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).[1] That is to say, in single winner elections no winner is harmed by up-ranking and no loser is helped by down-ranking. Douglas R. Woodall called the criterion mono-raise.

Visual examples of monotonicity in different voting systems. Colored areas correspond to the winner under each system when voter opinion shifts toward or away from the candidates (represented by dots). In monotonic systems, the colored areas have compact shapes, and shifting public opinion towards one of the candidate dots will either elect that candidate or have no effect on the results. In non-monotonic systems, the colored areas can have jagged or disjoint shapes, and shifting the center of public opinion toward a candidate may move from an area where that candidate would win to an area where the candidate loses after gaining additional support.

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 > z > y would violate the monotonicity criterion, while harming candidate x by changing some ballots from z > x > y to x > y > z would not.

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. 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.

Of the single-winner ranked voting systems, Borda, Schulze, ranked pairs, maximize affirmed majorities, descending solid coalitions,[2] and descending acquiescing coalitions[1][3] are monotonic, while Coombs' method, runoff voting, and instant-runoff voting (IRV) are not. The multi-winner single transferable vote (STV) system is also non-monotonic.

While Woodall articulated monotonicity in the context of ordinal voting systems, the property can be generalized to cardinal voting and plurality voting systems by evaluating whether reducing or removing support for a candidate can help that candidate win an election. In this context, first past the post, approval voting, range voting, majority judgment, as well as the multiple-winner systems single non-transferable vote, plurality-at-large voting (multiple non-transferable vote, bloc voting) and cumulative voting are monotonic. Party-list proportional representation using D'Hondt, Sainte-Laguë or the largest remainder method is monotonic in the same sense.