Autonomous Agents and Multi-Agent Systems

, Volume 31, Issue 3, pp 628–655 | Cite as

Positional scoring-based allocation of indivisible goods

  • Dorothea Baumeister
  • Sylvain Bouveret
  • Jérôme Lang
  • Nhan-Tam NguyenEmail author
  • Trung Thanh Nguyen
  • Jörg Rothe
  • Abdallah Saffidine


We define a family of rules for dividing m indivisible goods among agents, parameterized by a scoring vector and a social welfare aggregation function. We assume that agents’ preferences over sets of goods are additive, but that the input is ordinal: each agent reports her preferences simply by ranking single goods. Similarly to positional scoring rules in voting, a scoring vector \(s = (s_1, \ldots , s_m)\) consists of m nonincreasing, nonnegative weights, where \(s_i\) is the score of a good assigned to an agent who ranks it in position i. The global score of an allocation for an agent is the sum of the scores of the goods assigned to her. The social welfare of an allocation is the aggregation of the scores of all agents, for some aggregation function \(\star \) such as, typically, \(+\) or \(\min \). The rule associated with s and \(\star \) maps a profile to (one of) the allocation(s) maximizing social welfare. After defining this family of rules, and focusing on some key examples, we investigate some of the social-choice-theoretic properties of this family of rules, such as various kinds of monotonicity, and separability. Finally, we focus on the computation of winning allocations, and on their approximation: we show that for commonly used scoring vectors and aggregation functions this problem is NP-hard and we exhibit some tractable particular cases.


Computational social choice Resource allocation Fair division Indivisible goods Preferences 



We are grateful to the anonymous ECAI’14 and COMSOC’14 reviewers for their helpful comments. In particular, we thank the reviewer who pointed out a proof sketch of Theorem 6 for her or his consent to include the result and its proof. This work was supported in part by Deutsche Forschungsgemeinschaft under grants RO 1202/14-1, RO 1202/14-2, and RO 1202/15-1, by a project of the DAAD-PPP / PHC PROCOPE program entitled “Fair Division of Indivisible Goods: Incomplete Preferences, Communication Protocols and Computational Resistance to Strategic Behavior,” by COST Action IC1205 on Computational Social Choice, by ANR project CoCoRICo-CoDec, by a grant for gender-sensitive universities funded by the NRW Ministry for Innovation, Science, and Research, and by Vietnam’s National Foundation for Science and Technology Development (NAFOSTED Project No. 102.01-2015.33).


