# Monotonicity

The monotonicity criterion is a voting system criterion used to evaluate both single and multiple winner ranked voting systems. A ranked voting system 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). [1] 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.

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.

There are several variations of that criterion; e.g., what Douglas R. Woodall called mono-add-plump: A candidate x should not be harmed if further ballots are added that have x top with no second choice. 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. 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.

Of the single-winner ranked voting systems, Borda, Schulze, ranked pairs, maximize affirmed majorities, Descending Solid Coalitions, and Descending Acquiescing Coalitions[1] are monotone, while Coombs' method, runoff voting, and instant-runoff voting (IRV) are not.

Most variants of the single transferable vote (STV) proportional representations are not monotonic, especially all that are currently in use for public elections (which simplify to IRV when there is only one winner).

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.

## Statement of Monotonicity Criteria

``` 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
```

## 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:

Preference Voters
1st 2nd
Right Center 28
Right Left 5
Left Center 30
Left Right 5
Center Left 16
Center Right 16

According to the 1st 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 two rows of the table:

Preference Voters
1st 2nd
Right Left 3
Left Right 7

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;[2] however, Lepelley et al.[3] found a probability of 397/6912 = 5.74% for 3-candidate elections.

Another probability model, the "impartial culture", yields about 15% probability. 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). Estimates of 5–15% order are easily confirmed in any probability model with "Monte Carlo experiments" and the aid of the "was it monotonic?" tests stated in the Lepelley paper.[citation needed] Nicholas Miller also disputed Allard's conclusion and provided a different mathematical model.[4]

## 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.[5]

### 2009 Burlington, Vermont mayoral election

A real-life monotonicity violation was detected in the 2009 Burlington, Vermont mayor election 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.[6] 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

Template:Refimprove section 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. (The policy of Australia's election authorities not to release this data is justifiable on privacy grounds.[according to whom?] 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.)

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.[7]

## Other forms of Monotonicity

### Single-winner monotonicity

There are several variations or types of monotonicity:

MONOTONICITY. A candidate x should not be harmed if:

• l (MONO-RAISE) x is raised on some ballots without changing the orders of the other candidates;
• l (MONO-RAISE-DELETE) x is raised on some ballots and all candidates now below x on those ballots are deleted from them;
• l (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;
• l (MONO-APPEND) x is added at the end of some ballots that did not previously contain x;
• l (MONO--SUB-PLUMP) some ballots that do not have x top are replaced by ballots that have x top with no second choice;
• l (MONO-SUB-TOP) some ballots that do not have x top are replaced by ballots that have x lop (and are otherwise arbitrary);
• l (MONO-ADD-PLUMP) further ballots are added that have x top with no second choice;
• l (MONO-ADD-TOP) further ballots are added that have x top (and are otherwise arbitrary);
• l (MONO-REMOVE-BOTTOM) some ballots are removed, all of which have x bottom, below all other candidates. [8]

### 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.