Margins Sorted Top-Ratings
I think Margins Sorted Top Ratings would be a good relatively burial
resistant public proposal.
- Voters rank from the top however many candidates they wish.
Equal-ranking allowed.
Give each candidate a score according to the number of ballots on which
they are ranked below
no others.
Line them up in that order, highest to lowest.
Check to see if all the candidates above bottom in this order pairwise
beat the candidate immediately
below them.
If they do then elect the candidate highest in the order.
If not begin with the pair that is pairwise out of order by the highest
margin and swap them.
(if there is an exact tie in the size of the margin then swap the
tied-margin pair lowest in the order).
Repeat until no pair of adjacent candidates are pairwise out of order
and then elect the highest-ordered
candidate. *
This could also use ratings ballots.
This meets Condorcet, but can be at least be explained (if not sold)
without reference to Condorcet or Smith.
It would be as monotonic as it is possible for a Condorcet method to be.
For the sake of simplicity (and elegance) it has some short-comings.
When there is a top cycle, voters who
didn't top-rate (rank below no other candidates) any of the candidates
in the Smith set are disadvantaged by comparison
those that did. It would also fail Clone-Independence.
A much more complicated method idea I had (that would be the same thing
with three candidates):
*Voters rank from the top however many candidates they wish.
Equal-ranking allowed.
(1) Eliminate (drop from the ballots and henceforth ignore) all
candidates not in the Smith set.
(2) Score the remaining candidates according to their minimum pairwise
scores, with ballots that rank two candidates
equal-top contributing a whole vote to each of the two candidate's
scores against each other. Otherwise ballots that
rank two candidates equal below top contribute zero to their pairwise
scores against each other.
(A possible variation is that they contribute half a vote to each if
they are ranked below top and above bottom.)
(3) Eliminate all candidates that don't have a "short" (one or two
steps) beatpath to every candidate with a higher minimum
pairwise score.
(4). Repeat step 2. Then margins-sort the resulting scores and elect
the highest-ordered candidate.*
This is trying to meet Clone Independence, Mono-raise, Chicken Dilemma,
Non-Drastic Defense.
Chris Benham
Great methods that avoid memtioning the controversial name of Condorcet
Incorporating short beatpaths is an idea I.lke as well ... as in the
following method:
Elect the winner W of the greatest Borda margin defeat against any
candidate L that has a two step beatpath back to any candidate that
pairbeats it ... perhaps, for example, L beats X beats W.
The motivation is that a buried beats-all candidate will always have a two
step beattpath back to any candidate that directly defeated it via the very
candidate X that expected to benefit from the burial.
AND
When one candidate is buried by another, the Borda margin between th the
winner W and the loser L increases by lowering the Borda count of the
buried candidate and raising the count of the "bus".that was raised to
create the pairwise defeat in question.
It is useful to know that the Borda Count for a candidate is the same as
the sum of its pairwise supports ..the max plus te mim I'm the case of rwo
matchups candidates ... which is a recipe for "declined Norda" in general
..in the match plus mim pairwise support.... which should be used in place
of ordinary Borda where clone independence is desired.
Note that the winner W will be a member of Smith because it will have a
bestpath through L to any other candidate ...
even though no mention of Smth or Comdorcet was needed.
It turns out empirically to be quite resistant to truncation offensives, as
well.
What do you think?
On Sat, Jan 27, 2024, 2:36 PM C.Benham cbenham@adam.com.au wrote:
I think Margins Sorted Top Ratings would be a good relatively burial
resistant public proposal.
- Voters rank from the top however many candidates they wish.
Equal-ranking allowed.
Give each candidate a score according to the number of ballots on which
they are ranked below
no others.
Line them up in that order, highest to lowest.
Check to see if all the candidates above bottom in this order pairwise
beat the candidate immediately
below them.
If they do then elect the candidate highest in the order.
If not begin with the pair that is pairwise out of order by the highest
margin and swap them.
(if there is an exact tie in the size of the margin then swap the
tied-margin pair lowest in the order).
Repeat until no pair of adjacent candidates are pairwise out of order
and then elect the highest-ordered
candidate. *
This could also use ratings ballots.
