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 --- This email has been checked for viruses by AVG. https://www.avg.com ---- Election-Methods mailing list - see https://electorama.com/em for list info |
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:
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").
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.
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Ted, Is there a typo? Chris Benham On 4/06/2019 7:12 am, Ted Stern wrote:
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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:
The 04: B > F > C ballots are a typo. They should be 04: B > A > C
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.
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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:
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