# Statistics of Instant Runoff Voting

Has anyone published statistics on how many recounts (i.e. runoffs) are necessary for resolving an instant runoff election, versus the number of candidates?

Data from real life would be good, but simulations would also be interesting.

Also of interest: how often do ties occur?

• A link to some explanation about what is Runoff Voting would be nice. – Evargalo Jan 8 '18 at 15:31
• @Evargalo there's also a CGP Grey video. – SCH Jan 8 '18 at 15:49
• why would more than one runoff be required? the spanish senate is elected via IRV if you need data. – CptEric Jan 8 '18 at 15:56
• @CptEric I understood that as long as no candidate has more than 50% of the votes, the votes of the worst candidate get redistributed (equivalent to a runoff). So with 8 candidates, there could be up to 7 runoffs, right ? – SCH Jan 8 '18 at 16:02
• @S.Chevalier When you only want to elect one person and don't require a minimum number of votes for the winning candidate, then you can stop after the 6th runoff, because then you are left with two candidates, one of them (hopefully) having more votes than the other. When one needs a minimum number of votes, then you might need to do a 7th runoff to check if the last candidate reached it. – Philipp Jan 8 '18 at 16:46

I just ran the following analysis based on the 2016 House of Representatives Election in Australia. It shows a slight correlation, however it also shows a significant standard deviation within electorates with the same number of candidates.

Most candidate counts had at least one electorate decided on first preferences, and most had at least one decided on two-candidate preferred.

For the benefit of those unable to see the chart:

• 3 candidates: 1.00 counts (average based on 1 electorate)
• 4 candidates: 1.82 counts (11 electorates)
• 5 candidates: 2.24 counts (34 electorates)
• 6 candidates: 3.33 counts (36 electorates)
• 7 candidates: 4.75 counts (28 electorates)
• 8 candidates: 4.41 counts (17 electorates)
• 9 candidates: 7.62 counts (8 electorates)
• 10 candidates: 5.88 counts (8 electorates)
• 11 candidates: 10.0 counts (7 electorates)

Source of raw data: Australian Electoral Commission.

Method: The electorates were grouped based on the number of candidates nominated in the 2016 election. The number of counts required for a candidate to reach 50% of the vote was determined for each electorate, and then the average, min/max and standard deviation was calculated within each group. In Australia, each count results in the elimination of exactly one candidate. For the sake of avoiding ambiguity, the first count is the first-preference, and the second count is the first runoff.

EDIT: Breakdown table to supplement the above:

``````                     CANDIDATES
1   2   3   4   5   6   7   8   9   10  11
+---+---+---+---+---+---+---+---+---+---+---+
1|   |   | 1 | 6 | 17| 11| 5 | 5 |   | 3 |   |
+---+---+---+---+---+---+---+---+---+---+---+
2|   |   |   | 1 | 3 | 2 |   |   |   |   |   |
+---+---+---+---+---+---+---+---+---+---+---+
3|   |   |   | 4 | 3 | 3 | 1 | 1 |   |   |   |
+---+---+---+---+---+---+---+---+---+---+---+
4|   |   |   |   | 11| 4 | 2 | 2 |   |   |   |
C  +---+---+---+---+---+---+---+---+---+---+---+
O 5|   |   |   |   |   | 16| 3 | 1 |   |   |   |
U  +---+---+---+---+---+---+---+---+---+---+---+
N 6|   |   |   |   |   |   | 17| 2 | 1 |   |   |
T  +---+---+---+---+---+---+---+---+---+---+---+
S 7|   |   |   |   |   |   |   | 6 | 1 |   |   |
+---+---+---+---+---+---+---+---+---+---+---+
8|   |   |   |   |   |   |   |   | 6 | 1 |   |
+---+---+---+---+---+---+---+---+---+---+---+
9|   |   |   |   |   |   |   |   |   | 4 |   |
+---+---+---+---+---+---+---+---+---+---+---+
10|   |   |   |   |   |   |   |   |   |   | 7 |
+---+---+---+---+---+---+---+---+---+---+---+
``````
• Great answer ! So all the 11-candidate electorates needed 11 runoffs. Do you know if that's a coincidence ? – SCH Jan 9 '18 at 14:09
• What do the distributions of counts look like for the >10 electorate groups look like? I'd be interested in seeing the electorates that decided on first preference separated out – Caleth Jan 9 '18 at 14:17
• @Caleth I've posted the electorate-level summaries here, if you'd like to play around with it. – Joe C Jan 9 '18 at 18:29
• @JoeC Thanks! The theory I have is that there are ~two populations of electorates, one where there is a clear winner in round one, and one where most other candidates are eliminated before a winner emerges – Caleth Jan 9 '18 at 19:13
• @Caleth I just edited my answer to provide a more fine-grained breakdown, and it definitely seems to support that theory. – Joe C Jan 9 '18 at 19:14