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

1. 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.
2. Any rating above Bottom (zero) is considered as approval.
3. For each candidate X, find X's relevant rating with a series of rounds starting with the highest rating.
   1. If this is the first round, initialize X's round total to zero, and the current rating level R as MAXLEVEL
   2. 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.
   3. 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.
   4. 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.
   5. 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)).
   6. 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.
4. Each candidate will now have a primary and secondary relevant rating.
5. 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.
6. The candidate with the highest relevant rating is the winner. The sorted list of candidates is the Relevant Ratings ranking.

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.

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.

1. Initialize the rating level R to MAXRATING
2. Initialize candidate totals, T(X), to zero
3. Initialize TCA(X,C) to the highest approval for any candidate on ballots that rate X below R
4. Repeat until a winner is found:
   1. For each candidate X, add ballots rating X at level R to T(X)
   2. Is T(X) > TCA(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 T(Q) is the winner
   4. Otherwise, decrement R by one
   5. Set TCA(X,C) to the highest approval for any candidate on ballots that rate X below the new R
   6. 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.
   7. 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.