The monotonicity criterion, also called positive weight,[1][2] is a principle of social choice theory that says voters should never have a negative (that is, reversed) effect on an election's results. In other words, increasing a winning candidate's grade should not cause them to lose.[3]
Positive association rules out cases where a candidate loses an election simply because they got "too much support." Systems that fail positive responsiveness arguably violate the principle of one man, one vote by creating situations where there is "at least one man with negative-one votes."[1]
Most voting systems (including Borda, Schulze, ranked pairs, and descending solid coalitions) satisfy monotonicity,[3] as do all commonly-used rated voting methods (including approval, score, graduated majority judgment, and STAR voting).[note 1]
However, the criterion is violated by instant-runoff voting, the single transferable vote, two-round systems,[4] and occasionally by Hamilton's method.[1]
Because of its importance, monotonicity is included among the original conditions of Arrow's impossibility theorem and May's theorem[5] (though Arrow later showed monotonicity could be replaced with the weaker Pareto efficiency).
Runoff voting systems (including instant-runoff voting, two-round runoff, and nonpartisan blanket primaries) 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.[6]
An example with three parties (Top, Center, Bottom) is shown below. In this scenario, the Bottom party is defeated after a successful campaign and popular platform earns them more supporters from the Top party (shifting voters in their direction). As a result, instant-runoff voting can sometimes reward candidates for being extreme, incompetent, or unpopular.
Popular Bottom | 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% |
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 strategically vote for the FDP, a strategy that earned 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.[1]
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 events tend to be most common with instant-runoff voting, leading some researchers who study the issue to argue that in particular exhibits monotonicity violations (and similar pathologies) with an "unnaceptably high" frequency.[7]
Results using the impartial culture model estimate about 15% of elections with 3 candidates;[8][9][10][11][12] however, the true probability may be much higher, especially when restricting observation to close elections.[13][9][10] For moderate numbers of candidates, the probability of a monotonicity failure quickly approaches 100%.[8]
A 2013 study using a 2D spatial model with various voter distributions estimated that at least 15% of IRV elections are nonmonotonic in the best-case scenario (when only 3 candidates run), with substantially larger values for more than 3 candidates. The authors 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."[7]
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.
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.[14]
Because Australian elections are typically held "in the black" (without public knowledge of the votes cast for each candidate), most monotonicity violations go undetected in Australia, suggesting they may be far more common than otherwise assumed.
An analysis of all 2007 election results found that every election where the result differed from that of plurality suffered from a monotonicity or participation failure.[15]
An analysis of Louisiana's gubernatorial elections (conducted with runoff voting) estimated that around 20% of elections in the state suffered from monotonicity failures, while 40% suffered participation failures.[16]
let's consider only 3-candidate IRV elections ... In the "random elections model" ... monotonicity failure occurs once every 6.9 elections, i.e. 14.5% of the time. ... probability that the resulting IRV election is "non-monotonic" ... approaches 100% as N becomes large.
Phenomenon: Nonmonotonicity | REM: 15.2305%, Dirichlet: 5.7436%, Quas 1D: 6.9445%
Phenomenon: Nonmonotonicity | REM: 15.2304%, Dirichlet: 5.7435%, Quas 1D: 6.9444%
Impartial Culture Profiles: All, TMF: 15.1%
Impartial Culture Profiles: All, Total MF: 15.0%