We consider a function $f(s_{1:p}, a_{1:p})$, where $p>1$ is an integer, $s_{1:p}$ denotes $(s_1,\ldots,s_p)^\top \in R^p$, and $a_{1:p}$ denotes $(a_1,\ldots,a_p)^\top \in R^p$.

Question: What is the necessary condition such that $$a^*_{1:p}(s_{1:p}) \triangleq \arg\max_{a_{1:p}} f(s_{1:p}, a_{1:p})$$ admits the factorization $$a_i^*(s_{1:p}) \equiv a_i^* (s_{n(i)})~\text{for all }i\in\{1, \ldots, p\},$$ where $n(i) \subsetneq \{1, \ldots, p\}$ with $|n(i)|<p$.