[Paper Review] Generalized selfish bin packing

György Dósa, Leah Epstein|arXiv (Cornell University)|Feb 18, 2012

Game Theory and Applications22 references18 citations

TL;DR

This paper studies a generalized selfish bin packing game where items' costs are shared proportionally by weight, analyzing pure Nash equilibria, strong equilibria, and Pareto-optimal equilibria. It establishes that for general weights, all four price of anarchy (PoA) values are 1.7, while for unit weights, they are strictly below 1.7—with strong PoA ≈1.691 and strictly Pareto optimal PoA significantly lower—highlighting a key divergence from classical bin packing behavior.

ABSTRACT

Standard bin packing is the problem of partitioning a set of items with positive sizes no larger than 1 into a minimum number of subsets (called bins) each having a total size of at most 1. In bin packing games, an item has a positive weight, and given a valid packing or partition of the items, each item has a cost or a payoff associated with it. We study a class of bin packing games where the payoff of an item is the ratio between its weight and the total weight of items packed with it, that is, the cost sharing is based linearly on the weights of items. We study several types of pure Nash equilibria: standard Nash equilibria, strong equilibria, strictly Pareto optimal equilibria, and weakly Pareto optimal equilibria. We show that any game of this class admits all these types of equilibria. We study the (asymptotic) prices of anarchy and stability (PoA and PoS) of the problem with respect to these four types of equilibria, for the two cases of general weights and of unit weights. We show that while the case of general weights is strongly related to the well-known First Fit algorithm, and all the four PoA values are equal to 1.7, this is not true for unit weights. In particular, we show that all of them are strictly below 1.7, the strong PoA is equal to approximately 1.691 (another well-known number in bin packing) while the strictly Pareto optimal PoA is much lower. We show that all the PoS values are equal to 1, except for those of strong equilibria, which is equal to 1.7 for general weights, and to approximately 1.611824 for unit weights. This last value is not known to be the (asymptotic) approximation ratio of any well-known algorithm for bin packing. Finally, we study convergence to equilibria.

Motivation & Objective

• To analyze the existence and properties of multiple types of equilibria—Nash, strong, strictly and weakly Pareto optimal—in a generalized selfish bin packing game with proportional cost sharing.
• To compute the asymptotic price of anarchy (PoA) and price of stability (PoS) for four equilibrium types under both general and unit weights.
• To investigate convergence to equilibria and compare the performance of equilibria to optimal solutions.
• To identify whether the observed PoA and PoS values correspond to known approximation ratios of classical bin packing algorithms.

Proposed method

• Models bin packing as a strategic game where each item's payoff is its weight divided by the total weight in its bin, inducing proportional cost sharing.
• Defines and analyzes four equilibrium types: standard Nash equilibrium (NE), strong Nash equilibrium (SNE), strictly Pareto optimal NE (SPO-NE), and weakly Pareto optimal NE (WPO-NE).
• Uses potential function techniques to bound convergence time to equilibria, with distinct potential functions for different phases of the process.
• Applies asymptotic approximation ratio analysis to compute PoA and PoS, comparing results for general weights and unit weights.
• Employs combinatorial arguments and case analysis to derive upper bounds on the number of steps in convergence sequences.
• Leverages known results from bin packing (e.g., First Fit’s approximation ratio) to relate game-theoretic measures to classical algorithms.

Experimental results

Research questions

• RQ1 Does every instance of the generalized selfish bin packing game admit all four types of equilibria: NE, SNE, SPO-NE, and WPO-NE?
• RQ2 What are the asymptotic prices of anarchy (PoA) for each of the four equilibrium types under general and unit weights?
• RQ3 How do the PoA and PoS values for unit weights compare to those for general weights, and do they match known approximation ratios of classical algorithms?
• RQ4 Is the strong price of stability (PoS) for unit weights equal to a known approximation ratio of any standard bin packing algorithm?
• RQ5 Can convergence to equilibria be bounded using potential function methods, and what are the implications for algorithmic design?

Key findings

• For general weights, all four types of price of anarchy (PoA) are equal to 1.7, matching the known asymptotic approximation ratio of the First Fit algorithm.
• For unit weights, the strong PoA is approximately 1.691, a well-known constant in bin packing, but the strictly Pareto optimal PoA is significantly lower.
• The price of stability (PoS) is 1 for all equilibrium types except strong equilibria: for general weights, PoS(SNE) = 1.7, and for unit weights, PoS(SNE) ≈ 1.611824, a value not known to correspond to any standard algorithm’s approximation ratio.
• All PoS values are 1 except for strong equilibria, indicating that optimal solutions are often achievable in stable configurations, except in the strong equilibrium case.
• The paper shows that strictly Pareto optimal equilibria are significantly more efficient than standard Nash equilibria, especially in the unit weight case, where the SPO-PoA is much lower than the standard PoA.
• Convergence to equilibria is guaranteed and bounded via potential function analysis, with step counts derived through combinatorial summation and case-based reasoning on bin configurations.