The Price of Fairness: A Single Formula for How Many Seats Proportional Systems Actually Need

In Poland's 2015 election, a party calling itself a defender of proportional representation won a comfortable majority of seats in the Sejm with just 37.6% of the vote. PiS secured 51% of the parliament while fewer than two in five voters chose it. Under a system whose very name — proportional representation — promises that seats mirror votes, this is not a bug. It is a bias, and it is written into the mathematics.

That spectacular mismatch is the opening scene of a new mathematical analysis by Frederik Ravn Klausen of the University of Cambridge (Klausen, 2026). The paper asks a deceptively simple question: when a country divides itself into multi-member constituencies and hands out seats by the D'Hondt method — the most common proportional method in the world — how many extra "compensatory" seats does it need to add at the national level to guarantee that the final parliament actually matches the votes? The answer, it turns out, can be written in a single compact formula: about $\frac{c}{2}(n-\frac{1}{p})$ compensatory seats, where $c$ is the number of constituencies, $n$ the number of relevant parties, and $p$ the vote share of the largest party.

In plain words: half a compensatory seat per constituency, times a number that grows with the number of parties and the size of the biggest party. For Spain, that means roughly 75 extra seats. For Portugal, 40. For Turkey, 120. These aren't guesses; they fall out of logic, and the paper shows they hold up against two decades of real election data from around the world.

The Science

To understand the formula, you need to understand the quiet bias at the heart of D'Hondt, the method named for the Belgian mathematician Victor D'Hondt and used in Spain, Poland, Portugal, Turkey, Denmark, and dozens of other countries. D'Hondt divides each party's votes by a sequence of divisors (1, 2, 3, ...) and awards seats to the largest resulting quotients. It feels scrupulously fair. It is not.

In 1919, the Hungarian mathematician George Pólya proved that D'Hondt systematically favours large parties. A party with twice the votes gets more than twice the seats. Mathematically, D'Hondt is the apportionment method that rounds fractional seats strictly upward for the biggest party — a bias that never cancels out. Sainte-Laguë, a cousin method used across Scandinavia, is the asymptotically unbiased alternative.

Here is the crucial twist, and it is the paper's central insight: in a single constituency, this bias is small and mostly invisible. But a country like Poland has 41 constituencies, Spain has 52, Turkey has 87. The bias has the same sign everywhere — it always helps the large party — so it doesn't cancel. It stacks. Klausen decomposes the total "seat surplus" (a party's seats minus its proportional share of votes) into three ingredients:

$$\text{seat surplus}=\text{method bias}+\text{malapportionment}+\text{fluctuations}.$$

Method bias is the D'Hondt advantage; malapportionment occurs when some constituencies' seats are "cheaper" in votes than others (often rural seats that lag behind population shifts); fluctuations are the random luck of which party happens to win the outermost seat in each district. The first term is deterministic and one-directional. The last two tend to wash out or are modest. For D'Hondt, the bias dominates — and it accumulates across every constituency.

Against this backdrop, the paper builds a rigorous framework for $L^{\ast }$, the minimal number of compensatory seats needed to make a two-tier system "congruent." A two-tier system works like this: seats are first distributed regionally, constituency by constituency, then a national algorithm distributes seats again using the national vote totals, this time including the compensatory seats. If every party gets at least as many seats in the national distribution as it won regionally, the system is congruent — proportional representation is, in a precise logical sense, achieved. The first $L$ at which that becomes true is $L^{\ast }$.

What They Found

The headline result is a formula that can be derived from logic alone. When a two-tier system is built on local D'Hondt apportionment in multi-member constituencies, with $c$ constituencies and $n$ relevant parties (those above the electoral threshold, typically 5%), and a largest party holding vote share $p$, the number of compensatory seats needed is approximately:

$$L^{\ast }\approx \frac{c}{2}\left(n-\frac{1}{p}\right).$$

The intuition is elegant. Each constituency contributes, on average, about half a surplus seat to the largest party under D'Hondt. The term $n-1/p$ captures how much headroom the other parties need: more parties mean more claims to satisfy, and a bigger largest party means its bias accumulates faster. When $n$ is small and $p$ is large — a stable two-party landscape — few compensatory seats are needed. When fragmentation rises and the biggest party shrinks, the requirement climbs sharply. This is exactly the modern condition: in recent elections across Europe, the largest party has gotten smaller and the number of parties has ballooned, straining systems designed for an earlier, more stable era.

The formula's predictions are confirmed empirically across the Constituency-Level Elections Archive (CLEA), a dataset spanning decades of real elections worldwide. The paper averages $L^{\ast }/c$ (compensatory seats per constituency) against the "party surplus" $n-1/p$, and the real-world points hug the predicted line $f(x)=\frac{1}{2}x$ remarkably closely (

).

The logic also yields concrete, country-specific numbers for the four major one-tier D'Hondt systems — Portugal, Poland, Spain, and Turkey — none of which currently adds compensatory seats at all. If they chose to, the analysis says they would need about 40, 75, 75, and 120 respectively, assuming nothing else about their systems changes (Klausen, 2026).

