The positive response/association,[1] [2] monotonicity, or nonperversity criterion[3] is a principle of social choice theory that says that increasing a candidate's ranking or rating should not cause them to lose.[4] 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 also called perverse[5] and are said to suffer the more-is-less paradox. Such paradoxes are especially common in ranked-choice voting (RCV-IRV), a behavior which can lead to the elimination of moderate candidates and the election of extremists.[6]
Systems that violate positive response can create situations where a voter's ballot has a reversed effect on the election, making it "less than worthless"; as such, perversity is generally considered to be an exceptionally severe pathology,[7] and German courts have previously struck down nonmonotonic systems for violating the right to equal and direct suffrage.[8] [9]
Most ranked methods (including Borda and all common tournament solutions) satisfy positive response, as do all commonly-used rated voting methods (including approval, highest medians, and score).
However, the criterion is violated by instant-runoff voting,[10] the single transferable vote,[11] and Hamilton's apportionment method.[2]
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 has a reversed effect on the election.
Runoff-based voting systems, such as ranked choice voting (instant-runoff) fail the positive response 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.[12]
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.
| Unpopular Bottom | |||||||
| Round 1 | Round 2 | Round 1 | Round 2 | ||||
|---|---|---|---|---|---|---|---|
| Top | +6% | Top | 31% | 46% | |||
| Center | 30% | 55% | ↗ | Center | |||
| Bottom | 45% | 45% | -6% | Bottom | 39% | 54% |
A famous example of a less-is-more paradox can be seen in the 2022 Alaska at-large special election.
Proportional representation systems using largest remainders for apportionment do not pass the positive response 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.[13]
For electoral methods failing positive value, the frequency of less-is-more paradoxes 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.
Results using the impartial culture model estimate about 15% of elections with 3 candidates;[14] [15] however, the true probability may be much higher, especially when restricting observation to close elections.[16] For moderate numbers of candidates, the probability of a less-is-more paradoxes quickly approaches 100%.
A 2013 study using a two-dimensional spatial model of voting estimated at least 15% of IRV elections would be nonmonotonic in the best-case scenario (with only three equally-competitive candidates). 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."[17]
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.[18]
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.[12]
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.
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.[2]