  We study competitive location problems in a continuous setting, in which
facilities have to be placed in a rectangular domain $R$ of normalized
dimensions of $1$ and $\rho\geq 1$, and distances are measured according to the
Manhattan metric. We show that the family of 'balanced' facility configurations
(in which the Voronoi cells of individual facilities are equalized with respect
to a number of geometric properties) is considerably richer in this metric than
for Euclidean distances. Our main result considers the 'One-Round Voronoi Game'
with Manhattan distances, in which first player White and then player Black
each place $n$ points in $R$; each player scores the area for which one of its
facilities is closer than the facilities of the opponent. We give a tight
characterization: White has a winning strategy if and only if $\rho\geq n$; for
all other cases, we present a winning strategy for Black.
