  We consider a game-theoretical problem called selfish 2-dimensional bin
packing game, a generalization of the 1-dimensional case already treated in the
literature. In this game, the items to be packed are rectangles, and the bins
are unit squares. The game starts with a set of items arbitrarily packed in
bins. The cost of an item is defined as the ratio between its area and the
total occupied area of the respective bin. Each item is a selfish player that
wants to minimize its cost. A migration of an item to another bin is allowed
only when its cost is decreased. We show that this game always converges to a
Nash equilibrium (a stable packing where no single item can decrease its cost
by migrating to another bin). We show that the pure price of anarchy of this
game is unbounded, so we address the particular case where all items are
squares. We show that the pure price of anarchy of the selfish square packing
game is at least 2.3634 and at most 2.6875. We also present analogous results
for the strong Nash equilibrium (a stable packing where no nonempty set of
items can simultaneously migrate to another common bin and decrease the cost of
each item in the set). We show that the strong price of anarchy when all items
are squares is at least 2.0747 and at most 2.3605.
