What about optimisation power of x′ as a measure of outcome that have utility greater than the utility of x′?

Let Ux′ be the set of outcome with utility greater than x′ according to utility function u:

Ux′:={x∈X|u(x)≥u(x′)}

The set Ux′ is invariant under translation and non-zero rescaling of the utility function u and we define the optimisation power of the outcome x′ according to utility function u as:

OPu(x′):=−log(∫Ux′p)=−log(∫x∈Ux′p(x)dx)

This does not suffer from comparing w.r.t a worst case and seem to satisfies the same intuition as the original OP definition while referring to some utility function.

This is in fact the same measure as the original optimisation power measure with an order given by the utility function

Right, I got confused because I thought your problem was about trying to define a measure of optimisation power—for ex analogous to the Yudkowsky measure—that was also referring to a utility function, while being invariant from scaling and translation but this is different from asking

“what fraction of the default expected utility comes from outcomes at least as good as this one?’”

What about optimisation power of x′ as a measure of outcome that have utility greater than the utility of x′?

Let Ux′ be the set of outcome with utility greater than x′ according to utility function u:

Ux′:={x∈X|u(x)≥u(x′)}The set Ux′ is invariant under translation and non-zero rescaling of the utility function u and we define the optimisation power of the outcome x′ according to utility function u as:

OPu(x′):=−log(∫Ux′p)=−log(∫x∈Ux′p(x)dx)This does not suffer from comparing w.r.t a worst case and seem to satisfies the same intuition as the original OP definition while referring to some utility function.

This is in fact the same measure as the original optimisation power measure with an order given by the utility function

Yeah this is the expectation of the Yudkowsky measure I think?

Right, I got confused because I thought your problem was about trying to define a measure of optimisation power—for ex analogous to the Yudkowsky measure—that was also referring to a utility function, while being invariant from scaling and translation but this is different from asking

“what fraction of the default expected utility comes from outcomes at least as good as this one?’”