[EM] 4+slot IBIFA revision

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[EM] 4+slot IBIFA revision

C.Benham
IBIFA was conceived as an Irrelevant Ballot independent version of
Bucklin, with the added benefits of having a less
severe truncation and/or compress at the top incentive and also being
much more (and absolutely more) Condorcet-consistent.

Inspired by an example from Ted Stern of  his "Relevant Ratings" method
(which I gather is IBIFA
modified to more closely resemble Majority Judgement), I've come to
believe that if ratings ballots
with four or more slots (or grades) are used then a simple rule change
can make the method still
more Condorcet-consistent  at no cost.

My idea (originally my misunderstanding of Ted's Relevant Ratings
method) is that if at some
(quasi-Bucklin) IBIFA round after the first (but before we have reached
just counting total approval scores)
we find more than one candidate Q qualified to win then instead of 
(Bucklin-like) giving the win to the Q
with the highest score in that round we elect the Q with the highest
score in the round before.

A link to the electowiki entry on my original version of IBIFA:

https://wiki.electorama.com/wiki/IBIFA

And the EM post in which I first suggested it:
http://lists.electorama.com/pipermail/election-methods-electorama.com//2010-May/091807.html

Here is the description of the revised 4-slot version, referring to
A-B-C-D grading ballots:

*Voters fill out 4-slot ratings ballots, say with A B C D grades.
  Default rating/grade is D, signifying least preferred and unapproved.

Any grade above D is interpreted as Approval.

If any candidate/s X has an A score that is higher than any other
candidate's approval
score on ballots that  don't give X an A grade, elect the X with the
highest A score.

Otherwise, if any candidate/s X has a A+B score that is higher than any
other candidate's
approval score on ballots that don't give X an A or B grade, elect the X
with the highest
  A score.

Otherwise, elect the candidate with the highest Approval score.*

  35: A
  10: A=B
  30: B>C
  25: C

With my Condorcet hat on, in this example I've said that B is the
weakest candidate.  A bit unfortunately
IBIFA here elects B, but FBC is a bit more "expensive" than Condorcet,
and so does Winning Votes and Margins.
Bucklin elects the most approved candidate C, but at least B both
pairwise beats and is more top-rated than C.

Ted Stern's eye-opening example:

49: A > B
03: B > A > C
10: D > B > C
38: E > F > C
05: G > D > H

The Condorcet winner is A.  Ted's Relevant Ratings and my revised 4+
slot IBIFA elect A.
My original version of IBIFA  and  Median Ratings methods such as
Bucklin and MJ elect B.

Top Ratings (A) scores:  A49,  E38,  D10,  G5,  B3,  C0
A + B scores:                   A51,  E38,  D15,  G5,  B62,  C0

In the second round A and B both "qualify".   On ballots that don't 
give A one of the two
top grades the most approved candidate is E with a score of 38, lower
than 51 so A qualifies.

On ballots that don't give B one of the top two grades the most approved
candidate is again
E with again a score of 38, lower than 62 so B qualifies. In the "round
before" A  has the
higher score (49 versus 3) so revised IBIFA gives the win to A.

A>B 49-13,   A>E 51-38,  A>D 51-15,  A>G 51>5, A>C  51-48.

At the cost of being a quite a bit more complicated,  IBIFA can be
combined  with Kevin Venzke's
special "tied-at-the-top" rule used in his "Improved Condorcet Approval"
method to make
the method even more Condorcet-consistent  (possibly as much as it
possible for a FBC method
to be).
https://wiki.electorama.com/wiki/Improved_Condorcet_Approval

*If one candidate T pairwise beats all others by the tied-at-the-top
rule then T wins. If there is no
such T then we elect the (revised) IBIFA winner.
If there is more than one T then we elect the one that "qualifies"
(according to IBIFA) in the earliest
IBIFA round. If there is more than one of these, then elect the one with
the highest score in the previous
round if there was one, otherwise simply with the highest top-ratings
score.*

4: A>B
6: A>C
6: B>A
2: B>C
3: C>B

B is the narrow Condorcet winner:  B>A 11-10,  B>C  12-9. No ballots
have any candidates tied at the top,
so B wins.  Plain IBIFA elects A, which is positionally dominant:  Top
scores: A10, B8, C2. Approval scores: A16,  B13,  C10.

For the time being the name I suggest  for  this is  Quasi-Condorcet IBIFA.

Chris Benham




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Re: [EM] [RangeVoting] 4+slot IBIFA revision

Ted Stern
Hi Chris,

You are so close to Relevant Ratings in your proposal.  I just want to point out how close and why the one missing factor is important.

You write:
My idea (originally my misunderstanding of Ted's Relevant Ratings method) is that if at some (quasi-Bucklin) IBIFA round after the first (but before we have reached just counting total approval scores) we find more than one candidate Q qualified to win then instead of (Bucklin-like) giving the win to the Q with the highest score in that round we elect the Q with the highest  score in the round before.

