value systems = local maxima of f, but only those higher than global average E f(x)

No. value systems = local maxima of f, period.

You said "if agents are better than random" in headlines. I take "than random" to mean " > E f(x)". So do you assume so or not?

What? Can't parse.

Let me try again.

• A - set of all local maxima x
• B - set of all local maxima x, for which f(x) > E f(x)

In either case, avg(A) and avg(B) are arbitrary points, and we have no a priori reason to believe they will be special in any way, so E f(avg(A)) = E f(avg(B)) = E f(x). Is this right?

In case of set B - we assumed for all x \in B . f(x) > E f(x), so picking avg(B) which is essentially a random point makes it worse.

In case of set A - E f(x) for set of arbitrary local maxima is no better than set of arbitrary points, so E(f(x) | x \in A) = E(f(x)) = E f(avg(a)), so your entire argument fails.