[EM] Re: 3-slot method

[Chris Benham]

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:

>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).
>
Yes, in a very mild way, this method  does fail  Condorcet  and 
 mono-raise  (aka monotonicity).
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