Monotonicity Criterion

A VotingMethodCriterion

MonotonicityCriterion: if one set of preference ballots would lead to an an overall ranking of alternative X above alternative Y and if some preference ballots are changed in such a way that the only alternative that has a higher ranking on any preference ballots is X, the method should still rank X above Y.

This one seems really obvious. The trouble is it would lead to the rejection of transferable voting systems such as SingleTransferableVote and InstantRunoffVoting, which I happen to think are on balance the best way to go.

Denoting a ballot that ranks candidate A 1, candidate B 2, C 3, etc as ABC; and 6 votes cast as ABC, as 6 ABC, consider:

• 8 ABC
• 4 BCA
• 5 CAB

B comes last and is excluded, and B's votes transfer to C. C beats A 9-8.

But suppose we take 2 of the ABC votes and switch them to BAC:

• 6 ABC
• 2 BAC
• 4 BCA
• 5 CAB

C now comes last and is excluded, and C's votes transfer to A. A beats B 11-6.

Ouch! Going from the second case to the first, A is only moved up on ballots, but goes from winner to loser.

So why do I still support transferable voting? For 2 reasons -- firstly only transferable voting allows you to express a second preference without any risk of it harming the chance of your first preference being elected. Arguably, the occasional monotonicity failure is the price we pay for this.

Secondly, I don't think it is very clear in any of the above cases which winner an ideal system would pick. CondorcetsCriterion? won't help us in these cases. (Not a coincidence...) So perhaps in the above cases, any system will be somewhat unsatisfactory. Taking pairwise comparisons (since there should be no argument when there are only two candidates), we see that in the first case, A beats B 13-4, B beats C 12-5 and C beats A 9-8. In the second case, A beats B 11-6, B beats C 12-5 and C beats A 9-8.