[EM] EM] recommendations (single-winner)

[Chris Benham]

  Participants / anyone interested,

In my opinion, the most important/serious category of  single-winner 
method  is
(1)  plain  rankings-ballot methods in which voters are asked to simply 
rank the candidates. Truncation should be allowed, and allowing equal 
non-last ranking  is desirable but perhaps not absolutely essential. 
 Methods that ask voters to rank only those candidates they approve, 
although the ballots look the same, are hybrid  Approval/ 
limited-rankings methods  and in my view are in a different  (much less 
desirable) category.
The other category I  favour is
(2) high-resolution  ratings ballots, with many more available slots 
than there are candidates, so that rankings can be inferred from ratings.
In  this category my current favourite is  "Automated-Approval  Margins" 
(AAM),  a  Condorcet-completion method.
Inferring rankings from the ratings, eliminate (and henceforth 
completely ignore) the non-members of the Schwartz set.
If more than one candidate remains, then  each ballot approves the 
candidates they rate above average  (the arithmetical mean of  the
Schwartz-set candidates)  and  half-approves those they rate exactly 
average. The (inferred) rankings are used to determine the results
of  the pairwise comparisons, but the margins between the (derived) 
"automated" approval-scores are used to weigh these results
(the strengths of the "defeats").  On this basis pick the Ranked Pairs 
winner . (I may later decide that something other than RP is
slightly better for the last step, but in practice it would very rarely 
give a different winner).

Schwartz // SC-WMA.
My current favourite single-winner  plain ranked-ballot method is 
 Schwartz // SC-WMA.  "SC-WMA"  stands for
"Symetrically Completed- Weighted Median Approval".   I  think it is 
probably impossible for  Smith // SC-WMA to ever give a
different result. (In  Woodall terms, that would be  "CNTT, SC-WMA" 
 with  "CNTT" standing  for  "Condorcet(Net)  TopTier").

 Voters rank the candidates, truncation ok. Non-last equal prefernces 
also ok.  (If these are not allowed, compliance with the
Non-Drastic Defense criterion is lost,  but the method is still good and 
may be more of a practical propsition).
Eliminate the non-members of  the Schwartz set (and henceforth continue 
as though they had never stood).
Symetrically complete the ballots.
Now apply the "Weighted Median Approval" method to pick the winner.
Each (remaining) candidate is assigned a  "weight" which is equal to the 
number of  first-prefernces they get. The sum of the "weights"
is equal to the total number of  non-empty ballots.
Each ballot approves the candidate they rank in first-place. If  the 
weight of candidates so far approved by a ballot sums to less than half 
the total  weight of all the candidates, then that ballot also approves 
the candidate they rank second..
And so on until each ballot has approved at least half the candidates 
 "by weight".
The candidate with the highest total  (thus derived)  approval score wins.

This would only very rarely be anywhere near as complicated as it might 
appear. Usually there will be easy short-cuts.  For example,
if the Schwartz  set contains three members, then in practice that means 
 that each ballot  approves the candidates they rank first and second; 
and those which only rank one candidate, approve that candidate and 
half-approve the other two.

The justification of  this method is  that unlike Winning-Votes, it 
 meets the Sincere Expectation Criterion. There is no silly zero-information
random-fill incentive. Unlike Margins, it meets  Minimal Defense, 
 Non-Drastic Defense  and  Truncation Resistance.
Also, unlike Margins, it meets Woodall's  Plurality and  Weak 
Independence of  Irrelevant Alternatives criteria.
Unlike WV, it meets Woodall's  Symetric Completion criterion. Unlike 
MAM, it meets Independence of  Pareto-Dominated Alternatives.
Also it is independant of  any losers who are no voters' most preferred 
 Schwartz-set member.
The method, unlike Bucklin or  QLTD,  meets Clone Independence.

The (IMO) small price that is paid for all this is that unlike MAM, the 
method fails Immunity from Majority Complaints; and  the method
can fail Mono-raise when there are more than three candidates  in the 
Schwartz set.

35:BA/CD
30:CD/BA
05:CA/DB
15:AC/DB
15:DB/AC

A  wins. Replace 5  CADB with 5  ACDB ballots gives:

35:BA/CD
30:CDB/A
20:AC/DB
15:DB/AC

Now A loses,  violating Mono-raise.

In the above examples, all the candidates are in the Schwartz set, and 
each ballot approves the candidates before the slash.

Chris Benham
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