Preferential ballot.svg 

In STV, each voter ranks the list of candidates in order of preference. In other words (under the most common ballot design), they place a ‘1’ beside their most preferred candidate, a ‘2’ beside their second most preferred, and so on. The ballot paper submitted by the voter therefore contains an ordinal list of candidates. In the ballot paper shown in the image on the right, the preferences of the voter are as follows:

  1. John Citizen
  2. Mary Hill
  3. Jane Doe

Counting the votes

Setting the quota

In an STV election, a candidate requires a certain minimum number of votes – the quota (or threshold) – to be elected. A number of different quotas can be used; the most common is the Droop quota, given by the formula:

\mbox{votes needed to win} = \left({{\rm \mbox{valid votes cast}} \over {\rm \mbox{seats to fill}}+1}\right) + 1

The Droop quota is an extension of requiring a 50% + 1 majority in single winner elections. For example, at most 3 people can have 25% + 1 in 3 winner elections, 9 can have 10% + 1 in 9 winner elections, and so on.

Finding the winners

An STV election proceeds according to the following steps:

  1. Any candidate who has reached or exceeded the quota is declared elected.
  2. If a candidate has more votes than the quota, that candidate’s surplus votes are transferred to other candidates. Votes that would have gone to the winner instead go to the next preference listed on their ballot.
  3. If no one new meets the quota, the candidate with the fewest votes is eliminated and that candidate’s votes are transferred.
  4. This process repeats until either a winner is found for every seat or there are as many seats as remaining candidates.

There are variations in applying these STV rules, such as in how to transfer surplus votes from winning candidates and whether to transfer votes to already elected candidates. When the number of votes to transfer from a losing candidate is too small to change the ordering of remaining candidates, more than one candidate can be eliminated simultaneously.

Because votes cast for losing candidates and excess votes cast for winning candidates are transferred to voters’ next choice candidates, STV is said to minimize wasted votes.

An example

Suppose a food election is conducted to determine what to serve at a party. There are 5 candidates, 3 of which will be chosen. The candidates are: OrangesPearsChocolateStrawberries, and Sweets. The 20 guests at the party have their ballots marked according to the table below. In this example, a second choice is needed by only some of the voters, however with a different vote distribution additional preferences may be needed.

# of Guests x x x x x x x x x x
x x x x
x x x x x x
1st Preference Orange Pear Chocolate Chocolate Strawberry Candy
2nd Preference Orange Strawberry Candy

First, the quota is calculated. Using the Droop quota, with 20 voters and 3 winners to be found, the number of votes required to be elected is:

\left({\mbox{20 votes cast} \over {\mbox{3 seats to fill}+1}}\right) +1 = \mbox{6 votes required}

When ballots are counted the election proceeds as follows:

Candidate: Orange Pear Chocolate Strawberry Candy
Round 1 x x x x x x x x x x
x x x x

x x x x
x x Round 1: Chocolate is declared elected, since Chocolate has more votes than the quota
Round 2 x x x x x x x x x x
x x
x x x x
x x x Round 2: Chocolate’s surplus votes transfer to Strawberry and Sweets in proportion to the Chocolate voters’ second choice preferences. However, even with the transfer of this surplus no candidate has reached the quota. Therefore Pear, who has the fewest votes, is eliminated.
Round 3 x x x x
x x
x x x x
x x
x x x x
x x x Round 3: Pear’s votes transfer to their second preference, Oranges, causing Orange to reach the quota and be elected. Orange meets the quota exactly, and therefore has no surplus to transfer.
Round 4 x x x x
x x
x x x x
x x
x x x x
x x x Round 4: Neither of the remaining candidates meets the quota, so Sweets are eliminated. Strawberry is the only remaining candidate and so wins the final seat.

Result: The winners are ChocolateOranges and Strawberries.



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