Harsanyi’s theorem doesn’t start from any axiom I would call “separability”. See this post for non-technical summary. It also doesn’t imply separability in different number cases. For example, average utilitarianism is consistent with Harsanyi’s theorem, but the average welfare level of unaffected individuals matters when choosing between options with different numbers of individuals. Under average utilitarianism, it’s good to create an individual with higher than the average welfare without them and bad to create individuals with lower than the average welfare without them, assuming no one else is affected, but the average welfare without them is exactly the average welfare of the unaffected.
One axiom (quoting from the summary post) seems close to separability:
If every individual in the group is indifferent between two options, then the group as a whole is indifferent between those two options
This axiom rules out preferences for and against ex post equality, which is often non-separable. In particular, it means you have to be indifferent between these two options, which should conflict with your intuition about a backup planet:
Everyone on Earth 1 and Earth 2 dies, or both planets get utopia, each with probability 50%.
Everyone on Earth 1 dies and Earth 2 gets utopia, or everyone on Earth 2 dies and Earth 1 gets utopia, each with probability 50%.
Prioritarianism, just ∑ni=1f(ui) (with the ui matching the utilities from the vNM theorem for each individual), satisfies separability but not this other axiom. Average utilitarianism and other functions of the form an×∑ni=1ui satisfy this other axiom (in fixed population cases) but not separability in different population size cases.
Harsanyi’s theorem doesn’t start from any axiom I would call “separability”. See this post for non-technical summary. It also doesn’t imply separability in different number cases. For example, average utilitarianism is consistent with Harsanyi’s theorem, but the average welfare level of unaffected individuals matters when choosing between options with different numbers of individuals. Under average utilitarianism, it’s good to create an individual with higher than the average welfare without them and bad to create individuals with lower than the average welfare without them, assuming no one else is affected, but the average welfare without them is exactly the average welfare of the unaffected.
One axiom (quoting from the summary post) seems close to separability:
This axiom rules out preferences for and against ex post equality, which is often non-separable. In particular, it means you have to be indifferent between these two options, which should conflict with your intuition about a backup planet:
Everyone on Earth 1 and Earth 2 dies, or both planets get utopia, each with probability 50%.
Everyone on Earth 1 dies and Earth 2 gets utopia, or everyone on Earth 2 dies and Earth 1 gets utopia, each with probability 50%.
Prioritarianism, just ∑ni=1f(ui) (with the ui matching the utilities from the vNM theorem for each individual), satisfies separability but not this other axiom. Average utilitarianism and other functions of the form an×∑ni=1ui satisfy this other axiom (in fixed population cases) but not separability in different population size cases.
Thanks Michael, seems that I was just a bit confused here.