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

The paper introduces a ranked-candidate voting method based on Phragmen's procedure that generates a top-down proportional list satisfying the Droop proportionality criterion, house monotonicity, and coherence, while ensuring the top-ranked candidate is the Instant Runoff winner.

The Gist

Imagine you are organizing a massive, multi-day festival where you need to pick a team of organizers (winners) from a huge pool of candidates. The voters are the festival-goers, and they have strong opinions about who should be in charge. They don't just pick one person; they rank everyone from "My absolute favorite" to "I'd rather not have them."

The problem is: How do you pick a team that truly represents the crowd, while making sure the rules don't break when you change the number of spots available?

This paper by Ross Hyman introduces a new, clever way to solve this puzzle. Let's break it down using a simple story.

The Problem: The "Moving Goalposts"

In the past, voting systems had some weird glitches.

• The "Alabama Paradox": Imagine you decide to add more seats to the festival committee. You'd think everyone would be happier, right? But in old systems, adding a seat could actually cause a popular candidate to lose their spot. It's like adding more slices to a pizza, but somehow your favorite slice disappears.
• The "New State Paradox": If a new group of people joins the festival, it shouldn't randomly knock out an existing member from a different group just because the math got messy.

Old methods (like the standard "Single Transferable Vote" used in places like Ireland) are great at fairness, but they can suffer from these glitches. They are like a game where the rules change depending on how many players are in the room.

The Goal: A Perfect "Hall of Fame" List

The author wants to create a Master List of candidates.

1. House Monotonicity: If you need 3 winners, the top 3 on the list should be the winners. If you need 4 winners, the top 4 should be the winners. The list shouldn't shuffle around; the top spots should always be the best spots.
2. Coherence: If you have two separate groups of voters (say, "Music Lovers" and "Art Lovers") who don't overlap, the system should treat them fairly. The Music Lovers shouldn't lose a seat just because the Art Lovers joined the party.
3. Droop Proportionality: This is the golden rule of fairness. If a group of voters is big enough to deserve a seat (say, more than 1/3 of the crowd for 3 seats), they must get a seat. It's the "Sine Qua Non" (the essential ingredient) of a fair election.

The Solution: The "Top-Down Phragmén" Method

The author proposes a new method called Top-Down Phragmén. Think of it as building a tower, one block at a time, from the very top down.

How it works (The Analogy):

Imagine you are building a tower of blocks (candidates).

1. Start at the Top (The IRV Winner): First, you find the single most popular candidate using a standard "Instant Runoff" method (like a game of musical chairs where the least popular gets eliminated until one remains). This person is the Top Block. They are the undisputed leader.
2. Build the Second Block: Now, you need a second person. But here's the trick: You must keep the Top Block on the team. You look for the next best person who, when added to the Top Block, creates the fairest possible team of two.
3. The "Seat Load" Concept: Imagine every voter has a "backpack" with a certain weight. When a candidate is elected, they take some weight off the voters' backpacks. The goal is to balance the backpacks so no group of voters is carrying too much weight (feeling ignored) while another group is carrying almost nothing.
4. The "Imperfect" Coalition: Sometimes, voters don't agree on exactly who their top 3 favorites are, but they agree on the general vibe. The new method is smart enough to handle these "fuzzy" groups. It doesn't demand perfect agreement; it just looks for the best balance of support.

Why is this better than the old ways?

• Old Way (Bottom-Up): Imagine trying to build a tower by starting with the bottom block, then the next, and hoping the top looks good. Sometimes, the person who ends up at the very top (the leader) isn't actually the most popular single person. It's like picking a captain for a sports team by first picking the benchwarmers and hoping the captain falls out of the sky.
• New Way (Top-Down): This method starts with the Captain (the IRV winner) and builds the rest of the team around them. This guarantees that the #1 person on your list is actually the #1 choice of the voters.

The "Magic" of the Math

The author proves that this method:

• Never breaks the rules: If you add more seats, the old winners stay winners.
• Never plays favorites: Two separate groups of voters won't accidentally hurt each other's chances.
• Guarantees Fairness: If a group is big enough to deserve a seat, they get one.

The Bottom Line

Ross Hyman has designed a voting system that acts like a fair, self-correcting scale. It builds a ranked list of winners from the top down, ensuring that the most popular person is always at the top, and that every subsequent person added to the list makes the whole team more representative of the crowd.

It's like having a referee who not only knows the rules perfectly but also ensures that no matter how the game changes (more players, new teams), the final score always reflects the true will of the fans. It's a "House Monotone, Coherent, and Droop Proportional" way to pick the best team, ensuring that the people in charge actually represent the people they serve.

Technical Summary

1. Problem Statement

The paper addresses a fundamental gap in ranked-candidate voting systems: the inability to generate a proportional candidate list that simultaneously satisfies three critical properties:

1. Droop Proportionality: Ensures that if a solid coalition of voters (a group ranking a set of candidates above all others) exceeds $K$ Droop quotas, at least $K$ candidates from that set are elected. This is considered the "sine qua non" of fair proportional representation.
2. House Monotonicity: Ensures that the winners of an $N$-seat election are always a subset of the winners of an $(N+1)$-seat election. This allows for the construction of a ranked list where the top $N$ candidates are the winners for $N$ seats, avoiding the Alabama Paradox (where increasing total seats reduces a group's representation).
3. Coherence: Ensures that if an election is split into two disjoint sets of candidates and ballots, the winners of the combined election are consistent with the winners of the separate elections. This avoids the New State Paradox.

