[EM] Elimination by MinPO to the MMPO candidate.

[C.Benham]

On 24/06/2023 5:55 am, Forest Simmons wrote:

> Example 3
> 49 C
> 26 A>B
> 25 B
> In this example A wins the sincere comparison 51 to 49.

Forest,

You don't specify in the example that any of the votes are insincere.  
Does A win because all the B voters are defecting against A and will all 
show up for the runoff?

> Example 1.
>
> 45 A>B (sincere A>C)
> 40 B>C
> 15 C>A
> The sincere winner between A and C is C. 

Quite an impressive effort to save such a relatively forlorn sincere 
CW.  I wouldn't think too many people from FPP or IRV land would be 
overly disturbed by just electing
A in a one-round process.

Chris


On 24/06/2023 5:55 am, Forest Simmons wrote:
> 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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