A Note on Approximating Weighted Nash Social Welfare with Additive Valuations

Authors Yuda Feng, Shi Li

Thumbnail PDF


  • Filesize: 0.65 MB
  • 9 pages

Document Identifiers

Author Details

Yuda Feng
  • Department of Computer Science and Technology, Harbin Institute of Technology, Heilongjiang, China
Shi Li
  • Department of Computer Science and Technology, Nanjing University, Jiangsu, China


YF is an incoming PhD student at Nanjing University, and his work was a part of his undergraduate dissertation supervised by SL.

Cite AsGet BibTex

Yuda Feng and Shi Li. A Note on Approximating Weighted Nash Social Welfare with Additive Valuations. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 63:1-63:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We give the first O(1)-approximation for the weighted Nash Social Welfare problem with additive valuations. The approximation ratio we obtain is e^{1/e} + ε ≈ 1.445 + ε, which matches the best known approximation ratio for the unweighted case [Barman et al., 2018]. Both our algorithm and analysis are simple. We solve a natural configuration LP for the problem, and obtain the allocation of items to agents using a randomized version of the Shmoys-Tardos rounding algorithm developed for unrelated machine scheduling problems [Shmoys and Tardos, 1993]. In the analysis, we show that the approximation ratio of the algorithm is at most the worst gap between the Nash social welfare of the optimum allocation and that of an EF1 allocation, for an unweighted Nash Social Welfare instance with identical additive valuations. This was shown to be at most e^{1/e} ≈ 1.445 by Barman et al. [Barman et al., 2018], leading to our approximation ratio.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Nash Social Welfare
  • Configuration LP
  • Approximation Algorithms


