I have $\Pi_n:\mathbb R^{n+1}\rightarrow \mathbb R$ and $F_n:\mathbb R^2\rightarrow \mathbb R$ with $$F_n(x,a)=\Pi_n(x,...,x,a)$$ $$f_n(x)=\operatorname{ArgMax}_{a\in\mathbb R}\{F_n(x,a)\} $$ such that $\Pi_n$ is a twice differentiable concave function. Furthermore I have the following condition C:
For all $n\in \mathbb N$, there exists $k_n>1$ such that $$|f_n(x)-f_n(y)|>k_n|x-y|$$ with $k_n\in O(n)$, for all $x,y\in\mathbb R^n$ with $x=(x,...,x)$ and $y=(y,...,y)$.
My question is the following.
Is it true that $\operatorname{ArgMax}_{a\in\mathbb R}\{p^nF_n(u,a)+(1-p)^nF_n(f_n(u),a)\}\in O(n^2)$ for any $p\in(0,1)$ ? If not, what sufficiency condition on $\Pi_n$ would induce this?