Forest Simmons wrote (24 April 2010):
I want to thank Markus for keeping me from going too far off track.
And the link he
gave below to a great message of Chris Benham was valuable for more
than showing
us that Bucklin violates mono-add-top:
Chris also pointed out that WMA (weighted median approval) does
satisfy Participation.
I never fully appreciated before what a good method WMA was.
Forest,
In the April 2004 message you refer to I can't see that I claimed that
WMA satisfies
Participation. The closest to that is:
"It seems to me that WMA and WMA-STV meet Mono-add-top."
In 2004 I was still new to analysing single-winner voting methods, so
the opinions I voiced
then shouldn't be taken as authoritative pronouncements. :)
As I explained how WMA works in 2004:
Voters rank the candidates, equal preferences ok.
Each candidate is given a weight of 1 for each ballot on which that
candidate is ranked alone in first place, 1/2 for each ballot on
which that candidate is equal ranked first with one other candidate,
1/3 for each ballot on which that candidate is ranked equal first with
two other candidates, and so on so that the total of all the weights
equals the number of ballots.
Then approval scores for each candidate is derived thus: each ballot
approves all candidates that are ranked in first or equal first place
(and does not approve all candidates that are ranked last or equal
last). Subject to that, if the total weight of the approved candidates
is less than half the total of number of ballots, then the candidate/s
on the second preference-level are also approved, and the third, and so
on; stopping as soon as the total weight of the approved candidates
equals or exceeds half the total mumber of ballots.
Then the candidate with the highest approval score wins.
2: k>a=x
1: k>a>b
4: a>k>b
1: d>x=b
1: e>x=b
1: f>x=b
1: g>x=b
1: h>x=b
1: i>x=b
Weights: a4, k3, defghi 1 each, bx 0 each.
All ballots approve their top two preference-levels, giving these final
scores: x8, a7, k7, b6, defghi 1 each. The winner is x.
Now say we add two ballots that bullet-vote (plump) for x.
2: k>a=x
1: k>a>b
4: a>k>b
1: d>x=b
1: e>x=b
1: f>x=b
1: g>x=b
1: h>x=b
1: i>x=b
2: x
Weights: a4, k3, x2, defghi 1 each.
Now all ballots approve all their ranked candidates, giving these scores:
b11, x10, a7, k7, defghi 1 each. The new winner is b.
This is a failure of Mono-add-Plump and so also a failure of Mono-add-Top
and Participation.
I reject all methods that fail mono-add-plump as unacceptably silly.
Also Douglas Woodall has shown that WMA fails Clone-Winner, so I don't
consider WMA to be a "good method".
Chris Benham
----
Election-Methods mailing list - see http://electorama.com/em for list info
