I want to point out that both assumptions 2, and 1 and 3 together have been objected to by academic philosophers.
Assumption 2 is ex post consequentialism: maximize the expected value of a social welfare function. Ex ante prioriatarianism/egalitarianism means rejecting 2: we should be fair to individuals with respect to their expected utilities, even if this means overall worse expected outcomes. This is, of course, vNM irrational, but Diamond defended it (and see my other comment here). Essentially, even if two outcomes are equally valuable, a probabilistic mixture of them can be more valuable because it gives people fairer chances; this is equality of opportunity. This contradicts the independence axiom specifically for vNM rationality (and so does the Allais paradox).
Assumptions 1 and 3 together are basically a weaker version of ex ante Pareto, according to which it’s (also) better to increase the expected utility of any individual(s) if it comes at no expected cost to any other individuals. Ex post prioritarianism/egalitarianism means rejecting the conjunction of 1 and 3, and ex ante Pareto: we should be more fair to individuals ex post (we want more fair actual outcomes after they’re determined), even if this means worse individual expected outcomes.
When Rawls and Harsanyi appeal to their versions of Veil of Ignorance Contractualism, they claim that the Equal Chance Formula supports the Utilitarian Average Principle, which requires us to act in ways that would maximize average utility, by producing the greatest sum of expectable benefits per person. This is the principle whose choice would be rational, in self-interested terms, for people who have equal chances of being in anyone’s position.
We can plausibly reject this argument, because we can reject this version of contractualism. As Rawls points out, Utilitarianism is, roughly, self-interested rationality plus impartiality. If we appeal to the choices that would be rational, in self-interested terms, if we were behind some veil of ignorance that made us impartial, we would expect to reach conclusions that are, or are close to being, Utilitarian. But this argument cannot do much to support Utilitarianism, because this argument’s premises are too close to these conclusions. Suppose that I act in a way that imposes some great burden on you, because this act would give small benefits to many other people who are much better off than you. If you object to my act, I might appeal to the Equal Chance Formula. I might claim that, if you had equal chances of being in anyone’s position, you could have rationally chosen that everyone follows the Utilitarian Principle, because this choice would have maximized your expectable benefits. As Scanlon and others argue, this would not be a good enough reply.9 You could object that, when we ask whether some act would be wrong, we are not asking a question about rational self-interested choice behind a veil of ignorance. Acts can be wrong in other ways, and for other reasons.
He claimed that we can reject ex ante Pareto (“Probabilistic Principle of Personal Good”), in favour of ex post prioritarianism/egalitarianism:
Even if one of two possible acts would be expectably worse for people, this act may actually be better for these people. We may also know that this act would be better for these people if they are worse off. This fact may be enough to make this act what we ought to do.
Here, by “worse off” in the second sentence, he meant in a prioritarian/egalitarian way. The act is actually better for them, because the worse off people under this act are better off than the worse off people under the other act. He continued:
We can now add that, like the Equal Chance Version of Veil of Ignorance Contractualism, this Probabilistic Principle has a built-in bias towards Utilitarian conclusions, and can therefore be rejected in similar ways. According to Prioritarians, we have reasons to benefit people which are stronger the worse off these people are. According to Egalitarians, we have reasons to reduce rather than increase inequality between people. The Probabilistic Principle assumes that we have no such reasons. If we appeal to what would be expectably better for people, that is like appealing to the choices that it would be rational for people to make, for self-interested reasons, if they had equal chances of being in anyone’s position. Since this principle appeals only to self-interested or prudential reasons, it ignores the possibility that we may have impartial reasons, such as reasons to reduce inequality, or reasons to benefit people which are stronger the worse off these people are. We can object that we do have such reasons.
When Rabinowicz pointed out that, in cases like Four, Prioritarians must reject the Probabilistic Principle of Personal Good, he did not regard this fact as counting against the Priority View. That, I believe, was the right response. Rabinowicz could have added that similar claims apply to Egalitarians, and to cases like Two and Three.
I want to point out that both assumptions 2, and 1 and 3 together have been objected to by academic philosophers.
Assumption 2 is ex post consequentialism: maximize the expected value of a social welfare function. Ex ante prioriatarianism/egalitarianism means rejecting 2: we should be fair to individuals with respect to their expected utilities, even if this means overall worse expected outcomes. This is, of course, vNM irrational, but Diamond defended it (and see my other comment here). Essentially, even if two outcomes are equally valuable, a probabilistic mixture of them can be more valuable because it gives people fairer chances; this is equality of opportunity. This contradicts the independence axiom specifically for vNM rationality (and so does the Allais paradox).
Assumptions 1 and 3 together are basically a weaker version of ex ante Pareto, according to which it’s (also) better to increase the expected utility of any individual(s) if it comes at no expected cost to any other individuals. Ex post prioritarianism/egalitarianism means rejecting the conjunction of 1 and 3, and ex ante Pareto: we should be more fair to individuals ex post (we want more fair actual outcomes after they’re determined), even if this means worse individual expected outcomes.
There was a whole issue of Utilitas devoted to prioritarianism and egalitarianism in 2012, and, notably, Parfit defended prioritarianism in it, arguing against ex ante Pareto (and hence the conjunction of 1 and 3):
He claimed that we can reject ex ante Pareto (“Probabilistic Principle of Personal Good”), in favour of ex post prioritarianism/egalitarianism:
Here, by “worse off” in the second sentence, he meant in a prioritarian/egalitarian way. The act is actually better for them, because the worse off people under this act are better off than the worse off people under the other act. He continued: