Ted, Russ, Forest, James,Juho and others,

I think that Ted's draft public "Definite Majority
Choice" proposal is excellent, in the sense that
anything that might be slightly better would be more
complicated and/or less intuitive.

Two contending methods that use the same style of
ballot are  James G-A's Approval-Weighted Pairwise and
my Approval Margins.
I've found a couple of examples that illustrate
differences between the three methods.
The first is copied from a Sep.22,04 James G-A post.

3 candidates: Kerry, Dean, and Bush. 100 voters.
    Sincere preferences
19: K>D>>B
5: K>>D>B
4: K>>B>D
18: D>K>>B
5: D>>K>B
1: D>>B>K
25: B>>K>D
23: B>>D>K
    Kerry is a Condorcet winner.

    Altered preferences
19: K>D>>B
5: K>>D>B
4: K>>B>D
18: D>K>>B
5: D>>K>B
1: D>>B>K
21: B>>K>D
23: B>>D>K
4: B>D>>K (these are sincerely B>>K>D)
    There is a cycle now, K>B>D>K

On the "sincere preferences" ballots, the approval
scores are B48, K46, D43, while
on the "altered preferences" ballots, the approval
scores are B48, D47, K46.

Approval Margins uses a "defeat-dropper" method,
measuring the strengths of the defeats by the margins
between the approval scores (but like AWP,determines
their "directions" purely by the rankings.)

Approval Margins:
D>K   47-46  (m+1)
K>B   46-48  (m-2)
B>D   48-47  (m+1)

B's defeat, with an approval margin of -2, is the
weakest and so is "dropped". B, the Buriers' favourite
but the sincere (and voted) Approval winner,wins.
DMC gets the same result by eliminating D and K.

AWP differs from AM in the way that it weighs defeats.
Quoting James:

> For a given defeat A over B, the magnitude of the
defeat is defined by
> the number of voters who place A above their
approval cutoff and B below
> their approval cutoff.

Approval-Weighted Pairwise:
D>K   06
K>B   46
B>D   44

AWP elects the sincere CW, K!
I used to think that electing the voted approval loser
was absurd if we assume that the votes are sincere,
but by that logic we should resolve all top cycles by
electing the Approval winner. From that point of view,
sometimes electing the approval loser is only a degree
"worse" than not always electing the approval winner!

Still, I don't see this example being a great
advertisement for AWP versus AM because the winner is
the sincere
Approval winner. 
My next example is the one I used in my last post on

An old example given by Adam Tarr.
Sincere rankings:
49 R>C>L
12 C>R>L
12 C>L>R
27 L>C>R
C is the CW.

Suppose there is pre-polling and so the L supporters
decide to approve C, while the C supporters sincerely
divide their approvals.
Further suppose that the R supporters all decide to
completely Bury C.  Then we might get:

49 R>L>>C
06 C>R>>L
06 C>>R>L
06 C>>L>R
06 C>L>>R
27 L>C>>R

Now all the candidates are in the top cycle: L>C>R>L. 
The approval scores are  L82,  R55, C51.

Approval Margins:
L>C  82-51 = +31
C>R  51-55 =  -4
R>L  55-82 = -27

AM elects L, backfiring on the Buriers!
Unfortunately this time DMC eliminates C, and then the
Buriers' candidate R wins.

Approval-Weighted Pairwise:
L>C  49
C>R  45
R>L  06

AWP gives the same good result as AM!

Chris Benham

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