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A generalization of the AL method for fair allocation of indivisible objects

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  • Haris Aziz

    (NICTA and UNSW)

Abstract

We consider the assignment problem in which agents express ordinal preferences over m objects and the objects are allocated to the agents based on the preferences. In a recent paper Brams, Kilgour, and Klamler (Not AMS 61(2):130–141, 2014) , presented the AL method to compute an envy-free assignment for two agents. The AL method crucially depends on the assumption that agents have strict preferences over objects. We generalize the AL method to the case where agents may express indifferences and prove the axiomatic properties satisfied by the algorithm. As a result of the generalization, we also get a O(m) speedup on previous algorithms to check whether a complete envy-free assignment exists or not. Finally, we show that unless $$P=NP$$ P = N P , there can be no polynomial time extension of GAL to the case of arbitrary number of agents.

Suggested Citation

  • Haris Aziz, 2016. "A generalization of the AL method for fair allocation of indivisible objects," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 4(2), pages 307-324, October.
    • Handle: RePEc:spr:etbull:v:4:y:2016:i:2:d:10.1007_s40505-015-0089-1
    DOI: 10.1007/s40505-015-0089-1
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    References listed on IDEAS

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    1. Steven J. Brams & Todd R. Kaplan, 2002. "Dividing the Indivisible: Procedures for Allocating Cabinet Ministries to Political Parties in a Parliamentary System," Discussion Papers 0202, University of Exeter, Department of Economics.
    2. Steven Brams & D. Kilgour & Christian Klamler, 2012. "The undercut procedure: an algorithm for the envy-free division of indivisible items," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 39(2), pages 615-631, July.
    3. John Cullinan & Samuel Hsiao & David Polett, 2014. "A Borda count for partially ordered ballots," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 42(4), pages 913-926, April.
    4. Bogomolnaia, Anna & Moulin, Herve, 2001. "A New Solution to the Random Assignment Problem," Journal of Economic Theory, Elsevier, vol. 100(2), pages 295-328, October.
    5. Brams, Steven J. & Kilgour, D. Marc & Klamler, Christian, 2013. "Two-Person Fair Division of Indivisible Items: An Efficient, Envy-Free Algorithm," MPRA Paper 47400, University Library of Munich, Germany.
    6. Steven J. Brams & Paul H. Edelman & Peter C. Fishburn, 2003. "Fair Division Of Indivisible Items," Theory and Decision, Springer, vol. 55(2), pages 147-180, September.
    7. Steven J. Brams & Peter C. Fishburn, 2000. "Fair division of indivisible items between two people with identical preferences: Envy-freeness, Pareto-optimality, and equity," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 17(2), pages 247-267.
    8. Aziz, Haris & Brandt, Felix & Harrenstein, Paul, 2013. "Pareto optimality in coalition formation," Games and Economic Behavior, Elsevier, vol. 82(C), pages 562-581.
    9. Demko, Stephen & Hill, Theodore P., 1988. "Equitable distribution of indivisible objects," Mathematical Social Sciences, Elsevier, vol. 16(2), pages 145-158, October.
    10. Katta, Akshay-Kumar & Sethuraman, Jay, 2006. "A solution to the random assignment problem on the full preference domain," Journal of Economic Theory, Elsevier, vol. 131(1), pages 231-250, November.
    11. Steven J. Brams & Daniel L. King, 2005. "Efficient Fair Division," Rationality and Society, , vol. 17(4), pages 387-421, November.
    12. Steven J. Brams & Todd R. Kaplan, 2004. "Dividing the Indivisible," Journal of Theoretical Politics, , vol. 16(2), pages 143-173, April.
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    Cited by:

    1. Aziz, Haris & Schlotter, Ildikó & Walsh, Toby, 2024. "Computational complexity of necessary envy-freeness," Mathematical Social Sciences, Elsevier, vol. 127(C), pages 86-98.

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    More about this item

    Keywords

    Fair division; Envy-freeness; Pareto optimality; AL method;
    All these keywords.

    JEL classification:

    • C70 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - General
    • D61 - Microeconomics - - Welfare Economics - - - Allocative Efficiency; Cost-Benefit Analysis
    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations

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