# Monotonicity

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The monotonicity criterion (sometimes referred to as the "mono-raise criterion") is a voting system criterion used to evaluate both single and multiple winner election methods. An election method is monotonic if it is neither possible to prevent the election of a candidate by ranking them higher on some of the ballots, nor possible to elect an otherwise unelected candidate by ranking them lower on some of the ballots (while nothing else is altered on any ballot).

## Mono-raise criterion

The mono-raise criterion is one of several sub-cases of the monotonicity family of criteria. A voting system satisfies the Mono-raise criterion:

If an alternative X loses, and the ballots are changed only by placing X in lower positions, without changing the relative position of other candidates, then X must still lose.

A looser way of phrasing this is that in a non-monotonic system, voting for a candidate can cause that candidate to lose. Systems which fail the monotonicity criterion suffer a form of tactical voting where voters might try to elect their candidate by voting against that candidate.

Plurality voting, Majority Choice Approval, Borda count, Schulze, Maximize Affirmed Majorities, and Descending Solid Coalitions are monotonic, while Coombs' method and Instant-runoff voting are not. Approval voting is monotonic, using a slightly different definition, because it is not a preferential system: You can never help a candidate by not voting for them.[1][2]

### Details

In deterministic single winner elections that is to say no winner is harmed by up-ranking and no loser can win by down-ranking. If the method relies on chance, then up-ranking a candidate can not decrease that candidate's chance of winning, nor can down-ranking the candidate increase it. Douglas R. Woodall called the criterion mono-raise.[3]

Raising a candidate x on some ballots while changing the orders of other candidates does not constitute a failure of monotonicity. E.g., harming candidate x by changing some ballots from z > x > y to x > y > z isn't a violation of the monotonicity criterion.

The monotonicity criterion renders the intuition that there should be neither need to worry about harming a candidate by (nothing else than) up-ranking nor it should be possible to support a candidate by (nothing else than) counter-intuitively down-ranking.

The result of David Austen-Smith and Jeffrey Banks that monotonicity in individual preferences is impossible is a nonissue: For given voter preferences v=v_1...v_n and a winner x under voting scheme alpha, they investigate changes in v, where e.g. altering v_i from a,b,c,d,x to d,c,x,b,a is allowed, which can't be seriously named a monotonicity property. That allows random permutations even ahead of x, and is therefore even more rigid than Woodall's mono-raise-random, which is already incompatible with [majority AND later-no-help AND later-no-harm]. [4]

Noncompliance with the monotonicity criterion doesn't tell anything about the likelihood of monotonicity violations, failing in one of a million possible elections would be as well a violation as missing the criterion in any possible election. Nor does compliance tell anything about the effect of other candidates: it's possible for a method to change the winner from X to Y when Z is ranked higher on some ballots without failing the monotonicity criterion.

Of the single-winner ranked voting systems, Borda, Schulze, Ranked Pairs, Maximize Affirmed Majorities, Descending Solid Coalitions, and Descending Acquiescing Coalitions[3] are monotone, while Coombs' method, runoff voting, and instant-runoff voting (IRV) are not.

Most variants of the single transferable vote (STV) proportional representation methods are not monotonic, especially all that are currently in use for public elections (which simplify to IRV when there is only one winner). No ranked voting method has been proven to pass both Droop proportionality and monotonicity, though it is suspected that Schulze STV passes both.

All plurality voting systems are monotone if the ballots are treated as rankings where using more than two ranks is forbidden. In this setting first past the post and approval voting as well as the multiple-winner systems single non-transferable vote, plurality-at-large voting (multiple non-transferable vote, bloc voting) and cumulative voting are monotonic. Party-list proportional representation using D'Hondt, Sainte-Laguë or the largest remainder method is monotone in the same sense.

In elections via the single-winner methods range voting and majority judgment nobody can help a candidate by reducing or removing support for them. The definition of the monotonicity criterion with regard to these methods is disputed. Some voting theorists argue that this means these methods pass the monotonicity criterion; others say that, as these are not ranked voting systems, they are out of the monotonicity criterion's scope.

### Implications

It's impossible for a method to pass all of monotonicity, later-no-harm, later-no-help, and mutual majority,[3] but there do exist methods that pass three of the four. First past the post passes the first three, instant-runoff voting passes the last three, and Descending Acquiescing Coalitions and Descending Solid Coalitions pass one of the later-no-help/harm criteria as well as monotonicity and mutual majority.

## Definition of monotonicity criteria

The general pattern of monotonicity criteria is:

If X is a winner under a voting rule, and one or more voters change their preferences in a way favourable to X, then X should still be a winner.

or for methods that employ some element of chance,

If one or more voters change their preferences in a way favourable to X, then the chance that X is elected should never decrease.

