Monotonicity criterion

In the comparison of electoral systems, the monotonicity criterion says that ranking a candidate higher on some ballots should not cause them to lose. Douglas Woodall called the criterion mono-raise to distinguish it from participation (sometimes called population monotonicity).

More formally, monotonicity says that if a voter modifies their ballot to ranking a candidate higher, the social ordering function should respond only by promoting that same option or not changing, never by placing it lower than before.

The monotonicity criterion is a way to make formal the idea that social choice functions or electoral systems should not exhibit "spite" towards some voters in the sense that it actively seeks to frustrate their preferences. It also formalizes the idea that voters should not have to "lie" to the system about who they prefer just to keep it from spiting them.

Plurality, Borda, Schulze, ranked pairs, descending solid coalitions, and descending acquiescing coalitions are monotonic, while Coombs' method and instant-runoff voting (IRV) are not.

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.