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Single Transferable Voting - Max does the math
11-12-03

From Max Blanchet, KPFA area

CHOICE VOTING OR STV IN ACTION

After reviewing the descriptions of STV in action available on certain websites, I have concluded that they leave much to be desired due to a lack of clarity and/or completeness. I decided, therefore, to write my own description of the process in order to help those individuals who might be interested in more than a cursory description of the workings of STV.

I have chosen as an illustration an election in which voters cast 30 valid ballots to elect 3 candidates out of a field of 5 candidates. The results are illustrated in attached Table I. The candidates - A through E - are listed horizontally (light gray) and the ballots are listed vertically and numbered 1 through 30 (dark gray.) In this particular case, all voters rank each candidate 1 through 5 - ranking the candidates is a key feature of Choice Voting whose importance becomes clearer as the process unfolds -- although in practice not all candidates would necessarily be ranked 1 through 5.

Droop Quota

To arrive at this quota, which is the threshold that a candidate must meet in order to be elected, the following formula is used:

Votes required to be elected =

{Number of Valid Ballots Cast/(Number of Seats to be filled+1)}+1

Furthermore, the result must be rounded down to the next integer.

To illustrate the formula, let us say that the voters were electing one candidate to one seat. The formula would yield:

{30/(1+1)}+1 = (30/2)+1 = 15+1 = 16

This last result is obvious as it corresponds to the 50% plus 1 vote formula we are all familiar with.

In the current case in which the voters are filling 3 seats, the formula yields: {30/(3+1)}+1 = (30/4)+1 = 7.5 +1 = 8.5

Rounded down to the next integer, this yields a quota of 8 votes (crosshatched.)

In the coming elections at KPFA, if one assumes that one third of the 30,000 listener-members were to cast valid ballots, the threshold for each of the 18 seats to be filled would be:

{(1/3x30,000)/(18+1)}+1 = (10,000/19)+1 = 526.32 + 1 or 527.

Table II is the same as Table I except that Candidate A's ballots are ordered (crosshatched horizontally,) such that first place ballots are listed first, second place ballots follow, and so on.

The table also lists for each candidate how many first place ballots each received. In this instance A got 12 such ballots, B got 5 ballots, C got 3 ballots, D got 5 ballots and E got 5 (crosshatched vertically.)

A, and only A, is elected in the first round since he or she got 12 first place ballots, or more than the threshold of 8 votes. A's surplus of 4 (12-8) votes - and this is the essence of STV - can now be transferred to the other candidates and help them get elected.

This is done in proportion to the number of second place ballots that the others received. Only A's first place ballots are considered in this step (thus the need to reorder Table II in the first place):

 Candidate Second Place Ballots Transferred Vote B 0 0 C 5 (5/12) x 4 = 1.667 D 2 2/12) x 4 = 0.667 E 5 (5/12) x 4 = 1.667 Total 12 4.000

The recalculated first place ranking becomes (crosshatched obliquely):

 Initially Adjustment Recalculated A 12 - 4 8 B 5 0 5 C 3 + 1.667 4.667 D 5 + 0.667 5.667 E 5 + 1.667 6.667

According to this new distribution, no other candidate has met the quota. C can, however, be eliminated since it has the fewest first place votes, which can now be transferred to B, D and E.

Final Steps

Table III is the same as Table II except that Candidate C is listed first (crosshatched horizontally.) Of C's 3 first place ballots, all list candidate D in second place. Thus, D gets all of C's transferred votes. D's first place tally goes from 5.667 to 10.333 (5.667+4.667) and D is now elected since it now exceeds the quota.

D's surplus of 2.333 votes (10.333-8) can be transferred to B and E in similar fashion. According to Table III, they must all go to E, which now has 9 first place votes. E is therefore the third winner and B is eliminated (crosshatched vertically.)

In a typical election in which each voter votes for one person only - presumably the person who is selected first in the case illustrated above - Candidate A would be elected outright and a second round would be needed to choose among Candidates B, D and E.

