Monotonicity criterion

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The monotonicity criterion, also called positive response^[1] or positive vote weight,^[2] is a principle of social choice theory that says that increasing a candidate's ranking or rating should not cause them to lose.^[3] Positive response rules out cases where a candidate loses an election as a result of receiving too much support from voters (i.e. being "too popular to win").

Systems that violate positive response (such as instant-runoff) can create situations where a ballot has the opposite effect of what the voter intended. This runs counter to the intuition that increasing an option's popularity in a democratic election should only improve the chances of that option winning. As a result, German courts have previously struck down nonmonotonic systems for violating the right to equal and direct suffrage.^[4][5]

Most voting systems (including Borda and all common tournament solutions) satisfy monotonicity,^[3] as do all commonly-used rated voting methods (including approval and score).^{[note 1]}

However, the criterion is violated by instant-runoff voting,^[6] the single transferable vote, and Hamilton's apportionment method.^[2]

The participation criterion is a closely-related, but different, concept. While positive responsiveness deals with a voter changing their opinion (or vote), participation deals with situations where a voter choosing to cast a ballot can have a reversed effect on the election.

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