  We study novel variations of Voronoi games and associated random processes
that we call Voronoi choice games. These games provide a rich framework for
studying questions regarding the power of small numbers of choices in
multi-player, competitive scenarios, and they further lead to many interesting,
non-trivial random processes that appear worthy of study.
  As an example of the type of problem we study, suppose a group of $n$ miners
are staking land claims through the following process: each miner has $m$
associated points independently and uniformly distributed on an underlying
space, so the $k$th miner will have associated points
$p_{k1},p_{k2},\ldots,p_{km}$. Each miner chooses one of these points as the
base point for their claim. Each miner obtains mining rights for the area of
the square that is closest to their chosen base, that is, they obtain the
Voronoi cell corresponding to their chosen point in the Voronoi diagram of the
$n$ chosen points. Each player's goal is simply to maximize the amount of land
under their control. What can we say about the players' strategy and the
equilibria of such games?
  In our main result, we derive bounds on the expected number of pure Nash
equilibria for a variation of the 1-dimensional game on the circle where a
player owns the arc starting from their point and moving clockwise to the next
point. This result uses interesting properties of random arc lengths on
circles, and demonstrates the challenges in analyzing these kinds of problems.
We also provide several other related results. In particular, for the
1-dimensional game on the circle, we show that a pure Nash equilibrium always
exists when each player owns the part of the circle nearest to their point, but
it is NP-hard to determine whether a pure Nash equilibrium exists in the
variant when each player owns the arc starting from their point clockwise to
the next point.
