[EM] Sorted Approval Margins (plus two other Condorcet methods)

[Forest Simmons]

On Fri, Aug 11, 2023, 11:55 AM C.Benham <cbenham at adam.com.au> wrote:

>
> This suggests another method ...
> elect the candidate with the highest ratio given by the expression
>
> ((MaxPS+MinPS)/2)/MaxPO
>
> which is an estimate of the ratio of the approval the candidate would get
> if it were the approval cutoff candidate to the max approval any other
> candidate would get with the same cutoff.
>
> In other words, it is candidate expected to  bear up the best under
> Approval voting if it were the projected winner ... therefore (adjacent to)
> the approval cutoff in the next round of repeated voting, say.
>
> Restricting this to Smith should be good.
>
> Example:
>
> 48 C
> 28 A>B
> 24 B
>
> The respective ratios for A, B, and C are
>
> 26/48, [(52+24)/2]/48, 48/52
>
> So C wins.
>
>
> Is there a typo here?
>

Yes, it should be 28.

Where does the 26 in "26/48" come from?  Should it be 28?
>
> I'm not really switched on to the positive point of this relatively
> complicated method.
>

If we are limited to the "Universal Domain" ... so no explicit approval
cutoff allowed ... a reasonable way to do Approval DSV is to find the
Approval Cutoff Candidate that would get the highest fraction of approval
compared to its max opposition.

This MidrangePS/MaxPO score is a rough estimate of that max fraction.

Electing the candidate that maximizes that fraction seems to be resistant
to burial and truncation defections.

As you remarked below, the pairwise support matrix has all of the needed
info.

>
> In your example it fails Minimal Defense.  Does it meet Chicken Dilemma?
>

The example gives hope that it does meet CD.

>
> *Eliminate all the candidates not in the Smith set. Give each remaining
> candidate a score equal to the number of ballots
> on which it is ranked (among remaining candidates) below no other
> candidate minus the number of ballots on which it
> is ranked (among remaining candidates) above no other candidate.
>
> This score can be approximated as the average of the Max and Min Pairwise
> Supports (restricted to Smith) of the candidate.
>
>
> I can see that that would nearly always (or always?) be the same thing,
> and that it could be just read off the pairwise matrix
> (which might streamline the counting process a lot).
>
> Chris B.
>
>
> On 11/08/2023 1:37 am, Forest Simmons wrote:
>
>
>
> On Sun, Aug 6, 2023, 2:59 PM C.Benham <cbenham at adam.com.au> wrote:
>
>>
>> I think Condorcet methods that don't allow voters to enter an approval
>> threshold have to choose between trying to
>> minimise Compromise incentive or trying to reduce Defection incentive.
>>
>> The methods I like in this category allow voters to rank however many
>> candidates they like and also approve all but
>> one or only one or any number in between of the candidates (consistent
>> with their rankings). Equal-ranking is allowed.
>>
>> Default approval goes only to candidates ranked below no other candidate.
>>
>> I suggest that voters can just mark one of the candidates as the lowest
>> ranked one they approve (i.e. only that candidate
>> and those ranked higher or equal to it are approved).
>>
>> But other ways of doing it could be fine.
>>
>> Regarding which algorithm, I very much like Forest's  Sorted Approval
>> Margins.
>>
>> I also like another method of his, the exact name of which I've
>> forgotten (something about "Chain" building or climbing):
>>
>> *Begin the chain with the most approved candidate.  Then add the most
>> approved candidate that covers that candidate.
>> Then add the most approved candidate that covers all the candidates
>> already in the chain.
>>
>> Keep doing that as many times as possible, and then elect the last added
>> candidate*.
>>
>> I think nearly always this will elect the same candidate as
>> Smith//Approval, but is more elegant and ensures that the
>> winner is Uncovered.
>>
>> For a practicable Condorcet method that uses plain ranked ballots
>> (equal-ranking and truncation allowed), I like
>> Smith//Ranked below none minus ranked above none.
>>
>> *Eliminate all the candidates not in the Smith set. Give each remaining
>> candidate a score equal to the number of ballots
>> on which it is ranked (among remaining candidates) below no other
>> candidate minus the number of ballots on which it
>> is ranked (among remaining candidates) above no other candidate.
>>
>
> This score can be approximated as the average of the Max and Min Pairwise
> Supports (restricted to Smith) of the candidate.
>
> This suggests another method ...
> elect the candidate with the highest ratio given by the expression
>
> ((MaxPS+MinPS)/2)/MaxPO
>
> which is an estimate of the ratio of the approval the candidate would get
> if it were the approval cutoff candidate to the max approval any other
> candidate would get with the same cutoff.
>
> In other words, it is candidate expected to  bear up the best under
> Approval voting if it were the projected winner ... therefore (adjacent to)
> the approval cutoff in the next round of repeated voting, say.
>
> Restricting this to Smith should be good.
>
> Example:
>
> 48 C
> 28 A>B
> 24 B
>
> The respective ratios for A, B, and C are
>
> 26/48, [(52+24)/2]/48, 48/52
>
> So C wins.
>
>
>
>
>
>> Elect the candidate with the highest score."
>>
>> Given how rare top cycles will likely be, I think this is probably good
>> enough.
>>
>> Obviously it meets Plurality.  It fails both Minimal Defense and Chicken
>> Dilemma, but never both at once :)
>>
>> It looks fair and gives a pretty-enough winner.  I'll be back later with
>> some examples.
>>
>> Chris Benham
>>
>>
>>
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