  Let $\Xi$ be a set of centers chosen according to a Poisson point process in
$\mathbb R^d$. Consider the allocation of $\mathbb R^d$ to $\Xi$ which is
stable in the sense of the Gale-Shapley marriage problem, with the additional
feature that every center $\xi\in\Xi$ has a random appetite $\alpha V$, where
$\alpha$ is a nonnegative scale constant and $V$ is a nonnegative random
variable. Generalizing previous results by Freire, Popov and Vachkovskaia
(\cite{FPV}), we show the absence of percolation when $\alpha$ is small enough,
depending on certain characteristics of the moment of $V$.
