Let $(\Omega,\mathcal{F},P)$ be a probability space and $X:\Omega \to \mathcal{X}\subseteq \mathbb{R}^n$ ($n\in\mathbb{N}$) be a random variable with cdf $F$. Suppose that an agent chooses $a\in\mathcal{A}\subseteq \mathbb{R}^n$ to maximize $\mathbb{E}_X [u(a,X)],$ and is permitted to randomize over $a$.
Question: are there any non-trivial cases where randomizing over $a$ is optimal and yield a strictly higher payoff than any deterministic choice of $a$?
Stating this more (but not completely) precisely: Let $\mathcal{G}$ denote the set of non-degenerate probability distributions over $\mathcal{A}$. Is it ever possible that
$$ \max_{G\in\mathcal{G}} \mathbb{E}_{a\sim G}\big[ \mathbb{E}_{X\sim F}[u(a,X)] \big] > \max_{a\in\mathcal{A}} \mathbb{E}_X [u(a,X)] ?$$
Assuming you want $a$ and $X$ to be independent, no, this is not possible. Let $\mu$ be any probability measure over $\mathcal A$. Then \begin{align*} \mathbb{E}_{A \sim \mu} [\mathbb{E}[u(A,X)]] &= \int_{\mathcal A} \mathbb{E}[u(A,X)] d\mu(A) \\ &\le \int_{\mathcal A} \left(\max_{a \in \mathcal A}\mathbb{E}[u(a,X)]\right) d\mu(A) \\ &= \max_{a \in \mathcal A}\mathbb{E}[u(a,X)]. \end{align*} Since $\mu$ was arbitrary, we conclude $\max_{\mu \in \mathcal G}\mathbb{E}_{A \sim \mu} [\mathbb{E}[u(A,X)]] \le \max_{a \in \mathcal A}\mathbb{E}[u(a,X)]$.