Prices of anarchy of selfish 2D bin packing games

TL;DR

This paper studies a game-theoretic model of 2D bin packing with selfish rectangles, proving convergence to Nash equilibria and establishing bounds on the inefficiency of stable packings, especially for square items.

Contribution

It introduces the selfish 2D bin packing game, proves convergence to Nash equilibria, and provides bounds on the price of anarchy for square items under various equilibrium concepts.

Findings

Game always converges to a Nash equilibrium.

Pure price of anarchy for squares is between 2.3634 and 2.6875.

Strong price of anarchy for squares is between 2.0747 and 2.3605.

Abstract

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.