I think they probably are incompatible. Anyway, I’ve been looking at Hamilton methods a bit further, and the following gives a participation failure:

2 seats, max score 10

30 voters: A=10, B=0

9 voters: A=0, B=10

1 voter: A=1, B=10

Without the 1 voter giving a score of 1 to A, it would be a tie for the second seat so this score means that A gets both seats. But then if you have the following ballots:

30 voters: A=10, B=0

9 voters: A=0, B=10

1 voter: A=1, B=10

1 voter: A=1, C=10

The extra voter changes the quota size and causes B to get the second seat, even though they prefer A to B. I think that’s what would happen anyway. Obviously Hamilton normally refers to party list, and SSS works slightly differently. But in terms of raw quotas, A has 302/410*2 = 1.473. B has 100/410*2 = 0.488.

Anyway, what happens in Hamilton is that when two parties should have exactly something point 5 seats, then they would tie for a seat and it would be the same as in Sainte-Laguë. But when other voters are added/removed to move the fractional part to more than half (with the ratio of the two parties’ voters staying constant) this will favour the larger party, which sees its fractional part rise faster. However if the fractional part drops below 0.5, it will favour the smaller party, sees its fractional part drop at a slower rate. Take the following example with 3 seats:

Party A: 30 voters

Party B: 10 voters

Party C: 20 voters

Party A is due 1.5 seats, B is due 0.5 and C is due exactly 1. So A and B tie for the final seat. But then add a C voter:

Party A: 30 voters

Party B: 10 voters

Party C: 21 voters

Party A is due 1.475 seats, B is due 0.492 seats and C 1.033. In this case Party B claims that final seat over A because of the more slowly dropping fractional part. Now remove two C voters from this example:

Party A: 30 voters

Party B: 10 voters

Party C: 19 voters

Party A is due 1.525 seats, B is due 0.508 and C 0.966. So A gets that final seat over B because of the faster rising fractional part.

The problem is that Hamilton is a mixture of ratio and absolute difference and while it might look right intuitively, I don’t think it stands up to scrutiny. Sainte-Laguë party list would give a tie for the final seat in all of these cases, which I think is the right result.