The discrete Voronoi game in R2

In this paper we study the last round of the discrete Voronoi game in R, a problem which is also of independent interest in competitive facility location. The game consists of two players P1 and P2, and a finite set U of users in the plane. The players have already placed two disjoint sets of facilities F and S, respectively in the plane. The game begins with P1 placing a new facility followed by P2 placing another facility, and the objective of both the players is to maximize their own total payoffs. In this paper we propose polynomial time algorithms for the optimal strategies of both the players under arbitrary locations of the existing facilities F and S. We show that in the L1 and the L∞ metrics, the optimal strategy of P2, given any placement of P1, can be found in O(n log n) time, and the optimal strategy of P1 can be found in O(n log n) time. In the L2 metric, the optimal strategies of P2 and P1 can be obtained in O(n ) and O(n) times, respectively.