A bigger house, one seat fewer
Five towns share the seats of a council in proportion to their populations. Nobody moves, nobody is born, nobody dies — the only thing that changes is that the council decides to seat one more person. Under the most obvious rule there is, the rule anybody would write down first, that can take a seat away from a town. It happens three times in the first sixty council sizes here, and it is not a rounding curiosity: it is one half of a theorem that says you have to give up something, and the other half is waiting on the other side of the page.
The obvious rule
A town with a share s of the population deserves s × h of the h seats. That number is its quota, and it is almost never a whole number. So: give every town the whole part of its quota, count how many seats are left over, and hand those to the towns with the largest fractions. Every town gets its quota rounded down or rounded up, which is exactly as fair as whole seats allow. This is Hamilton's method, and it is what almost everybody invents when asked.
Every council size at once
Each line is one town's seats, plotted against the size of the council. The lines climb, because a bigger council has more to give out — except where they do not. A ringed step is a town that lost a seat when the council gained one.
These five populations are chosen, not found: they are round hundreds adding to a hundred thousand exactly, and I searched for a split that shows the loss more than once inside a council small enough to draw. A reader is owed that, because a picture with three of anything in it invites you to think three is normal. How often it really happens is two figures down, and the answer there is measured rather than chosen.
The smallest example there is
Three towns, of 3, 3 and 1 people. Seven people, and that is the whole of it.
That is not a small example I happened to like. Searching every
population of two, three, four or five towns adding to twelve people or
fewer, at every council size up to thirty, and throwing out any allocation
that needed a tie broken — seven people is the smallest total that can do
it at all, and 3, 3, 1 is the only split of seven that does. The
search runs on this page; tests/test_apportionment.py runs it
again in Python, over a wider net, from a different direction.
How often it happens
Populations drawn at random, councils swept from one seat per town up to two hundred and forty, and the question asked at every step: did anybody lose a seat? With two towns it can never happen. It rises with the number of towns and then flattens out — which is the honest shape of it, and it is a long way from rare.
The populations come from a seeded generator, named on the page, so this figure is the same every time it is drawn — and the test file rebuilds that generator in Python and replays it. A measurement you cannot repeat is an anecdote.
And it is not really about the house
Hold the council still at one size and let the populations move instead. Nobody adds a seat; people are born and people leave. The same rule can then take a seat off a town that grew and hand it to one that did not gain a single person.
That is the population paradox, and it is searched for here in its strong form — grew against did not grow — because that form needs no argument about percentages: either the seat went to the town that gained people or it did not. Sorted by the population before the change, then after it, then by the size of the council, over every population of two to four towns totalling twelve people or fewer before and sixteen or fewer after, at every council up to twenty.
The rule that cannot do it
There is a family of methods that cannot lose a seat, and the reason is worth more than the proof. Give the seats out one at a time: each seat goes to the town with the highest priority, and a town's priority falls as it collects seats. Then the council of h+1 seats is the council of h seats plus one more — the first h seats are handed out in the same order whatever h is, so nothing that has been given can be taken back. That is the whole argument, and this is the ladder it describes.
Choosing how fast priority falls gives you the named methods: D'Hondt divides by the seats already held plus one and quietly favours large towns; Sainte-Laguë divides by twice-the-seats plus one, which is rounding to the nearest whole seat and is the even-handed one; Huntington–Hill, the method the United States House has used since 1941, divides by the geometric mean of the seats held and one more, and hands every town a seat before it hands anybody a second.
What you gave up to get it
Nothing is free. A town of 4 people beside two towns of 1, on a council of 3 seats: its quota is exactly 2 seats and D'Hondt gives it all three. Priority handed out the first seat at 4, the second at 4/2 = 2, the third at 4/3 — and every one of those beat the small towns' 1, so the small towns got nothing.
The Hamilton column beside it is there for contrast and is the one number on this page settled by a convention rather than by arithmetic: towns B and C have the same population and the same half-seat quota, and one of them gets the spare seat because the tie-break says so. That is why both searches on this page throw a tie-broken allocation away — D'Hondt's answer here needs no such coin, and a smallest example that depended on one would be a fact about the coin.
Hamilton's method can never do that: every town gets its quota rounded down or up, by construction. Divisor methods can never lose a seat, by construction. Balinski and Young proved in 1982 that no method can do both — any rule that always stays inside quota can be made to take a seat away by growing the house, and any rule that never takes a seat away can be pushed outside quota. That theorem is quoted here, not derived; it is the one claim on this page that the page cannot check. What it can check, and does below, is that each half of the trade is real on numbers you can count on your fingers.