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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.
#* 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 ▼## If this is the first round, initialize X's round total to zero, and the current rating level R as MAXLEVEL
##* AddInitialize theeach candidate X's total number of ballots rating X atgreater levelthan Ror equal to X's previous round total. We denote this total asR, '''TATG(X,R)''', whichto iszero. defined''[technically, asthis theshould totalbe numberwritten ofas ballotsa ratingfunction, '''TG(R,X at or above rating R.)''']''
* 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 T'''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.
## IsFor 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 TTG(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, setfind C* such that '''TCA( R,X,C *)''' tohas the highest approval for any candidate on ballots that rate X below the new R rating level .▼## For each candidate X, is ''' TTG( 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 TTG( 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 . So a Majority Grade is a primary rating of median-rating, plus a secondary rating of either MR+1 or MR withand the associated total approval at andvotes above that secondary 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 and much simpler 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
# 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.
Note that after R is decremented to zero, the new TCA(X,C) will be zero. So the final step after R is zero will always find an approval winner.
[[Category:Monotonic electoral systems]]
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