Basing on the two-player Voronoi game introduced by Ahn et al. [1] and the multi-player diffusion game introduced by Alon et al. [2], we investigate the following one-round multi-player discrete Voronoi game on grids and trees. There are n players playing this game on a graph $G=(V,E)$. Each player chooses an initial vertex from the vertex set of the graph and tries to maximize the size of the nearest vertex set. As the main result, we give sufficient conditions for the existence/non-existence of a pure-strategy Nash equilibrium in 4-player games on grids and only a constant gap leaves unknown. We further consider this game with more than 4 players and construct a family of strategy profiles, which are pure-strategy Nash equilibria on sufficiently narrow graphs. Besides, we investigate the game with 3 players on trees and design a linear time/space algorithm to decide the existence of a pure-strategy Nash equilibrium.

基于Ahn等人介绍的两人Voronoi游戏。[1]和Alon等人介绍的多人扩散游戏。[2]，我们研究了以下基于网格和树的单轮多人离散Voronoi游戏。在图表上有n个玩家在玩这个游戏$G=（V，Ë）$。每个玩家都从图形的顶点集中选择一个初始顶点，并尝试最大化最近的顶点集中的大小。作为主要结果，我们为网格上的4人游戏中的纯策略纳什均衡的存在/不存在提供了充分的条件，并且只有一个恒定的差距尚不明确。我们进一步考虑具有超过4个玩家的这场比赛，并构建了一系列策略配置文件，这些策略配置文件在足够狭窄的图上是纯策略纳什均衡。此外，我们调查了具有3位玩家在树上的游戏，并设计了线性时间/空间算法来确定纯策略纳什均衡的存在。