<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<html>
<head>
<meta http-equiv="Content-Type" content="text/html;charset=ISO-8859-1">
<title></title>
</head>
<body>
<meta http-equiv="Content-Type" content="text/html;charset=ISO-8859-1">
<title></title>
Participants / anyone interested,<br>
<br>
In my opinion, the most important/serious category of single-winner method
is <br>
(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.<br>
The other category I favour is<br>
(2) high-resolution ratings ballots, with many more available slots than
there are candidates, so that rankings can be inferred from ratings.<br>
In this category my current favourite is "Automated-Approval Margins"
(AAM), a Condorcet-completion method.<br>
Inferring rankings from the ratings, eliminate (and henceforth completely
ignore) the non-members of the Schwartz set.<br>
If more than one candidate remains, then each ballot approves the candidates
they rate above average (the arithmetical mean of the<br>
Schwartz-set candidates) and half-approves those they rate exactly average.
The (inferred) rankings are used to determine the results<br>
of the pairwise comparisons, but the margins between the (derived) "automated"
approval-scores are used to weigh these results<br>
(the strengths of the "defeats"). On this basis pick the Ranked Pairs winner
. (I may later decide that something other than RP is <br>
slightly better for the last step, but in practice it would very rarely
give a different winner).<br>
<br>
Schwartz // SC-WMA.<br>
My current favourite single-winner plain ranked-ballot method is Schwartz
// SC-WMA. "SC-WMA" stands for <br>
"Symetrically Completed- Weighted Median Approval". I think it is probably
impossible for Smith // SC-WMA to ever give a<br>
different result. (In Woodall terms, that would be "CNTT, SC-WMA" with
"CNTT" standing for "Condorcet(Net) TopTier").<br>
<br>
Voters rank the candidates, truncation ok. Non-last equal prefernces also
ok. (If these are not allowed, compliance with the<br>
Non-Drastic Defense criterion is lost, but the method is still good and
may be more of a practical propsition).<br>
Eliminate the non-members of the Schwartz set (and henceforth continue
as though they had never stood).<br>
Symetrically complete the ballots.<br>
Now apply the "Weighted Median Approval" method to pick the winner. <br>
Each (remaining) candidate is assigned a "weight" which is equal to the
number of first-prefernces they get. The sum of the "weights"<br>
is equal to the total number of non-empty ballots.<br>
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..<br>
And so on until each ballot has approved at least half the candidates "by
weight".<br>
The candidate with the highest total (thus derived) approval score wins.<br>
<br>
This would only very rarely be anywhere near as complicated as it might
appear. Usually there will be easy short-cuts. For example,<br>
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.<br>
<br>
The justification of this method is that unlike Winning-Votes, it meets
the Sincere Expectation Criterion. There is no silly zero-information<br>
random-fill incentive. Unlike Margins, it meets Minimal Defense, Non-Drastic
Defense and Truncation Resistance.<br>
Also, unlike Margins, it meets Woodall's Plurality and Weak Independence
of Irrelevant Alternatives criteria.<br>
Unlike WV, it meets Woodall's Symetric Completion criterion. Unlike MAM,
it meets Independence of Pareto-Dominated Alternatives.<br>
Also it is independant of any losers who are no voters' most preferred Schwartz-set
member.<br>
The method, unlike Bucklin or QLTD, meets Clone Independence.<br>
<br>
The (IMO) small price that is paid for all this is that unlike MAM, the method
fails Immunity from Majority Complaints; and the method<br>
can fail Mono-raise when there are more than three candidates in the Schwartz
set.<br>
<br>
35:BA/CD<br>
30:CD/BA<br>
05:CA/DB<br>
15:AC/DB<br>
15:DB/AC<br>
<br>
A wins. Replace 5 CADB with 5 ACDB ballots gives:<br>
<br>
35:BA/CD<br>
30:CDB/A<br>
20:AC/DB<br>
15:DB/AC<br>
<br>
Now A loses, violating Mono-raise.<br>
<br>
In the above examples, all the candidates are in the Schwartz set, and each
ballot approves the candidates before the slash.<br>
<br>
Chris Benham<br>
</body>
</html>