# Monotonicity Criterion

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

A candidate x should not be harmed if x is raised on some ballots without changing the orders of the other candidates.

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 a property considered in Arrow's impossibility theorem. Some political scientists, however, doubt the value of monotonicity as an evaluative measure of voting systems. David Austen-Smith and Jeffrey Banks, for example, published an article in The American Political Science Review in which they argue that "monotonicity in electoral systems is a nonissue: depending on the behavioral model governing individual decision making, either everything is monotonic or nothing is monotonic."

Although all voting systems are vulnerable to tactical voting, systems which fail the monotonicity criterion suffer an unusual form, where voters with enough information about other voter strategies can support their candidate by counter-intuitively voting against that candidate.

Of the single-winner voting systems, plurality voting (first past the post), Borda count, Schulze method, and Ranked Pairs (Maximize Affirmed Majorities) are monotonic, while Coombs' method, runoff voting and instant-runoff voting are not. The single-winner methods of range voting, majority judgment 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 ranked voting systems.

Of the multiple-winner voting systems, all plurality voting methods are monotonic, such as plurality-at-large voting (bloc voting), cumulative voting, and the single non-transferable vote. Most versions of the single transferable vote, including all variants currently in use for public elections (which simplify to instant runoff when there is only one winner) are not monotonic.

Read more about Monotonicity Criterion:  Instant-runoff Voting and The Two-round System Are Not Monotonic, Real-life Monotonicity Violations

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