Wei Dai comments on Best utility normalisation method to date?

[Wei Dai]

Stuart, what’s your view on the problem I described in Is the potential astronomical waste in our universe too small to care about? Translated to this setting, the problem is that if you do a normalisation when you’re uncertain about the size of the universe (i.e., $E_{{\pi }_{rd}}$ is computed under this uncertainty), and then later find out the actual size of the universe (or just gets some information that shifts your expectation of the size of the universe or of how many lives or observer-moments it can support), you’ll end up putting almost all of your efforts into Total Utilitarianism (if the shift is towards the universe being bigger) or almost none of your efforts into it (if the shift is in the opposite direction).

[Stuart_Armstrong]

Hum… It seems that we can stratify here. Let $X$ represent the values of a collection of variables that we are uncertain about, and that we are stratifying on.

When we compute the normalising factor for utility $U$ under two policies $\pi$ and ${\pi }^{\prime }$, we normally do it as:

• $U\to U/N_U$, with $N_U={\sum }_{x}P(X=x)(E_{\pi ,X=x}U-E_{{\pi }^{\prime },X=x}U)$.

And then we replace $U$ with $U/N_U$.

Instead we might normalise the utility $U$ separately for each value of $x$:

• Conditional on $X=x$, then $U\to U/N_{U,x}$, with $N_{U,x}=E_{\pi ,X=x}U-E_{{\pi }^{\prime },X=x}U$.

The problem is that, since we’re dividing by the $N$, the expectation of $U/N_{U,x}$ is not the same $U/N_U$.

Is there an obvious improvement on this?

Note that here, total utilitarianism get less weight in large universes, and more in small ones.

I’ll think more...