# Monotonicity criterion

In the comparison of electoral systems, the monotonicity or mono-raise criterion says that ranking a candidate higher on some ballots should not cause them to lose.[1] The monotonicity criterion formalizes the idea that social choice functions and electoral systems should not exhibit "spite" towards some voters, in the sense of actively attempting to frustrate their preferences.

Plurality, Borda, Schulze, ranked pairs, and descending solid coalitions[1] are monotonic, while instant-runoff voting (IRV) is not.[1]

Cardinal systems typically satisfy an even stronger version of the criterion, that assigning a higher score to a candidate (without changing the order of the candidates) should never decrease the candidate's final placement. This criterion is satisfied by approval voting, score voting, STAR Voting, and graduated majority judgment.

Because of its importance, monotonicity was one of the original conditions for Arrow's impossibility theorem, before it was discovered not to be necessary and replaced by the weaker Pareto efficiency.

## By method

### Runoff voting

Runoff voting systems (including instant-runoff voting) fail the monotonicity criterion.

Suppose there are left, right, and center candidates competing for a total of 100 votes cast. Consider the two scenarios:

Scenario 1
Preference Voters
30 5 16 16 5 28
1st Left Left Center Center Right Right
2nd Center Right Left Right Left Center
Scenario 2
Preference Voters
30 7 16 16 3 28
1st Left Left Center Center Right Right
2nd Center Right Left Right Left Center
Tally
Round Votes
Left Center Right
1st 35 32 33
2nd 51 -- 49
Tally
Round Votes
Left Center Right
1st 37 32 31
2nd 40 60 --

According to 1st preferences in Scenario 1, 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.

Scenario 2 differs from Scenario 1 only by two voters- instead of ranking Right first, Left second, they rank Left first, Right second. Now Left gets 37 first preferences, Right receives 31 first preferences, and Center still receives 32 first preferences, and there is again no candidate with an absolute majority of first preferences. But now Right is eliminated, and Center remains in round 2 of IRV (or the actual runoff in the Two-round system). Center beats its opponent Left with a remarkable majority of 60 to 40 votes.

## Frequency of violations

For any election, the frequency of monotonicity violations will depend on the electoral method, the candidates, and the distribution of outcomes. Many (in fact, most practical) electoral methods, including score, approval, and STAR voting will never exhibit monotonicity failures. Researchers who study the issue have said that instant-runoff voting in particular exhibits monotonicity violations (and similar pathologies) with an "unnaceptably high" frequency.[2]

### Estimated likelihood of IRV lacking monotonicity

Results using the impartial culture model yields about 15% probability in elections with 3 candidates;[3][4][5][6][7] however, the true probability can be much higher, especially when restricting observation to close elections.[8][4][5] For moderate numbers of candidates and close elections, the probability of a monotonicity failure quickly approaches 100%.[3]

A 2013 study using a 2D spatial model with various voter distributions found that IRV was non-monotonic in at least 15% of competitive elections, increasing with number of candidates. The authors conclude 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."[2]

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.

Because full ballots are rarely released for ranked voting elections, monotonicity violations are likely more common than suggest

### Examples

#### 2009 Burlington election

In the 2009 Burlington mayoral election, incumbent Bob Kiss was re-elected despite losing in a head-to-head matchup with Democrat Andy Montroll (the Condorcet winner). However, if supporters of centrist candidate Kurt Wright had chosen to vote strategically by placing Kiss first on their ballots, Kiss would have lost.[9]

#### Australian elections and by-elections

Because Australian elections are typically held "in the black" (without public knowledge of the votes cast for each candidate), most examples of nonmonotonic reasoning go undetected in Australia, suggesting they may be far more common than otherwise assumed. In 2007, every Australian election where the result differed from that of plurality suffered a monotonicity or participation failure;[10] similar failures have been observed in state by-elections.[11]

#### Louisiana governor races

An analysis of Louisiana's gubernatorial elections (conducted with runoff voting) estimated around 20% of elections in the state suffered from monotonicity failures, while 40% suffered participation failures.[12]

## References

1. ^ a b c D R Woodall, "Monotonicity and Single-Seat Election Rules", Voting matters, Issue 6, 1996
2. ^ a b Ornstein, Joseph T.; Norman, Robert Z. (2014-10-01). "Frequency of monotonicity failure under Instant Runoff Voting: estimates based on a spatial model of elections". Public Choice. 161 (1–2): 1–9. doi:10.1007/s11127-013-0118-2. ISSN 0048-5829. S2CID 30833409.
3. ^ a b Smith, Warren D. (March 2009). "Monotonicity and Instant Runoff Voting". RangeVoting.org. Retrieved 2020-07-25. 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.
4. ^ a b Smith, Warren D. (August 2010). "IRV Paradox Probabilities in 3-candidate elections - Master List". RangeVoting.org. Retrieved 2020-07-25. Phenomenon: Nonmonotonicity | REM: 15.2305%, Dirichlet: 5.7436%, Quas 1D: 6.9445%
5. ^ a b Smith, Warren D. "Same IRV 3-candidate paradox probabilities from different random number generator". RangeVoting.org. Retrieved 2020-07-25. Phenomenon: Nonmonotonicity | REM: 15.2304%, Dirichlet: 5.7435%, Quas 1D: 6.9444%
6. ^ Miller, Nicholas R. (2016). "Monotonicity Failure in IRV Elections with Three Candidates: Closeness Matters" (PDF). University of Maryland Baltimore County (2nd ed.). Table 2. Retrieved 2020-07-26. Impartial Culture Profiles: All, TMF: 15.1%
7. ^ Miller, Nicholas R. (2012). MONOTONICITY FAILURE IN IRV ELECTIONS WITH THREE ANDIDATES (PowerPoint). p. 23. Impartial Culture Profiles: All, Total MF: 15.0%
8. ^ Quas, Anthony (2004-03-01). "Anomalous Outcomes in Preferential Voting". Stochastics and Dynamics. 04 (1): 95–105. doi:10.1142/S0219493704000912. ISSN 0219-4937.
9. ^ Burlington Vermont 2009 IRV mayor election
10. ^ "RangeVoting.org - Australia 2007 elections - IRV pathologies galore". www.rangevoting.org. Retrieved 2024-02-04.
11. ^ "An Example of Non-Monotonicity and Opportunites [sic] for Tactical Voting at an Australian Election". Antony Green's Election Blog. 2011-05-04. Archived from the original on 2011-05-08. Retrieved 2017-03-14.
12. ^ "RangeVoting.org - Louisiana Governor Races 1975-2007". www.rangevoting.org. Retrieved 2024-02-06.