You may be familiar with Improved Condorcet Approval, a variant of Condorcet//Approval that satisfies the Favorite Betrayal Criterion.
The essence of the method is to only consider defeats where
A[X,Y] > (A[Y,X] + Tied-At-Top[X,Y])
The method is neither Smith-efficient, Condorcet-winner compliant, or clone-independent.
However, the tied-at-top methodology is interesting, since it can easily be applied to Approval Sorted Margins in order to confer FBC-compliance.
Recall that in sorted margins, the ranking is initially seeded in descending order of approval, and then the out-of-order-pairwise pair with the smallest margin in approval is reversed.
To render ASM FBC-compliant, we simply change the out-of-order test from
A[X,Y] < A[Y,X]
A[X,Y] + Tied-at-top[X,Y] < A[Y,X]
The resulting method is now FBC-compliant, at the cost of Smith-efficiency and Condorcet-compliance. However, it is still clone-independent.
In preliminary tests, it appears that standard ASM handles situations ICA was created to address, without the Tied-at-Top modification. For example,
Smith//Approval winner: C
ASM winner: B (with approval cutoff at score 0)
FBC-ASM winner: B
ICA winner: B
Does anyone have examples of situations in which Condorcet//Approval or Smith//Approval fails FBC, but ICA finds a different winner than ASM?