One-Round Discrete Voronoi Game in ℝ2 in Presence of Existing Facilities

In this paper we consider a simplified variant of the discrete Voronoi Game in R, 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 sets of facilities F and S, respectively in the plane. The game begins by 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. When |F | = |S| = m, this corresponds to the last round of the (m + 1)-round discrete Voronoi Game in R. In this paper we propose polynomial time algorithms for obtaining optimal strategies of both the players under arbitrary locations of the existing facilities F and S. We show that the optimal strategy of P2, given any placement of P1, can be found in O(n) time, and the optimal strategy of P1 can be found in O(n) time.