Remainders

Every seat-allocation problem starts with a quota — some number expressing how many people (or votes) each seat is meant to represent. Divide each party's or state's share by the quota, and you get a number almost never divisible exactly. A delegation worth 4.73 seats cannot send four-and-three-quarters of a person. Something has to round.

The largest-remainder method handles this in the most transparent way imaginable. Each claimant first receives the whole-number part of its quota — the lower quota — and the fractional part is recorded as a remainder. Any seats still unallocated (there will be some, because the lower quotas do not add up to the chamber size) are then handed one at a time to whichever claimants have the largest remainders, in descending order, until the house is full.

The appeal is obvious. The rule is easy to explain, easy to audit, and it has a visible fairness property: no claimant ends up with fewer seats than its strict mathematical entitlement rounded down, nor more than that entitlement rounded up. In apportionment jargon, the method satisfies quota, meaning outcomes are always within one seat of the exact share. Methods that divide and round by formula — divisor methods — can and sometimes do violate quota, awarding a claimant a seat outside that range.

What the remainder method buys in transparency, though, it pays for in instability. The fractional parts interact in ways the simple rule conceals. Add one more seat to the chamber, and the quotas all shift slightly; the ranking of remainders can flip, and a claimant that gained a seat under the old size may lose one under the new. This is not a theoretical curiosity — it is the Alabama paradox, documented in the United States in the 1880s when a routine calculation showed that Alabama would drop from eight congressional seats to seven if the House were enlarged from 299 to 300. The discovery eventually drove American apportionment away from remainder methods entirely.

There is also a population paradox: one claimant's population can grow faster than another's in relative terms, and yet its remainder can shrink, costing it a seat while the slower-growing claimant gains one. The method contains no safeguard against this because it works snapshot-by-snapshot; it cannot see rates of change.

Remainder methods remain in use for legislative elections in many countries — they are common in proportional representation systems that allocate seats party-by-party from regional vote totals. There, the paradoxes are real but usually small in magnitude. For the higher-stakes task of apportioning a fixed chamber among units whose populations shift every decade, most systems have concluded that the instabilities are too sharp a price for elegance.