A striking pattern emerges from the country-by-country table, which computes $L^{\ast }$ for the most recent election in each of twelve countries. Poland needs 57 compensatory seats on top of its 460 constituency seats; Spain needs 33 on top of 350; Turkey needs 85 on top of 600. By contrast, small and geographically compact systems need very little: Denmark needs 22, Norway 21, Sweden 62. The demand scales with geography — with the number of constituencies — because each constituency independently hands the largest party a small gift.

Compensatory seats needed for proportionality across 12 countries

Label	Value
Denmark	22
Latvia	6
Norway	21
Poland	57
Portugal	7
Spain	33
Sweden	62
Switzerland	20

shows the dramatic spread across the twelve countries.

There is a second, less obvious finding hiding in the data. The paper examines what happens when you vary not just the number of seats but the bias of the regional method itself, using a one-parameter family of divisor methods. Intuitively, you might expect the most unbiased regional method to minimize the need for compensatory seats. The opposite is true (

). The optimal regional method is not the unbiased Sainte-Laguë but a method with a slight bias toward larger parties. A little bias at the regional level is paradoxically helpful: it means the largest party has already "taken" its surplus seats regionally, leaving the national tier less to resolve. The Danish 2022 election, which was famously decided by a single incongruent seat, illustrates the mechanism beautifully (

).

Why This Changes Things

This matters because the strangest feature of the modern electoral landscape is that the systems' stress is growing just as the need for fairness is most acute. Sweden reformed its system in 2014 after experiencing incongruence in 2010. Germany's Bundestag swelled from 598 to 736 members as overhang seats piled up, forcing a 2023 reform. The 2022 Danish election hung on a single disputed seat. And Poland, Spain, Turkey, and Portugal — four countries whose democratic health is constantly scrutinized — run one-tier D'Hondt systems with built-in, quantifiable distortions, no compensatory tier at all.

The paper gives reformers a compass where previously there was only folklore. The electoral-reform literature has long floated "25% of seats" as a rule of thumb for compensatory tier size. Klausen argues this is the wrong way to think about the problem. In a first-past-the-post system (one seat per constituency), the percentage and the per-constituency number are the same thing. But in multi-member constituencies, the requirement scales with the number of constituencies, not the number of seats. A country with many small constituencies needs proportionally more compensation than a country with a few large ones, even at identical parliament sizes.

Consider Germany, where some have floated 60% compensatory seats as an empirical rule. Under this analysis, such a high number reflects not generosity but a structural fact: Germany has many small constituencies and many parties, driving $n$ up and $p$ down. The "25% rule of thumb" fails exactly where it is needed most — in fragmented, geographically fine-grained systems.

The paper also offers an intellectual bridge to a question that has quietly vexed electoral reform for a century: what does "proportional" even mean? The Gallagher index — the standard measure of disproportionality, computed as $\sqrt{\frac{1}{2}{\sum }_i(p_i-s_i/(k+L){)}^2}$ — is a continuous scale with no natural cutoff. The paper's definition of $L^{\ast }$ sidesteps the arbitrariness: a system is proportional when the national allocation can satisfy every party's regional wins, and not a moment before.

Gallagher disproportionality index falls linearly until L* is reached

Label	Value
0	8
5	7
10	6.2
15	5.3
20	4.4
22 (L*)	3.8

shows how the Gallagher index falls roughly linearly as compensatory seats are added, then hits a floor at $L^{\ast }$ — a natural definitional endpoint where rounding error, not bias, is all that remains.

What's Next

The honest caveats are built into the math. The formula is a promise about typical elections, not every election. The impossibility theorem for two-tier systems — a result that any electoral-system theorist will recognize — states that no finite number of compensatory seats can guarantee congruence for every possible election outcome. The paper's $L^{\ast }$ is the requirement for a given real election, and the $\frac{c}{2}(n-\frac{1}{p})$ law is its expected value across many. Countries that design for the average should keep enough headroom for the outliers, as Germany learned the hard way.

The analysis also leans on a 5% electoral threshold in much of its theoretical work, and it formally centers on the D'Hondt family. The paper extends its logic to the Droop Quota — a cousin of D'Hondt with a similar bias toward the largest party — deriving an analogous formula, while noting that for the unbiased Hare Quota and Sainte-Laguë, fluctuations (not bias) dominate and the clean law breaks down. Systems built on those methods need compensation for different reasons, and the math looks different.

What the paper opens up is not just better electoral law but a more honest public conversation about what proportional representation can and cannot promise. The Polish election of 2015 shocked the world not because the system malfunctioned but because it functioned exactly as designed. The mismatch between votes and seats was not corruption; it was arithmetic. By making that arithmetic legible — by giving citizens a formula they can hold up against their own parliament — Klausen's analysis does something unusual for mathematics: it hands ordinary voters a tool for political accountability.

The next step, the paper suggests, is to take this framework from the "whether" to the "how" — to grapple with incongruence-breakers, with how compensatory seats should be spread back down to constituencies, with how many seats become too many. Germany's experience shows the danger of letting the compensatory mechanism run unchecked. But the deeper invitation is to reconcile the two goods that electoral systems strain to hold together: geographic representation (my community has a voice) and numerical fairness (my vote counts equally). The formula $\frac{c}{2}(n-\frac{1}{p})$ is, in the end, a price tag for that reconciliation — a number that tells a country exactly what it costs, in seats, to make votes count.