Where this differs from RR is as follows:
  • For each candidate Q qualified to win IBIFA, their total ballots from highest rating down to the current round rating exceed some highest total approval on complementary ballots excluding Q down to that rating.  Say that the highest total approval on such complementary ballots is TC.
  • Your modified IBIFA just looks at the Q totals from the previous round.
  • My Relevant Ratings method looks only at the previous round Q totals that are larger than their respective TC opposition in the current round!
In most situations, the Q you find with your modified IBIFA would be the same.  But it is possible that they might not be.  Let's carefully construct a 4 slot example, working backwards:

Say we want at least 3 candidates, ratings 3 = Excellent ("A"), 2 = Very Good ("B"), 1 = OK ("C"), 0 = disapproved ("D").
  • round 1 totals (scores at 3) of A48, B49, with other candidates below that (and not qualifying in any method)
  • round 2 totals (scores at 2 and above) of A52, B51, with other candidates below that (and not qualifying in any method)
  • round 3 totals (scores above 0) of A52, B52, and C > 54, with other candidates below that (only C qualifying under MCA or MJ)
  • In round 1, we want A48's most approved complementary candidate to be B or C with at least 49
  • In round 1, we want B49's most approved complementary candidate to be A or C with at least 50
  • In round 2, we want A52's most approved complementary candidate to be C with at most 47
  • In round 2, we want B51's most approved complementary candidate to be C with exactly 50.
  • We want A and B's total approval to be less than 50%, so there must be at least 105 ballots.  So we expect at least 5 irrelevant ballots.
Under this scenario, C will win both MCA and MJ in round 3.  B will win in modified IBIFA, as round 2 qualifier with the highest round 1 score.

But A will win both original IBIFA and relevant rating because while both A and B qualify in round 2, only A's round 1 score exceeds A's round 2 complementary approval winner C's approval of 47, while B's round 2 score of 49 is below B's complementary approval winner C's score of 50.

Here is a set of ballots that I think satisfies those constraints.

02: A > B > C
24: A > D > C
22: A > E > C
04: B > F > C
25: B > F > C
21: B > G > C
02: E > D=A > C
02: F > E=A > H
06: G > F > H

Round 1:  A48 vs complementary approval winner C with 51, B49 vs complementary approval winner C with 50.  Neither qualifies
Round 2:  A52 vs complementary approval winner C with 47, B51 vs complementary approval winner C with 48.  Both qualify in IBIFA-derived methods, but not in MCA or MJ with less than 50% of ballots
Round 3:  C99 passes 50% threshold, while A and B still less than 50% threshold for tiebreaker.

A52 pairwise beats B51 and is the Condorcet winner (Please check my arithmetic!)

The main point here is that while both IBIFA, modified IBIFA and Relevant Ratings can avoid electing a non-CW candidate C, the lowest level compromise approval winner elected by standard median ratings, your modified IBIFA will fail to choose the CW while relevant ratings and original IBIFA will find that candidate.

You suggestion of using undefeated tied-at-top winner first, then falling back to some IBI method, is an interesting one, however.

On Sun, Jun 2, 2019 at 8:16 PM Chris Benham [hidden email] [RangeVoting] <[hidden email]> wrote:

IBIFA was conceived as an Irrelevant Ballot independent version of
Bucklin, with the added benefits of having a less
severe truncation and/or compress at the top incentive and also being
much more (and absolutely more) Condorcet-consistent.

Inspired by an example from Ted Stern of?? his "Relevant Ratings" method
(which I gather is IBIFA
modified to more closely resemble Majority Judgement), I've come to
believe that if ratings ballots
with four or more slots (or grades) are used then a simple rule change
can make the method still
more Condorcet-consistent?? at no cost.

My idea (originally my misunderstanding of Ted's Relevant Ratings
method) is that if at some
(quasi-Bucklin) IBIFA round after the first (but before we have reached
just counting total approval scores)
we find more than one candidate Q qualified to win then instead of??
(Bucklin-like) giving the win to the Q
with the highest score in that round we elect the Q with the highest
score in the round before.

A link to the electowiki entry on my original version of IBIFA:

https://wiki.electorama.com/wiki/IBIFA

And the EM post in which I first suggested it:
http://lists.electorama.com/pipermail/election-methods-electorama.com//2010-May/091807.html

Here is the description of the revised 4-slot version, referring to
A-B-C-D grading ballots:

*Voters fill out 4-slot ratings ballots, say with A B C D grades.
 ??Default rating/grade is D, signifying least preferred and unapproved.

Any grade above D is interpreted as Approval.

If any candidate/s X has an A score that is higher than any other
candidate's approval
score on ballots that?? don't give X an A grade, elect the X with the
highest A score.

Otherwise, if any candidate/s X has a A+B score that is higher than any
other candidate's
approval score on ballots that don't give X an A or B grade, elect the X
with the highest
 ??A score.

