[EM] Quota Limited Weighted Approval

[Chris Benham]

  Participants,
Here is  my improvement  on  previous versions of  "Weighted Median 
Approval"  single-winner  ranked-ballot method.

Voters rank the candidates. Equal preferences and truncation ok.
(1) Symetrically complete the ballots.
(2) Based on these now symetrically completed ballots, give each 
candidate a weight of  1 for each ballot on which it is
ranked in first place. (The total weight of  the candidates will now be 
equal to the total number of  original before-step-1
ballots.  Any candidate with a weight  equal to or greater than half 
 the total weight of all the candidates wins).
(3) Each ballot fully approves the highest-ranked  candidates whose 
combined weight is less than half  the total weight of
all the candidates. Each ballot also fractionally approves the next 
highest-ranked candidate, so that  the combined weight
of  the candidates approved by each ballot  is equal to half  the total 
weight of all the candidates.
(4) The candidate with the highest  approval score wins.

This method meets (mutual) Majority, Independence of  Clones, 
 Participation,  Reverse Symetry, Symetric Completion,
Woodall's  Plurality criterion, and Independence of  Pareto-Dominated 
 Alternatives. It is independent of  any losers with
no first preferences.
It fails Condorcet, Later-no-harm, Later-no-help, and  Steve Eppley's 
 "resistance to truncation" criterion.
It might be ok regarding his other  two "defensive strategy" criteria: 
"minimal defense" and "non-drastic defense".

49:A>B
24:B
27:C>B
100 ballots. B is the CW and Borda winner.

Symetrically completing the ballots, this becomes:
49:A>B>C
12:B>A>C
12:B>C>A
27:C>B>A

These ballots give these approvals
49: 1xA, .04167xB
12: 1xB, .5306xA
12: 1xB, .963xC
27: 1xC, .583xB

This gives these final approval scores:   A: 55.367,    B:51.9158,   C: 
39.46,   so  A wins  (as in DSC,  Margins and IRV).

If  all  the 24 B voters vote B>C, then Majority says "not A", and B 
wins.(an example of the method failing Later-no-help.)

This example is not the best advertisement for the method, which I like.

Chris Benham










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