committee Monotonicity and Proportional Representation for Ranked Preferences.” arXiv. https://doi.org/10.48550/arXiv.2406.19689v3. Balinski, M and Young, H

work page doi:10.48550/arxiv.2406.19689v3

[9] [9]

What Is Just?

“What Is Just?” The American Mathematical Monthly 112 (6): 502–11. https://doi.org/10.1080/00029890.2005.11920221. Brill, Markus, Rupert Freeman, Svante Janson, and Martin Lackner

work page doi:10.1080/00029890.2005.11920221 2005

[10] [10]

Phragmén’s V oting Methods and Justified Representation

“Phragmén’s V oting Methods and Justified Representation.” Mathematical Programming 203 (1): 47–76. https://doi.org/10.1007/s10107-023-01926-8. Hyman, Ross, Deb Otis, Seamus Allen, and Greg Dennis

work page doi:10.1007/s10107-023-01926-8

[11] [11]

A Majority Rule Philosophy for Instant Runoff V oting

“A Majority Rule Philosophy for Instant Runoff V oting.” Constitutional Political Economy 35 (3): 425–36. https://doi.org/10.1007/s10602-024-09442-3. Otten, J

work page doi:10.1007/s10602-024-09442-3

[12] [12]

Tideman, Nicolaus

http://lists.electorama.com/pipermail/election-methods-electorama.com//2002-November/074151.html. Tideman, Nicolaus

work page 2002