Suppose $f(X,v_i), \textrm{ for } i=1,2$ are two probability distributions of the same parametric family. Moreover, I have a function L such that:
$$\sum_{x} I_{\{L(X)\leq \bar{\gamma}\}} f(X,v_1) \geq \sum_{x} I_{\{L(X)\leq \bar{\gamma}\}} f(X,v_2).$$
Alternatively, $P(L(X)\leq \bar{\gamma}|v_1) \geq P(L(X)\leq \bar{\gamma}|v_2)$, for some fixed value $\bar{\gamma}$. Under which conditions, $\rho^{th}$ percentile of the first distribution will be smaller than the $\rho^{th}$ percentile of the second distribution? In my viewpoint, if the dispersion of the first distribution is smaller than the second one, then, the claim will always hold true for any $\rho$. However, I can not show it formally.