[EM] Start of electowiki article on MDDAsc (typo correction)

[Forest Simmons]

I said (in the second to the last paragraph below) that when there are only
three candidates

 [X=Y>Bottom] = [X=Y<Top],

What I meant was

[X=Y>Bottom] = [X=Y=Top],

This is because there will always be at least one top and one bottom
candidate, which means there can never be two middle candidates in the case
of only three total.

On Thu, Dec 1, 2016 at 1:43 PM, Forest Simmons <fsimmons at pcc.edu> wrote:

> 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
>
>
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