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.
----
Election-Methods mailing list - see http://electorama.com/em for list info
