Monotonicity criterion Totally Explained

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ť 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:
ť A candidate x shouldn't be harmed [for example,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:
ť 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 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." (External Link

) 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 could theoretically try to elect their candidate by counter-intuitively 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 there's no evidence that voters actually pursue such counter-intuitive strategies in non-monotonic voting systems in real-world elections.
Of the single-winner voting systems, First Past the Post, 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're 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's only one winner are not monotonic.

Example

Instant-runoff voting

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:
ť
   Because of the votes Belinda loses, she's 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 appears to have caused her to lose. In fact, of course, it wasn't the increase in support for Andrea that hurt her.
   Non-monotionic scenarios for IRV are frequently miss-presented along the lines of; "Having more voters support candidate A can cause A to switch from being a winner to being a loser." Note that it's not the fact that A gets more votes that causes A to lose. In fact that, by itself, can never cause a candidate to lose with IRV. The actual cause is the shift of support among other candidates, (in the example above, the decline in support for Belinda) which changes which candidate A faces in the final match-up.
   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.
   Crispin Allard argues that the circumstances under which this could occur would be extremely rare, fewer than once per century under normal political conditions. (External Link

) Nicholas Miller disputes this conclusion and provides a different mathematical model. (External Link

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