Otherwise, elect the candidate with the highest Approval score.*

 ??35: A
 ??10: A=B
 ??30: B>C
 ??25: C

With my Condorcet hat on, in this example I've said that B is the
weakest candidate.?? A bit unfortunately
IBIFA here elects B, but FBC is a bit more "expensive" than Condorcet,
and so does Winning Votes and Margins.
Bucklin elects the most approved candidate C, but at least B both
pairwise beats and is more top-rated than C.

Ted Stern's eye-opening example:

49: A > B
03: B > A > C
10: D > B > C
38: E > F > C
05: G > D > H

The Condorcet winner is A.?? Ted's Relevant Ratings and my revised 4+
slot IBIFA elect A.
My original version of IBIFA?? and?? Median Ratings methods such as
Bucklin and MJ elect B.

Top Ratings (A) scores:?? A49,?? E38,?? D10,?? G5,?? B3,?? C0
A + B scores:???????????????????????????????????? A51,?? E38,?? D15,?? G5,?? B62,?? C0

In the second round A and B both "qualify".???? On ballots that don't??
give A one of the two
top grades the most approved candidate is E with a score of 38, lower
than 51 so A qualifies.

On ballots that don't give B one of the top two grades the most approved
candidate is again
E with again a score of 38, lower than 62 so B qualifies. In the "round
before" A?? has the
higher score (49 versus 3) so revised IBIFA gives the win to A.

A>B 49-13,???? A>E 51-38,?? A>D 51-15,?? A>G 51>5, A>C?? 51-48.

At the cost of being a quite a bit more complicated,?? IBIFA can be
combined?? with Kevin Venzke's
special "tied-at-the-top" rule used in his "Improved Condorcet Approval"
method to make
the method even more Condorcet-consistent?? (possibly as much as it
possible for a FBC method
to be).
https://wiki.electorama.com/wiki/Improved_Condorcet_Approval

*If one candidate T pairwise beats all others by the tied-at-the-top
rule then T wins. If there is no
such T then we elect the (revised) IBIFA winner.
If there is more than one T then we elect the one that "qualifies"
(according to IBIFA) in the earliest
IBIFA round. If there is more than one of these, then elect the one with
the highest score in the previous
round if there was one, otherwise simply with the highest top-ratings
score.*

4: A>B
6: A>C
6: B>A
2: B>C
3: C>B

B is the narrow Condorcet winner:?? B>A 11-10,?? B>C?? 12-9. No ballots
have any candidates tied at the top,
so B wins.?? Plain IBIFA elects A, which is positionally dominant: Top
scores: A10, B8, C2. Approval scores: A16,?? B13,?? C10.

For the time being the name I suggest?? for?? this is Quasi-Condorcet IBIFA.

Chris Benham






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Re: [EM] [RangeVoting] 4+slot IBIFA revision

C.Benham

Ted,

Why does your example have the "B>F>C" voters divided into two groups?

04: B > F > C
25: B > F > C
Is there a typo?

Chris Benham

On 4/06/2019 7:12 am, Ted Stern wrote:
Hi Chris,

You are so close to Relevant Ratings in your proposal.  I just want to point out how close and why the one missing factor is important.

You write:
My idea (originally my misunderstanding of Ted's Relevant Ratings method) is that if at some (quasi-Bucklin) IBIFA round after the first (but before we have reached just counting total approval scores) we find more than one candidate Q qualified to win then instead of (Bucklin-like) giving the win to the Q with the highest score in that round we elect the Q with the highest  score in the round before.

Where this differs from RR is as follows:
  • For each candidate Q qualified to win IBIFA, their total ballots from highest rating down to the current round rating exceed some highest total approval on complementary ballots excluding Q down to that rating.  Say that the highest total approval on such complementary ballots is TC.
  • Your modified IBIFA just looks at the Q totals from the previous round.
  • My Relevant Ratings method looks only at the previous round Q totals that are larger than their respective TC opposition in the current round!
In most situations, the Q you find with your modified IBIFA would be the same.  But it is possible that they might not be.  Let's carefully construct a 4 slot example, working backwards:

Say we want at least 3 candidates, ratings 3 = Excellent ("A"), 2 = Very Good ("B"), 1 = OK ("C"), 0 = disapproved ("D").
  • round 1 totals (scores at 3) of A48, B49, with other candidates below that (and not qualifying in any method)
  • round 2 totals (scores at 2 and above) of A52, B51, with other candidates below that (and not qualifying in any method)
  • round 3 totals (scores above 0) of A52, B52, and C > 54, with other candidates below that (only C qualifying under MCA or MJ)
  • In round 1, we want A48's most approved complementary candidate to be B or C with at least 49
  • In round 1, we want B49's most approved complementary candidate to be A or C with at least 50
  • In round 2, we want A52's most approved complementary candidate to be C with at most 47
  • In round 2, we want B51's most approved complementary candidate to be C with exactly 50.
  • We want A and B's total approval to be less than 50%, so there must be at least 105 ballots.  So we expect at least 5 irrelevant ballots.
Under this scenario, C will win both MCA and MJ in round 3.  B will win in modified IBIFA, as round 2 qualifier with the highest round 1 score.

