I think claims like “exactly twice as bad” are ill-defined.
Suppose you have some preference relation on possible states R, so that X is preferred to Y if and only if R(X, Y) holds. Next, suppose we have a utility function U, such that if R(X, Y) holds, then U(X) > U(Y). Now, take any monotone transformation of this utility function. For example, we can take the exponential of U, and define U’(X) = 2^(U(X)). Now, note that U(X) > U(Y) if and only if that U’(X) > U’(Y). Now, even if U is additive along some dimension of X, U’ won’t be.
But there’s no principled reason to believe that U is a “truer” reflection of the agent’s preferences than U’, since both of them are equally faithful to the underlying preference relation. So if you want to do meaningful comparisons of utility you have to do them in a way that’s invariant under monotone transformations. Since “twice as bad” isn’t invariant such a transformation, it’s not evidently a meaningful claim.
Now, there might be some additional principle you can advance to justify claims like that, but I haven’t seen it, or its justification, yet.
I think claims like “exactly twice as bad” are ill-defined.
Suppose you have some preference relation on possible states R, so that X is preferred to Y if and only if R(X, Y) holds. Next, suppose we have a utility function U, such that if R(X, Y) holds, then U(X) > U(Y). Now, take any monotone transformation of this utility function. For example, we can take the exponential of U, and define U’(X) = 2^(U(X)). Now, note that U(X) > U(Y) if and only if that U’(X) > U’(Y). Now, even if U is additive along some dimension of X, U’ won’t be.
But there’s no principled reason to believe that U is a “truer” reflection of the agent’s preferences than U’, since both of them are equally faithful to the underlying preference relation. So if you want to do meaningful comparisons of utility you have to do them in a way that’s invariant under monotone transformations. Since “twice as bad” isn’t invariant such a transformation, it’s not evidently a meaningful claim.
Now, there might be some additional principle you can advance to justify claims like that, but I haven’t seen it, or its justification, yet.