📈 economics

Seat Reversals and Candidate Status Changes under Biproportional Representation

This article analyzes the 2023 Zurich cantonal election to demonstrate that while biproportional representation significantly improves aggregate party proportionality, it creates a measurable trade-off by altering local candidate outcomes and district seat allocations through specific mechanical constraints.

Original authors: Sandro Lüscher

Published 2026-08-10

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Sandro Lüscher

Original paper licensed under CC BY 4.0 (https://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 massive school election where every student gets to vote for their favorite club. In a standard election, the school is divided into small neighborhoods (districts). Each neighborhood counts its own votes and sends a set number of representatives to the big student council. This is fair locally, but it can get messy. If a popular club gets most of its votes in just one neighborhood, it might end up with way too many seats overall, while a club with steady support everywhere gets too few. It's like if the "Science Club" won every single seat in the science wing but got zero seats in the art wing, even though they had fans in both.

To fix this, some schools use a special "two-step" rule called biproportional representation. First, they count all the votes across the whole school to decide exactly how many total seats each club should get. Then, they try to fit those numbers into the specific neighborhoods without changing the total number of seats each neighborhood sends. It's a mathematical puzzle: "How do we give the Science Club their fair share of seats across the whole school, while still respecting that the Art Wing needs to send exactly five people?" This system is designed to make the final council look exactly like the whole student body. But here's the tricky part: to make the big picture fair, the rules sometimes have to shuffle who gets a seat in a specific neighborhood, even if that neighborhood voted differently. This paper asks: when we fix the big picture, how many individual students get bumped off the council, and does it feel like a fair swap or a confusing surprise?

The Great Seat Shuffle: When Fairness Moves the Goalposts

This paper, written by Sandro Lüscher, dives into the 2023 election in the Swiss canton of Zurich to see exactly what happens when you use this "two-step" fairness rule. The author treats the election like a giant, pre-recorded video game. He takes the actual votes people cast, the actual lists of candidates, and the actual rankings, and then he runs the numbers twice: once with the real "two-step" rule (biproportionality) and once with a simpler rule where every neighborhood counts its own votes independently (like a standard local election).

By keeping everything else exactly the same and only changing the math rule, he can see exactly how many seats get swapped around just to make the total numbers fair.

The Main Discovery: It's Not Just About "Reversals"

For a long time, experts worried about a specific weird thing called a "seat reversal." This happens when a party that got more votes in a neighborhood ends up with fewer seats than a party that got fewer votes. It sounds like a glitch! Imagine the "Gaming Club" getting more votes than the "Art Club" in your neighborhood, but the Art Club gets the seat. That feels wrong.

The paper confirms that this does happen. In the 2023 Zurich election, there were two places where a party with fewer votes got more seats than a party with more votes. This is the "seat reversal" everyone talks about.

However, the paper's big "Aha!" moment is that seat reversals are only half the story.

The author found that biproportional representation changes the outcome for fourteen different candidates in total. But only four of those changes were due to those weird "seat reversals." The other ten changes happened even when the party with more votes still got more seats!

Think of it like this: Imagine a party list is a ladder. The voters decide who climbs the ladder and in what order. The "two-step" rule doesn't change the order of the climbers; it just changes how high the ladder is.

If the rule says the party gets 3 seats, the top 3 climbers get to sit on the council.
If the rule says the party gets 2 seats (because the big-picture math demanded it), the 3rd climber gets kicked off, even if they were still higher up the ladder than anyone else.

The paper shows that in Zurich, seven seats changed hands between parties. This meant fourteen candidates had their status changed (seven got elected, seven lost their seats). But only two of those seat swaps involved a "seat reversal." The other five swaps were just parties gaining or losing a seat while still staying in their usual order.

The "Vote Gap" Surprise

The author also looked at how close the races were for the people who got bumped. He measured the "vote gap"—the difference in votes between the person who got the seat and the person who just missed it.

He found a wild variety:

In one case (in a district called Dielsdorf), the difference was tiny: just 40 votes separated the winner from the loser. That's a razor-thin margin!
In other cases, the gap was huge—hundreds or even over a thousand votes.

