Non-Monotonicity in Australian Preference Voting

There are several methods of voting which allow voters to rank candidates in order of their preference, rather than just selecting a single desired candidate and then doing a single count (Plurality or "First Past the Post" voting). The system specifically used in Australian elections is "Instant-Runoff Voting" (IRV).

IRV is intended to allow for a variety of political parties of various sizes to flourish (unlike the famously two-party-dominated politics of the USA) as citizens who vote for a minor party as their first preference don't "waste" their vote; if their first preference is too obscure to get in, their vote goes to their second preference, and so on.

However, it is still possible for "vote-splitting" to have a negative effect on minor parties - in some cases, giving a candidate a higher preference can paradoxically cause them to lose, as they can be eliminated earlier.

Recap: How IRV Works

Determining the winner in IRV follows the following algorithm:

  1. Count all the first preferences for all candidates.
  2. If there is a candidate with more than half the votes, they are the winner.
  3. Else, eliminate the candidate with the lowest number of votes, then redistribute all the votes allocated to the eliminated candidate, according to the next listed preference on each vote.
  4. Repeat from step 2, adding the redistributed votes to the count.

Recap: How Monotonicity Works

In mathematics, a "Monotonic" function is a function which either always increases or always decreases as its independent variable increases. For example, all of the following are monotonic functions:

y = x             (y always increases as x increases)
y = x + 345       (y always increases as x increases)
y = -x + 12       (y always decreases as x increases)

Note y = x² is not monotonic, as y only increases as x increases above x=0, it is the opposite for values of x below 0.

Non-monotonicity: When IRV Doesn't Work

An ideal voting system would also be monotonic - as votes or preferences for a candidate increase, their chance of winning should always increase. However, it is possible that increasing the preference of a party can cause it to be eliminated in an earlier round, whereas decreasing a preference can cause them to survive later to win.

The following two examples (an electorate of 25,000 voters and somewhat arbitrary party names) shows this quirk of IRV:

Party 7000 Votes 8000 Votes 10000 Votes
Liberal Second First Third
Labour First Second Second
Green Third Third First

The vote counting process:

  1. No party has more than 12,500 votes for a win on first preferences. - The Labour candidate has the least first preference votes, so their 7000 votes are redistributed to the second preference; in this case, all to the Liberals.
  2. The Liberal candidate now has 15,000 votes on the second round, enough to win.

Note that the Labour candidate was eliminated before the second round, despite being either the first or second preference for all 25,000 voters. The Liberal winner was only first or second preference for 15,000 voters.

However, if 4000 of the largest group (Greens,Labour,Liberal) changed their preferences to put the Liberal party first and the Labour party last, this would turn the Liberal win into a Labour win:

Party 4000 Votes 6000 Votes 7000 Votes 8000 Votes
Liberal First Third Second First
Labour Third Second First Second
Green Second First Third Third

The votes are now counted as follows:

  1. No party has more than 12,500 votes for a win on first preferences. - The Greens now have the least first preference votes, so their 6000 votes are redistributed to the second preference, Labour.
  2. Labour now has 13,000 votes on first and second preferences, enough to win.

Note that even in this example, where some voters preferring the Liberals higher caused them to lose, the winner still had more combined first and second preferences than the other parties (21,000 Labour, 19,000 Liberal, 10,000 Green).

In more complex cases the winner of an IPV vote may actually have less combined preferences than a loser, as in the 2009 Burlington election discussed below.

Has this ever happened?

A quick search hasn't turned up any Aussie elections where this has been an issue. The only famous one internationally seems to be the 2009 Burlington, Vermont mayoral election. There were four candidates with a significant fraction of the total vote, and three rounds:

Candidate 1st Round 2nd Round 3rd Round
Bob Kiss 2585 28.8% 2981 33.8% 4313 51.5%
Kurt Wright 2951 32.9% 3294 37.3% 4061 48.5%
Andy Montroll 2063 23.0% 2554 28.9%  
Dan Smith 1306 14.5%    

33% of the voters put Kurt Wright first; most of his voters chose Andy Montroll as their second preference. However, Kurt Wright was still in the running in the second round, so those second preferences were not in effect, and Andy Montroll was eliminated.

In the final tally, 54% of the voters preferred Andy Montroll ahead of Bob Kiss, although Kiss won the election and Montroll was eliminated in the second-last round.

The controversy over the result triggered the repeal of IPV voting in the city, returning to a two round system.

Condorcet Voting Systems

The odd result in Burlington would not have occurred in a "Condorcet method" voting system, as popularized by 18th century mathematician Marquis de Condorcet. A Condorcet winner is a candidate who would win in a head-to-head against every single other candidate in the election, and can therefore be considered the clearly preferred candidate by the entire electorate.

Analysis of the Burlington election shows that by looking at the preferences of each combination of two candidates, Andy Montroll was preferred in every pair he was in, although in the IPV system he was eliminated before the last round of counting.

As preferences can be cyclic, there is a chance that an election will produce no Condorcet winner. However, there are multiple systems which guarantee that if there is a Condorcet winner, that candidate will be the one chosen by the system.