# Monotonicity

## Introduction

Monotonicity is an election method criterion that requires the following:

Ranking or rating a candidate higher should never cause that candidate to lose, nor should ranking or rating a candidate lower ever cause that candidate to win, assuming all other candidates remain rated or ranked the same.

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

- A winner can be changed to a loser by experiencing an increase in support, OR
- A loser can be changed to a winner by experiencing a decrease in support.

## Why Is Non-Monotonicity a Problem?

The problem with non-monotonicity is that we know IRV must be electing 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 a 50% chance that the wrong winner was elected.

## Non-Monotonicity with Instant Runoff Voting

One noteworthy non-monotonic election method is instant runoff voting (IRV). Here is a pair of simplified IRV elections which exhibit this problem:

**Before Additional Support for X**

**# of voters their ranking**

6 X > Y

6 Y > Z

5 Z > X

X wins

Candidate X wins this election. 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.

**After Additional Support for X**

**# of voters their ranking**

8 [+2] X > Y

4 [-2] Y > Z

5 Z > X

Z wins

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

## How Common Are Non-Monotonic IRV Elections?

It is impossible to empirically determine the frequency of non-monotonic IRV elections because full ballot sets are rarely published. For example, Australia has used IRV since 1918, but they do not release enough information in order to reconstruct the ballots.

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, the probability becomes quite large i.e. in which the IRV and Plurality winners differ. Interestingly, these are exactly the elections IRV advocates tend to cite as examples of the “success." See non-monotonicity rates of these elections below:

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 >Labor > independent) had decreased their support for Liberal, ranking (Labor> Laboral), it would have caused the Liberal to win the IRV election. That is, the Liberal Party lost because some voters ranked him too high.

## Misleads

“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." -Rob Richie, FairVote

"Bottom line for me on the Burlington nonmonotoncity debate is: 1) no loser could have turned themselves into a winner by getting fewer votes." -Rob Richie, FairVote

Richie’s second comment reveals a simple mistake. His point that, “no loser could have turned themselves into a winner by getting fewer votes,” misses half the equation.

Recall that non-monotonicity is when:

- A winner can be changed to a loser by experiencing an increase in support, OR
- A loser can be changed to a winner by experiencing a decrease in support.

And in Burlington Bob Kiss won. But he would have lost if some voters had ranked him higher. In other words, Kiss won because some voters ranked him too low. This is exactly what non-monotonicity is.

To determine if an election is non-monotonic, you must ask whether one of the following is true:

Could the winner have been made the loser by gaining support?

Could the loser have been made the winner by losing support?

In Burlington, case number 1 was true. And this is why the Burlington election was non-monotonic.