But A will win both original IBIFA and relevant rating because while both A and B qualify in round 2, only A's round 1 score exceeds A's round 2 complementary approval winner C's approval of 47, while B's round 2 score of 49 is below B's complementary approval winner C's score of 50.

Here is a set of ballots that I think satisfies those constraints.

02: A > B > C
24: A > D > C
22: A > E > C
04: B > F > C
25: B > F > C
21: B > G > C
02: E > D=A > C
02: F > E=A > H
06: G > F > H

Round 1:  A48 vs complementary approval winner C with 51, B49 vs complementary approval winner C with 50.  Neither qualifies
Round 2:  A52 vs complementary approval winner C with 47, B51 vs complementary approval winner C with 48.  Both qualify in IBIFA-derived methods, but not in MCA or MJ with less than 50% of ballots
Round 3:  C99 passes 50% threshold, while A and B still less than 50% threshold for tiebreaker.

A52 pairwise beats B51 and is the Condorcet winner (Please check my arithmetic!)

The main point here is that while both IBIFA, modified IBIFA and Relevant Ratings can avoid electing a non-CW candidate C, the lowest level compromise approval winner elected by standard median ratings, your modified IBIFA will fail to choose the CW while relevant ratings and original IBIFA will find that candidate.

You suggestion of using undefeated tied-at-top winner first, then falling back to some IBI method, is an interesting one, however.

On Sun, Jun 2, 2019 at 8:16 PM Chris Benham [hidden email] [RangeVoting] <[hidden email]> wrote:

IBIFA was conceived as an Irrelevant Ballot independent version of
Bucklin, with the added benefits of having a less
severe truncation and/or compress at the top incentive and also being
much more (and absolutely more) Condorcet-consistent.

Inspired by an example from Ted Stern of?? his "Relevant Ratings" method
(which I gather is IBIFA
modified to more closely resemble Majority Judgement), I've come to
believe that if ratings ballots
with four or more slots (or grades) are used then a simple rule change
can make the method still
more Condorcet-consistent?? at no cost.

My idea (originally my misunderstanding of Ted's Relevant Ratings
method) is that if at some
(quasi-Bucklin) IBIFA round after the first (but before we have reached
just counting total approval scores)
we find more than one candidate Q qualified to win then instead of??
(Bucklin-like) giving the win to the Q
with the highest score in that round we elect the Q with the highest
score in the round before.

A link to the electowiki entry on my original version of IBIFA:

https://wiki.electorama.com/wiki/IBIFA

And the EM post in which I first suggested it:
http://lists.electorama.com/pipermail/election-methods-electorama.com//2010-May/091807.html

Here is the description of the revised 4-slot version, referring to
A-B-C-D grading ballots:

*Voters fill out 4-slot ratings ballots, say with A B C D grades.
 ??Default rating/grade is D, signifying least preferred and unapproved.

Any grade above D is interpreted as Approval.

If any candidate/s X has an A score that is higher than any other
candidate's approval
score on ballots that?? don't give X an A grade, elect the X with the
highest A score.

Otherwise, if any candidate/s X has a A+B score that is higher than any
other candidate's
approval score on ballots that don't give X an A or B grade, elect the X
with the highest
 ??A score.

Otherwise, elect the candidate with the highest Approval score.*

 ??35: A
 ??10: A=B
 ??30: B>C
 ??25: C

With my Condorcet hat on, in this example I've said that B is the
weakest candidate.?? A bit unfortunately
IBIFA here elects B, but FBC is a bit more "expensive" than Condorcet,
and so does Winning Votes and Margins.
Bucklin elects the most approved candidate C, but at least B both
pairwise beats and is more top-rated than C.

Ted Stern's eye-opening example:

49: A > B
03: B > A > C
10: D > B > C
38: E > F > C
05: G > D > H

The Condorcet winner is A.?? Ted's Relevant Ratings and my revised 4+
slot IBIFA elect A.
My original version of IBIFA?? and?? Median Ratings methods such as
Bucklin and MJ elect B.

Top Ratings (A) scores:?? A49,?? E38,?? D10,?? G5,?? B3,?? C0
A + B scores:???????????????????????????????????? A51,?? E38,?? D15,?? G5,?? B62,?? C0

In the second round A and B both "qualify".???? On ballots that don't??
give A one of the two
top grades the most approved candidate is E with a score of 38, lower
than 51 so A qualifies.

On ballots that don't give B one of the top two grades the most approved
candidate is again
E with again a score of 38, lower than 62 so B qualifies. In the "round
before" A?? has the
higher score (49 versus 3) so revised IBIFA gives the win to A.

A>B 49-13,???? A>E 51-38,?? A>D 51-15,?? A>G 51>5, A>C?? 51-48.

