Ranked-choice or preference voting requires each winning candidate to meet a quota (threshold) of votes, based on the number of seats and the number of ballots. There are numerous possible quotas. These are definitions of a few (where Q = quota; S = number of seats; B = number of ballots):
The Droop quota is a quota widely used in public elections. Participation of a large number of voters reduces to insignificance its practice of rounding upward to an integer. Yet, in a small electorate, such as a membership organization or committee, rounding can change the results of an election. Wikipedia claims, "The Droop quota is smaller than the Hare quota. . . ." That can only be true if there are more than twice as many ballots as seats. Furthermore, when the number of ballots is smaller than the square of the number of seats, the Droop quota may be larger than the Hare quota.
Here are a few examples (out of thousands) where the Droop quota exceeds the Hare quota:
| Ballots | Seats | Hare quota | Droop quota | Excess of Droop over Hare | Percent by which Droop exceeds Hare |
|---|---|---|---|---|---|
| 8 | 3 | 2.67 | 3 | 0.33 | 13% |
| 10 | 4 | 2.50 | 3 | 0.50 | 20% |
| 12 | 5 | 2.40 | 3 | 0.60 | 25% |
| 13 | 5 | 2.60 | 3 | 0.40 | 15% |
| 18 | 5 | 3.60 | 4 | 0.40 | 11% |
| 14 | 6 | 2.33 | 3 | 0.67 | 29% |
| 15 | 6 | 2.50 | 3 | 0.50 | 20% |
| 16 | 6 | 2.67 | 3 | 0.33 | 13% |
| 21 | 6 | 3.50 | 4 | 0.50 | 14% |
| 22 | 6 | 3.67 | 4 | 0.33 | 9% |
| 28 | 6 | 4.67 | 5 | 0.33 | 7% |
The people who distribute OpenSTV define the Droop threshold (quota) as "the smallest number such that any candidate receiving this number of votes is guaranteed to be elected". It can be no such thing if the Hare threshold (quota) is smaller and guarantees election. In fact, any threshold higher than the Hagenbach-Bischoff quota gurantees election to candidates who reach it.
Because it is the highest number that can produce more winners than needed, we recommend that the Hagenbach-Bischoff quota (Q=B/(S+1)) be employed as the largest number of votes that does not score a win, that it be the threshold a candidate must exceed for election.
Questions? Comments? Send mail to aloe@rev.net.
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Last revised: 17 December 2011
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