Un-preferable investment within FSD set

Question

Given 3 different investments F, G and H, all of them are part of the FSD set (neither of them is superior FSD on the other). Is there any investment H so no investor with utility function U'>0 would prefer it?

Answer 1

There are plenty. Consider $M(x) = \max \{ F(x), G(x), H(x) \}$. You can show that $M(x)$ is also a CDF and that $M(x)$ is weakly FSD-dominated by any of $F, G, H$. To play safe, shift $M(x)$ leftwards by an amount $a > 0$ and consider $M_a(x) = M(x+a)$. Now $M_a(x)$ is strictly FSD-dominated by any of $F, G, H$ for any $a>0$.