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Abstract:
Recent interest in Nash equilibria led to a study of the {\it
price of anarchy} (poa) and the {\it strong price of anarchy}
(spoa) for scheduling problems. The two
measures express the worst case ratio between the cost of an
equilibrium (a pure Nash equilibrium, and a strong equilibrium,
respectively) to the cost of a social optimum.
The atomic players are the jobs, and the delay of a job is the
completion time of the machine running it, also called the load
of this machine. The social goal is to minimize the maximum delay
of any job, while the selfish goal of each job is to minimize its
own delay, that is, the delay of the machine running it.
We consider scheduling on uniformly related machines. While
previous studies either consider identical speed machines or an
arbitrary number of speeds, focusing on the number of machines as
a parameter, we consider the situation in which the number of
different speeds is small. We reveal a linear dependence between
the number of speeds and the poa. For a set of machines of at
most $p$ speeds, the poa turns out to be exactly $p+1$. The
growth of the poa for large numbers of related machines is
therefore a direct result of the large number of potential speeds.
We further consider a well known structure of processors, where
all machines are of the same speed except for one possibly faster
machine. We investigate the poa as a function of both the speed
ratio between the fastest machine and the number of slow machines.
