# Monotonicity criterion

A voting system is monotonic if it satisfies the monotonicity criterion, given below. In mathematics, monotonicity usually refers to the different concept of a monotonic function.

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.

Douglas Woodall's definition is:

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.

Both definitions are equivalent. A slicker, though looser, way of phrasing this is that in a non-monotonic system, voting for a candidate can cause that candidate to lose.

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.

Plurality voting, Majority Choice Approval, Borda count, Cloneproof Schwartz Sequential Dropping, and Maximize Affirmed Majorities are monotonic, while Coombs' method, Instant-runoff voting and the Single Transferable Vote are not. Approval voting is monotonic, using a slightly different definition as it is not a preferential system: you can never help a candidate by not voting for them.

## Example

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 transfering votes to Andrea, who is elected with a majority.

She then serves a full term, and does such a good job that she persuades some 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 elminated first this time, and her second preferences are transfered to Cynthia, who wins.

That is, in this case, between elections, Andrea's preferential ranking increased - more electors put her first - but this increase in support caused her to lose.

Obviously, in the real world, such problems would be difficult to detect, because there would be other movements of votes, and we would not know whether the same people cast the same votes, etc.

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] (http://www.mcdougall.org.uk/VM/ISSUE5/P1.HTM) - although this is subject to complex mathematical disputation [2] (http://research.umbc.edu/~nmiller/RESEARCH/DRAFT.POST.pdf).

It should also be noted that, in the example above, both outcomes are fair within the terms of a one off STV election.

## References

• 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

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