Advertisement

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

  • Pascal Lange
  • Nhan-Tam Nguyen
  • Jörg RotheEmail author
Article
  • 4 Downloads

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.

Keywords

Fair division Indivisible goods Social welfare Computational complexity 

Mathematics Subject Classification (2010)

91B32 68Q17 68Q25 68T42 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

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.

References

  1. 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)Google Scholar
  2. 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)Google Scholar
  3. 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)Google Scholar
  4. 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)Google Scholar
  5. 5.
    Brams, S., Taylor, A.: Fair division: From cake-cutting to dispute resolution. Cambridge University Press, Cambridge (1996)CrossRefzbMATHGoogle Scholar
  6. 6.
    Budish, E.: The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes. Journal of Political Economy 119(6), 1061–1103 (2011)CrossRefGoogle Scholar
  7. 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)Google Scholar
  8. 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)zbMATHGoogle Scholar
  9. 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)Google Scholar
  10. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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)Google Scholar
  12. 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)CrossRefGoogle Scholar
  13. 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)Google Scholar
  14. 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)Google Scholar
  15. 15.
    Dunne, P., Wooldridge, M., Laurence, M.: The complexity of contract negotiation. Artif. Intell. 164(1–2), 23–46 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Endriss, U., Maudet, N., Sadri, F., Toni, F.: Negotiating socially optimal allocations of resources. J. Artif. Intell. Res. 25, 315–348 (2006)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-completeness. W. H Freeman and Company (1979)Google Scholar
  18. 18.
    Golovin, D.: Max-min fair allocation of indivisible goods. Tech. Rep. CMU-CS-05-144, School of Computer Science Carnegie Mellon University (2005)Google Scholar
  19. 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)Google Scholar
  20. 20.
    Irving, R., Leather, P., Gusfield, D.: An efficient algorithm for the “optimal” stable marriage. Journal of the ACM 34(3), 532–543 (1987)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Karp, R.: Reducibility among Combinatorial Problems. In: Miller, R., Thatcher, J. (eds.) Complexity of computer computations, pp. 85–103. Plenum Press (1972)Google Scholar
  22. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 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)Google Scholar
  24. 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. 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. 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)Google Scholar
  27. 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)CrossRefzbMATHGoogle Scholar
  28. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 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)Google Scholar
  30. 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)Google Scholar
  31. 31.
    Sandholm, T.: Contract types for satisficing task allocation. In: Proceedings of the AAAI spring symposium: Satisficing models, pp. 23–25 (1998)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut für InformatikHeinrich-Heine-Universität DüsseldorfDüsseldorfGermany

Personalised recommendations