Abstract
We consider the bin packing problem in the 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 with a capacity of 1. Each item seeks to be packed in one of the bins so as to minimize its cost (payoff). The social cost is the number of bins used in the packing. Existing research has focused on three bin packing games with selfish items, namely the Unit game, the Proportional game, and the General Weight game, each of which uses a unique payoff rule. In this paper we propose a new bin packing game in which the payoff of an item is a function of its own size and the size of the maximum item in the same bin. We find that the new payoff rule induces the items to reach a better Nash equilibrium. We show that the price of anarchy of the new bin packing game is \(\frac{3}{2}\) and prove that any feasible packing can converge to a Nash equilibrium in \(n^2-n\) steps without increasing the social cost.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bil’o V (2006) On the packing of selfish items. In: Proceedings of the 20th international parallel and distributed processing symposium (IPDPS 2006), IEEE, New York
D’osa G (2007) The tight bound of First Fit Decreasing bin-packing algorithm is FFD(I) 11/9OPT(I) + 6/9. In: Proceedings of 1st international symposium on combinatorics, algorithms, probabilistic and experimental methodologies (ESCAPE). Lecture notes in computer science, vol 4614. Springer, pp 1–11
D’osa G, Epstein L (2012) Generalized selfish bin packing. CoRR, arXiv:1202.40809
D’osa G, Sgall J (2013) First Fit bin packing: a tight analysis. In: Proceedings of 30th international symposium on theoretical aspects of computer science (STACS 2013), pp 538–549
Epstein L (2013) Bin packing games with selfish items. Lect Notes Comput Sci 8087:8–21
Epstein L, Kleiman E (2011) Selfish bin packing. Algorithmica 60(2):368–394
Epstein L, Kleiman E, Mestre J (2009) Parametric packing of selfish items and the subset sum algorithm. In: Leonardi S (eds) WINE 2009. Lecture notes in computer science, vol 5929. Springer, Heidelberg, pp 67–78
Garey MR, Graham RL, Johnson DS, Yao ACC (1976) Resource constrained scheduling as generalized bin packing. J Comb Theory Ser A 21(3):257–298
Johnson DS, Demers AJ, Ullman JD, Garey MR, Graham RL (1974) Worst-case performance bounds for simple one-dimensional packing algorithms. SIAM J Comput 3(4):299–325
Ma RX, D’osa G, Han X, Ting HF, Ye D, Zhang Y (2013) A note on a selfish bin packing problem. J Glob Optim 56:1457–1462
Ullman JD (1971) The performance of a memory allocation algorithm. Technical Report 100, Princeton University, Princeton
Yu G, Zhang G (2008) Bin packing of selfish items. In: Proceedings of the 4th international workshop on internet and network economics, WINE 2008. Lecture notes in computer science, vol 5385. Springer, Heidelberg, pp 446–453
Acknowledgements
This research was supported in part by the National Natural Science Foundation of China under Grant Numbers 11201439 and 11271341. This work was also supported in part by the Shandong Provincial Natural Science Foundation under Grant Number ZR2012AQ12 and by the Doctoral Fund of the Ministry of Education of China (20120132120001).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nong, Q.Q., Sun, T., Cheng, T.C.E. et al. Bin packing game with a price of anarchy of \(\frac{3}{2}\) . J Comb Optim 35, 632–640 (2018). https://doi.org/10.1007/s10878-017-0201-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-017-0201-6