Sorted Approval Margins (plus two other Condorcet methods)
I think Condorcet methods that don't allow voters to enter an approval
threshold have to choose between trying to
minimise Compromise incentive or trying to reduce Defection incentive.
The methods I like in this category allow voters to rank however many
candidates they like and also approve all but
one or only one or any number in between of the candidates (consistent
with their rankings). Equal-ranking is allowed.
Default approval goes only to candidates ranked below no other candidate.
I suggest that voters can just mark one of the candidates as the lowest
ranked one they approve (i.e. only that candidate
and those ranked higher or equal to it are approved).
But other ways of doing it could be fine.
Regarding which algorithm, I very much like Forest's Sorted Approval
Margins.
I also like another method of his, the exact name of which I've
forgotten (something about "Chain" building or climbing):
*Begin the chain with the most approved candidate. Then add the most
approved candidate that covers that candidate.
Then add the most approved candidate that covers all the candidates
already in the chain.
Keep doing that as many times as possible, and then elect the last added
candidate*.
I think nearly always this will elect the same candidate as
Smith//Approval, but is more elegant and ensures that the
winner is Uncovered.
For a practicable Condorcet method that uses plain ranked ballots
(equal-ranking and truncation allowed), I like
Smith//Ranked below none minus ranked above none.
*Eliminate all the candidates not in the Smith set. Give each remaining
candidate a score equal to the number of ballots
on which it is ranked (among remaining candidates) below no other
candidate minus the number of ballots on which it
is ranked (among remaining candidates) above no other candidate.
Elect the candidate with the highest score."
Given how rare top cycles will likely be, I think this is probably good
enough.
Obviously it meets Plurality. It fails both Minimal Defense and Chicken
Dilemma, but never both at once :)
It looks fair and gives a pretty-enough winner. I'll be back later with
some examples.
Chris Benham
If I had to narrow it down to three for public proposal, I would probably
choose the same three. DMC would be right up there too ... in the form of
approval based Benham ... eliminate low approval candidates until an
undefeated candidate remains.
We called the chain building method
Uncovered Approval or unc(approval). Thanks for dusting it off!
For those not familiar with "chains" ...in the election methods context a
chain is a transitive beatpath ... so each member of the beatpath is beaten
by each of its predecessors ... not only by its immediate predecessor.
On Sun, Aug 6, 2023, 2:59 PM C.Benham cbenham@adam.com.au wrote:
I think Condorcet methods that don't allow voters to enter an approval
threshold have to choose between trying to
minimise Compromise incentive or trying to reduce Defection incentive.
The methods I like in this category allow voters to rank however many
candidates they like and also approve all but
one or only one or any number in between of the candidates (consistent
with their rankings). Equal-ranking is allowed.
Default approval goes only to candidates ranked below no other candidate.
I suggest that voters can just mark one of the candidates as the lowest
ranked one they approve (i.e. only that candidate
and those ranked higher or equal to it are approved).
But other ways of doing it could be fine.
Regarding which algorithm, I very much like Forest's Sorted Approval
Margins.
Or more commonly, "Approval Sorted Margins"
[A rose by any other name ...]
I also like another method of his, the exact name of which I've
forgotten (something about "Chain" building or climbing):
*Begin the chain with the most approved candidate. Then add the most
approved candidate that covers that candidate.
Then add the most approved candidate that covers all the candidates
already in the chain.
Keep doing that as many times as possible, and then elect the last added
candidate*.
I think nearly always this will elect the same candidate as
Smith//Approval, but is more elegant and ensures that the
winner is Uncovered.
For a practicable Condorcet method that uses plain ranked ballots
(equal-ranking and truncation allowed), I like
Smith//Ranked below none minus ranked above none.
*Eliminate all the candidates not in the Smith set. Give each remaining
candidate a score equal to the number of ballots
on which it is ranked (among remaining candidates) below no other
candidate minus the number of ballots on which it
is ranked (among remaining candidates) above no other candidate.
Elect the candidate with the highest score."
