Quota

I would like to break this page up into three or four pages (minimally: "Quota", Hare quota and Droop quota). I'm likely to be bold and just do it, but please leave comments on Talk:Quota if you object. -- RobLa (talk) 00:44, 5 December 2020 (UTC)

A quota is a number of votes (obtained by formula) often relevant to deciding who wins and how ballots are evaluated or modified in proportional voting methods.

The following quotas are listed from largest to smallest. See the PSC#Types of PSC article for more information.

Hare quota

The Hare quota may be given as:

${\frac {{\mbox{total}}\;{\mbox{votes}}}{{\mbox{total}}\;{\mbox{seats}}}}$

Where:

• ${\text{total votes}}$ = the total valid poll; that is, the number of valid (unspoiled) votes cast in an election.
• ${\text{total seats}}$ = the total number of seats to be filled in the election.

When there are 5 seats to be filled and 100 votes cast, the Hare quota is (100/5) = 20 votes.

In the single-winner case, a Hare quota is just all of the voters. In general, voting methods that are based on Hare quotas attempt to represent all voters, but don't guarantee that a majority of voters will get even half of the seats.

Droop quota

Sources differ as to the exact formula for the Droop quota. As used in the Republic of Ireland the formula is usually written:

$\left({\frac {\text{total valid poll}}{{\text{seats}}+1}}\right)+1$

but more precisely

$\operatorname {Integer} \left({\frac {\text{total valid poll}}{{\text{seats}}+1}}\right)+1$

where:

• ${\text{total valid poll}}$ = Total number of valid (unspoiled) votes cast in an election.
• ${\text{seats}}$ = total number of seats to be filled in the election.
• $\operatorname {Integer} ()$ refers to the integer portion of the number, sometimes written as $\operatorname {floor} ()$

One reason Droop quotas are used more often than Hare Quotas for ranked PR methods is because not only do they often help reduce the amount of vote-counting necessary, but they almost entirely eliminate the possibility of a majority of voters receiving a minority of seats compared to Hare Quotas. The Droop Quota is the smallest possible quota that guarantees that there will be as many quotas as there are winners desired.

When there are 5 seats to be filled and 100 votes cast, the Droop quota is 17 votes, which is calculated as: Integer((100/(5+1)) + 1) = Integer((100/6) + 1) = Integer(~16.667 + 1) = Integer(~17.667) = 17 votes.

In the single-winner case, a Droop quota is a majority. In general, Droop quota-based methods tend to leave at least just under a Droop quota unrepresented. See the utility article, as the debate between Hare and Droop quotas somewhat parallels and generalizes the utilitarianism vs. majority rule debate.

Hagenbach-Bischoff quota

The Hagenbach-Bischoff quota (HB quota) (known by a few other names as well) is:

$\left({\frac {\text{total valid poll}}{{\text{seats}}+1}}\right)$

Some sources call the HB Quota a Droop Quota instead. There will always be exactly one more HB quota than seats to be filled. Because of this, it will on rare occasion be necessary to break a tie between various candidates to decide who should win with PR methods that use the HB quota.

When there are 5 seats to be filled and 100 votes cast, the HB quota is (100/(5+1)) = ~16.667 votes.

In the single-winner case, an HB quota is half of the voters. In this case, two candidates could each have half of the votes, i.e. two candidates each have one quota, but only one seat can be allotted. Because of this, many PR methods that use HB quotas specify that a candidate must have more votes than k HB quotas to get k seats (i.e. over half of the votes, in the single-winner case).