  We give conditions for equilibria in the following Voronoi game on the
discrete hypercube. Two players position themselves in $\{0,1\}^d$ and each
receives payoff equal to the measure (under some probability distribution) of
their Voronoi cell (the set of all points which are closer to them than to the
other player). This game can be thought of as a discrete analogue of the
Hotelling--Downs spatial voting model in which the political spectrum is
determined by $d$ binary issues rather than a continuous interval.
  We observe that if an equilibrium does exist then it must involve the two
players co-locating at the majority point (ie the point representing majority
opinion on each separate issue). Our main result is that a sufficient condition
for an equilibrium is that on each issue the majority option is held by at
least $\frac{3}{4}$ of voters. The value $\frac{3}{4}$ can be improved slightly
in a way that depends on $d$ and with this improvement the result is best
possible. We give similar sufficient conditions for the existence of a local
equilibrium.
  We also analyse the situation where the distribution is a mix of two product
measures. We show that either there is an equilibrium or the best response to
the majority point is its antipode.
