[EM] Elimination by MinPO to the MMPO candidate.

[Forest Simmons]

Before the examples, note that the Condorcet Loser elimination step is
redundant because the removal of covered candidates will automatically take
care of it.

The method thus simplified can be cast as ...

0. Elect the Condorcet Winner if there is one. Else ...

1. Let K be the candidate with the smallest MaxPO among the remaining
candidates.
2. Let X be the candidate with the smallest PO against K.
3. Let S be the set of remaining candidates that are currently covered by X.
4. Eliminate every member of S.
5. Eliminate X.
6  Repeat steps one thru five until only two candidate remains.
7. Elect the winner of a sincere runoff between the two remaining
candidates.

Note that once any Smith candidate is eliminated, only Smith candidates
will remain ... since each Smith candidate covers every candidate outside
of Smith.

Example 1.

45 A>B (sincere A>C)
40 B>C
15 C>A

The MMPO candidate is A with a maxPO of only 55, frim C.

The candidate with the least PO against A is B, with a PO of only 40
against A.

Candidate B is the only one eliminated because it covers nobody else.

The sincere winner between A and C is C.

Example 2.

46 A>B
44 B>C
5 C>A
5 C>B

The MMPO candidate is B whose max opposition is only 51, from A.

The min opposition to B is only ten, from C.

Eliminating C leaves A and B for the runoff.

If the given ballots are sincere, then A will win.

But the cycle brings into question the sincerity of the ballots; false
cycles are more likely than sincere cycles.

 If the cycle was caused by B's unilateral burial of A under C...  by
changing  44 B>A to 44 B>C ... then A wins the sincere comparison, removing
the doubt.

If the cycle was caused by burial of B under A ... by changing 10 C>A to 5
each of C>A and C>B ... then B wins the sincere comparison, and is restored
to its rightful place as sincere CW.

Example 3
49 C
26 A>B
25 B

The MMPO candidate is A based on its second highest PO score of 25 (from
B), because A and B were tied for the smallest MaxPO (of 49 from C).

The candidate with min PO against A is B with 25, (compared to 49 from C).

So B is eliminated, leaving the sincere runoff between A and C.

In this example A wins the sincere comparison 51 to 49.

I hope these examples are sufficient to show how the method works.

fws


On Fri, Jun 23, 2023, 12:56 AM Forest Simmons <forest.simmons21 at gmail.com>
wrote:

> Chris has been keeping me humble ... shooting down my worst nonsense ...
> while Kristofer has been busy cleaning up code ... so watch out for an
> explosion of progress!
>
> Think of the MMPO candidate C as being located in the midst of the other
> candidates in such a way that the smallest neighborhood centered on C that
> is large enough to contain the other candidates is as small as possible.
>
> If you extend a line segment from C to another candidate X on the
> outskirts of that neighborhood, and then cut that segment with a
> perpendicular bisecting plane ... that plane will separate the voters
> closer to X from the voters closer to C.
>
> The number of voters closer to X is known as X's Pairwise Opposition to C
> in that pairwise contest.
>
> This imaginary schematic suggests that the candidate X with the least PO
> to C, would be a good one to eliminate in a single winner method.
>
> Accordingly ... when every candidate suffers at least one pairwise defeat,
> we suggest the following elimination procedure:
>
> While more than two candidates remain ... Eliminate the candidate that is
> defeated pairwise by all of the other candidates, if there is such a
> candidate ... Otherwise ...
> 1. let X be the candidate with the least PO to the candidate C whose Max
> PO is smallest among the remaining candidates. Then
> 2. Eliminate X, as well as any candidate(s) covered by X relative to the
> remaining candidates.
> EndWhile
>
> Finally, elect the sincere runoff winner between the two remaining
> candidates.
>
> Examples to follow soon ....
>
> For now, we only mention that the eliminations are designed to make it
> highly likely that when a sincere undefeated candidate exists; it will be
> one of the finalists ... and also to point out that the elimination of
> candidates covered by other candidates being eliminated ... this precaution
> ensures that no covered candidate will be elected ... thus avoiding a
> potential liability that plauges nearly all other extant methods.
>
> fws
>
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