Stuart_Armstrong comments on Example population ethics: ordered discounted utility

[Stuart_Armstrong]

a can be prioritized over b just by the ordering, even though they have identical utility.

Nope. Their ordering is only arbitrary as long as they have exactly the same utility. As soon as a policy would result in one of them having higher utility than the other, their ordering is no longer arbitrary. So if we ignore other people $u_a<u_b$ means the term in the sum is $u_a+\gamma u_b$. If $u_a>u_b$, it’s $u_b+\gamma u_a$. If $u_a=u_b$, it can be either term (and they are equal).

(I can explain in more detail if that’s not enough?)

[Evan Clark]

I have realized that I am coming off like I don’t understand algebra, which is a result of my failure to communicate. As unlikely as I am making it sound, I understand what you are saying and already knew it.

What I mean is this:

Despite a = b, it could “look like” a < b or b > a if you didn’t have access to the world but only to the (expanded) sum. If you can ask for the difference between the total sum and the sum ignoring a, but not for the actual value of a.

I can’t think of a non-pathological case where this would actually matter, but it seems like a desirable desideratum that a = b will always “look like” a = b regardless of what kind of (sufficiently fine-grained) information that you have.

EDIT : After reading your above comment about willingness to sacrifice elegance, I kind of wish I hadn’t said anything at all, considering my comments are all in the interest of what I would consider elegance. To be sure, I think elegance is a legitimate practical concern, but I wouldn’t have engaged with you initially had I known your view.

[Stuart_Armstrong]

Hum, not entirely sure what you’re getting at...

I’d say that $u_a=u_b$ always “looks like $u_a=u_b$”, in the sense that there is a continuity in the overall $U\left(W\right)$; small changes to our knowledge of $u_a$ and $u_b$ make small changes to our estimate of $U\left(W\right)$.

I’m not really sure what stronger condition you could want; after all, when $u_a=u_b$, we can always write

• $\dots +{\gamma }^n{u}_z+{\gamma }^{n+1}{u}_a+{\gamma }^{n+2}{u}_b+{\gamma }^{n+3}{u}_c+\dots$

as:

• $\dots +{\gamma }^n{u}_z+\frac{{\gamma }^{n+1}+{\gamma }^{n+2}}{2}(u_a+u_b)+{\gamma }^{n+3}{u}_c+\dots$.

We could equivalently define $U\left(W\right)$ that way, in fact (it generalises to larger sets of equal utilities).

Would that formulation help?

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