Let us consider an optimization problem over $[0,1]$. That is, we are given two continuous functions $$ f,g:[0,1]^2\to \Bbb R $$ such that
- $f(x,y)$ is non-decreasing in $x$ and non-increasing in $y$;
- $g(x,y)$ is non-increasing in $x$ and non-decreasing in $y$.
Denote $x^*(y):=\max\left( \operatorname{Argmax}\limits_{x\in [0,1]}f(x,y)\cdot g(x,y)\right)$. What are sufficient conditions for $x^*$ to be non-decreasing?
In economics the lattice approach is becoming popular. One sufficient condition for the maximizer to be nondecreasing in the parameters is that the objective function to be maximized is supermodular in $x$ and has increasing differences in $x,y$. See Theorem 5 (and others) in here. The techniques they describe are due to Topkis.