Suppose I define a function $f(A,B,C,D)$ as the probability of $\frac{A}{B}$ being greater than $\frac{C}{D}$. Thus $f(A,B,C,D) = Pr \left(\frac{A}{B}>\frac{C}{D}\right)$. The distributions of $A,B,C,D$ are unknown, we only know that $A,B,C,D \geq 1$.
My intuition is that $f$ is increasing in $A, D$ and decreasing in $B,C$, or $f_A>0$, $f_D>0$ etc. Is it possible to prove this if the distributions are unknown, and if so, how?