At the cost of being a quite a bit more complicated,?? IBIFA can be
combined?? with Kevin Venzke's
special "tied-at-the-top" rule used in his "Improved Condorcet Approval"
method to make
the method even more Condorcet-consistent?? (possibly as much as it
possible for a FBC method
to be).
https://wiki.electorama.com/wiki/Improved_Condorcet_Approval

*If one candidate T pairwise beats all others by the tied-at-the-top
rule then T wins. If there is no
such T then we elect the (revised) IBIFA winner.
If there is more than one T then we elect the one that "qualifies"
(according to IBIFA) in the earliest
IBIFA round. If there is more than one of these, then elect the one with
the highest score in the previous
round if there was one, otherwise simply with the highest top-ratings
score.*

4: A>B
6: A>C
6: B>A
2: B>C
3: C>B

B is the narrow Condorcet winner:?? B>A 11-10,?? B>C?? 12-9. No ballots
have any candidates tied at the top,
so B wins.?? Plain IBIFA elects A, which is positionally dominant: Top
scores: A10, B8, C2. Approval scores: A16,?? B13,?? C10.

For the time being the name I suggest?? for?? this is Quasi-Condorcet IBIFA.

Chris Benham






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Re: [EM] [RangeVoting] 4+slot IBIFA revision

Ted Stern
In reply to this post by Ted Stern
See inserted correction below:

On Mon, Jun 3, 2019 at 2:42 PM Ted Stern <[hidden email]> wrote:
Hi Chris,

You are so close to Relevant Ratings in your proposal.  I just want to point out how close and why the one missing factor is important.

You write:
My idea (originally my misunderstanding of Ted's Relevant Ratings method) is that if at some (quasi-Bucklin) IBIFA round after the first (but before we have reached just counting total approval scores) we find more than one candidate Q qualified to win then instead of (Bucklin-like) giving the win to the Q with the highest score in that round we elect the Q with the highest  score in the round before.

Where this differs from RR is as follows:
  • For each candidate Q qualified to win IBIFA, their total ballots from highest rating down to the current round rating exceed some highest total approval on complementary ballots excluding Q down to that rating.  Say that the highest total approval on such complementary ballots is TC.
  • Your modified IBIFA just looks at the Q totals from the previous round.
  • My Relevant Ratings method looks only at the previous round Q totals that are larger than their respective TC opposition in the current round!
In most situations, the Q you find with your modified IBIFA would be the same.  But it is possible that they might not be.  Let's carefully construct a 4 slot example, working backwards:

Say we want at least 3 candidates, ratings 3 = Excellent ("A"), 2 = Very Good ("B"), 1 = OK ("C"), 0 = disapproved ("D").
  • round 1 totals (scores at 3) of A48, B49, with other candidates below that (and not qualifying in any method)
  • round 2 totals (scores at 2 and above) of A52, B51, with other candidates below that (and not qualifying in any method)
  • round 3 totals (scores above 0) of A52, B52, and C > 54, with other candidates below that (only C qualifying under MCA or MJ)
  • In round 1, we want A48's most approved complementary candidate to be B or C with at least 49
  • In round 1, we want B49's most approved complementary candidate to be A or C with at least 50
  • In round 2, we want A52's most approved complementary candidate to be C with at most 47
  • In round 2, we want B51's most approved complementary candidate to be C with exactly 50.
  • We want A and B's total approval to be less than 50%, so there must be at least 105 ballots.  So we expect at least 5 irrelevant ballots.
Under this scenario, C will win both MCA and MJ in round 3.  B will win in modified IBIFA, as round 2 qualifier with the highest round 1 score.

But A will win both original IBIFA and relevant rating because while both A and B qualify in round 2, only A's round 1 score exceeds A's round 2 complementary approval winner C's approval of 47, while B's round 2 score of 49 is below B's complementary approval winner C's score of 50.

Here is a set of ballots that I think satisfies those constraints.

02: A > B > C
24: A > D > C
22: A > E > C
04: B > F > C

The 04: B > F > C ballots are a typo.  They should be 

04: B > A > C
 
25: B > F > C
21: B > G > C
02: E > D=A > C
02: F > E=A > H
06: G > F > H

Round 1:  A48 vs complementary approval winner C with 51, B49 vs complementary approval winner C with 50.  Neither qualifies
Round 2:  A52 vs complementary approval winner C with 47, B51 vs complementary approval winner C with 48.  Both qualify in IBIFA-derived methods, but not in MCA or MJ with less than 50% of ballots
Round 3:  C99 passes 50% threshold, while A and B still less than 50% threshold for tiebreaker.

A52 pairwise beats B51 and is the Condorcet winner (Please check my arithmetic!)

