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

[Chris Benham]

Kristofer Munsterhjelm  wrote (Sun. Aug.10):

"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."
Kristopher,
The intution or  "feeling" you refer to is based on the idea that the best method/s
must be mathematically elegant and that methods tend to be consistently good
or consistently bad.

But in the comparison among reasonable and  "good" methods, this idea is wrong.
Rather it is the case that many arguably desirable properties (criteria compliances)
are mutually incompatible. So on discovering that  method X has some mathematically
inelegant or paradoxical flaw one shouldn't immediately conclude that  X  must be
one of  the worst methods.  That "flaw" may enable X to have some other desirable
features.
To look at it the other way, Participation is obviously interesting and viewed in isolation
a desirable property. But I know that it is quite "expensive", so on discovering that method
Y meets Participation I know that it must fail other criteria (that I value) so  I don't expect
Y  to be one of my favourite methods.  

"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)."
That's right, but I think that Carey's method  (that I thought was called "Improved FPP")
is monotonic (meets mono-raise) when there are 3 candidates (and that is the point of it.)

"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."
Of course Schulze isn't a "positional function".  Obviously if there are just 3 candidates in
the Schwartz set then "Schulze-elimination" must equal Schulze, but maybe there is some
relatively complicted example where there are more than 3 candidates in the top cycle
where the two methods give a different result.
Chris Benham


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