[EM] Can someone point me at an example of the nonmonotonicity of IRV?

[Kristofer Munsterhjelm]

Kathy Dopp wrote:
>> From: "rob brown" <rob at karmatics.com>
>> Subject: Re: [EM] Can someone point me at an example of the
>>        nonmonotonicity of IRV?
> 
>> Are you aware that in going to a doctor to treat an injury, you can get in a
>> car accident and get injured some more?  Why would anyone go to a doctor if
>> doing so can actually make your health WORSE?
> 
> OK. So you are saying we must use voting methods where voting for our
> FIRST-Choice candidate as our LAST Choice helps our first choice
> candidate win, and when I go to the polls I have no idea if that is
> true or false because "I might get into an accident when I drive to
> the doctor when I'm sick?
> 
> I must have fallen down the rabbit hole when I joined this list.
> 
> 

I think what he means is that although the paradox is severe when it 
does happen (similarly to driving off the road), it happens very rarely, 
and in general, IRV gives a result that's better than say, Plurality 
applied to ranked ballots.

If it happens too often, though, one could get real paradoxes such as 
one that Ossipoff gave: a candidate being shown to be corrupt (so that 
many rank him lower) leads to that candidate's victory.

There's also the "it smells fishy" that nonmonotonicity - of any kind or 
frequency - evokes. I think that's stronger for nonmonotonicity than for 
things like strategy vulnerability because it's an error that appears in 
the method itself, rather than in the move-countermove "game" brought on 
by strategy, and thus one thinks "if it errs in that way, what more 
fundamental errors may be in there that I don't know of?". But that 
enters the realm of feelings and opinion.

A less "feelings-based" way of showing the oddities of IRV would be to 
point at Yee pictures: http://rangevoting.org/IEVS/Pictures.html
The disconnected regions in IRV pictures are a consequence of 
nonmonotonicity - moving towards a candidate leads to another winning. 
Note that a method may be nonmonotonic in general and still be monotonic 
in the subset that 2D Yee-pictures cover. Also, that doesn't resolve the 
problem of figuring out how severe a monotonicity failure is, but just 
how frequently they occur in "voting space".

>> Just because there is a non-zero chance of harm resulting from your choice
>> does not mean that you should be paralyzed from making a decision.
> 
> I am *not* paralyzed. I have DECIDED that I IRV voting is an insane
> voting method that would cause much more havoc with voting systems.

Out of curiosity, what voting system would you recommend? I'm not saying 
"don't say anything if you don't have an alternative", I'm just curious.

> Nope. Never said it was and I have no problem with voting methods that
> do such things, but you may have neglected to notice that with IRV,
> ranking my first-choice LAST could help my first-choice MORE than
> ranking my first-choice FIRST. In IRV, putting my candidate FIRST can
> help my LAST place candidate win and putting my candidate LAST can
> help my FIRST place candidate win.
> 
> Please identify all the other voting methods for me which have that
> property (that ranking or rating a candidate LAST can help that
> candidate MORE than ranking or rating a candidate FIRST) in addition
> to IRV so that I can oppose them as well because I am not familiar
> with any of these other methods that share that property with IRV.

I think that all methods that work by calculating the ranking according 
to a positional function, then eliminating one or more candidates, then 
repeating until a winner is found will suffer from nonmonotonicity. I 
don't know if there's a proof for this somewhere, though.

A positional function is one that gives a points for first place, b 
points for second, c for third and so on, and whoever has the highest 
score wins, or in the case of elimination, whoever has the lowest score 
is eliminated.

Less abstractly, these methods are nonmonotonic if I'm right: Coombs 
(whoever gets most last-place votes is eliminated until someone has a 
majority), IRV and Carey's Q method (eliminate loser or those with below 
average plurality scores, respectively), and Baldwin and Nanson (the 
same, but with Borda).

It may be that this can be formally proven or extended to other 
elimination methods. I seem to remember a post on this list saying that 
Schulze-elimination is just Schulze, but I can't find it. If I remember 
correctly, then that means that not all elimination methods are 
nonmonotonic.