Statement of Criterion
With the relative order or rating of the other candidates unchanged, voting a candidate higher should never cause the candidate to lose, nor should voting a candidate lower ever cause the candidate to win.
Complying Methods
All the methods listed in the compliance table above are monotonic except Instant Runoff Voting (IRV).
Commentary
In the ordinal methods (Condorcet, Borda, and IRV), a candidate is "voted higher" by being ranked higher. In Approval Voting, a candidate is "voted higher" by being "approved" rather than "disapproved." In a conventional plurality system, a candidate can be "voted higher" only by being voted for at all rather than not voted for.
Monotonicity is perhaps the most basic criterion for election methods. Common sense tells us that good election methods should be monotonic. Methods that fail to comply are erratic.
A simple example will prove that IRV is non-monotonic. Consider, for example, the following vote count with three candidates {A,B,C}:
In this example, eight voters ranked the candidates (A,C), five ranked them (B,A), and four ranked them (C,B). Candidate C was ranked first by the fewest voters and is eliminated. Since all the voters who ranked C first also ranked B second, B now has nine top-choice votes and wins.
Suppose, however, that two of the voters who had ranked A first reverse their first two preferences so their votes change from (A,C) to (C,A). Now the vote count is:
| 6: |
A,C |
| 5: |
B,A |
| 4: |
C,B |
| 2: |
C,A |
Candidate B is now ranked first by the fewest voters and is eliminated. Since the five voters who ranked B first also ranked A second, A now has eleven top-choice votes and wins. Hence, the two voters who demoted A from first to second choice caused A to win. That is, they caused A to win by ranking A lower, without changing the relative ordering of the other candidates. IRV therefore fails monotonicity.
For an even more bizarre example, consider the following vote count with four candidates {A,B,C,D}:
| 7: |
A,B,C |
| 6: |
B,A,C |
| 5: |
C,B,A |
| 3: |
D,C,B |
Applying the rules of IRV, candidate A wins. But suppose the three voters who voted (D,C,B) now promote A from last choice all the way up to first choice, without changing the relative order of the other candidates. Now B wins instead of A. So by promoting A from last to first choice, those voters caused A to lose instead of win. An election method that allows such nonsensical anomalies is erratic and should be rejected.
These are hardly contrived theoretical examples without practical relevance. IRV has serious problems both in theory and in practice. In practice, voters would soon realize, or be advised, that they cannot safely vote sincerely, and the political system would likely remain bogged down in a two-party duopoly just as it is today. And that is the optimistic scenario. If a third party somehow manages to become a strong contender, it could throw the entire political system into chaos, just as it could in our current plurality system. (See The Problem with IRV.)