  1. 1.
    Aziz, H., Gaspers, S., Mackenzie, S., & Walsh, T. (2015). Fair assignment of indivisible objects under ordinal preferences. Artificial Intelligence, 227, 71–92.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aziz, H., Walsh, T., & Xia, L. (2015). Possible and necessary allocations via sequential mechanisms. In Proceedings of the 24th International Joint Conference on Artificial Intelligence (pp. 468–474). AAAI Press/IJCAI.Google Scholar
  3. 3.
    Bansal, N., & Sviridenko, M. (2006). The Santa Claus problem. In Proceedings of the 38th ACM Symposium on Theory of Computing (pp. 31–40). ACM PressGoogle Scholar
  4. 4.
    Baumeister, D., & Rothe, J. (2015). Preference aggregation by voting. In J. Rothe (Ed.), Economics and computation. An introduction to algorithmic game theory, computational social choice, and fair division, chap 4 (pp. 197–325). Berlin: Springer.Google Scholar
  5. 5.
    Baumeister, D., Bouveret, S., Lang, J., Nguyen, N., Nguyen, T., Rothe, J., et al. (2014). Axiomatic and computational aspects of scoring allocation rules for indivisible goods. In A. Procaccia & T. Walsh (Eds.), Proceedings of the 5th international workshop on computational social choice. Pittsburgh, PA: Carnegie Mellon University.Google Scholar
  6. 6.
    Baumeister, D., Bouveret, S., Lang, J., Nguyen, T., Nguyen, N., & Rothe, J. (2014). Scoring rules for the allocation of indivisible goods. In Proceedings of the 21st European conference on artificial intelligence (pp. 75–80). IOS PressGoogle Scholar
  7. 7.
    Bouveret, S., & Lang, J. (2011). A general elicitation-free protocol for allocating indivisible goods. In Proceedings of the 22nd international joint conference on artificial intelligence (pp 73–78). AAAI Press/IJCAI.Google Scholar
  8. 8.
    Bouveret, S., & Lemaître, M. (2016). Characterizing conflicts in fair division of indivisible goods using a scale of criteria. Journal of Autonomous Agents and Multi-Agent Systems, 30(2), 259–290.CrossRefGoogle Scholar
  9. 9.
    Bouveret, S., Endriss, U., & Lang, J. (2010). Fair division under ordinal preferences: Computing envy-free allocations of indivisible goods. In Proceedings of the 19th European conference on artificial intelligence (pp. 387–392). IOS Press.Google Scholar
  10. 10.
    Bouveret, S., Chevaleyre, Y., & Maudet, N. (2016). Fair allocation of indivisible goods. In F. Brandt, V. Conitzer, U. Endriss, J. Lang, & A. Procaccia (Eds.), Handbook of computational social choice chap 12. Cambridge: Cambridge University Press.Google Scholar
  11. 11.
    Brams, S., & Fishburn, P. (2002). Voting procedures. In K. Arrow, A. Sen, & K. Suzumura (Eds.), Handbook of social choice and welfare chap 4 (pp. 173–236). Amsterdam: North-Holland.CrossRefGoogle Scholar
  12. 12.
    Brams, S., & King, D. (2005). Efficient fair division—help the worst off or avoid envy? Rationality and Society, 17(4), 387–421.CrossRefGoogle Scholar
  13. 13.
    Brams, S., & Taylor, A. (1996). Fair division: From cake-cutting to dispute resolution. Cambridge: Cambridge University Press.CrossRefzbMATHGoogle Scholar
  14. 14.
    Brams, S., Edelman, P., & Fishburn, P. (2004). Fair division of indivisible items. Theory and Decision, 5(2), 147–180.MathSciNetzbMATHGoogle Scholar
  15. 15.
    Budish, E. (2011). The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes. Journal of Political Economy, 119(6), 1061–1103.CrossRefGoogle Scholar
  16. 16.
    Budish, E., & Cantillon, E. (2012). The multi-unit assignment problem: Theory and evidence from course allocation at Harvard. The American Economic Review, 102(5), 2237–2271.CrossRefGoogle Scholar
  17. 17.
    Caragiannis, I., & Procaccia, A. (2011). Voting almost maximizes social welfare despite limited communication. Artificial Intelligence, 175(9–10), 1655–1671.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Caragiannis, I., Kurokawa, D., Moulin, H., Procaccia, A., Shah, N., & Wang, J. (2016). The unreasonable fairness of maximum Nash welfare. In Proceedings of the 17th ACM conference on economics and computation.Google Scholar
  19. 19.
    Darmann, A., & Schauer, J. (2015). Maximizing Nash product social welfare in allocating indivisible goods. European Journal of Operational Research, 247, 548–559.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Elkind, E., Faliszewski, P., Skowron, P., & Slinko, A. (2014). Properties of multiwinner voting rules. In Proceedings of the 13th international conference on autonomous agents and multiagent systems (pp. 53–60). IFAAMAS.Google Scholar
