Let $\Xi$ be a discrete set in $\rd$. Call the elements of $\Xi$ {\em centers}. The well-known Voronoi tessellation partitions $\rd$ into polyhedral regions (of varying volumes) by allocating each site of $\rd$ to the closest center. Here we study allocations of $\rd$ to $\Xi$ in which each center attempts to claim a region of equal volume $\alpha$. We focus on the case where $\Xi$ arises from a Poisson process of unit intensity. In an earlier paper by the authors it was proved that there is a unique allocation which is {\em stable} in the sense of the Gale--Shapley marriage problem. We study the distance $X$ from a typical site to its allocated center in the stable allocation. The model exhibits a phase transition in the appetite $\alpha$. In the critical case $\alpha=1$ we prove a power law upper bound on $X$ in dimension $d=1$. (Power law lower bounds were proved earlier for all $d$). In the non-critical cases $\alpha<1$ and $\alpha>1$ we prove exponential upper bounds on $X$.