# [EM] Is this another formulation of Benham's new method?

Forest Simmons fsimmons at pcc.edu
Sat May 3 17:39:40 PDT 2014

```Chris,

thanks for exploring this idea.

My thinking is that any good way of generating a social ordering of the
candidates cannot be harmed by (and might benefit from) the "Majority
Enhancement" technique.

For example, if we apply ME to the any of our lists after they have been
sorted pairwise whether by the bubble sort (as in regular DMC) or according
to margins, as in Approval Sorted Margins or (as another example)
MaxMinLV(erw)Margins.

Any list sorted pairwise will end up with a member of Smith at the top, and
whenever Smith consists of only a cycle of three candidates, all three will
be uncovered, so the ME process will not change the result.  It just
guarantees an uncovered winner in general, and will preserve monotonicity
and clone proofness if the list is already compliant.

Forest

Date: Thu, 01 May 2014 23:56:50 +0930
> From: "C.Benham" <cbenham at adam.com.au>
> To: election-methods at lists.electorama.com
> Subject: Re: [EM] Is this another formulation of Benham's new method?
> Content-Type: text/plain; charset="iso-8859-1"; Format="flowed"
>
> Forest,
> > Is Condorcet(MaxPO(tw)) equivalent to Condorcet(MinLV(eq rank
> > whole))?  If so the margins versions would be equivalent too.
> >
> > "PO" stands for "Pairwise Opposition,"  and "tw" for "Truncation
> > Whole," which means that if two candidates are truncated together on a
> > ballot they are both counted in opposition to each other.
>
> It looks like yes.
>
> 35 A>B
> 25 B
> 40 C
>
> C > A  40-35,    A>B 35-25,   B>C 60-40.      MinMax (Losing Votes)
> scores:  C40 > A35 > B25.   (Margins Sort keeps this order).
>
> Forest's suggested  "truncation whole" version:
> C>A 65-60,    A>B 75-65,   B>C 60-40.        Max Pairwise Opposition
> (truncation whole) scores:  C60 < A65 < B75 .
>
> The smaller a candidate's Max PO score the better for it.   I'll look at
> your other method suggestion later.
>
> Chris Benham
>
>
> On 5/1/2014 6:05 AM, Forest Simmons wrote:
> > Is Condorcet(MaxPO(tw)) equivalent to Condorcet(MinLV(eq rank
> > whole))?  If so the margins versions would be equivalent too.
> >
> > "PO" stands for "Pairwise Opposition,"  and "tw" for "Truncation
> > Whole," which means that if two candidates are truncated together on a
> > ballot they are both counted in opposition to each other.
> >
> > With this convention the Pairwise Opposition (tw) from candidate A
> > against candidate B is the number N of ballots on which A is ranked
> > strictly above B plus the number of ballots on which A and B are
> > truncated together.
> >
> > With the equal rank whole convention, the LV strength of the defeat of
> > B by A is the number M of ballots on  which B is ranked (but not
> > truncated!) above or equal to A.
> >
> > Careful consideration reveals N+M is a constant, namely the total
> > number of ballots, since no case was left out or counted more than once.
> >
> > This suggests a formulation of Benham's new method that we could call
> > MPO(tw) Sorted Pairwise Margins. in analogy to Approval Sorted
> > Pairwise Margins.
> >
> > List the candidates in order of MPO(tw) scores, and then adjust the
> > list by reversal of adjacent pairs that are out of pairwise defeat
> > order taking into account how close they are in their scores.
> >
> > I believe that the above discussion shows that this formulation is
> > equivalent to Benham's MaxMinLV(erw) Margins method.
> >
> > The truncation whole (tw) convention forces MMPO(tw) to comply with
> > Plurality.  The pairwise sorting feature makes it comply with Smith.
> >
> > But this brings up another method:  Majority Enhanced MPO(tv), in
> > analogy with Majority Enhanced Approval:
> >
> > Initiate a list L with the name of the candidate with the least
> > MPO(tv).  Then while there is any candidate that covers all of the
> > candidates listed, from among such candidates add to the list the name
> > of the one with the least MPO(tw). Elect the last candidate added to
> > the list.
> >
> > Forest
> >
> >
> >
> >
> > ----
> > Election-Methods mailing list - see http://electorama.com/em for list
> info
>
> -
>
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