Suppose that the function $F(x,y)$ is supermodular in $(x,y)$. We know that this implies that $x^* =\operatorname{arg\, max} F(x,y)$ is increasing in $y$. Under which conditions (if any) do we have that $x^*$ is a convex function of $y$?
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Interesting question. Is there a particular class of $F$ of interest? I have thought about this before but did not get anywhere at this high level of generality. – leslie townes Mar 04 '21 at 15:40
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Only thing we have is that F is concave in x. Did you get any results for a particular class of F? Thank you! – Filomena Garcia Mar 04 '21 at 17:15
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This may no longer be relevant but since the question is still open, here goes.
Let $X\subseteq \mathbb{R}$, and let $x^*(y)=\operatorname{arg\, max}_{x\in X}F(x, y)$. Suppose that $x^*(y)$ is a singleton (i.e., unique maximizer) and $x^*(y)<\sup X$ for all $y\in Y$.
The paper linked below shows that if $\partial F(x, y)/\partial x$ is a quasi-convex, then $x^*(y)$ is a convex function. The paper actually provides a more general result that allows for non-unique maximizers, multidimensional choice variables, and does not require differentiability. There are also no assumptions such as supermodularity imposed on $F$, which means that if you additionally assume supermodularity of $F$ along with quasi-convexity of $\partial F/\partial x$, then your $x^*(y)$ should be convex and increasing in $y$.
tsm
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