This is important because it shows that the "two-step" rule can change the outcome for a candidate who was clearly the most popular in their party, not just for someone in a tight race. The rule changes the number of seats available, which moves the "finish line" up or down the ladder. If the finish line moves, the person who crossed it changes, regardless of how far ahead they were.

The Trade-Off: Fairness vs. Local Stability

So, was it worth it? The paper measures the "fairness" of the election using two famous math scores (the Gallagher index and the Loosemore–Hanby index).

Without the special rule, the unfairness score was 2.64 (Gallagher) and 4.81 (Loosemore–Hanby).
With the special rule, the scores dropped to 1.73 and 2.55.

The special rule made the election much fairer for the parties as a whole. But the price was that 7 out of 180 seats (about 3.9%) ended up being held by a different person than they would have been in a simple local election.

The author ran computer simulations to see if this was a fluke. He shook up the vote numbers slightly (like simulating a slightly different day) and found that getting 7 seat swaps is actually a very normal, average result. It's not a disaster; it's just how the math works. The "seat reversals" (the weird ones) were also normal, happening in about 2 districts on average in these simulations.

What This Means for You

The paper concludes that while "seat reversals" are the most dramatic and visible part of this system, they are not the whole story. The system changes the personal makeup of the parliament in many more ways than just those dramatic reversals.

It's a trade-off. You get a council that perfectly matches the total votes of the whole region, but you lose a little bit of the guarantee that the local neighborhood's specific ranking will always be respected. The paper doesn't say this is "bad" or "good," but it does say that voters and candidates need to understand that their local ranking is only part of the equation. The "global math" can adjust the finish line, changing who wins, even if the local voters didn't change their minds.

In short: The system works to make the big picture fair, but it does so by adjusting the finish line for individual candidates, sometimes in surprising ways that go far beyond the famous "seat reversals."

Technical Summary: Seat Reversals and Candidate Status Changes under Biproportional Representation

Problem and Context
Proportional representation (PR) systems typically evaluate outcomes at the party level, focusing on the correspondence between vote shares and legislative seat shares. However, when PR is implemented within territorial districts, aggregate disproportionality can arise even if every district uses a proportional formula, due to variations in district magnitude and the geographic distribution of support. Biproportional representation addresses this by constraining district-level allocations to satisfy canton-wide party entitlements while maintaining fixed district magnitudes.

While the mathematical literature acknowledges that this global correction can produce "discordant seat allocations" (where a party with more local votes receives fewer seats than a weaker party, known as seat reversals), existing research has largely focused on party-level outcomes. This article addresses a gap in understanding how these global constraints affect the personal composition of parliament in personalized list systems. Specifically, it investigates how changes in a party's district seat entitlement—induced by biproportionality—alter the electoral threshold within local candidate rankings, thereby changing which specific individuals are elected, even in the absence of a classic seat reversal.

Methodology
The study employs a mechanical counterfactual analysis of the 2023 Zurich cantonal election. The design compares the observed biproportional seat matrix against a hypothetical scenario where seats are allocated independently within each district using the Webster/Sainte-Laguë method.

Constants: The analysis holds constant district magnitudes, party votes, the legal list-group quorum (5% in a district or 3% canton-wide), candidate votes, and candidate rankings.
Variable: The only changing element is the constraint linking local allocations to canton-wide party entitlements.
Mechanism: The study isolates the immediate mechanical consequence of removing the upper-apportionment constraint. It does not estimate behavioral responses (e.g., how parties or voters would adapt to a different system).
Sensitivity Analysis: To assess the stability of the results, the author utilizes a Dirichlet-based local sensitivity simulation. This perturbs the observed party vote shares within each district across three concentration parameters ( $\kappa \in \{40, 80, 160\}$ ) to generate a benchmark distribution of seat transfers and discordant allocations.
Metrics: The study measures cell-level changes, matrix distance (L1 norm), candidate status changes, discordant seat allocations (seat reversals), and aggregate proportionality using the Gallagher and Loosemore–Hanby indices.

