We study the multidimensional vector scheduling problem with selfish jobs, both in non-cooperative and in cooperative versions. We show existence of assignments that are Nash, strong Nash, weakly and strictly Pareto optimal Nash equilibria in these settings. We improve upon the previous bounds on the price of anarchy for the non-cooperative case, and find tight bounds for every number of machines and dimension. For the cooperative case we provide tight bounds on the strong prices of anarchy and stability, as well as tight bounds on weakly and strictly Pareto optimal prices of anarchy and stability, for every number of machines and dimension.