If an alternative X loses, and the ballots are changed only by placing X in lower positions, without changing the relative position of other candidates, then X must still lose.
A looser way of phrasing this is that in a non-monotonic system, voting for a candidate can cause that candidate to lose. Systems which fail the monotonicity criterion suffer a form of tactical voting where voters might try to elect their candidate by voting against that candidate.
Plurality voting, Majority Choice Approval, Borda count, Schulze, Maximize Affirmed Majorities, and Descending Solid Coalitions are monotonic, while Coombs' method and Instant-runoff voting are not. Approval voting is monotonic, using a slightly different definition, because it is not a preferential system: You can never help a candidate by not voting for them.
Some parts of this article are derived from text at http://condorcet.org/emr/criteria.shtml
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