Pairwise array (equal rated whole):

  ['A', 'B', 'C', 'D', 'E', 'F', 'G',  'H']
A [56.  52.   56.  56.  54.  54.  56.  56.]
B [50.  52.   52.  52.  52.  52.  52.  52.]
C [46.  48.  100.  74.  76.  75.  79. 100.]
D [ 2.  26.   26.  26.  24.  26.  26.  26.]
E [ 4.  26.   26.  26.  26.  24.  26.  26.]
F [33.   8.   33.  33.  33.  33.  27.  33.]
G [27.   6.   27.  27.  27.  27.  27.  27.]
H [ 6.   8.    8.   8.   6.   0.   2.   8.]

A is definitely the Condorcet Winner.
 

The main point here is that while both IBIFA, modified IBIFA and Relevant Ratings can avoid electing a non-CW candidate C, the lowest level compromise approval winner elected by standard median ratings, your modified IBIFA will fail to choose the CW while relevant ratings and original IBIFA will find that candidate.

You suggestion of using undefeated tied-at-top winner first, then falling back to some IBI method, is an interesting one, however.

On Sun, Jun 2, 2019 at 8:16 PM Chris Benham [hidden email] [RangeVoting] <[hidden email]> wrote:

IBIFA was conceived as an Irrelevant Ballot independent version of
Bucklin, with the added benefits of having a less
severe truncation and/or compress at the top incentive and also being
much more (and absolutely more) Condorcet-consistent.

Inspired by an example from Ted Stern of?? his "Relevant Ratings" method
(which I gather is IBIFA
modified to more closely resemble Majority Judgement), I've come to
believe that if ratings ballots
with four or more slots (or grades) are used then a simple rule change
can make the method still
more Condorcet-consistent?? at no cost.

My idea (originally my misunderstanding of Ted's Relevant Ratings
method) is that if at some
(quasi-Bucklin) IBIFA round after the first (but before we have reached
just counting total approval scores)
we find more than one candidate Q qualified to win then instead of??
(Bucklin-like) giving the win to the Q
with the highest score in that round we elect the Q with the highest
score in the round before.

A link to the electowiki entry on my original version of IBIFA:

https://wiki.electorama.com/wiki/IBIFA

And the EM post in which I first suggested it:
http://lists.electorama.com/pipermail/election-methods-electorama.com//2010-May/091807.html

Here is the description of the revised 4-slot version, referring to
A-B-C-D grading ballots:

*Voters fill out 4-slot ratings ballots, say with A B C D grades.
 ??Default rating/grade is D, signifying least preferred and unapproved.

Any grade above D is interpreted as Approval.

If any candidate/s X has an A score that is higher than any other
candidate's approval
score on ballots that?? don't give X an A grade, elect the X with the
highest A score.

Otherwise, if any candidate/s X has a A+B score that is higher than any
other candidate's
approval score on ballots that don't give X an A or B grade, elect the X
with the highest
 ??A score.

Otherwise, elect the candidate with the highest Approval score.*

 ??35: A
 ??10: A=B
 ??30: B>C
 ??25: C

With my Condorcet hat on, in this example I've said that B is the
weakest candidate.?? A bit unfortunately
IBIFA here elects B, but FBC is a bit more "expensive" than Condorcet,
and so does Winning Votes and Margins.
Bucklin elects the most approved candidate C, but at least B both
pairwise beats and is more top-rated than C.

Ted Stern's eye-opening example:

49: A > B
03: B > A > C
10: D > B > C
38: E > F > C
05: G > D > H

The Condorcet winner is A.?? Ted's Relevant Ratings and my revised 4+
slot IBIFA elect A.
My original version of IBIFA?? and?? Median Ratings methods such as
Bucklin and MJ elect B.

Top Ratings (A) scores:?? A49,?? E38,?? D10,?? G5,?? B3,?? C0
A + B scores:???????????????????????????????????? A51,?? E38,?? D15,?? G5,?? B62,?? C0

In the second round A and B both "qualify".???? On ballots that don't??
give A one of the two
top grades the most approved candidate is E with a score of 38, lower
than 51 so A qualifies.

On ballots that don't give B one of the top two grades the most approved
candidate is again
E with again a score of 38, lower than 62 so B qualifies. In the "round
before" A?? has the
higher score (49 versus 3) so revised IBIFA gives the win to A.

A>B 49-13,???? A>E 51-38,?? A>D 51-15,?? A>G 51>5, A>C?? 51-48.

At the cost of being a quite a bit more complicated,?? IBIFA can be
combined?? with Kevin Venzke's
special "tied-at-the-top" rule used in his "Improved Condorcet Approval"
method to make
the method even more Condorcet-consistent?? (possibly as much as it
possible for a FBC method
to be).
https://wiki.electorama.com/wiki/Improved_Condorcet_Approval

*If one candidate T pairwise beats all others by the tied-at-the-top
rule then T wins. If there is no
such T then we elect the (revised) IBIFA winner.
If there is more than one T then we elect the one that "qualifies"
(according to IBIFA) in the earliest
IBIFA round. If there is more than one of these, then elect the one with
the highest score in the previous
round if there was one, otherwise simply with the highest top-ratings
score.*

4: A>B
6: A>C
6: B>A
2: B>C
3: C>B

B is the narrow Condorcet winner:?? B>A 11-10,?? B>C?? 12-9. No ballots
have any candidates tied at the top,
so B wins.?? Plain IBIFA elects A, which is positionally dominant: Top
scores: A10, B8, C2. Approval scores: A16,?? B13,?? C10.