This meets Condorcet, but can be at least be explained (if not sold)
without reference to Condorcet or Smith.
It would be as monotonic as it is possible for a Condorcet method to be.
For the sake of simplicity (and elegance) it has some short-comings.
When there is a top cycle, voters who
didn't top-rate (rank below no other candidates) any of the candidates
in the Smith set are disadvantaged by comparison
those that did. It would also fail Clone-Independence.
A much more complicated method idea I had (that would be the same thing
with three candidates):
*Voters rank from the top however many candidates they wish.
Equal-ranking allowed.
(1) Eliminate (drop from the ballots and henceforth ignore) all
candidates not in the Smith set.
(2) Score the remaining candidates according to their minimum pairwise
scores, with ballots that rank two candidates
equal-top contributing a whole vote to each of the two candidate's
scores against each other. Otherwise ballots that
rank two candidates equal below top contribute zero to their pairwise
scores against each other.
(A possible variation is that they contribute half a vote to each if
they are ranked below top and above bottom.)
(3) Eliminate all candidates that don't have a "short" (one or two
steps) beatpath to every candidate with a higher minimum
pairwise score.
(4). Repeat step 2. Then margins-sort the resulting scores and elect
the highest-ordered candidate.*
This is trying to meet Clone Independence, Mono-raise, Chicken Dilemma,
Non-Drastic Defense.
Chris Benham
Election-Methods mailing list - see https://electorama.com/em for list
info
On Sat, Jan 27, 2024 at 2:36 PM C.Benham cbenham@adam.com.au wrote:
I think Margins Sorted Top Ratings would be a good relatively burial
resistant public proposal.
- Voters rank from the top however many candidates they wish.
Equal-ranking allowed.
Give each candidate a score according to the number of ballots on which
they are ranked below
no others.
Line them up in that order, highest to lowest.
Check to see if all the candidates above bottom in this order pairwise
beat the candidate immediately
below them.
If they do then elect the candidate highest in the order.
If not begin with the pair that is pairwise out of order by the highest
margin and swap them.
This is exactly the opposite of the rule for other Sorted Margins methods.
In Approval Sorted Margins (see electowiki), the pairwise out of order pair
with minimum margin is the pair that is swapped. The intent here is to
make the minimal change to the seed ranking in order to get pairwise
ordering.
Did you mean to introduce a new sorted margins method? I think swapping the
maximum margin pair would have very odd behavior.
(if there is an exact tie in the size of the margin then swap the
tied-margin pair lowest in the order).
Repeat until no pair of adjacent candidates are pairwise out of order
and then elect the highest-ordered
candidate. *
This could also use ratings ballots.
This meets Condorcet, but can be at least be explained (if not sold)
without reference to Condorcet or Smith.
It would be as monotonic as it is possible for a Condorcet method to be.
For the sake of simplicity (and elegance) it has some short-comings.
When there is a top cycle, voters who
didn't top-rate (rank below no other candidates) any of the candidates
in the Smith set are disadvantaged by comparison
those that did. It would also fail Clone-Independence.
A much more complicated method idea I had (that would be the same thing
with three candidates):
*Voters rank from the top however many candidates they wish.
Equal-ranking allowed.
(1) Eliminate (drop from the ballots and henceforth ignore) all
candidates not in the Smith set.
(2) Score the remaining candidates according to their minimum pairwise
scores, with ballots that rank two candidates
equal-top contributing a whole vote to each of the two candidate's
scores against each other. Otherwise ballots that
rank two candidates equal below top contribute zero to their pairwise
scores against each other.
So, a variation on equal-rated-whole, with only equal-top having erw,
otherwise equal-rated-zero.
It appears that your tied-at-top-erw array must be recounted after the
initial elimination down to the Smith set, so it's not summable on first
pass tabulation, correct?
(A possible variation is that they contribute half a vote to each if
they are ranked below top and above bottom.)
(3) Eliminate all candidates that don't have a "short" (one or two
steps) beatpath to every candidate with a higher minimum
pairwise score.
I understand the motivation, but it seems a bit complex for a public
proposal.
