[EM] Re: Weighted Median Approval

[Chris Benham]

Participants,
As Forest has pointed out, in my last post (Fri.Apr.16) I somehow made a 
simple calculation blunder.
Before giving my new demonstration, here is the relevant part of  that 
post, corrected.

Here is a simple example from Marcus Schulze:
2: A>B>C
3: B>C>A
4: C>A>B

The median rank of  all  the candidates is 2.  Using  Forest's formula 
to break this tie, C is the winner of  what  I  will
simply call  the "Highest Median Rank" (HMR) method. There are 9 
ballots, so the middle one is ballot 5 (with 4 below
and 4 above). For candidate A, spreading out the ballots in a row  in 
order from those on which A is ranked highest to
those on which A is ranked lowest, we see that on the middle ballot A is 
ranked second; so that is A's median rank.
Using Forest's formula to break the tie, we subtract the number of 
ballots on which A is ranked below second from
the number on which A is ranked above second, and then divide this sum 
by the number of ballots on which A is ranked
exactly second. For A this comes to (2-3)/4 =  -.25.  For B this is 
(3-4)/2 = -.5  and for C it is  (4-2)/3 = .333333.

I wrongly had  A winning, because I  erred in having  "(4-2)/3 = -.6666" 
, i.e  minus 2/3 instead of the correct plus 1/3.

I  have now found an example in which the HMR method described above 
gives a  different winner from QLTD.

20: A>B>C
04: A>C>B
20: B>A>C
17: B>C>A
39: C>A>B
100 ballots. All candidates in the Smith set.

All  3  candidates have a  median rank of  2, and  A wins with the only 
positive tie-breaking  "Q" value.
A: (24-17)/59 = .118644
B: (37-43)/20 = -.3
C: (39-40)/21 = -.047619
In this situation, where no candidate is ranked first on a majority of 
ballots and all the candidates are ranked first or second on a
majority of  ballots; then the QLTD winner is the first-preference 
winner . C is the QLTD winner.
Plain Bucklin agrees with HMR and picks  A.
Ranked Pairs, Beat Path, Min Max  and  Borda all pick A.

Weights    A:24      B:37     C:39
WMA  approvals
20: AB
04: AC
20: BA
17: BC
39: CA

WMA final scores     A: 83     B: 57    C: 60,    A wins.
WMA-STV  eliminates B, and then  elects C.

Chris Benham