For the time being the name I suggest?? for?? this is Quasi-Condorcet IBIFA.

Chris Benham






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Re: [EM] [RangeVoting] 4+slot IBIFA revision

Ted Stern
I've begun documenting Relevant Rating on electowiki:


I will add examples eventually as I get time, otherwise, if you wish to edit the page, please feel free to do so!

On Wed, Jun 12, 2019 at 5:39 PM Ted Stern <[hidden email]> wrote:
See inserted correction below:

On Mon, Jun 3, 2019 at 2:42 PM Ted Stern <[hidden email]> wrote:
Hi Chris,

You are so close to Relevant Ratings in your proposal.  I just want to point out how close and why the one missing factor is important.

You write:
My idea (originally my misunderstanding of Ted's Relevant Ratings method) is that if at some (quasi-Bucklin) IBIFA round after the first (but before we have reached just counting total approval scores) we find more than one candidate Q qualified to win then instead of (Bucklin-like) giving the win to the Q with the highest score in that round we elect the Q with the highest  score in the round before.

Where this differs from RR is as follows:
  • For each candidate Q qualified to win IBIFA, their total ballots from highest rating down to the current round rating exceed some highest total approval on complementary ballots excluding Q down to that rating.  Say that the highest total approval on such complementary ballots is TC.
  • Your modified IBIFA just looks at the Q totals from the previous round.
  • My Relevant Ratings method looks only at the previous round Q totals that are larger than their respective TC opposition in the current round!
In most situations, the Q you find with your modified IBIFA would be the same.  But it is possible that they might not be.  Let's carefully construct a 4 slot example, working backwards:

Say we want at least 3 candidates, ratings 3 = Excellent ("A"), 2 = Very Good ("B"), 1 = OK ("C"), 0 = disapproved ("D").
  • round 1 totals (scores at 3) of A48, B49, with other candidates below that (and not qualifying in any method)
  • round 2 totals (scores at 2 and above) of A52, B51, with other candidates below that (and not qualifying in any method)
  • round 3 totals (scores above 0) of A52, B52, and C > 54, with other candidates below that (only C qualifying under MCA or MJ)
  • In round 1, we want A48's most approved complementary candidate to be B or C with at least 49
  • In round 1, we want B49's most approved complementary candidate to be A or C with at least 50
  • In round 2, we want A52's most approved complementary candidate to be C with at most 47
  • In round 2, we want B51's most approved complementary candidate to be C with exactly 50.
  • We want A and B's total approval to be less than 50%, so there must be at least 105 ballots.  So we expect at least 5 irrelevant ballots.
Under this scenario, C will win both MCA and MJ in round 3.  B will win in modified IBIFA, as round 2 qualifier with the highest round 1 score.

But A will win both original IBIFA and relevant rating because while both A and B qualify in round 2, only A's round 1 score exceeds A's round 2 complementary approval winner C's approval of 47, while B's round 2 score of 49 is below B's complementary approval winner C's score of 50.

Here is a set of ballots that I think satisfies those constraints.

02: A > B > C
24: A > D > C
22: A > E > C
04: B > F > C

The 04: B > F > C ballots are a typo.  They should be 

04: B > A > C
 
25: B > F > C
21: B > G > C
02: E > D=A > C
02: F > E=A > H
06: G > F > H

Round 1:  A48 vs complementary approval winner C with 51, B49 vs complementary approval winner C with 50.  Neither qualifies
Round 2:  A52 vs complementary approval winner C with 47, B51 vs complementary approval winner C with 48.  Both qualify in IBIFA-derived methods, but not in MCA or MJ with less than 50% of ballots
Round 3:  C99 passes 50% threshold, while A and B still less than 50% threshold for tiebreaker.

A52 pairwise beats B51 and is the Condorcet winner (Please check my arithmetic!)

Pairwise array (equal rated whole):

  ['A', 'B', 'C', 'D', 'E', 'F', 'G',  'H']
A [56.  52.   56.  56.  54.  54.  56.  56.]
B [50.  52.   52.  52.  52.  52.  52.  52.]
C [46.  48.  100.  74.  76.  75.  79. 100.]
D [ 2.  26.   26.  26.  24.  26.  26.  26.]
E [ 4.  26.   26.  26.  26.  24.  26.  26.]
F [33.   8.   33.  33.  33.  33.  27.  33.]
G [27.   6.   27.  27.  27.  27.  27.  27.]
H [ 6.   8.    8.   8.   6.   0.   2.   8.]

A is definitely the Condorcet Winner.
 