(4). Repeat step 2. Then margins-sort the resulting scores and elect
the highest-ordered candidate.*
So, another re-tabulation, which again loses initial summability.
This is trying to meet Clone Independence, Mono-raise, Chicken Dilemma,
Non-Drastic Defense.
Chris Benham
Election-Methods mailing list - see https://electorama.com/em for list
info
(This is a re-send with an error corrected)
I think Margins Sorted Top Ratings would be a good relatively burial
resistant public proposal.
- Voters rank from the top however many candidates they wish.
Equal-ranking allowed.
Give each candidate a score according to the number of ballots on which
they are ranked below
no others.
Line them up in that order, highest to lowest.
Check to see if all the candidates above bottom in this order pairwise
beat the candidate immediately
below them.
If they do then elect the candidate highest in the order.
If not begin with the pair that is pairwise out of order by the lowest
margin and swap them.
(if there is an exact tie in the size of the margin then swap the
tied-margin pair lowest in the order).
Repeat until no pair of adjacent candidates are pairwise out of order
and then elect the highest-ordered
candidate. *
This could also use ratings ballots.
This meets Condorcet, but can be at least be explained (if not sold)
without reference to Condorcet or Smith.
It would be as monotonic as it is possible for a Condorcet method to be.
For the sake of simplicity (and elegance) it has some short-comings.
When there is a top cycle, voters who
didn't top-rate (rank below no other candidates) any of the candidates
in the Smith set are disadvantaged by comparison
those that did. It would also fail Clone-Independence.
A much more complicated method idea I had (that would be the same thing
with three candidates):
*Voters rank from the top however many candidates they wish.
Equal-ranking allowed.
(1) Eliminate (drop from the ballots and henceforth ignore) all
candidates not in the Smith set.
(2) Score the remaining candidates according to their minimum pairwise
scores, with ballots that rank two candidates
equal-top contributing a whole vote to each of the two candidate's
scores against each other. Otherwise ballots that
rank two candidates equal below top contribute zero to their pairwise
scores against each other.
(A possible variation is that they contribute half a vote to each if
they are ranked below top and above bottom.)
(3) Eliminate all candidates that don't have a "short" (one or two
steps) beatpath to every candidate with a higher minimum
pairwise score.
(4). Repeat step 2. Then margins-sort the resulting scores and elect
the highest-ordered candidate.*
This is trying to meet Clone Independence, Mono-raise, Chicken Dilemma,
Non-Drastic Defense.
Chris Benham
Chris,
My suggestion in the replyI gave to your message (quoted below) made use of
a "decloned Borda" score in the form of MaxPairwise Support plus
MinPairwise Support, which sum is identical to Total Pairwise Support in
the case of three candidate ... which... in turn, is the pairwise
formulation of ordinary Borda.
Additionally in the case of three candidates it is the same as your default
approval.
Even with many candidates MaxPS plus MimPS is the same as your default
approval ... when we adopt the interpretatiomn ... that MinPS(X) is the
pairwise support of the ballots fin favor of X over the virtual default
approval cutoff candidate as a virtual candidate
i.e.. .. the number of ballots on which Xi.e.anked below mobody.
Similarly, if we count as a virtual candidate the bottom count cutoff, then
MaxPS(X) is the same as the Implicit Approval of X.
Then MaxPS plus MinPS is the zImlicit Approval plus the Top Count.
So MinPlusMaxPS Is the same as Declomed Borda, which in turn, is the same
as Default Approval.
Then including the possibility of explicit approval, the method I proposed
generalizes to ...
For each candidate X, let Nemesis(X) be the most approved candidate that
defeats X pairwise.
And let diff(X) be the difference
Approval(NemesisX) ) - Approval(X).
Let L be the max diff((X) constrained by X having a two step beatpathback
to to Nemesis(L).
Elect W =Nemesis(L
On Sat, Jan 27, 2024, 7:51 PM Forest Simmons forest.simmons21@gmail.com
wrote:
Great methods that avoid memtioning the controversial name of Condorcet
Incorporating short beatpaths is an idea I.lke as well ... as in the
following method:
Elect the winner W of the greatest Borda margin defeat against any
candidate L that has a two step beatpath back to any candidate that
pairbeats it ... perhaps, for example, L beats X beats W.
