Abstract
A selfish bin packing game is a variant of the classical bin packing problem in a game theoretic setting. In our model the items have not only a size but also a positive weight. The cost of a bin is 1, and this cost is shared among the items being in the bin, proportionally to their weights. A packing is a Nash equilibrium (NE) if no item can decrease its cost by moving to another bin, and OPT means a packing where the items are packed optimally (into minimum number of bins). Without any misunderstanding we denote by NE both the packing and the number of bins in the packing, and the same holds for the OPT packing. We are interested in the Price of Anarchy (PoA), which is the limsup of NE/OPT ratios. Recently there is a growing interest for games where the PoA is low.
We give a setting for the weights where the PoA is between 1.4646 and 1.5. The lower bound is valid also for the special case of the game where the weight of any item is the same as its size, and any item has size at most one half. The previous bound was about 1.46457. Next we give another setting where the PoA is at most \(16/11\approx 1.4545\). This value is better than any previous, that was got for such games.
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Notes
- 1.
An existing example on the large scale is that different types of products include different percentages of VAT in their price, hence orienting the consumption habits of people to some extent.
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Acknowledgements
Gyorgy Dosa acknowledges the financial support of Széchenyi 2020 under the EFOP-3.6.1-16-2016-00015. Research of the first and third authors was supported in part by the National Research, Development and Innovation Office – NKFIH under the grant SNN 116095. All three authors are supported by Stiftung Aktion Österreich-Ungarn 99öu1.
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Dosa, G., Kellerer, H., Tuza, Z. (2018). Bin Packing Games with Weight Decision: How to Get a Small Value for the Price of Anarchy. In: Epstein, L., Erlebach, T. (eds) Approximation and Online Algorithms. WAOA 2018. Lecture Notes in Computer Science(), vol 11312. Springer, Cham. https://doi.org/10.1007/978-3-030-04693-4_13
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