[EM] WMA

[C.Benham]

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