Skip to main content
Log in

The price to pay for forgoing normalization in fair division of indivisible goods

  • Published:
Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

Abstract

We study the complexity of fair division of indivisible goods and consider settings where agents can have nonzero utility for the empty bundle. This is a deviation from a common normalization assumption in the literature, and we show that this inconspicuous change can lead to an increase in complexity: In particular, while an allocation maximizing social welfare by the Nash product is known to be easy to detect in the normalized setting whenever there are as many agents as there are resources, without normalization it can no longer be found in polynomial time, unless P = NP. The same statement also holds for egalitarian social welfare. Moreover, we show that it is NP-complete to decide whether there is an allocation whose Nash product social welfare is above a certain threshold if the number of resources is a multiple of the number of agents. Finally, we consider elitist social welfare and prove that the increase in expressive power by allowing negative coefficients again yields NP-completeness.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Bansal, N., Sviridenko, M.: The Santa Claus problem. In: Proceedings of the 38th ACM Symposium on Theory of Computing, pp. 31–40. ACM (2006)

  2. Bouveret, S.: Fair allocation of indivisible items: Modeling, computational complexity and algorithmics. Ph.D. Thesis, Institut Supérieur De L’Aéronautique Et De l’Espace, Toulouse, France (2007)

  3. Bouveret, S., Chevaleyre, Y., Maudet, N.: Handbook of computational social choice, chap. 12. In: Brandt, F., Conitzer, V., Endriss, U., Lang, J., Procaccia, A. (eds.) , pp 284–310. Cambridge University Press, Cambridge (2016)

  4. Bouveret, S., Lemaître, M., Fargier, H., Lang, J.: Allocation of indivisible goods: A general model and some complexity results (extended abstract). In: Proceedings of the 4th international joint conference on autonomous agents and multiagent systems, pp. 1309–1310. ACM Press (2005)

  5. Brams, S., Taylor, A.: Fair division: From cake-cutting to dispute resolution. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

  6. Budish, E.: The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes. Journal of Political Economy 119(6), 1061–1103 (2011)

    Article  Google Scholar 

  7. Caragiannis, I., Kurokawa, D., Moulin, H., Procaccia, A., Shah, N., Wang, J.: The unreasonable fairness of maximum Nash welfare. In: Proceedings of the 17th ACM conference on economics and computation, pp. 305–322. ACM (2016)

  8. Chevaleyre, Y., Dunne, P., Endriss, U., Lang, J., Lemaître, M., Maudet, N., Padget, J., Phelps, S., Rodríguez-Aguilar, J., Sousa, P.: Issues in multiagent resource allocation. Informatica 30(1), 3–31 (2006)

    MATH  Google Scholar 

  9. Chevaleyre, Y., Endriss, U., Estivie, S., Maudet, N.: Multiagent resource allocation with k-additive utility functions. In: Proceedings of the DIMACS-LAMSADE workshop on computer science and decision theory, Annales du LAMSADE, vol. 3, pp. 83–100 (2004)

  10. Chevaleyre, Y., Endriss, U., Estivie, S., Maudet, N.: Multiagent resource allocation in k-additive domains: Preference representation and complexity. Ann. Oper. Res. 163(1), 49–62 (2008)

    Article  MathSciNet  Google Scholar 

  11. Chevaleyre, Y., Endriss, U., Maudet, N.: Allocating goods on a graph to eliminate envy. In: Proceedings of the 22nd AAAI conference on artificial intelligence, pp. 700–705. AAAI Press (2007)

  12. Chevaleyre, Y., Endriss, U., Maudet, N.: Simple negotiation schemes for agents with simple preferences: sufficiency, necessity and maximality. Auton. Agent. Multi-Agent Syst. 20(2), 234–259 (2010)

    Article  Google Scholar 

  13. Cole, R., Devanur, N., Gkatzelis, V., Jain, K., Mai, T., Vazirani, V., Yazdanbod, S.: Convex program duality, fisher markets, and Nash social welfare. In: Proceedings of the 18th ACM conference on economics and computation, pp. 459–460. ACM (2017)

  14. Damamme, A., Beynier, A., Chevaleyre, Y., Maudet, N.: The power of swap deals in distributed resource allocation. In: Proceedings of the 14th international conference on autonomous agents and multiagent systems, pp. 625–633. IFAAMAS (2015)

  15. Dunne, P., Wooldridge, M., Laurence, M.: The complexity of contract negotiation. Artif. Intell. 164(1–2), 23–46 (2005)

    Article  MathSciNet  Google Scholar 

  16. Endriss, U., Maudet, N., Sadri, F., Toni, F.: Negotiating socially optimal allocations of resources. J. Artif. Intell. Res. 25, 315–348 (2006)

