Superadditivity of quantile quotients

Hey all,

I've been searching for some knowledge on ordering of sums of quantiles. I've found some requirements such that, for RVs X, Y:

C_q(X + Y) > Cq(X) + C_q(Y)

However, I need to find out for what RVs X, Y, Z the following holds:

C_q(X + Y + Z)/(C_q(X) + C_q(Y) + C_q(Z)) > C_q(X + Y)/(C_q(X) + C_q(Y))

I need this result for a game theoretic problem with quantile utility functions. Anyone has any idea for what RVs this would hold?


Any help is appreciated. Thanks!