Multistage interval scheduling games

We study a game theoretical model of multistage interval scheduling problems in which each job consists of exactly one task (interval) for each of t stages (machines). In the game theoretical model, the machine of each stage is controlled by a different selfish player who wants to maximize her total profit, where the profit for scheduling the task of a job j is a fraction of the weight of the job that is determined by the set of players that also schedule their corresponding task of job j. We provide criteria for the existence of pure Nash equilibria and prove bounds on the Price of Anarchy and the Price of Stability for different social welfare functions.