Given how rare top cycles will likely be, I think this is probably good
enough.
Obviously it meets Plurality. It fails both Minimal Defense and Chicken
Dilemma, but never both at once :)
It looks fair and gives a pretty-enough winner. I'll be back later with
some examples.
Chris Benham
Forest,
If I had to narrow it down to three for public proposal, I would
probably choose the same three. DMC would be right up there too ... in
the form of approval based Benham ... eliminate low approval
candidates until an undefeated candidate remains.
I am very pleased that we are in such agreement. But I think "DMC" is
quite a bit worse than ASM and UncApproval. I thought about it quite a
bit a few years ago.
Beyond being maybe a bit easier to explain and a bit more appealing to
those that love one-at-a-time eliminations, I can't see any argument
that it is better than Smith//Approval.
I used to sometimes suggest a method that featured trying to help voters
by sometimes "moving" their approval cut-offs. For example a version of
Smith//Approval where ballots
that make no approval distinction among the Smith set candidates have
their cut-offs moved the smallest distance so they do. (In other words
those that originally approved all of
them would now approve all but those they rank above none of the others
and and those that originally originally approved none of them would now
approve those they rank below
no others).
But now I think (at least for ASM and UncApp) it is better for a public
proposal to pretend that "approval" is sincere and on some absolute
scale and not relative and tactical.
Another idea I had was to use say 0-100 scoring ballots and interpret a
higher than average score (on the individual ballot) as approval and an
exactly average score as half-approval.
But needless to say, that would slow down hand-counting a lot. Also
probably a bit too fancy for a public proposal.
Chris B.
On 9/08/2023 9:43 am, Forest Simmons wrote:
If I had to narrow it down to three for public proposal, I would
probably choose the same three. DMC would be right up there too ... in
the form of approval based Benham ... eliminate low approval
candidates until an undefeated candidate remains.
We called the chain building method
Uncovered Approval or unc(approval). Thanks for dusting it off!
For those not familiar with "chains" ...in the election methods
context a chain is a transitive beatpath ... so each member of the
beatpath is beaten by each of its predecessors ... not only by its
immediate predecessor.
On Sun, Aug 6, 2023, 2:59 PM C.Benham cbenham@adam.com.au wrote:
I think Condorcet methods that don't allow voters to enter an
approval
threshold have to choose between trying to
minimise Compromise incentive or trying to reduce Defection incentive.
The methods I like in this category allow voters to rank however many
candidates they like and also approve all but
one or only one or any number in between of the candidates
(consistent
with their rankings). Equal-ranking is allowed.
Default approval goes only to candidates ranked below no other
candidate.
I suggest that voters can just mark one of the candidates as the
lowest
ranked one they approve (i.e. only that candidate
and those ranked higher or equal to it are approved).
But other ways of doing it could be fine.
Regarding which algorithm, I very much like Forest's Sorted Approval
Margins.
Or more commonly, "Approval Sorted Margins"
[A rose by any other name ...]
I also like another method of his, the exact name of which I've
forgotten (something about "Chain" building or climbing):
*Begin the chain with the most approved candidate. Then add the most
approved candidate that covers that candidate.
Then add the most approved candidate that covers all the candidates
already in the chain.
Keep doing that as many times as possible, and then elect the last
added
candidate*.
I think nearly always this will elect the same candidate as
Smith//Approval, but is more elegant and ensures that the
winner is Uncovered.
For a practicable Condorcet method that uses plain ranked ballots
(equal-ranking and truncation allowed), I like
Smith//Ranked below none minus ranked above none.
*Eliminate all the candidates not in the Smith set. Give each
remaining
candidate a score equal to the number of ballots
on which it is ranked (among remaining candidates) below no other
candidate minus the number of ballots on which it
is ranked (among remaining candidates) above no other candidate.
Elect the candidate with the highest score."
Given how rare top cycles will likely be, I think this is probably
good
enough.
