3.2: Note that the definition of intergenerational equity in Zame’s paper is what you call finite intergenerational equity (and his definition of an ethical preference relation involves the same difference), so his results are actually more general than what you have here. Also, I don’t think that “almost always we can’t tell which of two populations is better” is an accurate plain-English translation of “{X,Y: neither XY} has outer measure one”, because we don’t know anything about the inner measure. In fact, if the preference relation respects the weak Pareto ordering, then {X,Y: neither XY} has inner measure 0. So an ethical preference relation must be so wildly nonmeasurable that nothing at all can be said about the frequency with which we can’t tell which of two populations is better.
4.1:
Partial translation scale invariance: suppose after some time T, X and Y become the same. Then we can add any arbitrary utility vector A to both X and Y without changing the ordering. (I.e. X > Y iff X+A > Y+A)
X+A and Y+A won’t necessarily be valid utility vectors. I assume you also want to add the condition that they are.
4.3: What does “truncated at time T” mean? All utilities after time T replaced with some default value like 0?
4.5:
Weak sensitivity: for any utility vector, we can modify its first generation somehow to make it better
Since you defined utilities as being in the closed interval [0,1], if you have a utility vector starting with 1, you can’t get anything better just by modifying the first generation, so weak sensitivity should never hold in any sensible preference relation. I’m guessing you mean that we can modify its first generation to make it either better or worse (not necessarily both, unless you switch to open-interval-valued utilities).
4.7: Your definition of dictatorship of the present naively sounded to me like it’s saying “there’s some time T after which changing utilities of generations cannot affect the ordering of any pairs of utility vectors.” But from theorem 4.8, I take it you actually meant “for any pair of utility vectors X and Y such that X<Y, there exists a time T such that changing utilities of generations after T cannot reverse the preference to get X>=Y.”
3.2 good catch – I knew I was gonna mess those up for some paper. I’m not sure how to talk about the measurability result though; any thoughts on how to translate it?
4.3 basically, yeah. It’s easier for me to think about it just as a truncation though
4.5 yes you’re right – updated
4.7 yes, that’s what I mean. Introducing quantifiers seems to make things a lot more complicated though
Some kind of nitpicky comments:
3.2: Note that the definition of intergenerational equity in Zame’s paper is what you call finite intergenerational equity (and his definition of an ethical preference relation involves the same difference), so his results are actually more general than what you have here. Also, I don’t think that “almost always we can’t tell which of two populations is better” is an accurate plain-English translation of “{X,Y: neither XY} has outer measure one”, because we don’t know anything about the inner measure. In fact, if the preference relation respects the weak Pareto ordering, then {X,Y: neither XY} has inner measure 0. So an ethical preference relation must be so wildly nonmeasurable that nothing at all can be said about the frequency with which we can’t tell which of two populations is better.
4.1:
X+A and Y+A won’t necessarily be valid utility vectors. I assume you also want to add the condition that they are.
4.3: What does “truncated at time T” mean? All utilities after time T replaced with some default value like 0?
4.5:
Since you defined utilities as being in the closed interval [0,1], if you have a utility vector starting with 1, you can’t get anything better just by modifying the first generation, so weak sensitivity should never hold in any sensible preference relation. I’m guessing you mean that we can modify its first generation to make it either better or worse (not necessarily both, unless you switch to open-interval-valued utilities).
4.7: Your definition of dictatorship of the present naively sounded to me like it’s saying “there’s some time T after which changing utilities of generations cannot affect the ordering of any pairs of utility vectors.” But from theorem 4.8, I take it you actually meant “for any pair of utility vectors X and Y such that X<Y, there exists a time T such that changing utilities of generations after T cannot reverse the preference to get X>=Y.”
Thanks!
3.2 good catch – I knew I was gonna mess those up for some paper. I’m not sure how to talk about the measurability result though; any thoughts on how to translate it?
4.3 basically, yeah. It’s easier for me to think about it just as a truncation though
4.5 yes you’re right – updated
4.7 yes, that’s what I mean. Introducing quantifiers seems to make things a lot more complicated though
Unfortunately, I can’t think of a nice ordinary-language way of talking about such nonmeasurability results.