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Relevant Rating is an Irrelevant Ballot Independent adaptation of Majority Judgment, following the methodology of Chris Benham's IBIFA. It was proposed by Ted Stern.

Voting process

Voters use rated ballots. Any number of ratings may be allowed, but as an initial proposal, it is suggested that 3, 4 or 5 rating slots of approval and one slot of disapproval be used, as in Majority Judgment. To illustrate the method, it is not important what the rating levels are called, only that they can be tabulated as levels 0 (disapproved) through MAXRATING.

Rules

Voters fill out rated ballots, rating between zero (disapproved) to MAXRATING (most preferred), for a total of MAXRATING+1 slots. If only 2 slots are used, the method is equivalent to Approval voting, so we assume that at least 3 slots are used.
Any rating above Bottom (zero) is considered as approval.
For each candidate X, find X's relevant rating with a series of rounds starting with the highest rating.
Initialize the rating level R to MAXRATING
Initialize each candidate X's total number of ballots rating X greater than or equal to R, TG(X), to zero. [technically, this should be written as a function, TG(R,X)]
Assume each candidates total number of ballots rating X at R, T(R,X), has been found during tabulation.
Assume that the approval for every candidate C, TCA(R,X,C), on ballots that rate X below rating R, has been found during tabulation. Let C* be the candidate with the highest such complementary approval at rating R for candidate X [technically this should be written as a function, C*(R,X).
Repeat while R is greater than zero:

For each candidate X, add ballots rating X at level R, T(R,X), to TG(X)
For each candidate X, is T(X) > TCA(R,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 TG(Q) is the winner. Each member of the qualifying set has the relevant rating (R,R-1,TG(R,Q)).
Otherwise, decrement R by one
For each candidate X, find C* such that TCA(R,X,C*) has the highest approval for any candidate on ballots that rate X below the new R rating level.
For each candidate X, is TG(R+1,X) > TCA(R,X,C*) (using new R and new TCA(R,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 TG(R+1,Q') is the winner. And each member of the new qualifying set has relevant rating (R,R+1,TG(R+1,Q')).

If the method has continued to this point with some members not receiving relevant rating, then all remaining candidates have the relevant rating (0,0,TG(1,X)), and if no winner has been found, the candidate with highest total approval is the winner.

As iterations occur above, a full relevant ranking of candidates can be formed by appending every qualifying set of candidates to the ranking, in descending order of TG(X). The resulting order will have strictly decreasing associated relevant rating 3-tuples.

The comparison to Majority Judgment should be clear: in MJ, each candidate has a Majority Grade consisting of their median rating, with a secondary rating determined by removing median ballots until the rating either increases or decreases, and the votes above that rating.

In Relevant rating, when comparing the total number of ballots approving a candidate X at and above a rating R to a contrasting number, instead of using the total number of ballots rating X below R as that contrasting number (as in MJ), we use the maximum approval for any candidate on those complementary ballots. As in IBIFA, this is what renders the method independent of irrelevant ballots, because ballots that don't change the complementary approval winner won't change the relevant rating.

