Convergence Time to Nash Equilibrium in Selfish Bin Packing

Miyazawa, Flávio Keidi; Vignatti, André Luís

doi:10.1016/j.endm.2009.11.026

Cited by 8 publications

(4 citation statements)

References 5 publications

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“…In this section we find the exact worst case number of steps, which turns out to be Θ(n 3/2 ). Note that [19] showed using methods from [11] (where convergence for scheduling problems is studied) that for the case of proportional cost sharing, the number of steps can be exponential. Given an integer n ≥ 1, we let…”

Section: The Exact Convergence Timementioning

confidence: 99%

“…He also proved that any such bin packing game converges to an NE after a finite (but possibly exponentially long) sequence of steps, starting from any initial configuration of the items. The time of convergence for this type of cost sharing was also studied in [19,20]. Multiple papers studied the quality of NE and other types of equilibria [3,9,10,7,1].…”

Section: Introductionmentioning

confidence: 99%

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The Convergence Time for Selfish Bin Packing

Dósa¹,

Epstein

2018

Acta Cybern

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In classic bin packing, the objective is to partition a set of n items with positive rational sizes in (0, 1] into a minimum number of subsets called bins, such that the total size of the items of each bin at most 1. We study a bin packing game where the cost of each bin is 1, and given a valid packing of the items, each item has a cost associated with it, such that the items that are packed into a bin share its cost equally. We find tight bounds on the exact worst-case number of steps in processes of convergence to pure Nash equilibria. Those are processes that are given an arbitrary packing as an initial packing. As long as there exists an item that can reduce its cost by moving from its bin to another bin, in each step, a controller selects such an item and instructs it to perform such a beneficial move. The process converges when no further beneficial moves exist. The tight function of n that we find is in Θ(n 3/2). This improves the previous bound of Ma et al. [14], who showed an upper bound of O(n 2).

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Section: The Exact Convergence Timementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Convergence Time for Selfish Bin Packing

Dósa¹,

Epstein

2018

Acta Cybern

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“…In this section we find the exact worst case number of steps, which turns out to be Θ(n 3/2 ). Note that [34] showed using methods from [20] that for the case of general weights (in fact, for proportional weights) the number of steps can be exponential. Theorem 7.1 Let n be an integer, consider the integers i, j such that 0 ≤ j ≤ i − 1 and n = i(i + 1)/2 − j.…”

Section: The Exact Convergence Time For Unit Weightsmentioning

confidence: 99%

“…This last result implies that the PoS is equal to 1. The time of convergence was also studied in [34,35]. The quality of NE solutions was further investigated in [17], where nearly tight bounds for the PoA were given; an upper bound of 1.6428 and a lower bound of 1.6416.…”

Section: Introductionmentioning

confidence: 99%

Generalized selfish bin packing

Dósa¹,

Epstein²

2012

Preprint

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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 (PoAand 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.

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Selfish Bin Packing Problems

Epstein¹

2016

Encyclopedia of Algorithms

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Convergence Time to Nash Equilibrium in Selfish Bin Packing

Cited by 8 publications

References 5 publications

The Convergence Time for Selfish Bin Packing

The Convergence Time for Selfish Bin Packing

Generalized selfish bin packing

Selfish Bin Packing Problems

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