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The One-Round Voronoi Game

Abstract 
 In the one-round Voronoi game, the first player chooses
 an n-point set W in a square Q, and then the
 second player places another n-point set B into
 Q. The payoff for the second player is the fraction of
 the area of Q occupied by the regions of the points of
 B in the Voronoi diagram of W \cup B.
 We give a (randomized) strategy for the second player that always
 guarantees him a payoff of at least ½ + α,
 for a constant α > 0 and every large enough n. This
 contrasts with the one-dimensional situation, with
 Q=[0,1], where the first player can always win more
 than ½.

