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Seat Apportionment, Explained

After reading this you will be able to turn a column of party vote counts into a seat allocation under five different rules, and explain exactly why the same votes give different seats.

What seat apportionment does

You have vote counts for each party and a fixed number of seats. You need whole seats, not fractions. A party with 34% of the vote in a 12-seat chamber has earned 4.08 seats on paper, but nobody sits in 0.08 of a chair. Apportionment rules decide who gets rounded up and who gets rounded down.

The five rules on this page split into two families. Highest-averages methods (D'Hondt, Sainte-Laguë, and modified Sainte-Laguë) hand out seats one at a time, always to the party with the largest current quotient. Largest-remainder methods (with the Hare or Droop quota) give each party its whole quotas first, then distribute leftover seats by the size of the fractional remainders.

The hook: with 855,000 votes over five parties and 12 seats, D'Hondt gives the largest party 5 seats while Hare largest remainder gives it 4. Same votes, same seat total, one seat moves. The rest of this article shows you why.

When to use each rule

The choice is political, not merely arithmetic, so match the rule to the goal.

D'Hondt
Use when you want to reward larger parties and encourage stable majorities. It divides by 1, 2, 3, 4, ... and consistently rounds in favour of big parties.
Sainte-Laguë
Use when you want the closest thing to strict proportionality. It divides by 1, 3, 5, 7, ... and treats large and small parties almost even-handedly.
Modified Sainte-Laguë
Use when you like Sainte-Laguë but want to keep the smallest parties from grabbing a seat on a tiny vote. The first divisor becomes 1.4 instead of 1, so the first seat is harder to win.
Largest remainder (Hare)
Use when you want the plain intuition of "one quota per seat" and are comfortable with the quota paradoxes it can produce.
Largest remainder (Droop)
Use when you want a largest-remainder method that leans slightly toward larger parties, as it does in single transferable vote counts.

Do not use any of these for ranked ballots or head-to-head preferences. Those need a different toolset.

The highest-averages formula

Every highest-averages method builds a table of quotients. For party with votes v_i, the quotient at seat number s is:

q_{i,s} = \frac{v_i}{d_s}

Here v_i is the party's vote count and d_s is the divisor for the s-th seat. The only difference between the methods is the divisor sequence:

  • D'Hondt: d_s = s, so 1, 2, 3, 4, ...
  • Sainte-Laguë: d_s = 2s - 1, so 1, 3, 5, 7, ...
  • Modified Sainte-Laguë: same but the first divisor is 1.4, so 1.4, 3, 5, 7, ...

You compute every party's quotient for its next seat, hand the seat to the largest quotient, then recompute only that party's next quotient. Repeat until the seats run out.

The largest-remainder formula

Largest remainder starts with a quota, the number of votes one seat is "worth". Two quotas are common:

Q_{\text{Hare}} = \frac{V}{S}, \qquad Q_{\text{Droop}} = \left\lfloor \frac{V}{S + 1} \right\rfloor + 1

Here V is the total valid vote and S is the number of seats. Each party gets \lfloor v_i / Q \rfloor whole seats. Add up those whole seats: some seats are usually left over. Rank the parties by the fractional part of v_i / Q and give the leftover seats one each to the largest remainders.

The Droop quota is smaller than the Hare quota, so parties clear their whole quotas more easily and fewer seats are left to the remainder stage. That small shift is what makes Droop mildly friendlier to larger parties than Hare.

A worked example with the demo data

Five parties, 12 seats, 3% threshold

The demo data: Red 340,000, Blue 280,000, Green 160,000, Yellow 60,000, Purple 15,000. Total valid vote is 855,000. Seats: 12. Threshold: 3%.