  21. 21.
    Faliszewski, P., & Hemaspaandra, L. (2009). The complexity of power-index comparison. Theoretical Computer Science, 410(1), 101–107.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gardenfors, P. (1973). Assignment problem based on ordinal preferences. Management Science, 20(3), 331–340.CrossRefzbMATHGoogle Scholar
  23. 23.
    Garey, M., & Johnson, D. (1978). “Strong” NP-completeness results: Motivation, examples, and implications. Journal of ACM, 25(3), 499–508.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Garey, M., & Johnson, D. (1979). Computers and intractability: A guide to the theory of NP-completeness. San Francisco: W. H. Freeman and Company.zbMATHGoogle Scholar
  25. 25.
    Garg, N., Kavitha, T., Kumar, A., Mehlhorn, K., & Mestre, J. (2010). Assigning papers to referees. Algorithmica, 58(1), 119–136.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Golovin, D. (2005). Max-min fair allocation of indivisible goods. Tech. Rep. CMU-CS-05-144, School of Computer Science. Carnegie Mellon University.Google Scholar
  27. 27.
    Hemaspaandra, E., & Hemaspaandra, L. (2007). Dichotomy for voting systems. Journal of Computer and System Sciences, 73(1), 73–83.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Herreiner, D., & Puppe, C. (2002). A simple procedure for finding equitable allocations of indivisible goods. Social Choice and Welfare, 19(2), 415–430.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kohler, D., & Chandrasekaran, R. (1971). A class of sequential games. Operations Research, 19(2), 270–277.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Lang, J., & Rothe, J. (2015). Fair division of indivisible goods. In J. Rothe (Ed.), Economics and computation. An introduction to algorithmic game theory, computational social choice, and fair division chap 8 (pp. 493–550). Berlin: Springer.Google Scholar
  31. 31.
    Manlove, D. (2013). Algorithmics of matching under preferences. Series on theoretical computer science. Singapore: World Scientific Publishing.CrossRefGoogle Scholar
  32. 32.
    Moulin, H. (1995). Cooperative microeconomics: A game-theoretic introduction. Upper Saddle River, NJ: Prentice Hall.CrossRefGoogle Scholar
  33. 33.
    Moulin, H. (2004). Fair division and collective welfare. Cambridge, MA: MIT Press.Google Scholar
  34. 34.
    Nguyen, N., Baumeister, D., & Rothe, J. (2015). Strategy-proofness of scoring allocation correspondences for indivisible goods. In Proceedings of the 24th international joint conference on artificial intelligence (pp. 1127–1133). AAAI Press/IJCAI.Google Scholar
  35. 35.
    Pruhs, K., & Woeginger, G. (2012). Divorcing made easy. In: Proceedings of the 6th international conference on fun with algorithms (pp. 305–314). Springer.Google Scholar
  36. 36.
    Roth, A., & Sotomayor, M. (1990). Two-sided matching: A study in game-theoretic modeling and analysis. Econometric society monographs. Cambridge: Cambridge University Press.CrossRefzbMATHGoogle Scholar
  37. 37.
    Skowron, P., Faliszewski, P., & Slinko, A. (2013). Fully proportional representation as resource allocation: Approximability results. In Proceedings of the 23rd International Joint Conference on Artificial Intelligence (pp. 353–359). AAAI Press/IJCAI.Google Scholar
  38. 38.
    Thomson, W. (2011). Consistency and its converse: An introduction. Review of Economic Design, 15(4), 257–291.MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Wilson, L. (1977). Assignment using choice lists. Operational Research Quarterly, 28, 569–578.CrossRefzbMATHGoogle Scholar
  40. 40.
    Zwicker, W. (2016). Introduction to the theory of voting. In F. Brandt, V. Conitzer, U. Endriss, J. Lang, & A. Procaccia (Eds.), Handbook of computational social choice chap 2. Cambridge: Cambridge University Press.Google Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Heinrich-Heine Universität DüsseldorfDüsseldorfGermany
  2. 2.Univ. Grenoble Alpes, CNRS, LIGGrenobleFrance
  3. 3.LAMSADEUniversité Paris-Dauphine, Place du Maréchal de Lattre de TassignyParis Cedex 16France
  4. 4.Hai Phong UniversityHai PhongVietnam
  5. 5.University of New South WalesSydneyAustralia

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