Seats and Quotas

Quota Notes

It is surprising that some people get heated about which quota is “correct”. There are a half dozen in use and, except for the most bizarre, the choice makes little difference. The number of seats effects the accuracy of proportional representation more than the quota rule does. Recall that 5-seat PR districts create fewer “wasted votes” than 3-seat districts. That means fewer votes go to parties (or people) who fail to win a seat, and each party's share of seats is closer to its share of votes. Give the council enough seats so the quota is less than the voters in each significant faction.

(Many organizations find it hard to find enough skilled candidates to fill a 5-seat council – not to mention hard to reward the council members for long hours of study and meetings.)

A ballot with few preferences marked might have all of those candidates eliminated. The ballot may be treated as a tie for all remaining candidates, giving each of them a fraction of its weight; or the ballot and its weight may be ignored and the quota recalculated. A dynamic simple quota is Quota = (Weights Left / Seats Left) -0.01.

A Brief History of Quotas
Toward Inclusive Democracy

Plurality rule can leave a large majority of the voters unrepresented, as in the 3-way Korean election of 1987. Majority rules can leave close to half unrepresented, as is common in close, 2-way races.

Thomas Hare developed STV to end such failures by older rules. Hare's simple quota of ballots needed to win a seat made every vote count:
Quota = Voters / Seats.
If there are 100 voters and 5 seats, the quota is 20 and all 100 ballots are needed to elect 5 reps. This is called either the simple quota or the Hare quota. Sometimes the last candidate eliminated is the political opposite of the last candidate elected. Ballots then are forced to transfer to a very low preference or, if that preference is not marked, the ballot cannot be transferred, is thrown out and the last rep is elected with less than a quota. Critics found that simple quota can under represent a majority as shown in case 1 below.

Henry Richmond Droop designed a quota to avoid under representing a majority.
Quota = (Voters / Seats + 1) + 1 vote.
(The term "+ 1 vote" avoids a tie for the last seat.)
This quota has been refined further by Newland and Britton and by Irwin Mann. Most jurisdictions that use STV use one of these three quotas.

Critics point out that, unlike Hare's simple quota, they are designed to leave 1 quota of voters with no rep. The failure of Hare quota to empower the majority can lead to more serious political turmoil than the failure of Droop quota to represent a minority. But Hare's failure may be unlikely and Droop's failure almost certain in a particular electorate.

The easy quota, (Votes+1) / (Seats+1), is easy to remember. It is between Droop and Hare. So it does not fail in quota-borderline cases. (Such cases can make Droop or Hare elect too many or too few winners for the majority. Tideman, Dr. L. Bruce Anderson and others offer many examples to show how this happens. This danger led to the invention of better quota formulas.)

LERa may resolve the debate about which quota is best. It can use Droop quota for the chairperson and can use remaining voters / remaining seats to calculate the Hare quota for other seats. LERa then represents the majority properly while making every vote count. Thus LERa can get the best of both quotas.

Case 1, an election to fill 3 seats.

The candidates were A, B, R, and S and 36 ballots were cast.

Numbers of voters      12       7       9       8
Their preferences       AB     BA     RS     SR

Thus 19 voters, a majority, prefer A and B while 17 prefer R and S.

STV

The Hare quota is 36/3=12 votes to win a seat.
With that quota A is elected. No one else has the required quota so the weakest, B, is eliminated. Then R and S are elected. Thus the majority group with 19 voters who prefer A and B get only 1 rep while the 17 who prefer R and S get 2 reps.

The Droop quota is almost 36/4=9 votes.
With more than Droop quota, A is elected and her 3 excess votes transfer to B, who then has 10 and is elected. Then S is eliminated and R is elected. Thus Droop quota protects the majority's right to a majority of the reps.

LERa

The Condorcet winner in Case I (B) had fewer firsts (7) than the last candidate eliminated (S had 8). That causes LERa to switch to LERb if there are more than 3 seats or else to STV. The switch is needed to prevent the problem shown in Case 2. The majority voters gave second rank to the party leader, C, making her the Condorcet winner and chairperson. She is immune from elimination even though she has no firsts. Whether by chance or conspiracy, this shuts out the large minority group.

[The center is nearly always contested by 2 or more candidates. But in many organizations it is not divided or polarized by 2 factions. The voter preferences are more likely to run;