Monotonicity criterion

The positive response,   monotonicity, or nonperversity criterion is a principle of social choice theory that says that increasing a candidate's ranking or rating should not cause them to lose. Positive response rules out cases where a candidate loses an election as a result of receiving too much support from voters (i.e. being "too popular to win"); rules that violate positive response are called perverse.

Systems that violate positive response (such as instant-runoff and the two-round system) can create situations where a voter's ballot has a reversed effect on the election, making it "less than worthless". This runs counter to the basic principle that increasing an option's popularity in a democratic election should only improve the chances of that option winning; as a result, German courts have previously struck down nonmonotonic systems for violating the right to equal and direct suffrage.

Most voting systems (including Borda and all common tournament solutions) satisfy positive response, as do all commonly-used rated voting methods (including approval, score, and their proportional counterparts).

However, the criterion is violated by instant-runoff voting, the single transferable vote, and Hamilton's apportionment method.

The participation criterion is a closely-related, but different, concept. While positive responsiveness deals with a voter changing their opinion (or vote), participation deals with situations where a voter choosing to cast a ballot can have a reversed effect on the election.

Runoff voting
Runoff-based voting systems, such as ranked choice voting (instant-runoff) fail the monotonicity criterion. A notable example is the 2009 Burlington mayoral election, the United States' second instant-runoff election in the modern era, where Bob Kiss won the election as a result of 750 ballots ranking him in last place.

An example with three parties (Top, Center, Bottom) is shown below. In this scenario, the Bottom party initially loses. However, they are elected after running an unsuccessful campaign and adopting an unpopular platform, which pushes their supporters away from the party and into the Top party. This election is an example of a center-squeeze, a class of elections where instant-runoff and plurality have difficulties electing the majority-preferred candidate, because the first-round vote is split between an extremist and a moderate. Here, the loss of support for Bottom policies makes the Top party more popular, allowing it to defeat the Center party in the first round.

A famous example of a monotonicity failure is the 2022 Alaska at-large special election.

Quota rules
Proportional representation systems using largest remainders for apportionment do not pass the monotonicity criterion. This happened in the 2005 German federal election, when CDU voters in Dresden were instructed to vote for the FDP, a strategy that allowed the party an additional seat. As a result, the Federal Constitutional Court ruled that negative voting weights violate the German constitution's guarantee of equal and direct suffrage.

Frequency of violations
For electoral methods failing positive value, the frequency of monotonicity violations will depend on the electoral method, the candidates, and the distribution of outcomes. Negative voting weights tend to be most common with instant-runoff, with what some researchers have described as an "unacceptably high" frequency.

Theoretical models
Results using the impartial culture model estimate about 15% of elections with 3 candidates; however, the true probability may be much higher, especially when restricting observation to close elections. For moderate numbers of candidates, the probability of a monotonicity failure quickly approaches 100%.

A 2013 study using a 2D spatial model with various voter distributions estimated at least 15% of IRV elections are nonmonotonic in the best-case scenario (when only three candidates run). The researchers concluded that "three-way competitive races will exhibit unacceptably frequent monotonicity failures" and "In light of these results, those seeking to implement a fairer multi-candidate election system should be wary of adopting IRV."

Alaska 2022
Alaska's first-ever instant-runoff election resulted in negative vote weights for many Republican supporters of Sarah Palin, who could have defeated Mary Peltola by placing her first on their ballots.

Burlington, Vermont
In Burlington's second IRV election, incumbent Bob Kiss was re-elected, despite losing in a head-to-head matchup with Democrat Andy Montroll (the Condorcet winner). However, if Kiss had gained more support from Wright voters, Kiss would have lost.

Survey of nonmonotonic elections
A survey of 185 American instant-runoff elections where no candidate was ranked first by a majority of voters found five additional elections containing monotonicity failures.

2005 German Election in Dresden
A negative voting weight event famously resulted in the abolition of Hamilton's method for apportionment in Germany after the 2005 federal election. CDU voters in Dresden were instructed to strategically vote for the FDP, a strategy that allowed the party to earn an additional seat, causing substantial controversy. As a result, the Federal Constitutional Court ruled that negative voting weights violate the German constitution's guarantee of equal and direct suffrage.