Jobst, In response to me posting this suggestion for a 3-slot single-winner method,
(1) Voters put the candidates into one three slots, Preferred, Approved, or Disapproved. (Default is Disapproved).
(2) If not all the candidates are rated as Disapproved by a majority, then eliminate those that are.
(3) After the eliminations, candidates who are rated as Approved on ballots that rate no (remaining) candidate as Preferred
are promoted on those ballots to Preferred.
(4) Elect the winner of the pairwise comparison between the candidate who is (after steps 2 and 3) most Preferred and the
candidate who is least Disapproved.
you offered these comments:
Yes, in a very mild way, this method does fail Condorcet and mono-raise (aka monotonicity).First of all, it it not Condorcet, is it? It seems to me that in the following example, it will elect A though C is the CW: 2 A>C>B 4 A>B>C 4 C>B>A 1 C>A>B 2 B>C>A Defeats A>B(7:6), C>A(7:6), C>B(7:6), hence C is the CW. No candidate is disapproved by a majority (A:6, B:3, C:4). A is most preferred (6), B is least disapproved (3), hence A wins!
Did you check any other criteria? I always try Condorcet and monotonicity first. I fear the latter is quite improbable because of the detail "If not all the candidates are rated as Disapproved by a majority..." and because of the IRV-like promotion in step (3).
Douglas Woodall gives this example:
4 abc 6 acb 6 bac 2 bca 3 cba
Total votes 21. No candidate is rated as Disapproved by more than half the voters, so a (being both most Preferred and least Disapproved) is elected. However, if one of the bca ballots is replaced by bac, then c is Disapproved by more than half the voters, and so is excluded. Then b is promoted to Preferred on the 3 cba ballots and so b will beat a by 11 votes to 10.
What I find surprising and remarkable in this 3-candidate example is that a is both the most "Preferred" and the least "Disapproved", and yet is not in the top cycle! (b>c 12-9, b>a 11-10).
In response to your other question, I did more-or-less satisfy myself that it meets Minimal Defense, Trucation Resistance,
a kind of Mutual Majority (aka Majority for Solid Coalitions, aka Majority), No Zero-Information Strategy (and therfore
the weaker Sincere Expectation Criterion).
This method was meant to be simple and (maybe) saleable. I am sure that in public political elections it would work (99%)
as well as any other 3-slot method.
But on refllection, I now think the best 3-slot method is just:
"elect the least Disapproved member of the (3-slot) Schwartz set" !
Chris Benham
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