Monotonicity criterion

From Wikipedia, the free encyclopedia
A Yee diagram showing the winner for different electorates. The win region for each candidate is erratic, with random pixels dotting the image and jagged, star-shaped (convex) regions occupying much of the image. Moving the electorate to the left can cause a right-wing candidate to win, and vice versa.
A 4-candidate Yee diagram under IRV. Colors show the winning candidate when the median voter is located nearby. Black lines show the optimal solution (achieved by Condorcet or score). Note the erratic win regions under IRV.

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).

By method[edit]

Runoff voting[edit]

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 25% ☒N +6% Top 31% 46%
Center 30% 55% checkY Center 30% ☒N
Bottom 45% 45% -6% Bottom 39% 54% checkY

Quota rules[edit]

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]

Frequency of violations[edit]

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]

Theoretical models[edit]

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]

Real-world situations[edit]

Alaska 2022[edit]

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[edit]

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]

Australian elections and by-elections[edit]

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]

Louisiana governor races[edit]

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]

See also[edit]

Notes[edit]

  1. ^ Apart from majority judgment, these systems satisfy an even stronger form of positive responsiveness: if there is a tie, any increase in a candidate's rating will break the tie in that candidate's favor.

References[edit]

  1. ^ a b c d Pukelsheim, Friedrich (2014). Proportional representation : apportionment methods and their applications. Internet Archive. Cham ; New York : Springer. ISBN 978-3-319-03855-1.
  2. ^ May, Kenneth O. (1952). "A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision". Econometrica. 20 (4): 680–684. doi:10.2307/1907651. ISSN 0012-9682.
  3. ^ a b D R Woodall, "Monotonicity and Single-Seat Election Rules", Voting matters, Issue 6, 1996
  4. ^ 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.
  5. ^ May, Kenneth O. (1952). "A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision". Econometrica. 20 (4): 680–684. doi:10.2307/1907651. ISSN 0012-9682.
  6. ^ Graham-Squire, Adam T.; McCune, David (2023-06-12). "An Examination of Ranked-Choice Voting in the United States, 2004–2022". Representation: 1–19. arXiv:2301.12075. doi:10.1080/00344893.2023.2221689.
  7. ^ 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.
  8. ^ 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.
  9. ^ 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%
  10. ^ 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%
  11. ^ 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%
  12. ^ Miller, Nicholas R. (2012). MONOTONICITY FAILURE IN IRV ELECTIONS WITH THREE ANDIDATES (PowerPoint). p. 23. Impartial Culture Profiles: All, Total MF: 15.0%
  13. ^ Quas, Anthony (2004-03-01). "Anomalous Outcomes in Preferential Voting". Stochastics and Dynamics. 04 (1): 95–105. doi:10.1142/S0219493704000912. ISSN 0219-4937.
  14. ^ Burlington Vermont 2009 IRV mayor election
  15. ^ "RangeVoting.org - Australia 2007 elections - IRV pathologies galore". www.rangevoting.org. Retrieved 2024-02-04.
  16. ^ "RangeVoting.org - Louisiana Governor Races 1975-2007". www.rangevoting.org. Retrieved 2024-02-06.