The Ranked Approval Compromise Exception is a modification that can be applied to any method that collects information on both approval and pairwise rankings.
The motivation for its development was consideration of cases where "Everybody's Second Choice" seems to have more of a mandate than a frontrunning, polarized winner. Consider an election among candidates L, C, and R (Left, Center, and Right). We will use the pipe symbol to indicate the approval cutoff:
In this circumstance, some will see C as having a better mandate, and a better chance of governing effectively.
A slightly more complex case might look like:
In order to quantify what it means to be a compromise candidate, we start by simply identifying the winner of the election by its standard definition; we'll call this Standard Winner S. Next, identify the candidate with the highest approval score; we'll call the Approval Winner A. (In the above case, A is clearly C; S is presumably L, who would win under IRV, Borda, and any Condorcet method.)
Look at the pairwise race between S and A. The percentage of votes received by S in this race will be the Compromise Level. In this case, in the race between L and C, L receives 51% of the votes. (In races where equal rankings are allowed, we must choose whether to count 1, 0.5, or 0 votes for S when a ballot says S=A; no examination has been made of how this choice affects outcomes.)
We break the electorate into two groups: those whose ballots rank S>A, and those who rank S<A. (Again, if equal rankings are permitted we must decide whether to include voters saying S=A in one of the groups, count them as half-votes in each group, or not count them in this phase at all.) To win, A must receive approval from a percentage exceeding the Compromise Level. In the example above, we see...
Among voters who say L>C, A receives 36/51 = 70.6% approval, which exceeds the 51% Compromise Level.
Among voters who say C<L, A receives 35/49 = 71.4% approval, which exceeds the 51% Compromise Level.
Thus, in the example, C will win.
The logic here is that even though a slim majority might prefer the standard winner, we can tell because of their high approval level for the compromise, that they will not, as a group, be too upset if that compromise wins; we check the approval level among the remainder of the population — those who ranked the proposed compromise over the standard winner — to see whether they are sufficiently enthusiastic about the compromise to make it worthwhile.
One side effect of the compromise exception is to discourage one form of strategic voting in Definite Majority Choice, and potentially other methods based on ranking and approval data. It has been suggested by Jeff Fisher that voters in DMC would come to routinely approve any candidate they expect their favorite to defeat in pairwise rankings, because even if this leads the candidate to have no double-defeats, they will be beaten in the final selection by the favorite, and in the meantime they may produce a double-defeat against some other candidate that threatened the favorite. With the compromise exception, voters opting for this strategy would run a serious risk of inflating the dishonestly approved candidate's approval score to the point that the exception would come into play. Thus, it acts as a strong disincentive to approve dishonestly.
This concept was originally proposed by David Scotese, as a refinement of a method equivalent to Smith//Approval that had been suggested by R.M. "Auros" Harman. Scotese's "Condorcet Versus Approval" was refined by Harman to produce the version described above.