The one-round Voronoi game

Abstract

In the one-round Voronoi game, the first player chooses an n-point set IN in a square Q, and then the second player places another n-point set 8 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 U B. We give a (randomized) strategy for the second player that always guarantees him a payoff of at least (1) under bar2 + alpha, for a constant alpha > 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 (1) under bar2.