add an extra welfare state Ω to represent nonexistence, without saying how it compares to other welfare states at all (e.g. totalism or person-affecting views). Prospects can include nonexistence, so you (may be able to) compare prospects with different probabilities of nonexistence.
replace the finite constant population I with an infinite set I∞ of all possible individuals and assign welfare Ω (nonexistence) to individuals who don’t exist in a given welfare distribution.
generalize the Anteriority, Reduction to Prospects and Two-Stage Anonymity conditions. Only Reduction to Prospects looks different, since rather than defining lotteries for everyone in I∞ as a whole, you require it to hold for every finite non-empty subset of I∞.
define Omega Independence.
generalize Theorem 2.2 for Theorem 3.5.
For a given welfare state w, let 1w denote the prospect with definite welfare state w, with probability 1. In particular, 1Ω denotes definite nonexistence.
Omega Independence: For any two prospects P and P′, and any rational probability p∈[0,1]∩Q,
P≿P′ if and only if pP+(1−p)1Ω≿pP′+(1−p)1Ω
In other words, mixing with the same chance of nonexistence makes no difference to the ranking of two prospects.
Then Theorem 3.5 is basically the same as Theorem 2.2, with the corresponding definitions, but the social preorder only exists at all if Omega Independence is satisfied and the veil of ignorance comparisons are applied only to pairs of lotteries from a common finite subset of I∞ (which may have any individuals assigned nonexistence Ω, and any two finite sets can be expanded to their union, so prospects over finite sets of individuals can always be compared):
Theorem 3.5: Given an arbitrary individual preorder, there is at most one social preorder satisfying Anteriority, Reduction to Prospects, and Two-Stage Anonymity. When it exists, it is given by L≿L′ if and only if
∑i∈I1#IPi(L)≿∑i∈I1#IPi(L′)
according to the individual preorder for any finite non-empty I⊂I∞ such that L and L′ are lotteries in I. The social preorder exists if and only if the individual preorder satisfies Omega Independence.
Personally, I like the procreation asymmetry, so I might say that Ω is strictly better than some states (hence defined as negative), but either incomparable to or at least as good as all other states, so never worse than any other state.
For the variable population case, they
add an extra welfare state Ω to represent nonexistence, without saying how it compares to other welfare states at all (e.g. totalism or person-affecting views). Prospects can include nonexistence, so you (may be able to) compare prospects with different probabilities of nonexistence.
replace the finite constant population I with an infinite set I∞ of all possible individuals and assign welfare Ω (nonexistence) to individuals who don’t exist in a given welfare distribution.
generalize the Anteriority, Reduction to Prospects and Two-Stage Anonymity conditions. Only Reduction to Prospects looks different, since rather than defining lotteries for everyone in I∞ as a whole, you require it to hold for every finite non-empty subset of I∞.
define Omega Independence.
generalize Theorem 2.2 for Theorem 3.5.
For a given welfare state w, let 1w denote the prospect with definite welfare state w, with probability 1. In particular, 1Ω denotes definite nonexistence.
Omega Independence: For any two prospects P and P′, and any rational probability p∈[0,1]∩Q,
In other words, mixing with the same chance of nonexistence makes no difference to the ranking of two prospects.
Then Theorem 3.5 is basically the same as Theorem 2.2, with the corresponding definitions, but the social preorder only exists at all if Omega Independence is satisfied and the veil of ignorance comparisons are applied only to pairs of lotteries from a common finite subset of I∞ (which may have any individuals assigned nonexistence Ω, and any two finite sets can be expanded to their union, so prospects over finite sets of individuals can always be compared):
Theorem 3.5: Given an arbitrary individual preorder, there is at most one social preorder satisfying Anteriority, Reduction to Prospects, and Two-Stage Anonymity. When it exists, it is given by L≿L′ if and only if
according to the individual preorder for any finite non-empty I⊂I∞ such that L and L′ are lotteries in I. The social preorder exists if and only if the individual preorder satisfies Omega Independence.
Personally, I like the procreation asymmetry, so I might say that Ω is strictly better than some states (hence defined as negative), but either incomparable to or at least as good as all other states, so never worse than any other state.