The Conflict:

• Single Transferable Vote (STV), widely used in countries like Ireland and Australia, satisfies Droop proportionality but fails House Monotonicity and Coherence. Consequently, STV cannot reliably produce a proportional candidate list.
• Divisor Methods (used for party-list apportionment) satisfy House Monotonicity and Coherence but are traditionally designed for party lists, not ranked individual candidates.
• Existing Hybrid Attempts: Previous attempts to adapt Phragmén's method (a divisor method) to ranked ballots either relied on quotas (violating monotonicity/coherence) or required computationally expensive searches for "perfect" solid coalitions (making them impractical).

2. Methodology

Hyman proposes a Top-Down Phragmén method. Unlike "bottom-up" approaches (which determine the last place first) or quota-based methods, this algorithm constructs the candidate list from the top (1st place) down to the bottom.

Core Mechanism:
The method is based on Edvard Phragmén's divisor logic but adapts it for ranked ballots without using a fixed quota to trigger elections or exclusions. It operates iteratively for an $M$-winner election:

1. State Definition: Candidates are categorized as "previously elected" (the top $M-1$ winners already determined) or "hopeful" (remaining candidates).
2. Priority Calculation (The Innovation):
   • Standard Phragmén calculates priority based on the ratio of supporting ballots to seat loads.
   • Top-Down Modification: To ensure Droop compliance, the method accounts for "imperfect solid coalitions." It recognizes that a hopeful candidate $H$ might be supported by a coalition that includes various combinations of previously elected candidates.
   • The algorithm simulates the election of $H$ against different subsets of previously elected candidates to determine $H$'s "priority." Specifically, it creates a modified ballot set where $H$ is moved to the lowest rank of the coalition, and the previously elected candidates are promoted. It then runs a Phragmén simulation to see when $H$ would be elected.
   • The priority of a hopeful candidate is the maximum of these calculated priorities across all possible coalition configurations.
3. Iterative Exclusion:
   • The algorithm promotes candidates with the highest priority.
   • It reduces the rank of "hopeful" candidates on ballots to simulate their exclusion from the current coalition context.
   • If a candidate's priority never exceeds the threshold required to be elected given the current set of previously elected candidates, they are excluded.
   • This process repeats until only one hopeful candidate remains, who is then elected to the current position in the list.

Key Distinction: Unlike quota-based methods, this method does not rely on a fixed quota ($V_{total}/(N+1)$) to decide when to elect or exclude. Instead, it uses the dynamic priority calculation derived from Phragmén's load-balancing logic, ensuring that the election of the $M$-th winner is mathematically guaranteed to be consistent with the first $M-1$.

3. Key Contributions

• Algorithmic Innovation: The paper introduces a specific top-down algorithm that integrates Phragmén's divisor logic with ranked ballots without requiring the explicit identification of every solid coalition (which is computationally prohibitive).
• Theoretical Proof: The author provides formal proofs (Claims 1–9) demonstrating that the Top-Down Phragmén method satisfies:
  • House Monotonicity: By construction, previously elected candidates are never excluded in subsequent rounds.
  • Coherence: The method's priority calculations for disjoint candidate sets are independent of each other.
  • Droop Proportionality: The method guarantees that the IRV (Instant Runoff) winner is in at least one Droop-compliant set for any $N$, and by induction, the entire list satisfies Droop proportionality.
• Resolution of the "Top-Down" Paradox: Previous top-down methods (like Otten's) failed Droop proportionality. Hyman's method corrects this by properly accounting for seat loads of previously elected candidates within imperfect solid coalitions.

4. Results

• Example Validation: The paper demonstrates the method using a specific ballot set (Example 1: 60: ADCB, 51: BCDA, 45: CADB, 44: DCAB).
  • 1-Winner: The method elects C (consistent with IRV).
  • 2-Winners: The method elects A and C.
  • 3-Winners: The method elects A, B, and C.
  • Resulting List: C > A > B > D.
  • Comparison: This list satisfies Droop proportionality (e.g., the 149-ballot coalition for {A, C, D} ensures at least 2 of them are in the top 2, and 3 in the top 3).
• Property Verification:
  • The method successfully avoids the Alabama and New State paradoxes.
  • It satisfies "Later-no-harm" and "Later-no-help" criteria (changes to lower-ranked preferences do not harm higher-ranked candidates).
  • The highest-ranked candidate is always the IRV winner, ensuring the top of the list represents the broadest consensus.

5. Significance

• Bridging Theory and Practice: This method offers a viable solution for political parties and electoral commissions that need to generate ranked candidate lists for proportional representation elections. It solves the issue where STV produces a winner but fails to produce a consistent, proportional ranking for the rest of the list.
• Fairness Guarantees: By satisfying House Monotonicity and Coherence, the method ensures that the addition of new seats or new states (candidates) does not arbitrarily penalize existing groups, a critical requirement for stable democratic systems.
• Computational Feasibility: While the method is described as "aesthetically ugly" and complex due to the need to calculate priorities for multiple coalition permutations, it avoids the exponential complexity of searching for all perfect solid coalitions required by previous top-down approaches.
• Foundation for Future Research: The paper highlights the trade-off between "perfect" and "imperfect" solid coalitions, suggesting that future work could refine how minor deviations in voter preferences are handled to further improve proportionality without sacrificing computational efficiency.

In conclusion, Ross Hyman presents a rigorous, divisor-method-based voting system that successfully unifies the fairness of Droop proportionality with the structural stability of House Monotonicity and Coherence, providing a robust theoretical framework for ranked-candidate proportional representation.