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]

to

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,

20 A>B

20 A=B

15 B>C

45 C

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?