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.
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.