Selfish bin packing with cardinality constraints

Bin packing with cardinality constraints is a variant of bin packing. In this problem, items with sizes of at most 1 are to be partitioned (or packed) into subsets called bins, such that the total size of items packed into a bin would not exceed 1, and each bin would contain at most k items, for a given integer parameter k>1. We consider a class of games resulting from this problem by seeing the items as selfish players, trying to be packed into a (valid) bin where the total size of items is maximized. Such games always admit pure Nash equilibria. We analyze the Price of Anarchy (PoA) of such games, defined as the asymptotic worst case ratio between the maximum number of bins in a pure Nash equilibrium (NE), and the minimum number of bins in any valid solution, that is, the number of bins in a socially optimal solution. We provide a complete analysis of the PoA as a function of k; we prove that for k=2, any NE is an optimal solution, for k≥4, the PoA is exactly 2-1k, and for k=3, the PoA is exactly 117.