[EM] Max Approval Margin Defeat (was Re: Margins Sorted Top-Ratings)

[Forest Simmons]

Chris,

I just remember my default approval unlike yours has three levels ..
 Zero for unranked candidates, one for candidates ranked below the upper
cutoff, and two for candidates ranked above that cutoff .... this what
makes default approval the same as decloned Borda ... and is the approval I
have in mind for themax approval defeat margin method:

Lacking an undefeated candidate ... elect the winner W of the defeat pair
(W>L) that maximizes the difference between the approvals of W and L,
subject to the constraint that L repays W in two steps: L>X>W.

Sorry about the confusion!

fws

On Mon, Jan 29, 2024, 7:18 PM Forest Simmons <forest.simmons21 at gmail.com>
wrote:

> Chris,
>
> My suggestion in the replyI gave to your message (quoted below) made use
> of a "decloned Borda" score in the form  of MaxPairwise Support plus
> MinPairwise Support, which sum is identical to Total Pairwise Support in
> the case of three candidate ... which... in turn, is the pairwise
> formulation of ordinary Borda.
>
> Additionally in the case of three candidates it is the same as your
> default approval.
>
> Even with many candidates MaxPS plus MimPS is the same as your default
> approval ... when we adopt the interpretatiomn ... that MinPS(X) is the
> pairwise support of the ballots fin favor of X over the virtual default
> approval cutoff candidate as a virtual candidate
> i.e..  .. the number of ballots on which Xi.e.anked below mobody.
>
> Similarly, if we count as a virtual candidate the bottom count cutoff,
> then MaxPS(X) is the same as the Implicit Approval of X.
>
>
> Then MaxPS plus MinPS is the zImlicit Approval plus the Top Count.
>
> So MinPlusMaxPS Is the same as Declomed Borda, which in turn, is the same
> as Default Approval.
>
> Then including the possibility of explicit approval, the method I proposed
> generalizes to ...
>
> For each candidate X, let Nemesis(X) be the most approved candidate that
> defeats X pairwise.
>
> And let diff(X) be the difference
>
> Approval(NemesisX) ) - Approval(X).
>
> Let L be the max diff((X) constrained by X having a two step beatpathback
> to  to Nemesis(L).
>
>
> Elect W =Nemesis(L
>
> On Sat, Jan 27, 2024, 7:51 PM Forest Simmons <forest.simmons21 at gmail.com>
> wrote:
>
>> Great methods that avoid memtioning the controversial name of Condorcet
>>
>> Incorporating short beatpaths is an idea I.lke as well ... as in the
>> following method:
>>
>> Elect the winner W of the greatest Borda margin defeat against any
>> candidate L that has a two step beatpath back to any candidate that
>> pairbeats it ... perhaps, for example, L beats X beats W.
>>
>> The motivation is that a buried beats-all candidate will always have a
>> two step beattpath back to any candidate that directly defeated it via the
>> very candidate X that expected to benefit from the burial.
>>
>> AND
>>
>> When one candidate is buried by another, the Borda margin between th the
>> winner W and the loser L increases by lowering the Borda count of the
>> buried candidate and raising the count of the "bus".that was raised to
>> create the pairwise defeat in question.
>>
>> It is useful to know that the Borda Count for a candidate is the same as
>> the sum of its pairwise supports ..the max plus te mim I'm the case of rwo
>> matchups candidates ... which is a recipe for "declined Norda" in general
>> ..in the match plus mim pairwise support.... which should be used in place
>> of ordinary Borda where clone independence is desired.
>>
>> Note that the winner W will be a member of Smith because it will have a
>> bestpath through L to any other candidate ...
>>   even though no mention of Smth or Comdorcet was needed.
>>
>> It turns out empirically to be quite resistant to truncation offensives,
>> as well.
>>
>> What do you think?
>>
>> On Sat, Jan 27, 2024, 2:36 PM C.Benham <cbenham at adam.com.au> wrote:
>>
>>>
>>> I think  Margins Sorted Top Ratings would be a good  relatively burial
>>> resistant public proposal.
>>>
>>> * Voters rank from the top however many candidates they wish.
>>> Equal-ranking allowed.
>>>
>>> Give each candidate a score according to the number of ballots on which
>>> they are ranked below
>>> no others.
>>>
>>> Line them up in that order, highest to lowest.
>>>
>>> Check to see if all the candidates above bottom in this order pairwise
>>> beat the candidate immediately
>>> below them.
>>>
>>> If they do then elect the candidate highest in the order.
>>>
>>> If not begin with the pair that is pairwise out of order by the highest
>>> margin and swap them.
>>> (if there is an exact tie in the size of the margin then swap the
>>> tied-margin pair lowest in the order).
>>>
>>> Repeat until no pair of adjacent candidates are pairwise out of order
>>> and then elect the highest-ordered
>>> candidate. *
>>>
>>> This could also use ratings ballots.
>>>
>>> This meets Condorcet, but can be at least be explained (if not sold)
>>> without reference to Condorcet or Smith.
>>>
>>> It would be as monotonic as it is possible for a Condorcet method to be.
>>>
>>> For the sake of simplicity (and elegance) it has some short-comings.
>>> When there is a top cycle, voters who
>>> didn't top-rate (rank below no other candidates) any of the candidates
>>> in the Smith set are disadvantaged by comparison
>>> those that did.  It would also fail Clone-Independence.
>>>
>>> A much more complicated method idea I had (that would be the same thing
>>> with three candidates):
>>>
>>> *Voters rank from the top however many candidates they wish.
>>> Equal-ranking allowed.
>>>
>>> (1) Eliminate (drop from the ballots and henceforth ignore) all
>>> candidates not in the Smith set.
>>>
>>> (2) Score the remaining candidates according to their minimum pairwise
>>> scores, with ballots that rank two candidates
>>> equal-top contributing a whole vote to each of the two candidate's
>>> scores against each other. Otherwise ballots that
>>> rank two candidates equal below top contribute zero to their pairwise
>>> scores against each other.
>>>
>>> (A possible variation is that they contribute half a vote to each if
>>> they are ranked below top and above bottom.)
>>>
>>> (3) Eliminate all candidates that don't have a "short" (one or two
>>> steps) beatpath to every candidate with a higher minimum
>>> pairwise score.
>>>
>>> (4).  Repeat step 2.  Then margins-sort the resulting scores and elect
>>> the highest-ordered candidate.*
>>>
>>> This is trying to meet Clone Independence, Mono-raise, Chicken Dilemma,
>>> Non-Drastic Defense.
>>>
>>> Chris Benham
>>>
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>>> info
>>>
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