We consider a game-theoretic variant of an interval scheduling problem. Every job is associated with a length, a weight, and a color. Each player controls all the jobs of a specific color, and needs to decide on a processing interval for each of its jobs. Jobs of the same color can be processed simultaneously by the machine. A job is covered if the machine is configured to its color during its whole processing interval. The goal of the machine is to maximize the sum of weights of all covered jobs, and the goal of each player is to place its jobs such that the sum of weights of covered jobs from its color is maximized. The study of this game is motivated by several applications like antenna scheduling for wireless networks. We first show that given a strategy profile of the players, the machine scheduling problem can be solved in polynomial time. We then study the game from the players' point of view. We analyze the existence of Nash equilibria, its computation, and inefficiency. We distinguish between instances of the classical interval scheduling problem, in which every player controls a single job, and instances in which color sets may include multiple jobs.

翻译：我们考虑区间调度问题的一个博弈论变体。每个任务关联一个长度、一个权值以及一种颜色。每位玩家控制某特定颜色的所有任务，并需要为其每个任务确定一个处理区间。同种颜色的任务可由机器并行处理。若在任务的整个处理区间内，机器配置为与该任务颜色一致，则该任务被覆盖。机器的目标是最大化所有被覆盖任务权值之和，而每位玩家的目标是将其任务安排得使其控制颜色中被覆盖任务的权值之和最大化。对该博弈的研究源于无线网络天线调度等多种应用。我们首先证明，给定玩家的策略组合，机器调度问题可在多项式时间内求解。随后从玩家视角研究该博弈，分析纳什均衡的存在性、计算过程及其效率低下的问题。我们区分经典区间调度问题的实例（其中每位玩家仅控制单个任务）与颜色集合可能包含多个任务的实例。