The motivation is that a buried beats-all candidate will always have a two
step beattpath back to any candidate that directly defeated it via the very
candidate X that expected to benefit from the burial.
AND
When one candidate is buried by another, the Borda margin between th the
winner W and the loser L increases by lowering the Borda count of the
buried candidate and raising the count of the "bus".that was raised to
create the pairwise defeat in question.
It is useful to know that the Borda Count for a candidate is the same as
the sum of its pairwise supports ..the max plus te mim I'm the case of rwo
matchups candidates ... which is a recipe for "declined Norda" in general
..in the match plus mim pairwise support.... which should be used in place
of ordinary Borda where clone independence is desired.
Note that the winner W will be a member of Smith because it will have a
bestpath through L to any other candidate ...
even though no mention of Smth or Comdorcet was needed.
It turns out empirically to be quite resistant to truncation offensives,
as well.
What do you think?
On Sat, Jan 27, 2024, 2:36 PM C.Benham cbenham@adam.com.au wrote:
I think Margins Sorted Top Ratings would be a good relatively burial
resistant public proposal.
- Voters rank from the top however many candidates they wish.
Equal-ranking allowed.
Give each candidate a score according to the number of ballots on which
they are ranked below
no others.
Line them up in that order, highest to lowest.
Check to see if all the candidates above bottom in this order pairwise
beat the candidate immediately
below them.
If they do then elect the candidate highest in the order.
If not begin with the pair that is pairwise out of order by the highest
margin and swap them.
(if there is an exact tie in the size of the margin then swap the
tied-margin pair lowest in the order).
Repeat until no pair of adjacent candidates are pairwise out of order
and then elect the highest-ordered
candidate. *
This could also use ratings ballots.
This meets Condorcet, but can be at least be explained (if not sold)
without reference to Condorcet or Smith.
It would be as monotonic as it is possible for a Condorcet method to be.
For the sake of simplicity (and elegance) it has some short-comings.
When there is a top cycle, voters who
didn't top-rate (rank below no other candidates) any of the candidates
in the Smith set are disadvantaged by comparison
those that did. It would also fail Clone-Independence.
A much more complicated method idea I had (that would be the same thing
with three candidates):
*Voters rank from the top however many candidates they wish.
Equal-ranking allowed.
(1) Eliminate (drop from the ballots and henceforth ignore) all
candidates not in the Smith set.
(2) Score the remaining candidates according to their minimum pairwise
scores, with ballots that rank two candidates
equal-top contributing a whole vote to each of the two candidate's
scores against each other. Otherwise ballots that
rank two candidates equal below top contribute zero to their pairwise
scores against each other.
(A possible variation is that they contribute half a vote to each if
they are ranked below top and above bottom.)
(3) Eliminate all candidates that don't have a "short" (one or two
steps) beatpath to every candidate with a higher minimum
pairwise score.
(4). Repeat step 2. Then margins-sort the resulting scores and elect
the highest-ordered candidate.*
This is trying to meet Clone Independence, Mono-raise, Chicken Dilemma,
Non-Drastic Defense.
Chris Benham
Election-Methods mailing list - see https://electorama.com/em for list
info
Chris,
I just remember my default approval unlike yours has three levels ..
Zero for unranked candidates, one for candidates ranked below the upper
cutoff, and two for candidates ranked above that cutoff .... this what
makes default approval the same as decloned Borda ... and is the approval I
have in mind for themax approval defeat margin method:
Lacking an undefeated candidate ... elect the winner W of the defeat pair
(W>L) that maximizes the difference between the approvals of W and L,
subject to the constraint that L repays W in two steps: L>X>W.
Sorry about the confusion!
fws
On Mon, Jan 29, 2024, 7:18 PM Forest Simmons forest.simmons21@gmail.com
wrote:
Chris,
My suggestion in the replyI gave to your message (quoted below) made use
of a "decloned Borda" score in the form of MaxPairwise Support plus
MinPairwise Support, which sum is identical to Total Pairwise Support in
the case of three candidate ... which... in turn, is the pairwise
formulation of ordinary Borda.
