Monotonicity

WORK IN PROGRESS

Definition

Monotonicity is an election method property, defined by mathematicians as follows:

With the relative order or rating of the other candidates unchanged, voting a candidate higher should never cause the candidate to lose, nor should voting a candidate lower ever cause the candidate to win.

[Lynn Arthur Steen, professor of mathematics at St. Olaf College, Northfield, Minnesota]

In other words, an election method is non-monotonic if either of the following is possible:

  • A can be changed to a loser by experiencing an increase in support (“mono-raise”)
  • A loser can be changed to a winner by experiencing a decrease in support (“mono-lower”)

“Mono-raise” and “mono-lower” are logically identical

As the two following election scenarios demonstrate, “mono-raise” and “mono-lower” are actually logically identical criteria. For any pair of elections which demonstrate a failure of one of the criteria, you can simply swap the labels “before” and “after”, demonstrating a failure of the opposite criterion.

Example of monotonicity failure using Instant Runoff Voting

One noteworthy non-monotonic election method is Instant Runoff Voting. Here are is a pair of simplified IRV elections which exhibit this problem.
“before”
 # of voters
 their ranking
6 X > Y
6 Y > Z
5 Z > X
X wins
Candidate X wins this election, labeled “before”. But say that two of the voters who prefer Y hear of some important information which causes X to become their favorite, producing the following election, labeled “after”.
“after”
 # of voters
 their ranking
8 X > Y
4 Y > Z
5 Z > X
Z wins

Note the paradox. X changed from winner to loser by becoming more popular. Or, if you swap the arbitrary labels “before” and “after”, X changes from loser to winner by becoming less popular.

Why is non-monotonicity a problem?

The problem with non-monotonicity is that we know with mathematical certainty that IRV elects the wrong candidate in at least one of the two elections in the non-monotonic pair.
For instance, say we operate on the premise that X was indeed the rightful winner in the “before” election. Then X necessarily must also be the rightful winner in the “after” election, since X has even more support in the “after” election than in the “before” election. Yet IRV does not elect X in the “after” election.
Likewise, if we operate on the premise that X was indeed not the rightful winner of the “after” election, then X also cannot be the rightful winner of the “before” election, since X has even less support in the “before” election than in the “after” election. Yet IRV elects X in the “before” election.

This means that, statistically speaking, IRV elects the wrong winner in at least half of all non-monotonic elections. So if an election forms either half of a non-monotonic election pair, like either of the two above, we know there is at least an approximately 50% chance that the wrong winner was elected.

How common are non-monotonic elections with IRV?

It is impossible to empirically determine the frequency of non-monotonic IRV elections, due to the fact that full ballot sets are rarely published. For example, Australia has used IRV since 1918, and the Wikipedia’s monotonicity criterion article says the following:

every or almost every IRV election in Australia has been conducted in the black (i.e. not releasing enough information to reconstruct the ballots), nonmonotonicity is difficult to detect in Australia and apparently has never been detected, even though thanks to the Lepelley et al probability estimates it seems safe to say that it must have occurred in over 100 of their elections. (The policy of Australia's election authorities not to release this data is justifiable on privacy grounds. If rank-order ballots in an election with, say, 13 candidates, were released, even in a highly "anonymized" form, that would still provide enough information for a coercer to use to verify or deny that some voter had cast a pre-specified vote-pattern he'd demanded.)
Warren Smith, a Princeton math Ph.D. who has studied voting methods for over a decade, has calculated probability estimates of non-monotonic IRV elections using different models. (The results for the Direchlet model agree with the consensus of the preceding scientific literature.)

Random Election Model – 15.2305
Direchlet Model – 5.7436%
Quas 1-Dimensional Model – 6.9445%
But if we restrict attention to elections in which the IRV process matters, i.e. in which the IRV and Plurality winners differ (i.e. exactly the elections IRV advocates tend to cite as examples of the “success” of the IRV process), the probability becomes stunningly large:
Random Election Model – 35.8569%
Direchlet Model – 26.5477%
Quas 1-Dimensional Model – 9.7221%

Real world examples

Burlington, Vermont – 2009 mayoral election

The 2009 mayoral election in Burlington, Vermont formed one half of a non-monotonic election pair. Bob Kiss won, but would have lost if some voters had ranked him higher. Or in other words, Kiss won because some voters ranked him too low.

Frome electoral district, South Australia – 2009 House of Assembly by-election

In the 2009 Frome state by-election, the independent won. But if 31 to 321 of the voters who preferred Liberal over Labor over independent had decreased their support for Liberal, ranking Liberal lower than Labor, it would have caused the Liberal to win the IRV election. I.e. the Liberal lost because some voters ranked him too high.

Misleads

Some individuals, particularly those from The Center for Voting and Democracy (aka FairVote), have made a number of false and misleading statements about monotonicity in their defense of IRV. Here are some of them.

“Burlington did not exhibit monotonicity failure”

FairVote mistakenly claims that the 2009 Burlington mayoral election experienced no failure of monotonicity. Here are two quotes from FairVote executive director Rob Richie.

