  We study policies aiming to minimize the weighted sum of completion times of
jobs in the context of coordination mechanisms for selfish scheduling problems.
Our goal is to design local policies that achieve a good price of anarchy in
the resulting equilibria for unrelated machine scheduling. To obtain the
approximation bounds, we introduce a new technique that while conceptually
simple, seems to be quite powerful. With this method we are able to prove the
following results.
  First, we consider Smith's Rule, which orders the jobs on a machine in
ascending processing time to weight ratio, and show that it achieves an
approximation ratio of 4. We also demonstrate that this is the best possible
for deterministic non-preemptive strongly local policies. Since Smith's Rule is
always optimal for a given assignment, this may seem unsurprising, but we then
show that better approximation ratios can be obtained if either preemption or
randomization is allowed.
  We prove that ProportionalSharing, a preemptive strongly local policy,
achieves an approximation ratio of 2.618 for the weighted sum of completion
times, and an approximation ratio of 2.5 in the unweighted case. Again, we
observe that these bounds are tight. Next, we consider Rand, a natural
non-preemptive but randomized policy. We show that it achieves an approximation
ratio of at most 2.13; moreover, if the sum of the weighted completion times is
negligible compared to the cost of the optimal solution, this improves to \pi
/2.
  Finally, we show that both ProportionalSharing and Rand induce potential
games, and thus always have a pure Nash equilibrium (unlike Smith's Rule). This
also allows us to design the first \emph{combinatorial} constant-factor
approximation algorithm minimizing weighted completion time for unrelated
machine scheduling that achieves a factor of 2+ \epsilon for any \epsilon > 0.
