Quality of equilibria for selfish bin packing with cost sharing variants

Bin packing is the problem of splitting a set of items into a minimum number of subsets, called bins, of total sizes no larger than 1, where a solution is called a packing. We study bin packing games where an item also has a positive weight, and given a valid packing of the items, each item has a cost associated with it, such that the cost of an item is the ratio between its weight and the total weight of items packed in its bin. That is, the cost sharing is based linearly on the weights of items. We study several types of Nash equilibria: pure Nash equilibria, strong equilibria, strictly Pareto optimal equilibria, and weakly Pareto optimal equilibria, and show that any game of this class of bin packing games admits all these types of equilibria. We study the (asymptotic) prices of anarchy and stability (POA and POS) of these games with respect to the four types of equilibria. We find that the problem is strongly related to the well-known FIRST FIT algorithm, and all the four POA values are equal to 1.7. The POS values are shown to be equal to 1, except for strong equilibria, for which the POS value is 1.7. Additionally, we study the sub-class of bin packing games with equal weights, where the cost sharing of each bin is uniform in the sense that the cost of a bin is shared equally between its items. We analyze the price of anarchy, and find that this value is strictly smaller than 1.7 and in particular, it is not larger than 1.699396 and it is at least 1.69664.