Single Transferable Vote
STV
(Preferential Voting)
Sometimes used when more than one
option(candidate) is to be selected.
STV
• First proposed by Thomas Hare (English
barrister) and Carl Andrae (Danish
mathematician) in the 1850’s.
• If used to elect a single winner, it is just the
method of Hare elimination or sequential
run-off we studied earlier.
STV
How does it work?
An election using STV
Suppose that we have 4 seats to fill on a
local governing board, and that 50,000
votes are cast with 10 candidates running.
Voters will use ordinal ballots to rank all 10
candidates.
Step 1
• Set the quota or number of votes
necessary to be elected.
• This is done in such a way that no more
than the required number of candidates
can be elected.
• Here we want to set it at the smallest
value so that 4 candidates could be
elected, but 5 could not be.
Setting the Quota
• How could 5 be elected? If they each received
50, 000
 10, 000
votes. So we set the
5
quota at
50, 000
 1  10, 001
votes.
5
votes 

• In general quota (q) = int
where

1
 seats  1 


Int means ‘ignore any remainder’.
Step 2
• Count the first place votes.
• Elect any candidate(s) who meet or
exceed the quota.
• If no candidate meets the quota, eliminate
the one(s) with the fewest first place votes
and count again.
• Continue until one or more candidates
achieves quota.
Step 3
• Reassign ‘extra’ votes. Here’s how it
works…
An example of vote transfer
Suppose candidate A gets 900 more votes
than the quota of 10,001.
So that these voters don’t “waste” their
votes, we reassign these 900 votes to the
second place candidates on all the ballots
having A first.
Vote transfer (continued)
Suppose candidate B was second on 30%
of the ballots having A first, C was second
on 42% and D second on 28% of those
ballots. Then we would distribute 30% of
900 = .30(900) = 270 votes to B, 42% of
900 = .42(900) = 378 votes to C and 28%
of 900 = .28(900) = 252 votes to D!
Vote transfer (continued)
• Thinking in terms of total votes (10,901)
for A, each ‘extra’ vote is 900/10901 =
.08256 of a vote.
• Each voter who A ranked first would have
this fraction of his vote transferred to his
second choice.
• B was second on 30% of 10,901 or 3270
ballots and so gets 3270(.08256)=270
votes.
Vote transfer (continued)
• Note: If two or more candidates go over
quota, the larger surplus is distributed first.
• If the distribution will not 1)elect a
candidate or 2) keep a candidate from
elimination, then it is not made, and the
count progresses by eliminating the
candidate with the fewest votes.
• The process continues until all seats are
filled.
Why Use STV?
• It helps to assure proportional
representation for a voting constituency.
• In plurality, only your first place vote
counts, but with STV, if your first place
vote doesn’t help elect that candidate, part
of it will be transferred to help one of your
lower choices.
Midland Residents Association
• Two seats available
• 23 voters (13 conservative, 10 liberal)
• 4 candidates
Winston, Nadia—conservative, Winston
more so
Theo, Rudy—liberal, Rudy more so
The Vote
Conservative
7
Liberal
6
6
4
Winston
Nadia
Theo
Rudy
Nadia
Winston
Rudy
Theo
Theo
Theo
Nadia
Nadia
Rudy
Rudy
Winston
Winston