Bin Packing Games with Weight Decision: How to Get a Small Value for the Price of Anarchy

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 [Formula: see text] and at most [Formula: see text]. 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 [Formula: see text] and at most [Formula: see text].

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A bin packing game with cardinality constraints under the best cost rule

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830919500228 ◽

2019 ◽

Vol 11 (02) ◽

pp. 1950022

Author(s):

Qingqin Nong ◽

Jiapeng Wang ◽

Suning Gong ◽

Saijun Guo

Keyword(s):

Cooperative Game ◽

Bin Packing ◽

Social Cost ◽

Price Of Anarchy ◽

Packing Problem ◽

Cardinality Constraints ◽

Bin Packing Problem ◽

The Social ◽

The Cost

We consider the bin packing problem with cardinality constraints in a non-cooperative game setting. In the game, there are a set of items with sizes between 0 and 1, and a number of bins each of which has a capacity of 1. Each bin can pack at most [Formula: see text] items, for a given integer parameter [Formula: see text]. The social cost is the number of bins used in the packing. Each item tries to be packed into one of the bins so as to minimize its cost. The selfish behaviors of the items result in some kind of equilibrium, which greatly depends on the cost rule in the game. We say a cost rule encourages sharing if for an item, compared with sharing a bin with some other items, staying in a bin alone does not decrease its cost. In this paper, we first show that for any bin packing game with cardinality constraints under an encourage-sharing cost rule, the price of anarchy of it is at least [Formula: see text]. We then propose a cost rule and show that the price of anarchy of the bin packing game under the rule is [Formula: see text] when [Formula: see text].

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