Largest remainder method

The largest remainder method is one way of allocating seats proportionally for representative assemblies with party list voting systems. It is a contrast to the highest averages method.

The largest remainder method requires the number of votes for each party to be divided a quota representing the number of votes required for a seat, and this gives a notional number of seats to each, usually including an integer and either a fraction or alternatively a remainder. Each party receives seats equal to the integer. This will generally leave some seats unallocated: the parties are then ranked on the basis of the fraction or equivalently on the basis of the remainder, and parties with the larger fractions or reaminders are each allocated one additional seat until all the seats have been allocated. This gives the method its name.

There are several possibilities for the quotas. The most common are:
the Hare quota which is ; and
the Droop quota which is .

The Hare quota tends to be slightly more generous to less popular parties and the Droop quota to more popular parties. Which is more proportional depends on what measure of proportionality is used.

Examples of Hare Quota and Droop Quota in an election to allocate 10 seats

Hare Quota

Party	Yellows	Whites	Reds	Greens	Blues	Pinks	Total
Votes	47,000	16,000	15,800	12,000	6,100	3,100	100,000
Seats							10
Hare Quota							10,000
Votes/Quota	4.70	1.60	1.58	1.20	0.61	0.31
Automatic seats	4	1	1	1	0	0	7
Remainder	0.70	0.60	0.58	0.20	0.61	0.31
Highest Remainder Seats	1	1	0	0	1	0	3
Total Seats	5	2	1	1	1	0	10

Droop quota

Party	Yellows	Whites	Reds	Greens	Blues	Pinks	Total
Votes	47,000	16,000	15,800	12,000	6,100	3,100	100,000
Seats							10
Droop Quota							9,091
Votes/Quota	5.170	1.760	1.738	1.320	0.671	0.341
Automatic seats	5	1	1	1	0	0	8
Remainder	0.170	0.760	0.738	0.320	0.671	0.341
Highest Remainder Seats	0	1	1	0	0	0	2
Total Seats	5	2	2	1	0	0	10