Additionally in the case of three candidates it is the same as your
default approval.
Even with many candidates MaxPS plus MimPS is the same as your default
approval ... when we adopt the interpretatiomn ... that MinPS(X) is the
pairwise support of the ballots fin favor of X over the virtual default
approval cutoff candidate as a virtual candidate
i.e.. .. the number of ballots on which Xi.e.anked below mobody.
Similarly, if we count as a virtual candidate the bottom count cutoff,
then MaxPS(X) is the same as the Implicit Approval of X.
Then MaxPS plus MinPS is the zImlicit Approval plus the Top Count.
So MinPlusMaxPS Is the same as Declomed Borda, which in turn, is the same
as Default Approval.
Then including the possibility of explicit approval, the method I proposed
generalizes to ...
For each candidate X, let Nemesis(X) be the most approved candidate that
defeats X pairwise.
And let diff(X) be the difference
Approval(NemesisX) ) - Approval(X).
Let L be the max diff((X) constrained by X having a two step beatpathback
to to Nemesis(L).
Elect W =Nemesis(L
On Sat, Jan 27, 2024, 7:51 PM Forest Simmons forest.simmons21@gmail.com
wrote:
Great methods that avoid memtioning the controversial name of Condorcet
Incorporating short beatpaths is an idea I.lke as well ... as in the
following method:
Elect the winner W of the greatest Borda margin defeat against any
candidate L that has a two step beatpath back to any candidate that
pairbeats it ... perhaps, for example, L beats X beats W.
The motivation is that a buried beats-all candidate will always have a
two step beattpath back to any candidate that directly defeated it via the
very candidate X that expected to benefit from the burial.
AND
When one candidate is buried by another, the Borda margin between th the
winner W and the loser L increases by lowering the Borda count of the
buried candidate and raising the count of the "bus".that was raised to
create the pairwise defeat in question.
It is useful to know that the Borda Count for a candidate is the same as
the sum of its pairwise supports ..the max plus te mim I'm the case of rwo
matchups candidates ... which is a recipe for "declined Norda" in general
..in the match plus mim pairwise support.... which should be used in place
of ordinary Borda where clone independence is desired.
Note that the winner W will be a member of Smith because it will have a
bestpath through L to any other candidate ...
even though no mention of Smth or Comdorcet was needed.
It turns out empirically to be quite resistant to truncation offensives,
as well.
What do you think?
On Sat, Jan 27, 2024, 2:36 PM C.Benham cbenham@adam.com.au wrote:
I think Margins Sorted Top Ratings would be a good relatively burial
resistant public proposal.
- Voters rank from the top however many candidates they wish.
Equal-ranking allowed.
Give each candidate a score according to the number of ballots on which
they are ranked below
no others.
Line them up in that order, highest to lowest.
Check to see if all the candidates above bottom in this order pairwise
beat the candidate immediately
below them.
If they do then elect the candidate highest in the order.
If not begin with the pair that is pairwise out of order by the highest
margin and swap them.
(if there is an exact tie in the size of the margin then swap the
tied-margin pair lowest in the order).
Repeat until no pair of adjacent candidates are pairwise out of order
and then elect the highest-ordered
candidate. *
This could also use ratings ballots.
This meets Condorcet, but can be at least be explained (if not sold)
without reference to Condorcet or Smith.
It would be as monotonic as it is possible for a Condorcet method to be.
For the sake of simplicity (and elegance) it has some short-comings.
When there is a top cycle, voters who
didn't top-rate (rank below no other candidates) any of the candidates
in the Smith set are disadvantaged by comparison
those that did. It would also fail Clone-Independence.
A much more complicated method idea I had (that would be the same thing
with three candidates):
*Voters rank from the top however many candidates they wish.
Equal-ranking allowed.
(1) Eliminate (drop from the ballots and henceforth ignore) all
candidates not in the Smith set.
(2) Score the remaining candidates according to their minimum pairwise
scores, with ballots that rank two candidates
equal-top contributing a whole vote to each of the two candidate's
scores against each other. Otherwise ballots that
rank two candidates equal below top contribute zero to their pairwise
scores against each other.
