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 &frac;(symbol not translated) + 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 (symbol not translated)&frac;. [PUBLICATION ABSTRACT]