Obviously it meets Plurality. It fails both Minimal Defense and
Chicken
Dilemma, but never both at once :)
It looks fair and gives a pretty-enough winner. I'll be back
later with
some examples.
Chris Benham
I thought perhaps DMC with an explicit approval option would be better than
vanilla DMC based on Implicit Approval ... but I never looked at it closely.
On Wed, Aug 9, 2023, 6:31 PM C.Benham cbenham@adam.com.au wrote:
Forest,
If I had to narrow it down to three for public proposal, I would probably
choose the same three. DMC would be right up there too ... in the form of
approval based Benham ... eliminate low approval candidates until an
undefeated candidate remains.
I am very pleased that we are in such agreement. But I think "DMC" is
quite a bit worse than ASM and UncApproval. I thought about it quite a
bit a few years ago.
Beyond being maybe a bit easier to explain and a bit more appealing to
those that love one-at-a-time eliminations, I can't see any argument that
it is better than Smith//Approval.
I used to sometimes suggest a method that featured trying to help voters
by sometimes "moving" their approval cut-offs. For example a version of
Smith//Approval where ballots
that make no approval distinction among the Smith set candidates have
their cut-offs moved the smallest distance so they do.
This is a very appealing DSV idea that reminds me of an idea I once toyed
with ... to move the approval cutoff (on each ballot) the minimum amount to
make it adjacent to the ballot's lowest ranked candidate that pairwise
defeats every candidate ranked ahead of it on that ballot.
Now I would say just move the ballot's cutoff adjacent to the MMPO
candidate, which would be much simpler, but not simple enough, and would
still confuse the naive voter about the strategic vs intuitive role of the
cutoff.
(In other words those that originally approved all of
them would now approve all but those they rank above none of the others
and and those that originally originally approved none of them would now
approve those they rank below
no others).
But now I think (at least for ASM and UncApp) it is better for a public
proposal to pretend that "approval" is sincere and on some absolute scale
and not relative and tactical.
Very true!
Another idea I had was to use say 0-100 scoring ballots and interpret a
higher than average score (on the individual ballot) as approval and an
exactly average score as half-approval.
But needless to say, that would slow down hand-counting a lot. Also
probably a bit too fancy for a public proposal.
Chris B.
On 9/08/2023 9:43 am, Forest Simmons wrote:
If I had to narrow it down to three for public proposal, I would probably
choose the same three. DMC would be right up there too ... in the form of
approval based Benham ... eliminate low approval candidates until an
undefeated candidate remains.
We called the chain building method
Uncovered Approval or unc(approval). Thanks for dusting it off!
For those not familiar with "chains" ...in the election methods context a
chain is a transitive beatpath ... so each member of the beatpath is beaten
by each of its predecessors ... not only by its immediate predecessor.
On Sun, Aug 6, 2023, 2:59 PM C.Benham cbenham@adam.com.au wrote:
I think Condorcet methods that don't allow voters to enter an approval
threshold have to choose between trying to
minimise Compromise incentive or trying to reduce Defection incentive.
The methods I like in this category allow voters to rank however many
candidates they like and also approve all but
one or only one or any number in between of the candidates (consistent
with their rankings). Equal-ranking is allowed.
Default approval goes only to candidates ranked below no other candidate.
I suggest that voters can just mark one of the candidates as the lowest
ranked one they approve (i.e. only that candidate
and those ranked higher or equal to it are approved).
But other ways of doing it could be fine.
Regarding which algorithm, I very much like Forest's Sorted Approval
Margins.
Or more commonly, "Approval Sorted Margins"
[A rose by any other name ...]
I also like another method of his, the exact name of which I've
forgotten (something about "Chain" building or climbing):
*Begin the chain with the most approved candidate. Then add the most
approved candidate that covers that candidate.
Then add the most approved candidate that covers all the candidates
already in the chain.
Keep doing that as many times as possible, and then elect the last added
candidate*.
I think nearly always this will elect the same candidate as
Smith//Approval, but is more elegant and ensures that the
winner is Uncovered.