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Nima Anari, Shayan Oveis Gharan, Amin Saberi, and Mohit Singh. Nash Social Welfare, Matrix Permanent, and Stable Polynomials. In Christos H. Papadimitriou, editor, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017), volume 67 of Leibniz International Proceedings in Informatics (LIPIcs), pages 36:1-36:12, Dagstuhl, Germany, 2017. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2017.36.
  2. Julius Barbanel and Alan Taylor. The Geometry of Efficient Fair Division. The Geometry of Efficient Fair Division, pages 1-462, January 2005. URL: https://doi.org/10.1017/CBO9780511546679.
  3. Siddharth Barman, Sanath Kumar Krishnamurthy, and Rohit Vaish. Finding Fair and Efficient Allocations. In Proceedings of the 2018 ACM Conference on Economics and Computation, EC '18, pages 557-574, New York, NY, USA, 2018. Association for Computing Machinery. URL: https://doi.org/10.1145/3219166.3219176.
  4. Steven J. Brams and Alan D. Taylor. Fair Division: From Cake-Cutting to Dispute Resolution. Cambridge University Press, 1996. Google Scholar
  5. Felix Brandt, Vincent Conitzer, Ulle Endriss, Jérôme Lang, and Ariel D. Procaccia. Handbook of Computational Social Choice. Cambridge University Press, USA, 1st edition, 2016. Google Scholar
  6. Adam Brown, Aditi Laddha, Madhusudhan Reddy Pittu, and Mohit Singh. Approximation Algorithms for the Weighted Nash Social Welfare via Convex and Non-Convex Programs. In Proceedings of the Thirty-Fifth ACM-SIAM Symposium on Discrete Algorithms, 2024. Google Scholar
  7. Suchan Chae and Hervé Moulin. Bargaining Among Groups: An Axiomatic Viewpoint. International Journal of Game Theory, 39(1):71-88, 2010. URL: https://doi.org/10.1007/s00182-009-0157-6.
  8. Richard Cole, Nikhil Devanur, Vasilis Gkatzelis, Kamal Jain, Tung Mai, Vijay V. Vazirani, and Sadra Yazdanbod. Convex Program Duality, Fisher Markets, and Nash Social Welfare. In Proceedings of the 2017 ACM Conference on Economics and Computation, EC '17, pages 459-460, New York, NY, USA, 2017. Association for Computing Machinery. URL: https://doi.org/10.1145/3033274.3085109.
  9. Richard Cole and Vasilis Gkatzelis. Approximating the Nash Social Welfare with Indivisible Items. In Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing, STOC '15, pages 371-380, New York, NY, USA, 2015. Association for Computing Machinery. URL: https://doi.org/10.1145/2746539.2746589.
  10. Dagmawi Mulugeta Degefu, Weijun He, Liang Yuan, An Min, and Qi Zhang. Bankruptcy to Surplus: Sharing Transboundary River Basin’s Water under Scarcity. Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), 32(8):2735-2751, 2018. URL: https://EconPapers.repec.org/RePEc:spr:waterr:v:32:y:2018:i:8:d:10.1007_s11269-018-1955-z.
  11. Shahar Dobzinski, Wenzheng Li, Aviad Rubinstein, and Jan Vondrak. A Constant Factor Approximation for Nash Social Welfare with Subadditive Valuations. ArXiv, abs/2309.04656, 2023. URL: https://arxiv.org/abs/2309.04656.
  12. Jugal Garg, Martin Hoefer, and Kurt Mehlhorn. Satiation in Fisher Markets and Approximation of Nash Social Welfare. Mathematics of Operations Research, 0(0):null, 2023. URL: https://doi.org/10.1287/moor.2019.0129.
  13. Jugal Garg, Edin Husić, Wenzheng Li, László A. Végh, and Jan Vondrák. Approximating Nash Social Welfare by Matching and Local Search. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, pages 1298-1310, New York, NY, USA, 2023. Association for Computing Machinery. URL: https://doi.org/10.1145/3564246.3585255.
  14. Jugal Garg, Edin Husić, and László A. Végh. Approximating Nash Social Welfare under Rado Valuations. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, pages 1412-1425, New York, NY, USA, 2021. Association for Computing Machinery. URL: https://doi.org/10.1145/3406325.3451031.
  15. Jugal Garg, Pooja Kulkarni, and Rucha Kulkarni. Approximating Nash Social Welfare under Submodular Valuations through (Un)Matchings. ACM Trans. Algorithms, 19(4), September 2023. URL: https://doi.org/10.1145/3613452.
  16. John C. Harsanyi and Reinhard Selten. A generalized nash solution for two-person bargaining games with incomplete information. Management Science, 18(5):P80-P106, 1972. URL: http://www.jstor.org/stable/2661446.
  17. Harold Houba, Gerard van der Laan, and Yuyu Zeng. Asymmetric Nash Solutions in the River Sharing Problem. Game Theory & Bargaining Theory eJournal, 2013. URL: https://api.semanticscholar.org/CorpusID:17619205.
  18. E. Kalai. Nonsymmetric Nash Solutions and Replications of 2-Person Bargaining. Int. J. Game Theory, 6(3):129-133, September 1977. URL: https://doi.org/10.1007/BF01774658.
  19. Mamoru Kaneko and Kenjiro Nakamura. The Nash Social Welfare Function. Econometrica, 47(2):423-435, 1979. URL: http://www.jstor.org/stable/1914191.
  20. Frank Kelly. Charging and Rate Control for Elastic Traffic. European Transactions on Telecommunications, 8(1):33-37, 1997. URL: https://doi.org/10.1002/ett.4460080106.
  21. Annick Laruelle and Federico Valenciano. Bargaining in Committees as An Extension of Nash’s Bargaining Theory. Journal of Economic Theory, 132(1):291-305, 2007. URL: https://doi.org/10.1016/j.jet.2005.05.004.
  22. Euiwoong Lee. APX-Hardness of Maximizing Nash Social Welfare with Indivisible Items. ArXiv, abs/1507.01159, 2015. URL: https://arxiv.org/abs/1507.01159.
  23. W. Li and J. Vondrak. A Constant-Factor Approximation Algorithm for Nash Social Welfare with Submodular Valuations. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 25-36, Los Alamitos, CA, USA, February 2022. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS52979.2021.00012.
  24. Wenzheng Li and Jan Vondrák. Estimating the Nash Social Welfare for Coverage and Other Submodular Valuations. In Proceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '21, pages 1119-1130, USA, 2021. Society for Industrial and Applied Mathematics. Google Scholar
  25. Herve Moulin. Fair Division and Collective Welfare. The MIT Press, 2004. Google Scholar
  26. John F. Nash. The Bargaining Problem. Econometrica, 18(2):155-162, 1950. URL: http://www.jstor.org/stable/1907266.
  27. Nhan-Tam Nguyen, Trung Thanh Nguyen, Magnus Roos, and Jörg Rothe. Complexity and Approximability of Social Welfare Optimization in Multiagent Resource Allocation. In Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems - Volume 3, AAMAS '12, pages 1287-1288, Richland, SC, 2012. International Foundation for Autonomous Agents and Multiagent Systems. Google Scholar
  28. Jack Robertson and William Webb. Cake-Cutting Algorithms: Be Fair if You Can. A K Peters/CRC Press, 1998. Google Scholar
  29. Jrg Rothe. Economics and Computation: An Introduction to Algorithmic Game Theory, Computational Social Choice, and Fair Division. Springer Publishing Company, Incorporated, 1st edition, 2015. Google Scholar
  30. David B. Shmoys and Éva Tardos. An Approximation Algorithm for the Generalized Assignment Problem. Mathematical Programming, 62(1):461-474, 1993. URL: https://doi.org/10.1007/BF01585178.
  31. William Thomson. Replication Invariance of Bargaining Solutions. Int. J. Game Theory, 15(1):59-63, March 1986. URL: https://doi.org/10.1007/BF01769276.
  32. Hal R Varian. Equity, Envy, and Efficiency. Journal of Economic Theory, 9(1):63-91, 1974. URL: https://doi.org/10.1016/0022-0531(74)90075-1.
  33. H. Peyton Young. Equity: In Theory and Practice. Princeton University Press, 1994. URL: http://www.jstor.org/stable/j.ctv10crfx7.
  34. S. Yu, E. C. van Ierland, H. P. Weikard, and X. Zhu. Nash Bargaining Solutions for International Climate Agreements under Different Sets of Bargaining Weights. International Environmental Agreements: Politics, Law and Economics, 17(5):709-729, 2017. URL: https://doi.org/10.1007/s10784-017-9351-3.