Relevant rating

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:
 * 1) For each candidate X, add ballots rating X at level R, T(R,X), to TG(X)
 * 2) For each candidate X, is T(X) > TCA(R,X,C*)?  If so X is a member of the current qualifying set
 * 3) 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)).
 * 4) Otherwise, decrement R by one
 * 5) 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.
 * 6) 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.
 * 7) 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.