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This article is about a voting system criterion. See monotonic function for a mathematical notion.

The monotonicity criterion is a voting system criterion used to analyze both single and multiple winner voting systems. A voting system is monotonic if it satisfies one of the definitions of the monotonicity criterion, given below.

Douglas Woodall, calling the criterion mono-raise, defines it as:

A candidate x should not be harmed [i.e., change from being a winner to a loser] if x is raised on some ballots without changing the orders of the other candidates.

Mike Ossipoff defines the monotonicity criterion as:

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.

The definitions are logically equivalent. Note that the references to orders and relative positions concern the rankings of candidates other than X, on the set of ballots where X has been raised. So, if changing a set of ballots voting "A > B > C" to "B > C > A" causes B to lose, this does not constitute failure of Monotonicity, because in addition to raising B, we changed the relative positions of A and C.

This criterion may be intuitively justified by reasoning that in any fair voting system, no vote for a candidate, or increase in the candidate's ranking, should instead hurt the candidate. It is an important property considered in Arrow's impossibility theorem.

It is generally considered a good thing if a voting system is monotonic. Clearly, non-monotonicity is very counter-intuitive, although some do promote such systems (see Instant-runoff voting). Furthermore, although all voting systems are vulnerable to tactical voting, systems which fail the monotonicity criterion suffer an unusual form, where voters might try to elect their candidate by voting against that candidate. Tactical voting in this way presents an obvious risk if a voter's information about other ballots is wrong, however, and because of this non-monotonic voting systems may be somewhat discouraging of tactical voting.

Of the single-winner voting systems, First Past the Post, Majority Choice Approval, Borda count, Schulze method, and Maximize Affirmed Majorities are monotonic, while Coombs' method and Instant-runoff voting are not. The single-winner methods of range voting and approval voting are also monotonic as one can never help a candidate by reducing or removing support for them, but these require a slightly different definition of monotonicity as they are not preferential systems.

Of the multiple-winner voting systems, all plurality voting methods are monotonic, such as bloc voting, cumulative voting, and the single non-transferable vote. Those versions of the Single Transferable Vote which simplify to Instant Runoff when there is only one winner are not monotonic.

Suppose a president were being elected by instant runoff. Also suppose there are 3 candidates, and 100 votes cast. The number of votes required to win is therefore 51.

Suppose the votes are cast as follows:

Number of votes	1st Preference	2nd Preference
39	Andrea	Belinda
35	Belinda	Cynthia
26	Cynthia	Andrea

Cynthia is eliminated, thus transferring votes to Andrea, who is elected with a majority. She then serves a full term, and does such a good job that she persuades ten of Belinda's supporters to change their votes to her at the next election.

This election looks thus:

Number of votes	1st Preference	2nd Preference
49	Andrea	Belinda
25	Belinda	Cynthia
26	Cynthia	Andrea

Because of the votes Belinda loses, she is eliminated first this time, and her second preferences are transferred to Cynthia, who now wins 51 to 49. In this case Andrea's preferential ranking increased between elections - more electors put her first - but this increase in support caused her to lose.

In a real election, however, such problems may be more difficult to detect because there would be other movements of votes, and it may not be easily determined whether the same people cast the same votes.

Further, it has been argued that the circumstances where this could occur would be extremely rare, fewer than once per century under normal political conditions [1] - although this is subject to complex mathematical disputation [2].

• Woodall, Douglas R. "Monotonicity and Single-Seat Election Rules" Voting Matters, Issue 6, 1996.

Some parts of this article are derived from text at http://condorcet.org/emr/criteria.shtml

• Voting system
• Voting system criterion