The Single-Transferable-Vote System

Illustration

An illustration may help to make the STV system clear. Suppose thirty members of an organization vote to elect their president from among four candidates: Paul Berg, Aeron Glynn, Mónica Rodríguez, and Kim Tran. Each voter indicates his or her first, second, third and fourth choice by the numbers 1, 2, 3, and 4.

The voter marking the ballot below voted for Mónica Rodríguez as his or her first choice, Kim Tran as second choice, and Paul Berg as third choice. This voter voted “NO” for Aeron Glynn because he or she did not want Aeron Glynn as president under any circumstances.

Candidate	Choice
Paul Berg	3
Aeron Glynn	NO
Mónica Rodríguez	1
Kim Tran	2

An equivalent method is to mark a blank ballot by writing the names down in the preferred order from top to bottom. Thus the above ballot would appear as below:

The ballots are arranged in piles by the tellers according to the voters’ indications of their first choice candidates: all of the Paul Berg 1^st choice ballots are placed in one pile, all of the Aeron Glynn 1^st choice ballots in another pile, all of the Mónica Rodríguez 1^st choice ballots in a third pile, and the remaining ballots, which show Kim Tran as their 1^st choice, in a final pile.

Suppose that when these thirty ballots are counted according to the first choice votes, the count of the four piles made by the tellers is as indicated in the first count column of the table below (the leftmost column with vote numbers).

With this first count, Mónica Rodríguez leads the race with ten 1^st choice votes, while the next two candidates are tied with eight each, and the last candidate, Kim Tran, trails with four.

The minimum number for a majority of the 30 votes is 16. Since no candidate has obtained a majority of all the votes cast, the candidate with the smallest vote count, Kim Tran, is declared “out of the race”, and her four 1^st choice ballots are redistributed according to their 2^nd choice indication. The distribution of these four ballots is as shown in the second column of votes in the table.

Adding these 1^st and 2^nd choices together gives the second count, which is 9, 8, and 13, respectively, for the three remaining candidates, as indicated in the third vote column of the table. Still no candidate has obtained a majority of all the votes cast, since 16 is still the minimum needed for a majority.

Consequently, the candidate with the smallest vote count in this second round of counting, Aeron Glynn, must be dropped. Glynn’s eight 1^st choice ballots are then redistributed among the remaining candidates according to their 2^nd choice indication (or 3^rd choice, if the 2^nd choice on any of these ballots was Kim Tran, who has already been eliminated).

The redistribution is shown in the fourth column of vote numbers. Five of the first choice Glynn votes now pass to Mónica Rodríguez and two to Paul Berg. As it happens, the last Glynn 1^st choice ballot has no 2^nd or other choices indicated, and therefore it cannot pass on to any other candidates. When Glynn is declared “out of the race,” this ballot becomes an “exhausted ballot” and is indicated as such in the table.

(The “NO” vote for Glynn, on the sample ballot shown above in the first table, never came into play, because the vote from that ballot remains with Mónica Rodríguez. Once Glynn is eliminated, it is irrelevant, even if Rodríguez’ votes were to be later redistributed.)

The 2^nd (or 3^rd) choice ballots for Berg and Rodríguez are added to their previous totals of 9 and 13, respectively, to yield a 3^rd count of 11 votes for Paul Berg and 18 for Mónica Rodríguez, shown in the last column.

Mónica Rodríguez now has 18 votes, which is 3 more than the minimum majority now required, which is 15. 15 is a majority of the new total vote of 29, which is the total vote cast minus the exhausted ballot. Therefore, Mónica Rodríguez is declared elected to the office of president.

Filling other positions with the same ballots

The organization can use the same ballots to elect another candidate for a position. For example, the organization may decide to elect a vice president using the same ballots as were used for president.

To do this with the ballots above, first Mónica Rodríguez’ name is stricken from all the ballots, since she is not eligible to be vice-president as well as president. All three other candidates are back in the running, including Kim Tran and Aeron Glynn: their having been eliminated in the counting for president is irrelevant for this second race. The ballots are distributed in piles according to the highest-ranked names, and counting proceeds as before.

Note that it is by no means a given that Paul Berg will win this race, even though he came in second in the final count in the race for president. For instance, if eight or nine of Mónica Rodríguez’ voters have listed Aeron Glynn as their second choice (we already know that one voter did not), then Aeron Glynn will win the vice-presidency on the first count.

Updated 8 Nov 2007

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