
Chris and Steve, you might find it interesting to compare my restatement of Relevant Ratings with a similar formulation for Majority Judgment:
 Initialize the rating level R to MAXRATING
 Initialize candidate totals, T(X), to zero [T(X) is a shorthand for the total number of ballots voting for X at ratings >= R]
 Assume TR(X,R) [the total number of ballots rating candidate X at rating R] has been tabulated while counting ballots
 Repeat until a winner is found and R is greater than zero.
 For each candidate X, add TR(X,R), the number of ballots rating X at level R, to T(X)
 Is T(X) > 50%? If so X is a member of the current qualifying set
 If the current qualifying set has at least one member Q, the candidate with the highest T(Q) is the winner
 Otherwise, decrement R by one
 For each candidate X, is T(X) > (50%  (TR(X,R)/2)) (using new R)? If so, then X is a member of a new qualifying set.
 If the new qualifying set has at least one member Q', then the candidate with the highest T(Q') is the winner.
 If no winner has been found, the candidate with highest T(X) is the winner.
In Majority Judgment, each candidate has a majority grade 3tuple, consisting of their median rating, followed by the rating that occurs after removing median ballots, followed by the number of ballots that break the secondary tie. The median rating is found by following the above process for each candidate separately.
If X satisfies the comparison in step 4.2, X's majority grade is (R, R1, TR(X,R)).
If X satisfies the comparison in step 4.5, X's majority grade (using the initial R before decrementing) is (R1, R, TR(X,R))
If X falls through to step 5, X's majority grade is (0,0,T(X)).
Hopefully it should be clear how this is the same as relevant rating when every ballot is relevant  the maximum approved candidate on ballots that rate X below rating R will simply be the total number of ballots (100%) minus the ballots rating X at and above R. So for T(X) > TCA(X,R) is equivalent to T(X) > 100%  T(X), == 2*T(X) > 100%, == T(X) > 50%. Similarly for the other comparison. I've modified my electowiki page to include a simpler calculation of Relevant Rating:
 Initialize the rating level R to MAXRATING
 Initialize candidate totals, T(X), to zero
 Initialize TCA(X,C) to the highest approval for any candidate on ballots that rate X below R
 Repeat until a winner is found:
 For each candidate X, add ballots rating X at level R to T(X)
 Is T(X) > TCA(X,C)? If so X is a member of the current qualifying set
 If the current qualifying set has at least one member Q, the candidate with the highest T(Q) is the winner
 Otherwise, decrement R by one
 For each candidate X, set TCA(X,C) to the highest approval for any candidate on ballots that rate X below the new R rating level
 For each candidate X, is T(X) > TCA(X,C) (using new TCA(X,C))? If so, then X is a member of a new qualifying set.
 If the new qualifying set has at least one member Q', then the candidate with the highest T(Q') is the winner.
You could easily augment this algorithm with your Condercet modification.
IBIFA very naturally meets Plurality, Minimal Defense and
NonDrastic Defense so it occurred to
me to use it as a "Condorcetcompletion" method thus:
*Voters fill out out either unlimited rankings ballots or
multislot ratings ballots.
A pairwisebeatsall candidate wins. Otherwise carry on the IBIFA
process until a
Smithset member qualifies. If only one does that candidate is
elected.
If more than one does in the same round, then simplest and
probably good enough
is just to elect the one with highest score in that round.*
(That last provision is Bucklinlike as in original IBIFA. Using
the Smith set the more
complex Relevant Ratings and the possibly a bit arbitrarylooking
"revised IBIFA"
I think would be very unlikely to give different winners.)
I think this is my favourite method that meets both Condorcet and
Minimal Defense.
Also it can be used with an approval cutoff to meet what Forest
was asking for on 30 May 2019.
All rankings/ratings would be used to identify the Smith set, but
for the IBIFA stage ballots would
be treated as if they truncate all their unapproved candidates.
The default should be approval of
all candidates voted above at least one candidate.
Chris Benham
Forest Simmons [hidden email]
Thu May 30
In the example profiles below 100 = P+Q+R,
and 50>P>Q>R>0.
I am interested in simple methods that always ...
(1) elect candidate A given the following profile:
P: A
Q: B>>C
R: C,
and
(2) elect candidate C given
P: A
Q: B>C>>
R: C,
and
(3) elect candidate B given
P: A
Q: B>>C (or B>C)
R: C>>B. (or C>B)

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