International Affairs and Democracy

If You Really Must!
STV, a Worked Example

Here is a hypothetical worked example of Single Transferable Vote (STV).

Suppose the following results are obtained in two large northern ridings where those results are combined to elect two Members to the Legislature (for simplicity we are considering only 4 candidates, though in practice there would probably be more):

First we need to calculate the "Droop quota" (named after the person who devised it), the number of votes a candidate must obtain to be elected. It is calculated according to the formula
Droop quota = 1+ (Total no. of valid votes/(1+ number of members to be elected))
= 1 + (25454/3) = 8,486

What this formula says is that if votes were evenly distributed among all candidates, they would all be on the verge of winning, and any candidate would need only a few more votes to win (given by the formula). The Droop quota indicates that Oak Leaf has 14,588 - 8,486 = 6,102 "surplus votes", i.e., 6,102 more votes than needed to be elected. However, none of the other candidates meets the quota, so we have only one elected member so far. We need a second.

Since more people voted for Oak Leaf than needed for her election, those excess votes are "wasted" unless we can make use of those voters' second choice for other candidates. We need to take into account the second choices of all 14,588 votes, but in aggregate they must not equal more than 6,102 votes (the "surplus"). This can be achieved by applying a "transfer value" = number of surplus votes/total number of votes for O. Leaf = 6,102/14,588 = 0.4183.

Now, the second choices of the 14,588 votes for Oak Leaf are distributed as follows:

When Oak Leaf's voters' second choices are transferred, the total votes are then as follows:

Unfortunately, we still have only one winner as no one but Oak Leaf has yet achieved the Droop Quota.

At this stage we have to eliminate the candidate with the fewest first choice votes (Pussy Willow) and assign the second choices on those ballots among the other candidates (at full value). Of the 411 first-choice ballots for Pussy Willow, the second choices break down as follows:

In this example, the additional surplus of 123 votes is insufficient to bring Fir Bark's total above 8,486 (the Droop Quota) even if all the third choices on these ballots (the first choice was Pussy Willow, the second choice Oak leaf) were for Fir Bark (i.e., 7,276 + 123 = 7,399 < 8,486). We know, therefore, that a second candidate will have to be dropped, and the votes re-allocated (at full value). That second candidate will be Cherry Blossom, with 851 votes less than Fir Bark at this stage, leaving Fir Bark as the second winning candidate, along with Oak Leaf.

If the outcome were not obvious at this point, however, we would have had to calculate a new "transfer value" for all of Oak Leaf's "surplus", including the new additional surplus of 123. Using the same formula as above, the new "transfer value" is 6225/14,588 = 0.4267.

Fir Bark obtained 5,837 second-choice votes from Oak Leaf (see B. above), and if, as suggested, he obtained all 123 third choices on the ballots whose second choice was Oak Leaf, the votes trans-ferred to Fir Bark would be 0.4267 (the new transfer value) x (5,837 + 123 = 5,960) = 2,543. The new total for Fir Bark will be the votes he originally won (A.) (4,666) + the votes transferred at full value from eliminated candidate Pussy Willow (D.) (168) + the new transferred surplus from Oak Leaf (2,543) = 7,377. This, as we already predicted, is insufficient to achieve the Droop Quota, necessitating the elimination of Cherry Blossom-which settles the election in this example.

If, however, more than two candidates had remained, necessitating the need for further analysis, we would have had to re-distribute Cherry Blossom's votes (at full value), among the three (or more) remaining candidates, and recalculate a third "transfer value" for Oak Leaf, etc.

There's no question that the analysis of votes under the STV system is complex, but it's perfectly logical and fair, and easily accomplished when performed by computer. People can either mark a ballot as we do now, which is then scanned into a computer, or, if voting is done with touch-screen computers, those computers can be programmed to produce a paper print-out of the ballot that can be checked by the voter (and corrected if necessary), and this paper ballot can then be placed in a ballot box, as now. In both cases vote results can later be verified and checked independently, if required, avoiding the voting fiascoes that occurred in the recent Presidential election in the U.S.A.

Philip Symons, Feb., 2005