Hi Vasco. I think Figure 3 here, and the surrounding discussion of how imprecision works, might answer your objection.
The idea is:
Suppose two actions have precise EVs. Youāll presumably grant that a tiny change in the (expected) location of electrons can flip the difference in EV from +epsilon to -epsilon.
If so, then a tiny change in the (expected) location of electrons can flip the lower bound of an imprecise difference in EV from +epsilon to -epsilon.
What makes two actions incomparable, under the imprecise EV model, is that the interval of EV differences crosses zero.
So, itās unsurprising that a tiny change in the (expected) location of electrons can flip the two actions from ācomparableā to āincomparableā.
Can you say which step in this argument you reject, and why?
What makes two actions incomparable, under the imprecise EV model, is that the interval of EV differences crosses zero.
Imagine 2 states of the world which are exactly the same, and have an imprecice expected welfare of ā1 to 1. The difference between their imprecise expected welfare is ā2 (= ā1 ā 1) to 2 (1 - (-1)), which crosses 0. So their expected welfare would be incomparable under your framework? I would say their expected welfare would be comparable, and exactly the same.
The intervals are supposed to represent imprecise expected value in the way you define it, which allows for the 1st case you described above leading to āA and B are incomparableā? In my mind, if 2 states of the world are exactly the same, they should be comparable, and exactly as valuable no matter what.
Ah I missed the ā2 states of the world which are exactly the sameā part, sorry. Then yeah the EVs would be the same. Iām not sure how this is supposed to support your original commentās argument though.
What makes two actions incomparable, under the imprecise EV model, is that the interval of EV differences crosses zero.
What exactly do you mean by āinterval of EV differencesā? Imagine A = [a1, a2], and B = [b1, b2] are intervals representing the imprecise expected welfare of 2 states of the world, and that b2 >= a2. What would be the āinterval of EV differencesā between B and A in terms of a1, a2, b1, and b2? I thought it would be BāA = [b1 - a2, b2 - a1].
I see. I agree an infinitesimal change to one of 2 exactly identical states could make their expected welfare incomparable under your framework. However, it does not follow that any 2 interventions are incomparable with respect to how much they change expected welfare (across all space and time). I think intervals representing the expected change in welfare are sufficiently narrow for any decision-relevant comparisons to be feasible, although very often with lots of (standard) uncertainty involved.
Hi Vasco. I think Figure 3 here, and the surrounding discussion of how imprecision works, might answer your objection.
The idea is:
Suppose two actions have precise EVs. Youāll presumably grant that a tiny change in the (expected) location of electrons can flip the difference in EV from +epsilon to -epsilon.
If so, then a tiny change in the (expected) location of electrons can flip the lower bound of an imprecise difference in EV from +epsilon to -epsilon.
What makes two actions incomparable, under the imprecise EV model, is that the interval of EV differences crosses zero.
So, itās unsurprising that a tiny change in the (expected) location of electrons can flip the two actions from ācomparableā to āincomparableā.
Can you say which step in this argument you reject, and why?
Imagine 2 states of the world which are exactly the same, and have an imprecice expected welfare of ā1 to 1. The difference between their imprecise expected welfare is ā2 (= ā1 ā 1) to 2 (1 - (-1)), which crosses 0. So their expected welfare would be incomparable under your framework? I would say their expected welfare would be comparable, and exactly the same.
Depends on the details of what the intervals are supposed to represent. E.g.:
Say you have a representor (imprecise probabilities) where EV_P(A) = EV_P(B) = [-1, 1].
On one hand:
If:
for p1 in P, EV_p1(A) = ā1 while EV_p1(B) = 1, and
for p2 in P, EV_p2(A) = 1 while EV_p2(B) = ā1,
then A and B are incomparable.
OTOH:
If for all p in P, EV_p(A) = EV_p(B), then A and B are comparable.
(Ofc there are lots of other cases.)
The intervals are supposed to represent imprecise expected value in the way you define it, which allows for the 1st case you described above leading to āA and B are incomparableā? In my mind, if 2 states of the world are exactly the same, they should be comparable, and exactly as valuable no matter what.
Ah I missed the ā2 states of the world which are exactly the sameā part, sorry. Then yeah the EVs would be the same. Iām not sure how this is supposed to support your original commentās argument though.
What exactly do you mean by āinterval of EV differencesā? Imagine A = [a1, a2], and B = [b1, b2] are intervals representing the imprecise expected welfare of 2 states of the world, and that b2 >= a2. What would be the āinterval of EV differencesā between B and A in terms of a1, a2, b1, and b2? I thought it would be BāA = [b1 - a2, b2 - a1].
Given that the intervals are both derived from a representor P, the interval of EV diffs is {EV_p(A) - EV_p(B) | p in P}. See also here.
I see. I agree an infinitesimal change to one of 2 exactly identical states could make their expected welfare incomparable under your framework. However, it does not follow that any 2 interventions are incomparable with respect to how much they change expected welfare (across all space and time). I think intervals representing the expected change in welfare are sufficiently narrow for any decision-relevant comparisons to be feasible, although very often with lots of (standard) uncertainty involved.
Iām still not sure I understand why you find the arguments in the linked post, and post #3 of the sequence, uncompelling. Can you say more on that?
Do you have more thoughts on this comment? Feel free to follow up there.