Let $X$ and $Y$ be random variables, and $f(y, a):R^2\rightarrow R$ be a linear function that is increasing in both arguments.
Also assume that at any $a$ the following holds
$X - f(Y,a) + E'[X|a] = 0$
Taking $E[\cdot|a]$ on both sides of above equality yields
$E[X|a] - E[f(Y,a)|a] + E'[X|a] = 0$
Main Question: Under what additional restrictions on the joint distribution of $X$ and $Y$ and the functional form of $f$ is
$E[X|a] := E[X|X - f(Y,a) + E'[X|a] = 0]$
increasing in $a$?
If $X$ and $Y$ are independent, then I believe $E[X|a]$ should be increasing. However, I have trouble proving this, or even figuring out what concepts or terms to look for. So any help is appreciated!
Side Question: What terms or concepts would I look for, if I wanted to tackle this problem?
Context: $(X, Y)$ represent types of decision makers. I know that given their type $(X, Y)$ a decision maker will choose $a$ such that $X - f(Y,a) + E'[X|a] = 0$, where the decision maker takes $E'[X|a]$ as given. The expectations are taken over different types.