Elimination by MinPO to the MMPO candidate.
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 ...
- let X be the candidate with the least PO to the candidate C whose Max PO
is smallest among the remaining candidates. Then - 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
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 ...
-
Elect the Condorcet Winner if there is one. Else ...
-
Let K be the candidate with the smallest MaxPO among the remaining
candidates. -
Let X be the candidate with the smallest PO against K.
-
Let S be the set of remaining candidates that are currently covered by X.
-
Eliminate every member of S.
-
Eliminate X.
6 Repeat steps one thru five until only two candidate remains. -
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@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 ...
- let X be the candidate with the least PO to the candidate C whose Max
PO is smallest among the remaining candidates. Then - 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
It looks like the simplest procedure is to start by eliminating all
candidates outside of Smith.
Then elect the sincere winner between the Smith MMPO candidate K and the
Smth candidate X with the MaxPO against K.
I should have listened to Chris in the first place!
To make this method Landau compliant, if K is covered, repeatedly update
the candidate variablenK as the candidate that covers K with the least
pairwise opposition from the previous value of K.
Then initialize variable X as the candidate with the greatest PO against
this final value of K, and repeatedly update it as the candidate that
covers its previous value with the greatest pairwise opposition to K.
The sincere runoff is between these two Landau candidates.
None of this changes the results of our examples, because all candidates
were members of the Landau set ... which is the case whenever Smith has
fewer than four candidates.
fws
On Fri, Jun 23, 2023, 1:25 PM Forest Simmons forest.simmons21@gmail.com
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 ...
-
Elect the Condorcet Winner if there is one. Else ...
-
Let K be the candidate with the smallest MaxPO among the remaining
candidates. -
Let X be the candidate with the smallest PO against K.
-
Let S be the set of remaining candidates that are currently covered by
X. -
Eliminate every member of S.
-
Eliminate X.
6 Repeat steps one thru five until only two candidate remains. -
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@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 ...
- let X be the candidate with the least PO to the candidate C whose Max
PO is smallest among the remaining candidates. Then - 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
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 ...
-
Elect the Condorcet Winner if there is one. Else ...
-
Let K be the candidate with the smallest MaxPO among the remaining
candidates. -
Let X be the candidate with the smallest PO against K.
-
Let S be the set of remaining candidates that are currently covered
by X. -
Eliminate every member of S.
-
Eliminate X.
6 Repeat steps one thru five until only two candidate remains. -
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@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