  We consider the well-studied game-theoretic version of machine scheduling in
which jobs correspond to self-interested users and machines correspond to
resources. Here each user chooses a machine trying to minimize her own cost,
and such selfish behavior typically results in some equilibrium which is not
globally optimal: An equilibrium is an allocation where no user can reduce her
own cost by moving to another machine, which in general need not minimize the
makespan, i.e., the maximum load over the machines.
  We provide tight bounds on two well-studied notions in algorithmic game
theory, namely, the price of anarchy and the strong price of anarchy on machine
scheduling setting which lies in between the related and the unrelated machine
case. Both notions study the social cost (makespan) of the worst equilibrium
compared to the optimum, with the strong price of anarchy restricting to a
stronger form of equilibria. Our results extend a prior study comparing the
price of anarchy to the strong price of anarchy for two related machines
(Epstein, Acta Informatica 2010), thus providing further insights on the
relation between these concepts. Our exact bounds give a qualitative and
quantitative comparison between the two models. The bounds also show that the
setting is indeed easier than the two unrelated machines: In the latter, the
strong price of anarchy is $2$, while in ours it is strictly smaller.