In fact, no such failure occurred. ..non-monotonicity could have affected the election, but did not.
Bottom line for me on the Burlington nonmonotoncity debate is: 1) no loser could have turned themselves into a winner by getting fewer votes
http://dmarron.com/2010/09/19/the-feud-over-the-2009-burlington-mayoral-election/#comment-4123
Richie’s second comment reveals a simple mistake. Richie is confusing the concept of a non-monotonic election method with the concept of a non-monotonic election. His point that “no loser could have turned themselves into a winner by getting fewer votes” is an obvious reference to the
  • the question of whether an election method (e.g. IRV) is monotonic, and
  1. The question of whether an individual election is monotonic (i.e. whether the election is part of a non-monotonic election pair)
Lynn Arthur Steen’s definition explicitly lists both criteria. But because they are logically identical, one may arbitrarily use either criterion alone in order to judge whether an election method is monotonic. This is precisely what a mathematician named Douglas R. Woodall has done, coining the term “mono-raise”, but never (to our knowledge) invoking the equally valid “mono-lower”:

 apply a test for monotonicity to a particular election rather than to an election method. This simply does not make sense. An electionmethod (namely, IRV) is non-monotonic if there are cases where a candidate can go from winner to loser by gaining support. Again, this is precisely identical to saying that an election method is non-monotonic if there are cases where a candidate can go from loser to winner bylosing support. The definition merely depends on which election you treat as the “before” example, and which election you treat as the “after” example.
When looking at a specific election in the context of monotonicity, the only meaningful question to ask is, “is this election part of a non-monotonic election pair?” In order to determine that, one must ask both questions:
  • Could the winner have been made the loser by gaining support?
  • Could the loser have been made the winner by losing support?
FairVote’s second mistake, once they decided to apply a monotonicity test to a specific IRV election rather than to IRV itself, was to only apply the “mono-raise” test, when
Their secondary mistake was to then apply that test using only the “mono-raise” criterion, and to ignore the equally valid “mono-lower” criterion. This is apparently a result of misunderstanding the “mono-raise” definition given by Douglas R. Woodall.
To explain a little further, in the Burlington election, Bob Kiss could have been made a loser if some voters had ranked him higher. In other words, Kiss actually won because some voters “didn’t like him enough”. This is sufficient to show that the Burlington election formed one half of a non-monotonic election pair, and thus likely had the wrong outcome. (Alternate lines of evidence actually say that it did have the wrong outcome; Montroll was likely the true favorite of the electorate, and in fact a majority of Burlington voters preferred Montroll to Kiss.)
applies the mono-raise criterion to an individual election, rather than to the IRV election method. They point out that no candidate could have been changed from winner to loser by experiencing an increase in support.

But as we explained in the previous section, the mono-raise criterion can be equivalently stated as “mono-lower”. And indeed, Mayor Bob Kiss won by being ranked too low on some ballots. Here is an excerpt from FairVote’s page on the Burlington election.
Here FairVote has not only made the mistake of applying the criterion to the wrong thing (the Burlington election instead of the IRV system), but they’ve also conveniently picked the particular direction (mono-raise) that works for the election in question.

“Non-monotonicity has no effect”

[FairVote's] position is that such paradoxes are too rare to worry about. "We've had thousands of [IRV] elections and it's not an issue," said Rob Richie. Steven Hill, a senior analyst with [FairVote] dismisses "these mathematical paradoxes that while in theory are interesting for mathematicians to doodle around on their sketch pads, in fact have no basis in reality... it's also possible that a meteorite will strike Earth and wipe out life... but probably not for a few more million years."

In terms of the frequency of non-monotonicity in real-world elections: there is no evidence that this has ever played a role in any IRV election – not the IRV presidential elections in Ireland, nor the literally thousands of hotly contested IRV federal elections that have taken place for generations in Australia, nor in any of the IRV elections in the United States.

– FairVote webpage titled “Monotonicity and IRV – Why the Monotonicity Criterion is of Little Import” downloaded 24 August 2010; our emphasis added.
 …Ifjust over 25% of the supporters of Republican Wright had abandoned their true first choice and instead voted for any other candidate (although to meet the non-monotonicity definition they would need to switch to Kiss), they could have kept their favorite candidate, Wright, from making it into the runoff, and allowed the Democrat to face off against Kiss in the final runoff, where the Democrat would beat Kiss. But this did not happen, and there is nobody who thinks it was a sensible strategy for any voters or candidates to advocate. Certainly it had no impact on how candidates campaigned nor ever would have.

Another way that non-monotonicity could have occurred would be if Wright or Montroll lost the election because they got too many first choices that might have gone to some other candidate instead. Since Montroll didn’t even make it into the final runoff there is no way this could apply to him (any fewer first choices would just confirm his elimination). That leaves Wright. Again, even if some of his supporters had voted for any other candidate first, Wright would still lose the runoff between either Kiss or Montroll. Smith would need� 1,279 first choices� that actually went to Wright, to get Smith into the final runoff. But� at that point Wright would be in fourth place and have no chance of advancing to the final runoff.� So it is mathematically impossible for a switch of first choices away from Wright to have made him a winner. Thus, despite the Smith-Gierzynsnki analysis, there was, in fact, no non-monotonicity event in the Burlington election.

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