The main point here is that while both IBIFA, modified IBIFA and Relevant Ratings can avoid electing a non-CW candidate C, the lowest level compromise approval winner elected by standard median ratings, your modified IBIFA will fail to choose the CW while relevant ratings and original IBIFA will find that candidate.

You suggestion of using undefeated tied-at-top winner first, then falling back to some IBI method, is an interesting one, however.

On Sun, Jun 2, 2019 at 8:16 PM Chris Benham [hidden email] [RangeVoting] <[hidden email]> wrote:

IBIFA was conceived as an Irrelevant Ballot independent version of
Bucklin, with the added benefits of having a less
severe truncation and/or compress at the top incentive and also being
much more (and absolutely more) Condorcet-consistent.

Inspired by an example from Ted Stern of?? his "Relevant Ratings" method
(which I gather is IBIFA
modified to more closely resemble Majority Judgement), I've come to
believe that if ratings ballots
with four or more slots (or grades) are used then a simple rule change
can make the method still
more Condorcet-consistent?? at no cost.

My idea (originally my misunderstanding of Ted's Relevant Ratings
method) is that if at some
(quasi-Bucklin) IBIFA round after the first (but before we have reached
just counting total approval scores)
we find more than one candidate Q qualified to win then instead of??
(Bucklin-like) giving the win to the Q
with the highest score in that round we elect the Q with the highest
score in the round before.

A link to the electowiki entry on my original version of IBIFA:

https://wiki.electorama.com/wiki/IBIFA

And the EM post in which I first suggested it:
http://lists.electorama.com/pipermail/election-methods-electorama.com//2010-May/091807.html

Here is the description of the revised 4-slot version, referring to
A-B-C-D grading ballots:

*Voters fill out 4-slot ratings ballots, say with A B C D grades.
 ??Default rating/grade is D, signifying least preferred and unapproved.

Any grade above D is interpreted as Approval.

If any candidate/s X has an A score that is higher than any other
candidate's approval
score on ballots that?? don't give X an A grade, elect the X with the
highest A score.

Otherwise, if any candidate/s X has a A+B score that is higher than any
other candidate's
approval score on ballots that don't give X an A or B grade, elect the X
with the highest
 ??A score.

Otherwise, elect the candidate with the highest Approval score.*

 ??35: A
 ??10: A=B
 ??30: B>C
 ??25: C

With my Condorcet hat on, in this example I've said that B is the
weakest candidate.?? A bit unfortunately
IBIFA here elects B, but FBC is a bit more "expensive" than Condorcet,
and so does Winning Votes and Margins.
Bucklin elects the most approved candidate C, but at least B both
pairwise beats and is more top-rated than C.

Ted Stern's eye-opening example:

49: A > B
03: B > A > C
10: D > B > C
38: E > F > C
05: G > D > H

The Condorcet winner is A.?? Ted's Relevant Ratings and my revised 4+
slot IBIFA elect A.
My original version of IBIFA?? and?? Median Ratings methods such as
Bucklin and MJ elect B.

Top Ratings (A) scores:?? A49,?? E38,?? D10,?? G5,?? B3,?? C0
A + B scores:???????????????????????????????????? A51,?? E38,?? D15,?? G5,?? B62,?? C0

In the second round A and B both "qualify".???? On ballots that don't??
give A one of the two
top grades the most approved candidate is E with a score of 38, lower
than 51 so A qualifies.

On ballots that don't give B one of the top two grades the most approved
candidate is again
E with again a score of 38, lower than 62 so B qualifies. In the "round
before" A?? has the
higher score (49 versus 3) so revised IBIFA gives the win to A.

A>B 49-13,???? A>E 51-38,?? A>D 51-15,?? A>G 51>5, A>C?? 51-48.

At the cost of being a quite a bit more complicated,?? IBIFA can be
combined?? with Kevin Venzke's
special "tied-at-the-top" rule used in his "Improved Condorcet Approval"
method to make
the method even more Condorcet-consistent?? (possibly as much as it
possible for a FBC method
to be).
https://wiki.electorama.com/wiki/Improved_Condorcet_Approval

*If one candidate T pairwise beats all others by the tied-at-the-top
rule then T wins. If there is no
such T then we elect the (revised) IBIFA winner.
If there is more than one T then we elect the one that "qualifies"
(according to IBIFA) in the earliest
IBIFA round. If there is more than one of these, then elect the one with
the highest score in the previous
round if there was one, otherwise simply with the highest top-ratings
score.*

4: A>B
6: A>C
6: B>A
2: B>C
3: C>B

B is the narrow Condorcet winner:?? B>A 11-10,?? B>C?? 12-9. No ballots
have any candidates tied at the top,
so B wins.?? Plain IBIFA elects A, which is positionally dominant: Top
scores: A10, B8, C2. Approval scores: A16,?? B13,?? C10.

For the time being the name I suggest?? for?? this is Quasi-Condorcet IBIFA.

Chris Benham






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