Authors: Richard Cole, Vasilis Gkatzelis, and Vahab Mirrokni Title: Coordination Mechanisms for Weighted Sum of Completion Times Abstract: 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. In short, we present the first constant-factor-approximate coordination mechanisms for this model. First, we present a generalization of the ShortestFirst policy for weighted jobs, called SmithRule; we prove that it achieves an approximation ratio of 4 and we show that any set of non-preemptive ordering policies can result in equilibria with approximation ratio at least 3 even for unweighted jobs. Then, we present ProportionalSharing, a preemptive strongly local policy that beats this lower bound of 3; we show that this policy achieves an approximation ratio of 2.61 for the weighted sum of completion times and that the EqualSharing policy achieves an approximation ratio of 2.5 for the (unweighted) sum of completion times. Furthermore, we show that ProportionalSharing induces potential games (in which best-response dynamics converge to pure Nash equilibria). All of our upper bounds are for the robust price of anarchy, defined by Roughgarden [36], so they naturally extend to mixed Nash equilibria, correlated equilibria, and regret minimization dynamics. Finally, we prove that our price of anarchy bound for ProportionalSharing can be used to design a new combinatorial constant-factor approximation algorithm minimizing weighted completion time for unrelated machine scheduling.