[EM]Definite Majority Choice, AWP, AM

[Chris Benham]

Jobst,
You wrote (Thur.Mar.24):

"First, I'd like to emphasize that DMC, AWP, and AM
can be thought of as being essentially the same method
with only different definition of defeat strength, so
it seems quite natural to compare them in detail as
you started.

Recall that the DMC winner is the unique immune
candidate when defeat strength is defined as the
approval of the defeating candidate, so with
that definition, Beatpath, RP, and River become
equivalent to DMC.

Perhaps it is helpful to look at the defeat strength
like this: When A defeats B, then the defeat strength
is composed as a linear combination
of the following three components:"


                                     AM  AWP  DMC
                    
no. of voters approving A but not B    +   +   +
                  
no. of voters approving A and B        0   0   +
              
no. of voters approving B but not A    -   0   0
                 


                                                      
        
CB: I like this table. Doesn't AM look like the most
"natural" and  "balanced"? 
 
I was wondering if it is possible in AM for a
candidate who is both the sincere CW and sincere AW to
successfully Buried, and I've come up with an example
that shows that unfortunately it is, but AWP and DMC
likewise fail in the same example.

Sincere preferences:
48: A>B>>C
01: A>>C>B
03: B>>A>C
48: C>>B>A
B is the CW and AW.

Then 45 of the 48 A>B voters Bury B "strongly", i.e.
with both rankings and approval, while the other 3 of
the 48 only Bury with their rankings (not approving
C).
This gives:
45: A>C>>B
04: A>>C>B
03: B>>A>C
48: C>>B>A
C>B>A>C.

Approvals: A49,  B3,  C93.  AM and AWP both elect A,
while DMC elects C
Approval Margins             AWP     DMC  
C>B  93-3   (m+90)          93       B is eliminated
B>A  3-49   (m-46)           3
A>C  49-93  (m-44)           4

A's defeat is the weakest in both AM(-46) and  AWP(3),
while DMC eliminates B. All three methods elect the
Burier's candidate, A.

When there are three candidates in the top cycle, AM
has the property that the candidate with the lowest
voted approval score can't win.
Suppose there are three candidates in the top cycle.
>From highest to lowest, in order of approval scores
they are A,B,C.
If  A>C, that defeat will be positive number. Then
C>B, a negative number; and B>A, another negative
number. The weakest defeat will of
course be one of the negative numbers, so C can't win.
If on the other hand B>C, that defeat will be a
positive number. Then C>A, a negative number; and A>B,
a positive number. Obviously the
weakest defeat will be the negative number (A's) and
so again C can't win.

This 3-candidate analysis also shows that AM always
elects a member of  P (i.e. it never elects a
candidate who is pairwise beaten by a more
approved candidate).
If, in descending order of approval score, candidates
A,B,C are in a cycle; then the following is true in
AM:
(1) If  B>A,  and B's approval score is closer to A's
than to C's; then B wins
(2) Otherwise A wins.

To sum up the case for AM versus Approval-Weighted
Pairwise (AWP):
AWP  is only somewhat better than AM at electing the
sincere CW (both being good but fallible)  and that
seems to be balanced by it being worse at electing
the sincere AW. Therefore AWP's  ability to give a
majority an unanswerable complaint, by electing a
candidate from outside P, cannot be justified.

Jobst wrote:
"Here you state the obvious problem when looking at
both approval and defeat information. Forest's
ingenious argument was that we should at least not
elect a candidate where both kinds of information
agree that the candidate is defeated, leaving us with
his set P of candidates which are not strongly
defeated.

But when we take both kinds of information serious, it
does not seem appropriate to me to always elect a
candidate from the two extremes of P like Approval and
DMC do. Still, DMC has the obvious advantage of
extreme simplicity.

I would find it much more natural if the winner was
somewhere in the "middle" of P!"

Doesn't  Approval Margins fill the bill?  Welcome to
the AM fan club!


Chris  Benham



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