    Article  MathSciNet  Google Scholar 

  17. Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-completeness. W. H Freeman and Company (1979)

  18. Golovin, D.: Max-min fair allocation of indivisible goods. Tech. Rep. CMU-CS-05-144, School of Computer Science Carnegie Mellon University (2005)

  19. Heinen, T., Nguyen, N., Rothe, J.: Fairness and rank-weighted utilitarianism in resource allocation. In: Proceedings of the 4th international conference on algorithmic decision theory, pp. 521–536. Springer-Verlag Lecture Notes in Artificial Intelligence #9346 (2015)

  20. Irving, R., Leather, P., Gusfield, D.: An efficient algorithm for the “optimal” stable marriage. Journal of the ACM 34(3), 532–543 (1987)

    Article  MathSciNet  Google Scholar 

  21. Karp, R.: Reducibility among Combinatorial Problems. In: Miller, R., Thatcher, J. (eds.) Complexity of computer computations, pp. 85–103. Plenum Press (1972)

  22. Kash, I., Procaccia, A., Shah, N.: No agent left behind: Dynamic fair division of multiple resources. J. Artif. Intell. Res. 51, 579–603 (2014)

    Article  MathSciNet  Google Scholar 

  23. Lang, J., Rothe, J.: Fair division of indivisible goods. In: Rothe, J. (ed.) Economics and computation. An introduction to algorithmic game theory, computational social choice, and fair division, springer texts in business and economics, Chap. 8. Springer-Verlag (2015)

  24. Lange, P., Nguyen, N., Rothe, J.: The price to pay for forgoing normalization in fair division of indivisible goods. In: Nonarchival website proceedings of the 15th International Symposium on Artificial Intelligence and Mathematics. http://isaim2018.cs.virginia.edu/papers/ISAIM2018_Lange_etal.pdf (2018)

  25. Lange, P., Nguyen, N., Rothe, J.: The price to pay for forgoing normalization in fair division of indivisible goods. In: Nonarchival website proceedings of the 11th Multidisciplinary Workshop on Advances in Preference Handling. http://www.mpref-2018.preflib.org/wp-content/uploads/2017/12/paper_10.pdf (2018)

  26. Lipton, R., Markakis, E., Mossel, E., Saberi, A.: On approximately fair allocations of indivisible goods. In: Proceedings of the 5th ACM conference on electronic commerce, pp. 125–131. ACM Press (2004)

  27. Nguyen, N., Nguyen, T., Roos, M., Rothe, J.: Computational complexity and approximability of social welfare optimization in multiagent resource allocation. Journal of Autonomous Agents and Multi-Agent Systems 28(2), 256–289 (2014)

    Article  Google Scholar 

  28. Nguyen, T., Roos, M., Rothe, J.: A survey of approximability and inapproximability results for social welfare optimization in multiagent resource allocation. Ann. Math. Artif. Intell. 68(1–3), 65–90 (2013)

    Article  MathSciNet  Google Scholar 

  29. Ramezani, S., Endriss, U.: Nash social welfare in multiagent resource allocation. In: Agent-mediated electronic commerce. Designing trading strategies and mechanisms for electronic markets, pp. 117–131. Springer-Verlag Lecture Notes in Business Information Processing #79 (2010)

  30. Roos, M., Rothe, J.: Complexity of social welfare optimization in multiagent resource allocation. In: Proceedings of the 9th international conference on autonomous agents and multiagent systems, pp. 641–648. IFAAMAS (2010)

  31. Sandholm, T.: Contract types for satisficing task allocation. In: Proceedings of the AAAI spring symposium: Satisficing models, pp. 23–25 (1998)

Download references

Acknowledgements

We thank the anonymous AMAI, ISAIM 2018, and MPREF 2018 reviewers for helpful comments. This work was supported in part by DFG grant RO 1202/14-2.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jörg Rothe.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Preliminary versions of this paper have been presented at the15th International Symposium on Artificial Intelligence and Mathematics (ISAIM 2018) in Fort Lauderdale, USA, and at the11th Multidisciplinary Workshop on Advances in Preference Handling (M-PREF 2018), co-located with AAAI 2018 in New Orleans, USA, both with nonarchival proceedings.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lange, P., Nguyen, NT. & Rothe, J. The price to pay for forgoing normalization in fair division of indivisible goods. Ann Math Artif Intell 88, 817–832 (2020). https://doi.org/10.1007/s10472-019-09659-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10472-019-09659-1

Keywords

Mathematics Subject Classification (2010)

Navigation