As you’ve stated the view, I think it would violate transitivity. Consider the following three populations, where each position in the vector denotes the wellbeing of a specific person, and a dash represents the case where that person does not exist: A: (2, 1) A’: (1, 2) B: (2, -)
A is better than (or perhaps equally as good as?) B, because we match the first person with themselves, then settle by comparing the second person at 1 (in A) to non-existence in B. You didn’t say how exactly to do this, but I assume A is supposed to be at least as good as B, since that’s what you wanted to say (and I guess you mean to say that it’s better).
However, A and A’ are equally good.
Transitivity would entail that A’ is therefore at least as good as B, but on the procedure you described, A is worse than B because we compare them first according to wellbeing levels for those who exist in both, and the first person exists in both and is better off in B.
I don’t doubt that the view can be modified to solve this problem, but it’s common in population ethics that solving one problem creates another.
I probably won’t reply further, by the way—just because I don’t go on EA forums much. Best of luck.
Sorry, perhaps I wasn’t clear: I didn’t mean matching by the identity of the individual, I meant matching on just their utility values (doesn’t matter who is happy/suffering, only the unordered collection of utility values matters). So in your example, A and A’ would be identical worlds (modulo ethical preference).
Formally: Let a,b:UI→N be multisets of utilities (world states). (Notice that I’m using multisets and not vectors on purpose to indicate that the identities of the individuals don’t matter.) To compare them, define the multiset a∩b as (a∩b)(u)=min(a(u),b(u)), and define a′=a−a∩b and b′=b−a∩b (pointwise). Then we compare a′ and b′ with leximin.
However, this still isn’t transitive, unfortunately. E.g: A: {{2}} B: {{1, 3, 3}} C: {{3}} Then A≳B and B≳C but A≳/C .
Right now I think the best solution is use plain leximin (as defined in my post) and reject the Mere Addition Principle.
As you’ve stated the view, I think it would violate transitivity. Consider the following three populations, where each position in the vector denotes the wellbeing of a specific person, and a dash represents the case where that person does not exist:
A: (2, 1)
A’: (1, 2)
B: (2, -)
A is better than (or perhaps equally as good as?) B, because we match the first person with themselves, then settle by comparing the second person at 1 (in A) to non-existence in B. You didn’t say how exactly to do this, but I assume A is supposed to be at least as good as B, since that’s what you wanted to say (and I guess you mean to say that it’s better).
However, A and A’ are equally good.
Transitivity would entail that A’ is therefore at least as good as B, but on the procedure you described, A is worse than B because we compare them first according to wellbeing levels for those who exist in both, and the first person exists in both and is better off in B.
I don’t doubt that the view can be modified to solve this problem, but it’s common in population ethics that solving one problem creates another.
I probably won’t reply further, by the way—just because I don’t go on EA forums much. Best of luck.
Sorry, perhaps I wasn’t clear: I didn’t mean matching by the identity of the individual, I meant matching on just their utility values (doesn’t matter who is happy/suffering, only the unordered collection of utility values matters). So in your example, A and A’ would be identical worlds (modulo ethical preference).
Formally: Let a,b:UI→N be multisets of utilities (world states). (Notice that I’m using multisets and not vectors on purpose to indicate that the identities of the individuals don’t matter.) To compare them, define the multiset a∩b as (a∩b)(u)=min(a(u),b(u)), and define a′=a−a∩b and b′=b−a∩b (pointwise). Then we compare a′ and b′ with leximin.
However, this still isn’t transitive, unfortunately. E.g:
A: {{2}}
B: {{1, 3, 3}}
C: {{3}}
Then A≳B and B≳C but A≳/C .
Right now I think the best solution is use plain leximin (as defined in my post) and reject the Mere Addition Principle.