  We study an N-player game where a pure action of each player is to select a
non-negative function on a Polish space supporting a finite diffuse measure,
subject to a finite constraint on the integral of the function. This function
is used to define the intensity of a Poisson point process on the Polish space.
The processes are independent over the players, and the value to a player is
the measure of the union of its open Voronoi cells in the superposition point
process. Under randomized strategies, the process of points of a player is thus
a Cox process, and the nature of competition between the players is akin to
that in Hotelling competition games. We characterize when such a game admits
Nash equilibria and prove that when a Nash equilibrium exists, it is unique and
comprised of pure strategies that are proportional in the same proportions as
the total intensities. We give examples of such games where Nash equilibria do
not exist. A better understanding of the criterion for the existence of Nash
equilibria remains an intriguing open problem.
