The convergence time for selfish bin packing

In classic bin packing, the objective is to partition a set of n items with positive rational sizes in (0,1] into a minimum number of subsets called bins, such that the total size of the items of each bin at most 1. We study a bin packing game where the cost of each bin is 1, and given a valid packing of the items, each item has a cost associated with it, such that the items that are packed into a bin share its cost equally. We find tight bounds on the exact worst-case number of steps in processes of convergence to pure Nash equilibria. Those are processes that are given an arbitrary packing. As long as there exists an item that can reduce its cost by moving from its bin to another bin, in each step, a controller selects such an item and instructs it to perform such a beneficial move. The process terminates when no further beneficial moves exist. The function of n that we find is Θ(n '), improving the previous bound of Han et al., who showed an upper bound of O(n).