Deriving Utilitarianism from First Principles

(revised May 14, May 16, June 2, in light of the objection that the argument doesn't have several people's small gains justifying one person's big loss; that characteristics shouldn't matter)

I heard Professor Terence Irwin talk on 'Prudence, morality, and the importance of persons: a dilemma for Sidgwick' yesterday. He said that Sidgwick does a poor job of moving from his two axioms to utilitarianism, which is correct. Even the axioms aren't spelled out very clearly, it seems. Here's a fix-up.

Axiom A1. Pareto Improvements Are Good. If you can make one person better off without hurting anybody else, do it.

Axiom A2. Impartiality. Whether a change in welfare is good or bad shouldn't depend on the identity of the particular person affected or any personal characteristics. more precisely, whether an action that changes welfare by amount A affects person i instead of person j does not affect the action's moral goodness.

Result R1. By A1, if Jones can take an action that increases his welfare by 800 utils, he should do it.

Result R2. Suppose Jones can either do nothing or take the trio of actions T1:
Action X reduces Jones's welfare by 2000 utils.
Action Y1 increases Jones's welfare by 700 utils.
Action Z1 increases Jones's welfare by 500 utils.

By R1, Jones should take the trio of actions T1.

Result R3. Suppose Jones can either do nothing or take the trio of actions T2:
Action X reduces Jones's welfare by 2000 utils.
Action Y2 increases Smith's welfare by 700 utils.
Action Z2 increases Lee's welfare by 500 utils.

By A2 and R2, Jones should take this trio of actions T2.

Result R4. R3 would remain true for any trio of numbers (a,b,c) such that a is less than b+c. Thus, we have utilitarianism.

A possible flaw: Trio T1 has the same identity label for both actions, whereas Trio T2 has a different identity label for each action. Does A2 really require them to be treated in the same way?

Axiom 2 is different from saying that welfare pairs (2,3) and (3,2) are equivalent, and stronger. Even if (2,3) and (3,2) are equivalent, that does not imply that (3,3) and (2,4) are equivalent. Using Axiom 2, though, if start by saying (2,3) and (3,2) are equivalent, then the actions of "give 1 to person 1" and "give 1 to person 2" are equivalent, so we do get the implication that (3,3) and (2,4) are equivalent. Probably we can derive that (x,y) and (y,x) are equivalent too, from Axiom 2, though I don't see how immediately.

Now that I think about it, Axiom 2 is not so different from the contractarian axiom that if a person is willing to accept a gamble, then he should not complain if he is the loser. A contractarian introduces probability, though, and so needs expected utility perhaps-- or at least some comment on what happens to non-expected-utility maximizers.