[EM] Re: Weighted Median Approval
Chris Benham
chrisbenham at bigpond.com
Sun Apr 18 08:51:01 PDT 2004
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
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