For a practicable Condorcet method that uses plain ranked ballots
(equal-ranking and truncation allowed), I like
Smith//Ranked below none minus ranked above none.
*Eliminate all the candidates not in the Smith set. Give each remaining
candidate a score equal to the number of ballots
on which it is ranked (among remaining candidates) below no other
candidate minus the number of ballots on which it
is ranked (among remaining candidates) above no other candidate.
Elect the candidate with the highest score."
Given how rare top cycles will likely be, I think this is probably good
enough.
Obviously it meets Plurality. It fails both Minimal Defense and Chicken
Dilemma, but never both at once :)
It looks fair and gives a pretty-enough winner. I'll be back later with
some examples.
Chris Benham
On Sun, Aug 6, 2023, 2:59 PM C.Benham cbenham@adam.com.au wrote:
I think Condorcet methods that don't allow voters to enter an approval
threshold have to choose between trying to
minimise Compromise incentive or trying to reduce Defection incentive.
The methods I like in this category allow voters to rank however many
candidates they like and also approve all but
one or only one or any number in between of the candidates (consistent
with their rankings). Equal-ranking is allowed.
Default approval goes only to candidates ranked below no other candidate.
I suggest that voters can just mark one of the candidates as the lowest
ranked one they approve (i.e. only that candidate
and those ranked higher or equal to it are approved).
But other ways of doing it could be fine.
Regarding which algorithm, I very much like Forest's Sorted Approval
Margins.
I also like another method of his, the exact name of which I've
forgotten (something about "Chain" building or climbing):
*Begin the chain with the most approved candidate. Then add the most
approved candidate that covers that candidate.
Then add the most approved candidate that covers all the candidates
already in the chain.
Keep doing that as many times as possible, and then elect the last added
candidate*.
I think nearly always this will elect the same candidate as
Smith//Approval, but is more elegant and ensures that the
winner is Uncovered.
For a practicable Condorcet method that uses plain ranked ballots
(equal-ranking and truncation allowed), I like
Smith//Ranked below none minus ranked above none.
*Eliminate all the candidates not in the Smith set. Give each remaining
candidate a score equal to the number of ballots
on which it is ranked (among remaining candidates) below no other
candidate minus the number of ballots on which it
is ranked (among remaining candidates) above no other candidate.
This score can be approximated as the average of the Max and Min Pairwise
Supports (restricted to Smith) of the candidate.
This suggests another method ...
elect the candidate with the highest ratio given by the expression
((MaxPS+MinPS)/2)/MaxPO
which is an estimate of the ratio of the approval the candidate would get
if it were the approval cutoff candidate to the max approval any other
candidate would get with the same cutoff.
In other words, it is candidate expected to bear up the best under
Approval voting if it were the projected winner ... therefore (adjacent to)
the approval cutoff in the next round of repeated voting, say.
Restricting this to Smith should be good.
Example:
48 C
28 A>B
24 B
The respective ratios for A, B, and C are
26/48, [(52+24)/2]/48, 48/52
So C wins.
Elect the candidate with the highest score."
Given how rare top cycles will likely be, I think this is probably good
enough.
Obviously it meets Plurality. It fails both Minimal Defense and Chicken
Dilemma, but never both at once :)
It looks fair and gives a pretty-enough winner. I'll be back later with
some examples.
Chris Benham
This suggests another method ...
elect the candidate with the highest ratio given by the expression
((MaxPS+MinPS)/2)/MaxPO
which is an estimate of the ratio of the approval the candidate would
get if it were the approval cutoff candidate to the max approval any
other candidate would get with the same cutoff.
In other words, it is candidate expected to bear up the best under
Approval voting if it were the projected winner ... therefore
(adjacent to) the approval cutoff in the next round of repeated
voting, say.
Restricting this to Smith should be good.
Example:
48 C
28 A>B
24 B
The respective ratios for A, B, and C are
26/48, [(52+24)/2]/48, 48/52
So C wins.
Is there a typo here? Where does the 26 in "26/48" come from? Should
it be 28?
