Abstract
We study an online version of linear Fisher market. In this market there are \(m\) buyers and a set of \(n\) dividable goods to be allocated to the buyers. The utility that buyer \(i\) derives from good \(j\) is \(u_{ij}\). Given an allocation \(\hat{U}\) in which buyer \(i\) has utility \(\hat{U}_i\) we study a quality measure that is based on taking an average of the ratios \(U_{i}/\hat{U}_i\) with respect to any other allocation \(U\). Market equilibrium allocation is the optimal solution with respect to this measure. Our setting is online and so the allocation of each good should be done without any knowledge of the upcoming goods. We design an online algorithm for the problem that is only worse by a logarithmic factor than any other solution with respect to this quality measure, and in particular competes with the market equilibrium allocation. We prove a tight lower bound which shows that our algorithm is optimal up to constants. Our algorithm uses a primal dual convex programming scheme. To the best of our knowledge this is the first time that such a scheme is used in the online framework.
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Notes
We remark that it is desirable to have small value \(\frac{U_i}{\hat{U_i}}\) for each buyers \(i\) (and not just the average). However, it is easy to construct an example of one item in which any online algorithm has performance ratio \(m\) with respect to this harder measure.
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Acknowledgments
We would like to thank Gagan Goel for suggesting the wireless router motivation, and also acknowledge to attend a talk presented by Nitish Karula. A discussion among the audience members with the speaker inspired the application mentioned in the discussion section. Yossi Azar is supported in part by the Israel Science Foundation (Grant No. 1404/10) and by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11). Niv Buchbinder is supported by ISF Grant 954/11 and BSF Grant 2010426.
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A preliminary version appeared in the Proceedings of the 18th Annual European Symposium on Algorithms (2010), pp. 51–62.
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Azar, Y., Buchbinder, N. & Jain, K. How to Allocate Goods in an Online Market?. Algorithmica 74, 589–601 (2016). https://doi.org/10.1007/s00453-014-9964-7
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DOI: https://doi.org/10.1007/s00453-014-9964-7