Monotonicity is undoubtedly a desirable feature of an electoral method. I do however feel that no method can be perfect and that other features are more important ( proportional representation of parties, proportional representation of opinion,
I'm re-stating others here, but anyway...
I'd agree that proportionality in parties or opinion is more important than whether or not there is a bizarre result here or there. But this is meaningless when considering a single winner, where there is no "proportionality" to be had. In that case, you just want the candidate that best represents everyone in the district. (That may constitute a sort of "degenerate case" of proportionality, I suppose.)
[and] maximum freedom of voter choice regarding the individuals who represent you, etc).
Does a voting system that has been shown to support a two-party duopoly really provide this? We can debate until the cows come home about how much choice approval/Concorcet/etc. will provide, but from what I can see there's no reason to be satisfied with IRV in that regard.
As to the non-monotonicity of IRV and STV let me give an alternative definition of non-monotonicity
" Non-monotonicity -something that occurs more frequently in theoretical examples than real elections "
This is true, since monotonicity violations can only really occur when there are more than two viable candidates. While this is a desirable situation (because it increases voter choice) but it is vanishingly rare in places where IRV is used.
That said, the lack of frequent monotonicity violations doesn't get IRV off the hook in my mind. The fact that a method violates monotonicity means that it is strongly vulnerable to strategic manipulation. Sure enough, IRV has all sorts of situations where it makes sense to move your compromise up on the ballot, and it can be extremely hard for the voter to figure out when to do this and when not to.
I consider monotonicity one of the two most important "academic" criteria, among those that are violated by any reasonable methods. The other one is idependence from clones. Clone-independence means that the election can effectively merge multiple candidates that always appear next to each other on the ballot. Adding or removing a candidate that always appears just after a given candidate on every ballot will not change the result.
If a method fails clone-independence, then it will either have a bad "spoiler" problem (like plurality does), or will have the bizarre effect of making parties want to run as many candidates as possible (like Borda does). IRV actually passes clone-independence, which is one point in its favor.
Again, like monotonicity, it's not the failure of the precise definition of clone-independence that I have a problem with. It's that failing to satisfy that criteria is a red flag that other problems exist.
As a side note, only a few methods discussed here actually pass both clone-independence AND monotonicity. Beatpath, ranked pairs, approval, cardinal rankings, median ranking, and extended MCA (Bucklin with equal rankings allowed) all satisfy both. These are the only methods I know of that I'd work to get adopted for single-winner reform.
-Adam
