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From the bankruptcy problem and its Concede-and-Divide solution to the assignment problem and its Fair Division solution

Author

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  • Christian Trudeau

    () (Department of Economics, University of Windsor)

Abstract

We revisit two classic problems: the assignment problem, in which agents create value when matched with a partner, and the bankruptcy problem, in which we need to share an endowment among agents with conflicting claims. We show that since Core Selection constrains us to exactly divide the value created by a pair of matched agents, the assignment problem can be seen as a two-player bankruptcy problem. This interpretation allows us to show that the classic Concede-and-Divide (Aumann and Maschler, 1985) sharing method for the bankruptcy problem is equivalent to the Fair Division solution (Thompson, 1981) for the assignment problem, itself the average of the extreme points of the core of Demange (1982) and Leonard (1983). We then exploit the link between the two problems to offer two characterizations of the Fair Division solution. The key property is an adapation of the Minimal Rights First property (Curiel, Maschler and Tijs, 1987) for the bankruptcy problem. The minimal rights of a claimant is what is left of the endowment, if any, when all claimants but himself have received their full claims. The property states that we obtain the same shares if we distribute the minimal rights first, adjust the claims and endowment and proceed on the reduced problem or simply ignore them and proceed on the original problem. In assignment problems, the conceptual equivalent of minimal rights are the minimal core allocations. Given the important role that minimal core allocations play in this link between assignment and bankruptcy problems, it is important to be able to compute them efficiently. We provide a new algorithm to compute them.

Suggested Citation

  • Christian Trudeau, 2015. "From the bankruptcy problem and its Concede-and-Divide solution to the assignment problem and its Fair Division solution," Working Papers 1506, University of Windsor, Department of Economics.
  • Handle: RePEc:wis:wpaper:1506
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    File URL: http://web2.uwindsor.ca/economics/RePEc/wis/pdf/1506.pdf
    File Function: Second version, 2016
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    References listed on IDEAS

    as
    1. van den Brink, René & Pintér, Miklós, 2015. "On axiomatizations of the Shapley value for assignment games," Journal of Mathematical Economics, Elsevier, vol. 60(C), pages 110-114.
    2. Hamers, Herbert & Klijn, Flip & Solymosi, Tamas & Tijs, Stef & Pere Villar, Joan, 2002. "Assignment Games Satisfy the CoMa-Property," Games and Economic Behavior, Elsevier, vol. 38(2), pages 231-239, February.
    3. Leonard, Herman B, 1983. "Elicitation of Honest Preferences for the Assignment of Individuals to Positions," Journal of Political Economy, University of Chicago Press, vol. 91(3), pages 461-479, June.
    4. Nunez, Marina & Rafels, Carles, 2003. "Characterization of the extreme core allocations of the assignment game," Games and Economic Behavior, Elsevier, vol. 44(2), pages 311-331, August.
    Full references (including those not matched with items on IDEAS)

    More about this item

    Keywords

    assignment problems; bankruptcy problems; core; fair division; concede-and-divide; minimal rights.;

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • D63 - Microeconomics - - Welfare Economics - - - Equity, Justice, Inequality, and Other Normative Criteria and Measurement

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