The Coin-Flip Approval Transformation (CFAT) was added to PAMSAC to improve monotonicity and specifically fix the following example:

2 to elect

x voters: ABC

x voters: ABD

1 voters: C

1 voters: D

Under Ebert’s Method CD would be elected for any value of x. By applying CFAT, AB would be elected for high enough x, and it also changes it from Sainte-Laguë equivalent to D’Hondt equivalent for party voting. However, there is another (probably better) way of doing this. Instead of having to split voters into parts as CFAT does, the weight of a voter is increased by an amount that depends on the number of candidates that they have approved in the set.

If a voter has approved c candidates in a set, that voter’s weight when that set is being considered becomes 2c/(c+1). This, like CFAT, has the effect of turning Ebert’s Method from Sainte-Laguë into D’Hondt equivalent (I can put a proof in another post but will leave it for now). And in the example above AB would be elected for x above a certain amount because there would effectively be more voters when AB is elected.

It would still not be monotonic as things stand. For example:

2 to elect

1 voter: AB

1 voter: AC

Ebert’s Method with this modification would still elect BC and would therefore require approval removal to elect AB (or AC). However, this is where it has an advantage over CFAT. To elect AB, ordinarily you’d have to remove the A approvals from the AB voters. This would make AB and AC equivalent, and AB would win on a tie-break of most approvals. But then it would be impossible for PAMSAC to award the win to AB for any x in the following case even with approval removal:

x voters: AB

x voters: AC

1 voter: B

1 voter: C

But under this new transformation, while approvals would still be removed as before, the weight of a voter can still be based on the number of approvals pre-removal. So:

1 voter: AB

1 voter AC

When considering the AB result, we’d have:

4/3 voters: AB

1 voter: A

And after approval removal this can become:

7/6 voters: A

7/6 voters: B

This beats BC without the need for a tie-break, making the victory far less fragile and it isn’t as susceptible to the single B and single C voters as before. This means that the method would now be strictly monotonic (every approval for a set of candidates would improve it’s Ebert score) than than weakly (where an approval can’t harm it).