3A) I find ApC ≻ BpC implies A ≻ B to be simple and obvious.
3B) I find A ≻ B implies ApC ≽ BpC to be simple and obvious.
3C) I find A ≻ B implies ApC ≩[1] BpC to be complex and non-obvious.
(In particular, consider an agent that has a non-zero cost of calculating the relation between A and B. Then the optimum for said agent may be to return A ~ B even if A and B aren’t precisely the same value, because the cost of calculating if A or B is truly better is higher than the utility loss by just saying A ~ B. As a concrete made-up example—if I know that min(A,B)=0;max(A,B)=3, and that my cost of calculating which way around is 1, it’s worth it to me to calculate and return A ≻ B (or B ≻ A, or A ~ B, as appropriate), but if, say, p=0.1, it’s not worth it to me to do the calculation, and so I should return ApC ~ BpC.)
Can we split this please?
3A) I find ApC ≻ BpC implies A ≻ B to be simple and obvious.
3B) I find A ≻ B implies ApC ≽ BpC to be simple and obvious.
3C) I find A ≻ B implies ApC ≩[1] BpC to be complex and non-obvious.
(In particular, consider an agent that has a non-zero cost of calculating the relation between A and B. Then the optimum for said agent may be to return A ~ B even if A and B aren’t precisely the same value, because the cost of calculating if A or B is truly better is higher than the utility loss by just saying A ~ B. As a concrete made-up example—if I know that min(A,B)=0;max(A,B)=3, and that my cost of calculating which way around is 1, it’s worth it to me to calculate and return A ≻ B (or B ≻ A, or A ~ B, as appropriate), but if, say, p=0.1, it’s not worth it to me to do the calculation, and so I should return ApC ~ BpC.)
There’s a similar split with axiom 4.
I’m using this for clarity, though honestly this may have made this less clear. read: ≻ and very explicitly not ~.