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 ==Rules==
-# Voters fill out rated ballots, rating between zero (disapproved) to MAXRATING (most preferred), for a total of MAXRATING+1 slots. If only 2 slots are used, the method is equivalent to [[Approval voting]], so we assume that at least 3 slots are used.
+* Voters fill out rated ballots, rating between zero (disapproved) to MAXRATING (most preferred), for a total of MAXRATING+1 slots. If only 2 slots are used, the method is equivalent to [[Approval voting]], so we assume that at least 3 slots are used.
-# Any rating above Bottom (zero) is considered as approval.
+* Any rating above Bottom (zero) is considered as approval.
-# For each candidate X, find X's relevant rating with a series of rounds starting with the highest rating.
+*  For each candidate X, find X's relevant rating with a series of rounds starting with the highest rating.
+* Initialize the rating level R to MAXRATING
-## If this is the first round, initialize X's round total to zero, and the current rating level R as MAXLEVEL
-## Add the total number of ballots rating X at level R to X's previous round total.  We denote this total as '''TA(X,R)''', which is defined as the total number of ballots rating X at or above rating R.
+* Initialize each candidate X's total number of ballots rating X greater than or equal to R, '''TG(X)''', to zero. ''[technically, this should be written as a function, '''TG(R,X)''']''
+* Assume each candidates total number of ballots rating X at R, '''T(R,X)''', has been found during tabulation.
-## For all ballots that rate X below R, there is a most approved candidate C, with total complementary approval '''TCA(X,C,R)''', where TCA(X,C,R) is defined as the total approval for candidate C on ballots that rate X below rating R.
+* Assume that the approval for every candidate C, '''TCA(R,X,C)''', on ballots that rate X below rating R, has been found during tabulation. Let '''C*''' be the candidate with the highest such complementary approval at rating R for candidate X [technically this should be written as a function, '''C*(R,X)'''.
-## '''Threshold check:''' If '''TA(X,R) > TCA(X,C,R)''', that is If X's round total exceed's the maximum approval for any candidate C on ballots that rate X below rating R, then X's primary relevant rating is R.  Otherwise, decrement R by one and go back to step 3.2 above.
+* Repeat while R is greater than zero:
-## If there was a previous round (in other words, R for the current round is less than MAXRATING), and X's previous round total [that is, TA(X,R+1)] also exceed's C's approval ''for the current round'' (that is, '''TA(X,R+1) > TCA(X,C,R)'''], then X's secondary relevant rating is the 2-tuple '''(R+1,TA(X,R+1))'''.
+# For each candidate X, add ballots rating X at level R, '''T(R,X)''', to '''TG(X)'''
-## Otherwise, if R is greater than zero, X's secondary relevant rating is the 2-tuple '''(R,TA(X,R))'''.  If we have exhausted all approved rating levels and R is now zero, X's secondary relevant rating is the tuple '''(0,TA(X,1))''', because it makes no sense to count ballots on which X is disapproved.
+# For each candidate X, is '''T(X) > TCA(R,X,C*)'''?  If so X is a member of the current qualifying set
-# Each candidate will now have a primary and secondary relevant rating.
+# If the current qualifying set has at least one member Q, the candidate with the highest TG(Q) is the winner.  Each member of the qualifying set has the relevant rating '''(R,R-1,TG(R,Q))'''.
-# Sort the candidates in descending order by their primary and secondary relevant ratings -- if the primary relevant ratings are equal, break the tie using the secondary rating.  Note that in comparing the tuple (R+1,*) to (R,*), the tuple beginning with R+1 is greater.
+# Otherwise, decrement R by one
-# The candidate with the highest relevant rating is the winner.  The sorted list of candidates is the Relevant Ratings ranking.
+# For each candidate X, find C* such that '''TCA(R,X,C*)''' has the highest approval for any candidate on ballots that rate X below the new R rating level.
+# For each candidate X, is '''TG(R+1,X) > TCA(R,X,C*)''' (using new R and new TCA(R,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 TG(R+1,Q') is the winner.  And each member of the new qualifying set has relevant rating '''(R,R+1,TG(R+1,Q'))'''.
+* If the method has continued to this point with some members not receiving relevant rating, then all remaining candidates have the relevant rating '''(0,0,TG(1,X))''', and if no winner has been found, the candidate with highest total approval is the winner.
+As iterations occur above, a full relevant ranking of candidates can be formed by appending every qualifying set of candidates to the ranking, in descending order of TG(X).  The resulting order will have strictly decreasing associated relevant rating 3-tuples.
-The comparison to [[Majority Judgment]] should be clear: in MJ, each candidate has a Majority Grade consisting of their median rating, with a secondary rating determined by removing median ballots until the rating either increases or decreases.  So a Majority Grade is a primary rating of median-rating, plus a secondary rating of either MR+1 or MR with the associated total approval at and above that secondary rating.
+The comparison to [[Majority Judgment]] should be clear: in MJ, each candidate has a Majority Grade consisting of their median rating, with a secondary rating determined by removing median ballots until the rating either increases or decreases, and the votes above that rating.
 In Relevant rating, when comparing the total number of ballots approving a candidate X at and above a rating R to a contrasting number, instead of using the ''total number of ballots'' rating X below R as that contrasting number (as in MJ), we use the maximum approval for ''any candidate'' on those complementary ballots.  As in [[IBIFA]], this is what renders the method independent of irrelevant ballots, because ballots that don't change the complementary approval winner won't change the relevant rating.
-Using a similar process, we could also find the Relevant Rating winner without having to create a full Relevant Ratings ranking of candidates.
-# Initialize the rating level R to MAXRATING
-# Find all qualifying candidates Q whose total approval at and above rating R ('''TA(Q,R)''') is greater than the maximum approval for any candidate C on ballots that rate Q below R ('''TCA(Q,C,R)''').  If there are no qualifying candidates, then decrement R and repeat.
-# If there is more than one qualifying candidate Q, then additionally check whether each Q's total approval ''above'' rating R (that is, Q's total approval for the previous round's rating, TA(Q,R+1)) also exceeds their corresponding current round TCA(Q,C,R).  The additionally qualifying candidate with the highest TA(Q,R+1) is the winner.  Otherwise [that is, the additional qualification is not met], the candidate with the highest TA(Q,R) is the winner.
-# If all approval ratings above zero have been exhausted, the candidate with the highest total approval is the winner.
 [[Category:Monotonic electoral systems]]