The One-Round Voronoi Game
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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.