How not to value wagers

Given the Axiom of Choice, there is a rotationally invariant finitely additive probability measure defined for all subsets of a circle. We can use such a finitely probability measure to define an expected value Ef or integral of a bounded function f on the circle, and we might want to have a decision theory based on this expected value. Given a wager that pays f(z) at a uniformly randomly chosen location z on the circle, we are indifferent to buying the wager at price Ef, we must accept the wager at lower prices, and we must reject it at higher prices.

This procedure, however, leads to the following interesting thing: There will be bounded wagers that pay more than y no matter what, but where one is indifferent with respect to buying the wager at price y. To see this, let x be an irrational number, and as in my previous post, let u be a bounded function on the circle such that u(ρz) > u(z) for all z where ρ is rotation by x degrees. Then let f(z) = u(ρz) − u(z). Because of the additivity of integrals with respect to finitely additive measures and rotational invariance, we have Ef = ∫f(ρz)dP(z) − ∫f(z)dP(z) = 0. But f(z) > 0 for all z. So the decision theory tells us to be indifferent to the game where you get payoff f(z) at z when the game is offered for free, even though no matter what the outcome of the game, you will received a strictly positive amount.

More generally, given the Axiom of Choice, there is no finitely-additive rotationally-invariant expected value assignment for bounded utilities that respects the principle that any gamble that is sure to pay more than y ought to be accepted at price y.

Integration

It sure seems that:

1. A good human life is an integrated human life.

But suppose we have a completely non-religious view. Wouldn't it be plausible to think that there is a plurality of incommensurable human goods and the good life encompasses a variety of them, but they do not integrate into a unified whole? There is friendship, professional achievement, family, knowledge, justice, etc. Each of these constitutively contributes to a good human life. But why would we expect that there be a single narrative that they should all integrally fit into? The historical Aristotle, of course, did have a highest end, the contemplation of the gods, available in his story, and that provides some integration. But that's religion (though natural religion: he had arguments for the gods' existence and nature).

Nathan Cartagena pointed out to me that one might try to give a secular justification for (1) on empirical grounds: people whose lives are fragmented tend not to do well. I guess this might suggest that if there is no narrative that fits the various human goods into a single story, then one should make one, say by expressly centering one's life on a personally chosen pattern of life. But I think this is unsatisfactory. For I think that the norms that are created by our own choices for ourselves do not bear much weight. They are not much beyond hobbies, and hobbies do not bear much of the meaning of human life.

So all in all, I think the intuition behind (1) requires something like a religious view of life.