Different definitions of "favourable to X" lead to different monotonicity criteria. The primary monotonicity criterion, mono-raise, is:

If X is a winner under a voting rule, and one or more voters change their preferences by ranking or rating X higher without otherwise changing their ballots, then X should still be a winner.

## Instant-runoff voting and the two-round system are not monotonic

Using an example that applies to instant-runoff voting (IRV) and to the two-round system, it is shown that these voting systems violate the mono-raise criterion. Suppose a president were being elected among three candidates, a left, a right, and a center candidate, and 100 votes cast. The number of votes for an absolute majority is therefore 51.

Suppose the votes are cast as follows:

```28: Right > Center
5: Right > Left
30: Left > Center
5: Left > Right
16: Center > Left
16: Center > Right
```

According to first preferences, Left finishes first with 35 votes, Right gets 33 votes, and Center 32 votes, thus all candidates lack an absolute majority of first preferences. In an actual runoff between the top two candidates, Left would win against Right with 30+5+16=51 votes. The same happens (in this example) under IRV, Center gets eliminated, and Left wins against Right with 51 to 49 votes.

But if at least two of the five voters who ranked Right first, and Left second, would raise Left, and vote 1st Left, 2nd Right; then Left would be defeated by these votes in favor of Center. Let's assume that two voters change their preferences in that way, which changes the following ballots:

```4: Right > Left
7: Left > Right
```

Now Left gets 37 first preferences, Right only 31 first preferences, and Center still 32 first preferences, and there is again no candidate with an absolute majority of first preferences. But now Right gets eliminated, and Center remains in round 2 of IRV (or the actual runoff in the Two-round system). And Center beats its opponent Left with a remarkable majority of 60 to 40 votes.

### Estimated likelihood of IRV lacking monotonicity

Crispin Allard argued, based on a mathematical model that the probability of monotonicity failure actually changing the result of an election for any given constituency would be 1 in 4000;[5] however, Lepelley et al.[6] found a probability of 397/6912 = 5.74% for 3-candidate elections.

Warren Smith argued that in the impartial culture model, IRV fails monotonicity with 14.5% probability in three-candidate elections.[7] In elections with more than 3 candidates, these probabilities tend to increase eventually toward 100% (in some models this limit has been proven, in others it is only conjectured). Nicholas Miller also disputed Allard's conclusion and provided a different mathematical model.[8]

## Real-life monotonicity violations

If the ballots of a real election are released, it is fairly easy to prove if

• election of a candidate could have been circumvented by raising them on some of the ballots, or
• election of an otherwise unelected candidate by lowering them on some of the ballots

would have been possible (nothing else is altered on any ballot). Both events can be considered as real-life monotonicity violations.

However, the ballots (or information allowing them to be reconstructed) are rarely released for ranked voting elections, which means there are few recorded monotonicity violations for real elections.

### 2005 German federal election

A party-list strategy exploiting something similar (down-ranking CDU and additionally up-ranking another party, e.g. FDP) happened in the German federal election of 2005, in which conservative voters in Dresden deliberately voted against the CDU, their party of choice, in order to maximize that party's number of seats in the federal parliament. This was possible due to Germany's voting system (mixed member proportional with overhang seats computed independently for each federal state) and the fact that the vote in Dresden took place a week after the rest of the country due to the death of a candidate, enabling voters in Dresden to vote tactically in full knowledge of the results already achieved elsewhere. As a result of this, the German Constitutional Court ruled on July 3 2008 that the German voting system must be reformed to eliminate its non-monotonicity.[9]

### 2009 Burlington, Vermont mayoral election

[[2009 Burlington mayoral election]]

A real-life monotonicity violation was detected in the 2009 Burlington mayoral election (in Vermont) under instant-runoff voting (IRV), where the necessary information is available. In this election, the winner Bob Kiss could have been defeated by raising him on some of the ballots. For example, if all voters who ranked Kurt Wright over Bob Kiss over Andy Montroll, would have ranked Kiss over Wright over Montroll, and additionally some people who ranked Wright but not Kiss or Montroll, would have ranked Kiss over Wright, then these votes in favor of Kiss would have defeated him.[10] The winner in this scenario would have been Andy Montroll, who was also the Condorcet winner according to the original ballots, i.e. for any other running candidate, a majority ranked Montroll above the competitor.

