Largest remainder

Every proportional system has to solve the same basic problem: dividing a finite number of seats among parties whose vote shares do not come out to whole numbers. The largest remainder method attacks this in two steps. First, divide each party's vote total by a quota — a number representing what one seat is notionally "worth" — to get an automatic entitlement. Take the whole-number part of each result and award those seats immediately. Second, count what is left over: the fractional parts, ranked from largest to smallest. Distribute the remaining seats one by one down that list until the chamber is full.

The appeal is transparency. Every voter can see that a party needed, say, 50,000 votes for a full seat, received 2.7 quotas' worth of votes, was immediately awarded two seats, and then competed on the strength of its remainder — 0.7 of a quota — against every other party's remainder. No mysterious divisors, no successive rounds of division. One quota, one ranking, done.

What makes the method contentious is the choice of quota, because that choice is not neutral. The two most common variants are the Hare quota and the Droop quota, and they point in systematically different directions.

The Hare quota divides total valid votes by the exact number of seats available. It is the natural, intuitive choice and is used, for instance, in elections to the lower houses of several countries including Namibia and pre-reform electoral systems in a number of Eastern European states. Its weakness is that it tends to leave large remainders distributed among many parties, which means smaller parties can collect remainder seats even when their overall vote share is modest. The Hare quota therefore leans toward smaller parties relative to larger ones.

The Droop quota — total votes divided by the number of seats plus one, with one added to the result and any fraction dropped — was originally developed for single-transferable-vote counting but has been adapted for list systems. Because the Droop quota is smaller than the Hare quota, each party's initial entitlement is a larger whole number, fewer seats go to the remainder round, and large parties tend to do slightly better. The Droop quota's origins in multi-seat preference counting reflect a different objective — eliminating the possibility of a majority party failing to win a majority of seats — but its arithmetic carries over.

A third option, the Imperiali quota — votes divided by seats plus two — goes further still toward advantaging large parties, to the point where the initial entitlements can sum to more than the chamber has seats, making it mathematically unstable without a fallback rule.

The largest remainder method carries a structural peculiarity worth understanding: adding seats to the chamber does not guarantee that every party's seat count stays the same or rises. A party holding two seats in a 100-seat chamber might find itself holding only one in a 101-seat chamber, because the addition of a seat changes every party's remainder simultaneously and in ways that cannot be predicted without working through the full calculation. This is a specific instance of the Alabama paradox, first discovered in the context of apportioning seats among American states in the 1880s. The paradox is inherent to the remainder logic, not a bug in any particular implementation.

The highest-averages methods — D'Hondt, Sainte-Laguë — avoid the Alabama paradox by abandoning quotas and remainders altogether, trading one set of biases for another. Neither family of methods is strictly fairer; they embody different judgements about what proportionality should mean. Knowing which quota a legislature has written into its electoral law, and why, is part of reading any proportional result accurately.