The calculation is just based what C gets if their offer is accepted, and on what B thinks is fair; it doesn’t matter if the split is fair according to C, it’s just in the calculation B does so that the expected payout of C is not higher than what’s fair according to B due to them offering this split. If C thought that the fair split is 5:1, but offered an C!unfair (unfair-according-to-C) split of 4:2, B does the same calculation; the goal is to make C get a bit less than 3 in expectation in all B!unfair splits, because the B!fair split is 3:3
The high-level purpose is that this enables people with different notions of fairness to mutually cooperate almost always; and if you do this procedure, you incentivize fair-according-to-you splits. Others don’t have to do what you think is fair; and even if they offer what you consider unfair, you’ll often accept (you just need to reject enough for it you!unfair splits to not be worth offering just for the purpose of exploiting you, as it won’t work).
The calculation is just based what C gets if their offer is accepted, and on what B thinks is fair; it doesn’t matter if the split is fair according to C, it’s just in the calculation B does so that the expected payout of C is not higher than what’s fair according to B due to them offering this split. If C thought that the fair split is 5:1, but offered an C!unfair (unfair-according-to-C) split of 4:2, B does the same calculation; the goal is to make C get a bit less than 3 in expectation in all B!unfair splits, because the B!fair split is 3:3
The high-level purpose is that this enables people with different notions of fairness to mutually cooperate almost always; and if you do this procedure, you incentivize fair-according-to-you splits. Others don’t have to do what you think is fair; and even if they offer what you consider unfair, you’ll often accept (you just need to reject enough for it you!unfair splits to not be worth offering just for the purpose of exploiting you, as it won’t work).