  1. Apply the threshold first. 3% of 855,000 is 25,650. Purple has 15,000 votes, which is 1.75%, so Purple is removed before any seat is allocated. The effective total for the methods stays at 855,000 for quota purposes on this tool, but Purple wins zero seats.
  2. Run D'Hondt. The first quotients are the raw votes: Red 340,000 wins seat 1. Divide Red by 2 (170,000); Blue 280,000 now leads and wins seat 2. Continue. Red's sequence is 340,000, 170,000, 113,333, 85,000, 68,000. Blue's is 280,000, 140,000, 93,333, 70,000. Green's is 160,000, 80,000, 53,333. Yellow's is 60,000, 30,000. Taking the 12 largest quotients across all parties gives Red 5, Blue 4, Green 2, Yellow 1.
  3. Run Sainte-Laguë. Divisors are 1, 3, 5, 7, 9. Red: 340,000, 113,333, 68,000, 48,571. Blue: 280,000, 93,333, 56,000, 40,000. Green: 160,000, 53,333, 32,000. Yellow: 60,000, 20,000. The 12 largest give Red 4, Blue 4, Green 3, Yellow 1.
  4. Run Hare largest remainder. Quota is 855000 / 12 = 71250. Whole quotas: Red 4.77 gives 4, Blue 3.93 gives 3, Green 2.25 gives 2, Yellow 0.84 gives 0. That is 9 seats placed, 3 left. Remainders are Red 0.77, Blue 0.93, Green 0.25, Yellow 0.84. The three largest remainders (Blue 0.93, Yellow 0.84, Red 0.77) each take a seat: Red 5, Blue 4, Green 2, Yellow 1.

Notice how Green wins 3 seats under Sainte-Laguë but only 2 under D'Hondt and Hare. Green is the mid-size party that D'Hondt squeezes and Sainte-Laguë protects.

Green gains the disputed seat under Sainte-Laguë at Red's expense; D'Hondt and Hare agree here.

With the demo votes and 12 seats, changing the divisor rule moves one seat between Red and Green. D'Hondt gives Red 5 and Green 2; Sainte-Laguë gives Red 4 and Green 3. The modified 1.4 divisor matches pure Sainte-Laguë here because every party already clears its first seat comfortably.

Reading the results

Look past the seat counts to the effective votes per seat, which is a party's votes divided by the seats it won. Under D'Hondt in the example, Red has 340000 / 5 = 68000 votes per seat while Yellow has 60000 / 1 = 60000. A lower number means the party got more seats per vote, so Yellow is slightly "cheaper" per seat here. Under Sainte-Laguë, Green's figure is 160000 / 3 \approx 53333, the cheapest of all, which is the proportionality that Sainte-Laguë buys.

When two methods disagree, the disputed seat almost always sits between a large and a mid-size party. Track that one seat rather than staring at the whole table.

Common mistakes

Do not apply the threshold after allocating seats. The threshold removes parties from the count first. In the example, Purple's 15,000 votes are gone before seat 1 is handed out, so Purple cannot win a leftover seat even if its remainder would have been large.

A second trap is expecting largest remainder to be monotone. It is not. Adding seats to the chamber can, in rare cases, cost a party a seat under Hare (the Alabama paradox). Highest-averages methods never do this, which is one reason many parliaments prefer them.

A third: rounding the votes-per-seat figure and then comparing. Compare the raw quotient. The difference between 68,000 and 68,001 can decide the last seat, so let the tool keep full precision and only round for display.

Related tools

Seat apportionment assumes you already have a vote count per party. If instead you have ranked ballots and want to compare outcomes under plurality, instant-runoff, Borda and Condorcet, use the Voting Method Calculator. That tool answers "who wins" from preferences; this one answers "how many seats" from totals.

Frequently asked questions

Why does D'Hondt favour large parties?

Because it divides by 1, 2, 3, a large party's quotients stay high for longer. In the example Red's fifth quotient (68,000) still beats Green's third (53,333), so Red takes a fifth seat that pure Sainte-Laguë hands to Green.

Is Sainte-Laguë always perfectly proportional?

No. It is the closest of these rules on average, but whole seats can never match fractions exactly. With 12 seats Red's exact share is 4.77, and Sainte-Laguë rounds it to 4, not the 5 that D'Hondt gives.

What is the point of the 1.4 first divisor?

It raises the bar for a party's very first seat. Dividing by 1.4 instead of 1 lowers a small party's opening quotient by about 29%, so a party needs materially more votes to break in. Once a party holds one seat the sequence rejoins the normal 3, 5, 7.

When do Hare and Droop give different answers?

When the number of leftover seats after the whole-quota stage differs. Droop's smaller quota lets parties clear more whole seats up front, leaving fewer seats for the remainder step and often shifting one seat toward a larger party.

Does the threshold change the quota?

On this tool the excluded party's votes are removed from contention for seats, but you should read the effective votes per seat for the parties that remain, since those are the numbers that decide the outcome.

Try it yourself in the tool

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