[EM] Start of electowiki article on MDDAsc
Forest Simmons
fsimmons at pcc.edu
Thu Dec 1 13:43:00 PST 2016
Kevin,
I’m not sure if you saw the email where I demonstrated the equivalence of
ICA and MDDAsc in the case of implicit approval. So here is the proof:
Both methods tart out by disqualifying “dominated” or “strongly beaten”
candidates unless that would eliminate all of them. Then they elect the
most approved qualified candidate.
In MDDA “strongly beaten” means majority beaten. In MDDAsc it still means
more than fifty percent of the ballots opposed, but the “opposed” count
takes symmetric completion of equal (unapproved) rankings into account.
In ICA (and ICT) “dominated” means the pairwise winning margin is greater
than the number of equal top rankings:
[X>Y]-[Y>X]-[X=Y=Top] > 0
If we add to this inequality the identity [X>Y] + [Y>X] +[X=Y] = 100% , we
get
2[X>Y] + ([X=Y]-[X=Y=Top]) > 100%
If we replace the difference in the parentheses with the equivalent
expression [X=Y<Top] , and divide both sides by two, we get
[X>Y]+(1/2)[X=Y<Top] = 50%.
This is precisely what we mean by majority beaten with symmetric completion
of equal rankings below Top.
So "strongly beaten" and "dominated" mean the same thing in ICA/ICT and
MDDAsc as long as the symmetric completion affects all ranks below Top.
In other words, ICT and MDDTsc are identical methods.
But the convention in MDDAsc is that symmetric completion happens only
among the unapproved candidates. If all ranked candidates are approved,
then symmetric completion takes place only at Bottom.
That’s why in my version of ICA candidate X dominates candidate Y iff
[X>Y] – [Y>X] > [X=Y>Bottom]
In the case of only three candidates we always have [X=Y>Bottom] =
[X=Y<Top], so in that case my version of ICA is the same as yours. That is
why your simulations always give the same result for ICA and MDDAsc when
approval is taken as implicit approval.
In general the following definition of “dominates” makes ICA precisely
equivalent to MDDAsc no matter where the approval cutoff is:
Candidate X dominates candidate Y iff the pairwise margin of defeat of Y by
X is greater than the number of ballots on which X and Y are both approved
and ranked equal to each other.
My Best,
Forest
On Thu, Nov 24, 2016 at 7:25 AM, Kevin Venzke <stepjak at yahoo.fr> wrote:
> Hi Forest,
>
> I think references to MDDA's motivating criteria should be reduced because
> the new method MDDAsc shouldn't satisfy SFC or SDSC anymore.
>
> I'm not sure it's natural to say that MDDA itself is a variant of ICA. I
> would say both are Condorcet//Approval variants that satisfy FBC.
>
> I'm not sure if a different sort of reference to ICA would make enough
> sense... The apparent closeness between MDDAsc and ICA's results may be too
> obscure to be worth noting. For one thing, the similarity assumes that ICA
> uses an explicit approval cutoff, which isn't intended to be an option with
> ICA.
>
> The draft doesn't mention the Plurality criterion. I'm not sure about it
> yet. My hunch is that to the extent that you aren't using symmetric
> completion below top, you're still vulnerable to whatever factors cause
> MDDA(implicit) to violate Plurality with 4+ candidates.
>
> Kevin
>
>
> ------------------------------
> *De :* Forest Simmons <fsimmons at pcc.edu>
> *À :* Michael Ossipoff <email9648742 at gmail.com>; Chris Benham <
> cbenhamau at yahoo.com.au>; Kevin Venzke <stepjak at yahoo.fr>; Forest Simmons <
> fsimmons at pcc.edu>
> *Envoyé le :* Mercredi 23 novembre 2016 18h54
> *Objet :* Start of electowiki article on MDDAsc
>
> I've made a start on the electowiki article for MDDA(symmetric completion).
>
> Please check my statements and examples for accuracy, and make suggestions
> for improved exposition, etc.
>
> See the attachment.
>
> Thanks,
>
> Forest
>
>
>
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