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

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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 \(P=NP\), there can be no polynomial time extension of GAL to the case of arbitrary number of agents.

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Notes

  1. The notion of envy-freeness that they use is equivalent to SD (stochastic dominance) envy-freeness (Aziz et al. 2014) and necessary envy-freeness (Bouveret et al. 2010).

  2. This feasibility check is phrased in a different way in the original description of AL but is equivalent to checking for EF.

  3. The view of EF as being defined with respect to the SD relation makes it easy to argue for a maximal EF assignment.

  4. The argument in Theorem 3 of Brams et al. (2014) only shows that for strict preferences, AL finds maximally EF assignment. It does not show that for strict preferences, AL efficiently computes a complete EF assignment if a complete EF assignment exists.

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Acknowledgments

The author thanks Steven Brams for sharing the paper on the AL method with him. He also appreciates Sajid Aziz, Steven Brams, Christian Klamler, and Chun Ye for their useful feedback and to the reviewers for their detailed comments. NICTA is funded by the Australian Government through the Department of Communications and the Australian Research Council through the ICT Centre of Excellence Program.

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  1. NICTA and UNSW, Sydney, 2033, Australia

    Haris Aziz

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  1. Haris Aziz
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Aziz, H. A generalization of the AL method for fair allocation of indivisible objects. Econ Theory Bull 4, 307–324 (2016). https://doi.org/10.1007/s40505-015-0089-1

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