Max-min Online Allocations with a Reordering Puffer

We consider a scheduling problem where each job is controlled by a selfish agent, who is only interested in minimizing its own cost, which is defined as the total load on the machine that its job is assigned to. We consider the objective of maximizing the minimum load (cover) over the machines. Unlike the regular makespan minimization problem, which was extensively studied in a game theoretic context, this problem has not been considered in this setting before. We study the price of anarchy (\poa) and the price of stability (\pos). Since these measures are unbounded already for two uniformly related machines, we focus on identical machines. We show that the \pos is 1, and we derive tight bounds on the \poa for $m\leq6$ and nearly tight bounds for general $m$. In particular, we show that the \poa is at least 1.691 for larger $m$ and at most 1.7. Hence, surprisingly, the \poa is less than the \poa for the makespan problem, which is 2. To achieve the upper bound of 1.7, we make an unusual use of weighting functions. Finally, in contrast we show that the mixed \poa grows exponentially with $m$ for this problem, although it is only $\Theta(\log m/\log \log m)$ for the makespan. In addition we consider a similar setting with a different objective which is minimizing the maximum ratio between the loads of any pair of machines in the schedule. We show that under this objective for general $m$ the \pos is 1, and the \poa is 2