[EM] 4+slot IBIFA revision
C.Benham
cbenham at adam.com.au
Sun Jun 2 20:03:55 PDT 2019
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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