Max Blanchet
November 12, 2003

 TABLE I -- Thirty Ballots Cast to Elect 3 of 5 Candidates Voter Candidate A Candidate B Candidate C Candidate D Candidate E 1 1 3 2 4 5 2 5 4 3 2 1 3 4 3 2 1 5 4 3 1 2 4 5 5 2 1 3 5 4 6 1 3 2 5 4 7 1 5 3 4 2 8 2 3 4 5 1 9 3 4 5 1 2 10 4 5 1 2 3 11 5 1 2 3 4 12 1 5 3 4 2 13 4 3 2 5 1 14 3 4 5 1 2 15 4 5 1 2 3 16 5 1 2 3 4 17 1 4 3 2 5 18 1 3 2 5 4 19 1 5 4 3 2 20 1 3 2 4 5 21 1 3 2 5 4 22 1 5 4 3 2 23 3 4 5 2 1 24 1 5 4 2 3 25 1 4 5 3 2 26 5 1 2 3 4 27 5 4 3 2 1 28 5 3 4 1 2 29 5 4 1 2 3 30 4 4 3 1 2 Droop Quota 8

 TABLE II -- Same as Table I Except That Candidate A's Ballots Are Ordered Voter Candidate A Candidate B Candidate C Candidate D Candidate E 1 1 3 2 4 5 6 1 3 2 5 4 7 1 5 3 4 2 12 1 5 3 4 2 17 1 4 3 2 5 18 1 3 2 5 4 19 1 5 4 3 2 20 1 3 2 4 5 21 1 3 2 5 4 22 1 5 4 3 2 24 1 5 4 2 3 25 1 4 5 3 2 5 2 1 3 5 4 8 2 3 4 5 1 4 3 1 2 4 5 9 3 4 5 1 2 14 3 4 5 1 2 23 3 4 5 2 1 3 4 3 2 1 5 10 4 5 1 2 3 13 4 3 2 5 1 15 4 5 1 2 3 30 4 4 3 1 2 2 5 4 3 2 1 11 5 1 2 3 4 16 5 1 2 3 4 26 5 1 2 3 4 27 5 4 3 2 1 28 5 3 4 1 2 29 5 4 1 2 3 Droop Quota 8 Candidate A Candidate B Candidate C Candidate D Candidate E Total 1s 12 5 3 5 5 Accordingly, A is elected and its surplus (4) will be transferred to B, C, D and E. This will reflect how the voters who ranked A number 1 ranked B, C, D and E number 2. See next line. Total 2s n/a 0 5 2 5 Adjusted 1s 8 5 4.667 5.667 6.667 According to the above, no other candidate has met the quota; C can, however, be eliminated since it has the fewest 1 votes and its votes transferred to B, D and E. See Table III for next steps.

 TABLE III -- Same as Table II Except that Candidate C's Votes Are First Listed Voter Candidate C Candidate A Candidate B Candidate D Candidate E 10 1 4 5 2 3 15 1 4 5 2 3 29 1 5 4 2 3 1 2 1 3 4 5 6 2 1 3 5 4 18 2 1 3 5 4 20 2 1 3 4 5 21 2 1 3 5 4 4 2 3 1 4 5 3 2 4 3 1 5 13 2 4 3 5 1 11 2 5 1 3 4 16 2 5 1 3 4 26 2 5 1 3 4 7 3 1 5 4 2 12 3 1 5 4 2 17 3 1 4 2 5 5 3 2 1 5 4 30 3 4 4 1 2 2 3 5 4 2 1 27 3 5 4 2 1 19 4 1 5 3 2 22 4 1 5 3 2 24 4 1 5 2 3 8 4 2 3 5 1 28 4 5 3 1 2 25 5 1 4 3 2 9 5 3 4 1 2 14 5 3 4 1 2 23 5 3 4 2 1 Droop Quota 8 Candidate C Candidate A Candidate B Candidate D Candidate E Adjusted 1s 4.667 8 5 5.667 6.667 Total 2s n/a n/a 0 3 0 Since those who rank C number 1 all rank D number 2, D gets all of C's transferred votes. See below. D is now elected and E is also elected since it gets all of D's surplus votes. Readjusted 1s n/a n/a 5 10.333 6.667 Final 1s n/a n/a 5 8 9

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