The competitive facility location problem is the problem of determining facility locations involving multiple players to optimize their various gains. The Voronoi game is a competitive facility location problem on a given arena played by two players, the server and the adversary. The players alternately take turns, one or more times, to place their facilities in the arena with a predetermined set of n clients, where both facilities and clients are denoted by points, to maximize some resource gain. The Voronoi game on a polygon $\mathcal{P}$ is a type of competitive facility location problem where n clients are located on the boundary of $\mathcal{P}$. The server, Alice, and adversary, Bob, are respectively in the interior and the exterior of the polygon $\mathcal{P}$ at locations $\mathcal{A}$ and $\mathcal{B}$, respectively. Additionally, the metrics for Alice and Bob are the internal and external geodesic distances for the polygon $\mathcal{P}$, respectively. In this paper, we present some surprising results on the Voronoi games on polygons.

We prove lower and upper bounds of $\lceil n/3\rceil$ and $n-1$ respectively in the single-round game for the number of clients won by the server for n clients. Both bounds are tight. In the process, we show that in some convex polygons, the adversary wins no more than k clients in a k-round Voronoi game for any $k\le n$. Consequentially, the adversary Bob does not have a guaranteed good winning strategy even for the simpler case of convex polygons, i.e., there exist convex polygons such that no placement of $\mathcal{B}$ guarantees more than k clients in the k-round game. We also design $O(n{\log }^{2}n+m\log n)$ and $O(n+m)$ time algorithms to compute the optimal locations for the server and the adversary respectively to maximize their client counts where the convex polygon has size m. Moreover, we present an $O(n\log n)$ time algorithm to compute the common intersection of a set of n ellipses. This is needed in our algorithm and may be of independent interest.

Lastly, we present some results on the Voronoi games, where the arena is a convex polytope. The server and adversary are respectively in the interior and exterior of $\mathcal{P}$, and the clients are on the polytope boundary.

在有竞争力的工厂选址问题是确定涉及多个玩家来优化他们的各种增益设施地点的问题。该沃罗诺伊游戏是由两个球员，起到了给定的舞台上有竞争力的工厂选址问题服务器和对手。玩家交替轮流，一次或多次，将他们的设施放置在具有一组预定n 个客户端的竞技场中，其中设施和客户端都用点表示，以最大化一些资源收益。多边形上的Voronoi 游戏 $\mathcal{磷}$是一种竞争性设施选址问题，其中n 个客户位于$\mathcal{磷}$. 服务器 Alice 和对手 Bob 分别位于多边形的内部和外部$\mathcal{磷}$ 在地点 $\mathcal{一种}$ 和 $\mathcal{乙}$， 分别。此外，Alice 和 Bob 的度量是多边形的内部和外部测地距离$\mathcal{磷}$， 分别。在本文中，我们在多边形的 Voronoi 游戏中展示了一些令人惊讶的结果。

我们证明下界和上界 $\lceil n/3\rceil$ 和 $n-1$分别在单轮比赛中服务器为n个客户端赢得的客户端数量。两个边界都很紧。在这个过程中，我们证明了在一些凸多边形中，对手在k轮 Voronoi 博弈中赢得不超过k 个客户$克\le n$. 因此，即使对于凸多边形的简单情况，对手 Bob 也没有保证好的获胜策略，即存在凸多边形使得没有放置$\mathcal{乙}$保证在k轮游戏中超过k 个客户。我们还设计$哦(n{\mathrm{日志}}^{2}n+米\mathrm{日志}n)$ 和 $哦(n+米)$时间算法分别计算服务器和对手的最佳位置，以最大化其客户端数量，其中凸多边形的大小为m。此外，我们提出了一个$哦(n\mathrm{日志}n)$时间算法来计算一组n 个椭圆的公共交集。这在我们的算法中是必需的，并且可能是独立的。

最后，我们展示了 Voronoi 游戏的一些结果，其中竞技场是一个凸多面体。服务器和对手分别在内部和外部$\mathcal{磷}$，并且客户端位于多面体边界上。