On the facility location problem: One-round weighted Voronoi game

Voronoi game presents a simple geometric model for Competitive facility location problems with two players. Voronoi game is played by two players known as White and Black, they play in a continuous arena. In the one-round game, White starts game and places all his points in the arena then Black places her points. Then the arena is divided by distance between two players and the player with the largest arena is the winner. In this paper, the one-round weighted Voronoi game is considered in both one and two dimensional regions. In the weighed Voronoi game, the clients can also consider the quality of the facilities in addition to the distance to choose a facility. Accordingly, in one dimension, we will study two different models of facilities  and we will show that Black has a winning strategy in the one-round weighted Voronoi game.