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

[Kristofer Munsterhjelm]

Chris Benham wrote:
>  
> *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. 

Looking at this further, I think part of the intuition is also one of 
the frequency of the situations that would bring about the paradox. In 
the case of Participation, you'd have to have two districts that later 
join into one, which is not frequent; but for monotonicity, voters just 
have to change their opinions (which voters often do). That's not the 
entire picture, though; perhaps I consider monotonicity an inexpensive 
criterion, and thus one that reasonable methods should follow, or 
perhaps the degree of paradox (winner becomes loser) along with Yee-type 
visualization, makes nonmonotonicity seem all the worse.

The frequency idea is also related to the explanation of criteria 
failure conditions. If a person says that this method can cause winners 
to become losers when voters change their minds in favor of the 
now-loser, that appears completely ridiculous. On the other hand, 
LNHarm/LNHelp failure could be explained as a consequence of the method 
finding a common acceptable compromise, and so there's at least a 
"natural" reason for why it'd exist. Participation would be more 
difficult, but maybe one could draw parallels to the Simpson paradox of 
statistics like one would with Consistency failure.

This is like the IRV "method-focus" versus Condorcet "goal-focus", in 
reverse. Criterion failure that is the necessary consequence of some 
desirable trait can work (and even more so when one can easily see that 
there's no way to have both), but criterion failure that's based on how 
the method works rather than what it aims to achieve doesn't pass as easily.

> "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.)

Yes, Carey's method is called IFPP, as defined on 3 candidates. I think 
he used the name "Q method" for IFPP generalized to more than three 
candidates. The Q method is nonmonotonic - see 
http://listas.apesol.org/pipermail/election-methods-electorama.com/2001-September/006656.html

Carey later tried to patch Q for 4 candidates. The first patch failed, 
and he later came up with P4 
(http://listas.apesol.org/pipermail/election-methods-electorama.com/2001-September/006721.html 
) which I haven't tested. While Carey said that he didn't get around to 
rewriting it in stages (elimination) form, if that is possible, it's 
monotonic, and it's possible (in theory) to patch to five candidates, 
then patch that to six and so on up to infinity, the statement would 
have to be rephrased to "positional loser/below-average elimination 
methods are nonmonotonic". That's a lot of ifs, but to be charitable, 
I'll use that phrasing next.

> "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.

Of course it isn't, but if Schulze-elimination is Schulze, then that 
shows that one can't generalize the "all positional 
loser/average-elimination methods are nonmonotonic" to "all elimination 
methods are...", since we know that Schulze itself is monotonic.