I'm not really switched on to the positive point of this relatively
complicated method.
In your example it fails Minimal Defense. Does it meet Chicken Dilemma?
*Eliminate all the candidates not in the Smith set. Give each remaining
candidate a score equal to the number of ballots
on which it is ranked (among remaining candidates) below no other
candidate minus the number of ballots on which it
is ranked (among remaining candidates) above no other candidate.
This score can be approximated as the average of the Max and Min
Pairwise Supports (restricted to Smith) of the candidate.
I can see that that would nearly always (or always?) be the same thing,
and that it could be just read off the pairwise matrix
(which might streamline the counting process a lot).
Chris B.
On 11/08/2023 1:37 am, Forest Simmons wrote:
On Sun, Aug 6, 2023, 2:59 PM C.Benham cbenham@adam.com.au wrote:
I think Condorcet methods that don't allow voters to enter an
approval
threshold have to choose between trying to
minimise Compromise incentive or trying to reduce Defection incentive.
The methods I like in this category allow voters to rank however many
candidates they like and also approve all but
one or only one or any number in between of the candidates
(consistent
with their rankings). Equal-ranking is allowed.
Default approval goes only to candidates ranked below no other
candidate.
I suggest that voters can just mark one of the candidates as the
lowest
ranked one they approve (i.e. only that candidate
and those ranked higher or equal to it are approved).
But other ways of doing it could be fine.
Regarding which algorithm, I very much like Forest's Sorted Approval
Margins.
I also like another method of his, the exact name of which I've
forgotten (something about "Chain" building or climbing):
*Begin the chain with the most approved candidate. Then add the most
approved candidate that covers that candidate.
Then add the most approved candidate that covers all the candidates
already in the chain.
Keep doing that as many times as possible, and then elect the last
added
candidate*.
I think nearly always this will elect the same candidate as
Smith//Approval, but is more elegant and ensures that the
winner is Uncovered.
For a practicable Condorcet method that uses plain ranked ballots
(equal-ranking and truncation allowed), I like
Smith//Ranked below none minus ranked above none.
*Eliminate all the candidates not in the Smith set. Give each
remaining
candidate a score equal to the number of ballots
on which it is ranked (among remaining candidates) below no other
candidate minus the number of ballots on which it
is ranked (among remaining candidates) above no other candidate.
This score can be approximated as the average of the Max and Min
Pairwise Supports (restricted to Smith) of the candidate.
This suggests another method ...
elect the candidate with the highest ratio given by the expression
((MaxPS+MinPS)/2)/MaxPO
which is an estimate of the ratio of the approval the candidate would
get if it were the approval cutoff candidate to the max approval any
other candidate would get with the same cutoff.
In other words, it is candidate expected to bear up the best under
Approval voting if it were the projected winner ... therefore
(adjacent to) the approval cutoff in the next round of repeated
voting, say.
Restricting this to Smith should be good.
Example:
48 C
28 A>B
24 B
The respective ratios for A, B, and C are
26/48, [(52+24)/2]/48, 48/52
So C wins.
Elect the candidate with the highest score."
Given how rare top cycles will likely be, I think this is probably
good
enough.
Obviously it meets Plurality. It fails both Minimal Defense and
Chicken
Dilemma, but never both at once :)
It looks fair and gives a pretty-enough winner. I'll be back
later with
some examples.
Chris Benham
On Fri, Aug 11, 2023, 11:55 AM C.Benham cbenham@adam.com.au wrote:
This suggests another method ...
elect the candidate with the highest ratio given by the expression
((MaxPS+MinPS)/2)/MaxPO
which is an estimate of the ratio of the approval the candidate would get
if it were the approval cutoff candidate to the max approval any other
candidate would get with the same cutoff.
In other words, it is candidate expected to bear up the best under
Approval voting if it were the projected winner ... therefore (adjacent to)
the approval cutoff in the next round of repeated voting, say.
Restricting this to Smith should be good.
Example:
48 C
28 A>B
24 B
The respective ratios for A, B, and C are
26/48, [(52+24)/2]/48, 48/52
So C wins.
