[EM] What are some simple methods that accomplish the following conditions?

[Kevin Venzke]

Hi Forest,

I had two ideas.

Idea 1:
1. If there is a CW using all rankings, elect the CW.
2. Otherwise flatten/discard all disapproved rankings.
3. Use any method that would elect C in scenario 2. (Approval, Bucklin, MinMax(WV).)

So scenario 1 has no CW. The disapproved C>A rankings are dropped. A wins any method.
In scenario 2 there is no CW but nothing is dropped, so use a method that picks C.
In both versions of scenario 3 there is a CW, B.

If step 3 is Approval then of course step 2 is unnecessary.

In place of step 1 you could find and apply the majority-strength solid coalitions (using all rankings)
to disqualify A, instead of acting based on B being a CW. I'm not sure if there's another elegant way
to identify the majority coalition.

Idea 2:
1. Using all rankings, find the strength of everyone's worst WV defeat. (A CW scores 0.)
2. Say that candidate X has a "double beatpath" to Y if X has a standard beatpath to Y regardless
of whether the disapproved rankings are counted. (I don't know if it needs to be the *same* beatpath,
but it shouldn't come into play with these scenarios.)
3. Disqualify from winning any candidate who is not in the Schwartz set calculated using double
beatpaths. In other words, for every candidate Y where there exists a candidate X such that X has a
double beatpath to Y and Y does not have a double beatpath to X, then Y is disqualified.
4. Elect the remaining candidate with the mildest WV defeat calculated earlier.

So in scenario 1, A always has a beatpath to the other candidates, no matter whether disapproved
rankings are counted. The other candidates only have a beatpath to A when the C>A win exists. So
A has a double beatpath to B and C, and they have no path back. This leaves A as the only candidate
not disqualified.

In scenario 2, the defeat scores from weakest to strongest are B>C, A>B, C>A. Every candidate has
a beatpath to every other candidate no matter whether the (nonexistent) disapproved rankings are
counted. So no candidate is disqualified. C has the best defeat score and wins.

In scenario 3, the first version: B has no losses. C's loss to B is weaker than both of A's losses. B
beats C pairwise no matter what, so B has a double beatpath to C. However C has no such beatpath
to A, nor has A one to B, nor has B one to A. The resulting Schwartz set disqualifies only C. (C needs
to return B's double beatpath but can't, and neither A nor B has a double beatpath to the other.)
Between A and B, B's score (as CW) is 0, so he wins.  

Scenario 3, second version: B again has no losses, and also has double beatpaths to both of A and
C, neither of whom have double beatpaths back. So A and C are disqualified and B wins.

I must note that this is actually a Condorcet method, since a CW could never get disqualified and
would always have the best worst defeat. That observation would simplify the explanation of
scenario 3.

I needed the defeat strength rule because I had no way to give the win to B over A in scenario 3
version 1. But I guess if it's a Condorcet rule in any case, we can just add that as a rule, and greatly
simplify it to the point where it's going to look very much like idea 1. I guess all my ideas lead me to
the same place with this question.

Oh well, I think the ideas are interesting enough to post.

Kevin



>Le jeudi 30 mai 2019 à 17:32:42 UTC−5, Forest Simmons <fsimmons at pcc.edu> a écrit : 
>
>In the example profiles below 100 = P+Q+R, and  50>P>Q>R>0.  One consequence of these constraints is that in all three profiles below the cycle >A>B>C>A will obtain.
>
>I am interested in simple methods that always ...
>
>(1) elect candidate A given the following profile:
>
>P: A
>Q: B>>C
>R: C,
>and 
>(2) elect candidate C given
>P: A
>Q: B>C>>
>R: C,
>and 
>(3) elect candidate B given

>
>P: A
>Q: B>>C  (or B>C)
>R: C>>B. (or C>B)
>
>I have two such methods in mind, and I'll tell you one of them below, but I don't want to prejudice your creative efforts with too many ideas.
>
>Here's the rationale for the requirements:
>
>Condition (1) is needed so that when the sincere preferences are

>
>P: A
>Q: B>C
>R: C>B,
>the B faction (by merely disapproving C without truncation) can defend itself against a "chicken" attack (truncation of B) from the C faction.
>
>Condition (3) is needed so that when the C faction realizes that the game of Chicken is not going to work for them, the sincere CW is elected.
>
>Condition (2) is needed so that when  sincere preferences are

>
>P: A>C
>Q: B>C
>R: C>A,
>then the C faction (by proactively truncating A) can defend the CW against the A faction's potential truncation attack.
>
>Like I said, I have a couple of fairly simple methods in mind. The most obvious one is Smith\\Approval where the voters have 
>control over their own approval cutoffs (as opposed to implicit approval) with default approval as top rank only. The other 
>method I have in mind is not quite as 
>simple, but it has the added advantage of satisfying the FBC, while almost always electing from Smith.


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