### Australian elections and by-elections

Since 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, 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.[nb 1]

However, for the Australian federal election, 2010, one article was aware of the non-monotonicity possibility: Why Labor Voters In Melbourne Need To Vote Liberal. In 2009, the theoretical disadvantage of non-monotonicity worked out in practice in a state by-election in the South Australian seat of Frome. The eventual winner, an Independent who was a town mayor, scored only third on the primaries with about 21% of the vote. But since the National Party of Australia scored 4th place, their preferences were distributed beforehand, allowing the Independent to overtake the Australian Labor Party Candidate by 31 votes. Thus Labor was pushed into third place, and their preference distribution favoured the Independent, who overtook the leading Australian Liberal Party candidate to win the election. However, had anywhere between 31 and 321 of the voters who preferred Liberal over Labor and Independent switched their support from Liberal to Labor, it would have allowed the Liberal to win the IRV election. This is classic monotonicity violation: the 321 who voted for the Liberals took part in hurting their own candidate.[11]

## Other forms of monotonicity

There are several variations of the "monotonicity criterion". For example, there's what Douglas R. Woodall called "mono-add-plump". These are described in the following section. Agreement with such rather special properties is the best any ranked voting system may fulfill: The Gibbard–Satterthwaite theorem shows, that any meaningful ranked voting system is susceptible to some kind of tactical voting, and Arrow's impossibility theorem shows that individual rankings can't be meaningfully translated into a community-wide ranking where the order of candidates x and y is always independent of irrelevant alternatives z.

### Woodall's monotonicity criteria

[[Douglas Woodall ]]

Douglas Woodall defined several variations or types of monotonicity.[3] In Woodall's definitions, a candidate x should not be harmed if:

• (mono-raise) x is raised on some ballots without changing the orders of the other candidates;
• (mono-raise-delete) x is raised on some ballots and all candidates now below x on those ballots are deleted from them;
• (mono-raise-random) x is raised on some ballots and the positions now below x on those ballots are filled (or left vacant) in any way that results in a valid ballot;
• (mono-append) x is added at the end of some ballots that did not previously contain x;
• (mono-sub-plump) some ballots that do not have x top are replaced by ballots that have x top with no second choice;
• (mono-sub-top) some ballots that do not have x top are replaced by ballots that have x lop (and are otherwise arbitrary);
• (mono-add-plump): A candidate x should not be harmed if further ballots are added that have x top with no second choice.
• (mono-add-top) further ballots are added that have x top (and are otherwise arbitrary);
• (mono-remove-bottom) some ballots are removed, all of which have x bottom, below all other candidates.

### Multi-winner monotonicity

Monotonicity would be more aptly named endorsement monotonicity since it is the preservation of monotonicity relative to endorsement. Since it is the most important form of monotonicity is bears the simple naming. There are however two other important forms of monotonicity for multi-winner voting systems, Population monotonicity and House monotonicity.

Multi-winner monotonicity could also be considered in a weaker and stronger sense: the weak form is satisfied whenever, if A is one of the winners, ranking A higher does not kick A out of the winning set; whereas the stronger form is satisfied whenever, if A is one of the winners, ranking A higher does not kick anyone out of the winning set. In a single winner election, these criteria are the same, but the stronger form is harder to satisfy for multi-winner. Woodall's definition of mono-raise corresponds to the weak form.

## Footnotes

### Notes

1. If the Australian election authorities were to publish the full ranking of every voter, this could be used to facilitate vote buying or coercion. However, anonymization could close the hole, e.g. by truncating the reported ballots so that for every reported ranking, at least 0.1% of the voters began their ballots that way.

### References

1. Some parts of this article are derived from text at https://web.archive.org/web/20090610060543/http://condorcet.org/emr/criteria.shtml which was released to the public domain.
2. Mono-raise test copied from https://electowiki.org/w/index.php?title=Mono-raise_criterion&oldid=15949
3. a b c d Woodall, Douglas R. (1996). "Monotonicity and single-seat election rules". Voting matters. 6: 9–14.
4. Austen-Smith, David; Banks, Jeffrey (2014-08-01). "Monotonicity in Electoral Systems - American Political Science Review". American Political Science Review. 85 (2): 531–537. doi:10.2307/1963173. ISSN 1537-5943. Retrieved 2020-02-03.
5. Estimating the Probability of Monotonicity Failure in a UK General Election
6. Lepelley, Dominique; Chantreuil, Frédéric; Berg, Sven (1996-06-01). "The likelihood of monotonicity paradoxes in run-off elections". Mathematical Social Sciences. 31 (3): 133–146. doi:10.1016/0165-4896(95)00804-7. ISSN 0165-4896. Retrieved 2020-02-03.
7. Smith, Warren D. "Monotonicity and Instant Runoff Voting". rangevoting.org.
8. "Monotonicity failure under STV and related voting systems" (PDF). 2002-03-22. Retrieved 2020-02-03.
9. See e.g. [1]
10. Burlington Vermont 2009 IRV mayor election
11. http://blogs.abc.net.au/antonygreen/2011/05/an-example-of-non-monotonicity-and-opportunites-for-tactical-voting-at-an-australian-election.html