Is there a typo here?
Yes, it should be 28.
Where does the 26 in "26/48" come from? Should it be 28?
I'm not really switched on to the positive point of this relatively
complicated method.
If we are limited to the "Universal Domain" ... so no explicit approval
cutoff allowed ... a reasonable way to do Approval DSV is to find the
Approval Cutoff Candidate that would get the highest fraction of approval
compared to its max opposition.
This MidrangePS/MaxPO score is a rough estimate of that max fraction.
Electing the candidate that maximizes that fraction seems to be resistant
to burial and truncation defections.
As you remarked below, the pairwise support matrix has all of the needed
info.
In your example it fails Minimal Defense. Does it meet Chicken Dilemma?
The example gives hope that it does meet CD.
*Eliminate all the candidates not in the Smith set. Give each remaining
candidate a score equal to the number of ballots
on which it is ranked (among remaining candidates) below no other
candidate minus the number of ballots on which it
is ranked (among remaining candidates) above no other candidate.
This score can be approximated as the average of the Max and Min Pairwise
Supports (restricted to Smith) of the candidate.
I can see that that would nearly always (or always?) be the same thing,
and that it could be just read off the pairwise matrix
(which might streamline the counting process a lot).
Chris B.
On 11/08/2023 1:37 am, Forest Simmons wrote:
On Sun, Aug 6, 2023, 2:59 PM C.Benham cbenham@adam.com.au wrote:
I think Condorcet methods that don't allow voters to enter an approval
threshold have to choose between trying to
minimise Compromise incentive or trying to reduce Defection incentive.
The methods I like in this category allow voters to rank however many
candidates they like and also approve all but
one or only one or any number in between of the candidates (consistent
with their rankings). Equal-ranking is allowed.
Default approval goes only to candidates ranked below no other candidate.
I suggest that voters can just mark one of the candidates as the lowest
ranked one they approve (i.e. only that candidate
and those ranked higher or equal to it are approved).
But other ways of doing it could be fine.
Regarding which algorithm, I very much like Forest's Sorted Approval
Margins.
I also like another method of his, the exact name of which I've
forgotten (something about "Chain" building or climbing):
*Begin the chain with the most approved candidate. Then add the most
approved candidate that covers that candidate.
Then add the most approved candidate that covers all the candidates
already in the chain.
Keep doing that as many times as possible, and then elect the last added
candidate*.
I think nearly always this will elect the same candidate as
Smith//Approval, but is more elegant and ensures that the
winner is Uncovered.
For a practicable Condorcet method that uses plain ranked ballots
(equal-ranking and truncation allowed), I like
Smith//Ranked below none minus ranked above none.
*Eliminate all the candidates not in the Smith set. Give each remaining
candidate a score equal to the number of ballots
on which it is ranked (among remaining candidates) below no other
candidate minus the number of ballots on which it
is ranked (among remaining candidates) above no other candidate.
This score can be approximated as the average of the Max and Min Pairwise
Supports (restricted to Smith) of the candidate.
This suggests another method ...
elect the candidate with the highest ratio given by the expression
((MaxPS+MinPS)/2)/MaxPO
which is an estimate of the ratio of the approval the candidate would get
if it were the approval cutoff candidate to the max approval any other
candidate would get with the same cutoff.
In other words, it is candidate expected to bear up the best under
Approval voting if it were the projected winner ... therefore (adjacent to)
the approval cutoff in the next round of repeated voting, say.
Restricting this to Smith should be good.
Example:
48 C
28 A>B
24 B
The respective ratios for A, B, and C are
26/48, [(52+24)/2]/48, 48/52
So C wins.
Elect the candidate with the highest score."
Given how rare top cycles will likely be, I think this is probably good
enough.
Obviously it meets Plurality. It fails both Minimal Defense and Chicken
Dilemma, but never both at once :)
It looks fair and gives a pretty-enough winner. I'll be back later with
some examples.
Chris Benham