Krishnaswami: 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.
Utility functions over outcomes have additional structure beyond tehir ordering, because of how utilities interact with scalar probabilities to produce expected utilities that imply preferences over actions (as distinct from preferences over outcomes).
Taking the exponential of a positive utility function will produce the same preference ordering over outcomes but not the same preference ordering over actions (which is itself a quite interesting observation!) given fixed beliefs about conditional probabilities.
So when I say that two punches to two faces are twice as bad as one punch, I mean that if I would be willing to trade off the distance from the status quo to one punch in the face against a billionth (probability) of the distance between the status quo and one person being tortured for one week, then I would be willing to trade off the distance from the status quo to two people being punched in the face against a two-billionths probability of one person being tortured for one week. (“If...then” because I don’t necessarily defend this as a good preference—the actual comparison here is controversial even for utilitarians, since there are no overwhelming quantities involved.)
Any positive affine transformation of the utility function preserves the preference ordering over actions. The above statement is invariant under positive affine transformations of the utility function over outcomes, and thus exposes the underlying structure of the utility function. It’s not that events have some intrinsic number of utilons attached to them—a utility function invariantly describes the ratios of intervals between outcomes. This is what remains invariant under a positive affine transformation.
(I haven’t heard this pointed out anywhere, come to think, but surely it must have been observed before.)
Krishnaswami: 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.
Utility functions over outcomes have additional structure beyond tehir ordering, because of how utilities interact with scalar probabilities to produce expected utilities that imply preferences over actions (as distinct from preferences over outcomes).
Taking the exponential of a positive utility function will produce the same preference ordering over outcomes but not the same preference ordering over actions (which is itself a quite interesting observation!) given fixed beliefs about conditional probabilities.
So when I say that two punches to two faces are twice as bad as one punch, I mean that if I would be willing to trade off the distance from the status quo to one punch in the face against a billionth (probability) of the distance between the status quo and one person being tortured for one week, then I would be willing to trade off the distance from the status quo to two people being punched in the face against a two-billionths probability of one person being tortured for one week. (“If...then” because I don’t necessarily defend this as a good preference—the actual comparison here is controversial even for utilitarians, since there are no overwhelming quantities involved.)
Any positive affine transformation of the utility function preserves the preference ordering over actions. The above statement is invariant under positive affine transformations of the utility function over outcomes, and thus exposes the underlying structure of the utility function. It’s not that events have some intrinsic number of utilons attached to them—a utility function invariantly describes the ratios of intervals between outcomes. This is what remains invariant under a positive affine transformation.
(I haven’t heard this pointed out anywhere, come to think, but surely it must have been observed before.)