The monotonicity criterion is a fairness test for voting systems. It says that ranking a candidate higher on your ballot should never cause that candidate to lose, and ranking a candidate lower should never cause them to win. If a voting method can produce either of those paradoxical outcomes, it fails the monotonicity criterion.
This concept comes from social choice theory, the branch of mathematics that studies how groups make collective decisions. It sounds like something every voting system would satisfy automatically, but several widely used methods don’t.
How the Criterion Works
Imagine a candidate is winning an election. Now suppose some voters change their ballots to rank that candidate even higher, and nothing else changes. Under a monotonic voting system, the candidate still wins. The extra support can’t backfire. The same logic works in reverse: if a losing candidate picks up fewer votes or gets ranked lower, that shift can’t somehow push them into winning.
The criterion applies specifically to ranked voting systems, where voters list candidates in order of preference rather than just picking one. In these systems, the method used to tally those rankings determines whether monotonicity holds. Some tallying methods create chain reactions during elimination rounds that can produce counterintuitive results, where gaining support actually hurts a candidate.
Which Voting Methods Pass and Fail
Of the most commonly discussed ranked voting methods, three satisfy monotonicity and one does not:
- Plurality (first-past-the-post): Satisfies the criterion. More first-place votes always help.
- Borda count: Satisfies. Higher rankings translate directly into more points.
- Pairwise comparison (Condorcet methods): Satisfies. Ranking a candidate higher strengthens their head-to-head matchups.
- Ranked-choice voting (instant-runoff): Violates. The round-by-round elimination process can cause a candidate to lose precisely because they gained additional support.
Approval voting, where voters simply approve or disapprove of each candidate without ranking them, also satisfies monotonicity. More approvals always help.
Why Ranked-Choice Voting Fails
Ranked-choice voting (also called instant-runoff voting, or IRV) eliminates the candidate with the fewest first-place votes each round and redistributes their supporters’ votes to whatever candidate those voters ranked next. This elimination sequence is where the problem lives. Shifting votes to a candidate can change who gets eliminated in earlier rounds, which reshuffles how votes redistribute, which can ultimately knock out a stronger competitor who would have beaten the original winner.
There are two types of failure. “Upward” monotonicity failure happens when a winning candidate loses after gaining additional support. “Downward” monotonicity failure happens when a losing candidate wins after losing support. Both are logically possible under IRV’s elimination rules.
The Burlington 2009 Election
The most cited real-world example comes from the 2009 mayoral race in Burlington, Vermont, which used instant-runoff voting with three main candidates: Bob Kiss, Andy Montroll, and Kurt Wright.
When voters’ full rankings were examined, Andy Montroll was preferred over Kurt Wright by a margin of 933 voters and preferred over Bob Kiss by 588 voters. He beat every other candidate in a head-to-head matchup. But IRV’s elimination rounds never gave him the chance. Montroll was eliminated earlier in the process, and the final round came down to Kiss versus Wright. Kiss won by just 252 votes out of 8,374 cast.
The monotonicity problem showed up in the structure of the result. Of the voters who ranked Wright first, 1,510 had listed Montroll as their second choice and didn’t rank Kiss at all. If just 371 of those voters had instead ranked Montroll first (raising Montroll while lowering Wright), the elimination sequence would have changed and Montroll would have won. In other words, Wright’s strong first-round showing actually prevented the election of the candidate his own supporters preferred over Kiss. The extra support for Wright backfired on the voters who gave it.
Burlington repealed instant-runoff voting after this election.
How Often Monotonicity Failures Occur
Monotonicity failure isn’t just a theoretical curiosity that rarely shows up in practice. Research from the University of Maryland, Baltimore County, building on earlier analytical work, found that the risk scales dramatically with how competitive an election is.
Under one statistical model that treats all ballot orderings as equally likely, about 4.5% of three-candidate election profiles are vulnerable to upward monotonicity failure and about 2% to downward failure. But that model includes many lopsided, uncompetitive elections where the problem can’t arise simply because one candidate dominates.
When researchers simulated more realistic, competitive elections, the numbers climbed sharply. In closely contested races, upward monotonicity failure appeared in 45% or more of ballot profiles. When elections were most tightly contested, more than 50% of profiles were vulnerable to some form of monotonicity failure. Even in moderately competitive simulations, vulnerability ranged from about 12% to 36% depending on how closely clustered voter preferences were.
The pattern is clear: the more competitive the race, the more likely the elimination sequence can be disrupted by small shifts in support. Monotonicity failure is most likely to strike in exactly the elections where it matters most.
Connection to Broader Voting Theory
Monotonicity is one of several fairness criteria that mathematicians use to evaluate voting systems. It’s closely related to a concept called “positive association,” which Kenneth Arrow included in his foundational work on social choice. Positive association says that if a voting system outputs a result where candidate X beats candidate Y, then voters raising X in their rankings should never reverse that outcome. Combined with other basic requirements, positive association implies the Pareto principle: if every single voter prefers X over Y, the system must rank X above Y.
Arrow’s impossibility theorem proved that no ranked voting system can satisfy all desirable fairness criteria simultaneously. Every system involves tradeoffs. A method like plurality satisfies monotonicity but fails other tests, such as the independence of irrelevant alternatives (the idea that adding or removing a third candidate shouldn’t change whether X beats Y). Ranked-choice voting handles some spoiler effects better than plurality but sacrifices monotonicity in the process.
Understanding these tradeoffs is the practical value of criteria like monotonicity. No voting system is perfect, but knowing exactly how each one can fail helps communities choose the system whose failure modes they find least acceptable.