(A possible variation is that they contribute half a vote to each if
they are ranked below top and above bottom.)
(3) Eliminate all candidates that don't have a "short" (one or two
steps) beatpath to every candidate with a higher minimum
pairwise score.
(4). Repeat step 2. Then margins-sort the resulting scores and elect
the highest-ordered candidate.*
This is trying to meet Clone Independence, Mono-raise, Chicken Dilemma,
Non-Drastic Defense.
Chris Benham
Election-Methods mailing list - see https://electorama.com/em for list
info
Election-Methods mailing list - see https://electorama.com/em for list
info
On Mon, Jan 29, 2024 at 10:55 AM C.Benham cbenham@adam.com.au wrote:
(This is a re-send with an error corrected)
I think Margins Sorted Top Ratings would be a good relatively burial
resistant public proposal.
- Voters rank from the top however many candidates they wish.
Equal-ranking allowed.
Give each candidate a score according to the number of ballots on which
they are ranked below
no others.
Line them up in that order, highest to lowest.
Check to see if all the candidates above bottom in this order pairwise
beat the candidate immediately
below them.
If they do then elect the candidate highest in the order.
If not begin with the pair that is pairwise out of order by the lowest
margin and swap them.
(if there is an exact tie in the size of the margin then swap the
tied-margin pair lowest in the order).
Repeat until no pair of adjacent candidates are pairwise out of order
and then elect the highest-ordered
candidate. *
This could also use ratings ballots.
This meets Condorcet, but can be at least be explained (if not sold)
without reference to Condorcet or Smith.
It would be as monotonic as it is possible for a Condorcet method to be.
For the sake of simplicity (and elegance) it has some short-comings.
When there is a top cycle, voters who
didn't top-rate (rank below no other candidates) any of the candidates
in the Smith set are disadvantaged by comparison
those that did. It would also fail Clone-Independence.
Could you explain how Top-rated Sorted Margins fails Independence from
Clones? I'm not seeing how that would happen, unless you have unusual
restrictions on top-rating
A much more complicated method idea I had (that would be the same thing
with three candidates):
*Voters rank from the top however many candidates they wish.
Equal-ranking allowed.
(1) Eliminate (drop from the ballots and henceforth ignore) all
candidates not in the Smith set.
(2) Score the remaining candidates according to their minimum pairwise
scores, with ballots that rank two candidates
equal-top contributing a whole vote to each of the two candidate's
scores against each other. Otherwise ballots that
rank two candidates equal below top contribute zero to their pairwise
scores against each other.
(A possible variation is that they contribute half a vote to each if
they are ranked below top and above bottom.)
This idea is interesting, but I'd like to see a more nuanced incorporation
of the strength of equal ratings.
The simplest way to do that would be Equal-Rated-Score-fraction. In other
words, when two candidates are equally rated, they both receive
score/maxscore of a vote. So an equal top pair would each receive 1, and
equal bottom pair would receive zero, and equal-in-between would receive
something in between.
And of course, the scores would have to be recalculated after the Smith
elimination step. Say you had Smith candidates A3=B3 > C1=D1 on a 0-5
rating, and A, B, C were in Smith. Then you'd shift to A5=B5 > C3=D3, and
score A>B:1, B>A:1, C>D:0.6, D>C:0.6.
(3) Eliminate all candidates that don't have a "short" (one or two
steps) beatpath to every candidate with a higher minimum
pairwise score.
This is an interesting step, but I don't think it can be included without
some kind of philosophical motivation. And to keep it simple, I'd limit it
to one step. That is, if the ranking is A, B, C, D, E, E would be included
only if it requires only one indirect step to defeat each of A, B, C, and D.
(4). Repeat step 2. Then margins-sort the resulting scores and elect
the highest-ordered candidate.*
If you're using the ERSf method above, scores would have to be
re-normalized before counting.
This is trying to meet Clone Independence, Mono-raise, Chicken Dilemma,
Non-Drastic Defense.
An interesting idea. But why wouldn't sorted margins after the first
re-scoring work?