Monotonicity criterion

The monotonicity criterion is a voting system criterion used to evaluate both single and multiple winner ranked voting systems. A ranked voting system is monotonic if it is neither possible to prevent the election of a candidate by ranking them higher on some of the ballots, nor possible to elect an otherwise unelected candidate by ranking them lower on some of the ballots (while nothing else is altered on any ballot). In single winner elections that is to say no winner is harmed by up-ranking and no loser can win by down-ranking. Douglas R. Woodall called the criterion mono-raise.

Raising a candidate x on some ballots while changing the orders of other candidates does not constitute a failure of monotonicity. E.g., harming candidate x by changing some ballots from z > x > y to x > y > z isn't a violation of the monotonicity criterion.

The monotonicity criterion renders the intuition that there should be neither need to worry about harming a candidate by (nothing else than) up-ranking nor it should be possible to support a candidate by (nothing else than) counter-intuitively down-ranking. There are several variations of that criterion; e.g., what Douglas R. Woodall called mono-add-plump: A candidate x should not be harmed if further ballots are added that have x top with no second choice. Agreement with such rather special properties is the best any ranked voting system may fulfill: The Gibbard–Satterthwaite theorem shows, that any meaningful ranked voting system is susceptible to some kind of tactical voting, and Arrow's impossibility theorem shows that individual rankings can't be meaningfully translated into a community-wide ranking where the order of candidates x and y is always independent of irrelevant alternatives z. Noncompliance with the monotonicity criterion doesn't tell anything about the likelihood of monotonicity violations, failing in one of a million possible elections would be as well a violation as missing the criterion in any possible election.

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