Key Contributions
The article makes four primary contributions to the literature on apportionment and personalized PR:

Extension to Candidate Outcomes: It extends the concept of discordant seat allocations from party-level outcomes to the personal composition of parliament.
Conceptual Distinction: It distinguishes three non-equivalent phenomena:
- Classic seat reversals: A pairwise discordance where a locally stronger party receives fewer seats than a weaker party.
- Cross-rule entitlement changes: A party receiving a different number of seats under biproportionality compared to independent district PR (which may or may not involve a reversal).
- Allocation-induced candidate status changes: The change in which individual occupies a mandate resulting from the shift in the party's district entitlement.
Sequential Mechanism: It identifies the sequential mechanism linking global constraints to local outcomes: global constraints alter district seat entitlements, which in turn shift the electoral threshold within pre-determined local candidate rankings. The study uses the "candidate-threshold margin" (the vote gap across the shifted rank boundary) to characterize whether a status change crosses a narrow or clearly differentiated point in the ranking.
Trade-off Measurement: It quantifies the exchange between aggregate proportionality and local candidate continuity, benchmarking both seat transfers and reversals under perturbations of the vote matrix.

Results
The analysis of the 2023 Zurich election yields the following empirical findings:

Seat Transfers and Status Changes: Fourteen of the 188 party–district cells differ between the biproportional and independent district PR allocations. These differences constitute seven seat transfers, resulting in fourteen candidate status changes (seven candidates gaining status, seven losing it). Consequently, seven of the 180 parliamentary seats (3.9%) would have been occupied by a different candidate under independent district PR.
Seat Reversals: Only two of the seven transfers produce classic pairwise seat reversals (in the districts of Hinwil and Winterthur-Land). In these cases, a locally weaker party receives more seats than a stronger party. The remaining five transfers change candidate outcomes without reversing the local ordering of parties by votes and seats.
Aggregate Proportionality: Biproportionality improves aggregate proportionality, reducing the Gallagher index from 2.64 to 1.73 and the Loosemore–Hanby index from 4.81 to 2.55.
Decomposition of Transfers: Six of the seven transfers are required to reconcile the aggregate party totals produced by independent district PR with the canton-wide entitlements. The seventh transfer is an additional territorial rearrangement necessary to distribute the corrected totals while preserving district magnitudes.
Threshold Gaps: The vote gaps across the shifted candidate thresholds vary substantially. Five changes involve transitions between zero and one seat (no interior threshold). For the remaining nine, the normalized vote gap ranges from 0.67% to 42.8% of the mean candidate votes on the list, with a median of 20.1%. This indicates that mechanically equivalent status changes can cross either narrowly tied or clearly differentiated ranking boundaries.
Robustness: The perturbation analysis shows that the observed seven transfers and two reversals are not unusually high under nearby vote configurations (falling between the 23rd and 41st percentiles of simulated distributions). However, the observed improvement in proportionality (0.91 point reduction in the Gallagher index) is comparatively modest, falling in the lower percentiles of simulated gains.

Significance and Claims
The paper claims to reveal a measurable trade-off between aggregate proportionality and local allocation continuity. It argues that the concept of "seat reversals" captures only a subset of the local consequences of biproportionality. The broader incidence of global proportionality includes candidate status changes that occur even when the local ordering of parties remains concordant.

The study clarifies that candidate votes determine the order of candidates within a list, but the biproportional matrix determines the number of mandates available to a party in a district. Therefore, the "electoral threshold" moves within a ranking that voters have already determined. The paper posits that while biproportionality improves aggregate proportionality, it does so at the cost of local allocation autonomy, where the election of a local candidate may depend on the distribution of party support across the entire canton rather than solely on local votes.

The author maintains a modest stance, noting that the study establishes the institutional structure of local allocation conditionality rather than observing behavioral accountability effects. The findings suggest that the local incidence of global proportionality is broader than previously recognized, extending from visible party-level reversals to the personal composition of parliament, even in cases where no party overtakes another in seat rankings.