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Weak and Strong Consistency in Additive Cost Sharing

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  • Eric Friedman

    (Rutgers University)

Abstract

In this paper we study consistency in the context of additive cost sharing mechanisms. We contrast an extremely weak notion of consistency with the standard definition, which we denote strong consistency. First we show that many well known CSMs are consistent in both senses: Aumann-Shapley, random order methods, Shapley-Shubik, Serial cost, and also the weighted versions of these. We also provide general conditions which characterize the different types of consistency-- all methods generated by separable paths are weakly consistent, while those generated by associative paths are strongly consistent. Using this characterization, we show that any weakly consistent method which is demand monotonic is also strongly consistent.Next, we analyze the conditions under which a cost sharing method (CSM) for an arbitrary number of agents is uniquely defined by its behavior in the two agent case. We show that under weak (resp. strong) consistency all CSMs generated by a single separable (resp. associative) path are uniquely defined by their behavior on two agent problems. These include Aumann-Shapley, random order methods, Serial cost, and also the weighted versions of these. Shapley-Shubik, which is generated by multiple paths has a unique symmetric extensions, but also other nonsymmetric extensions as do many CSMs which are not generated by a single path.

Suggested Citation

  • Eric Friedman, 1997. "Weak and Strong Consistency in Additive Cost Sharing," Departmental Working Papers 199707, Rutgers University, Department of Economics.
    • Handle: RePEc:rut:rutres:199707
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    References listed on IDEAS

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    1. Thomson, W., 1996. "Consistent Allocation Rules," RCER Working Papers 418, University of Rochester - Center for Economic Research (RCER).
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    4. Robert J. Weber, 1977. "Probabilistic Values for Games," Cowles Foundation Discussion Papers 471R, Cowles Foundation for Research in Economics, Yale University.
    5. Sergiu Hart & Dov Monderer, 1997. "Potentials and Weighted Values of Nonatomic Games," Mathematics of Operations Research, INFORMS, vol. 22(3), pages 619-630, August.
    6. Louis J. Billera & David C. Heath, 1982. "Allocation of Shared Costs: A Set of Axioms Yielding A Unique Procedure," Mathematics of Operations Research, INFORMS, vol. 7(1), pages 32-39, February.
    7. Friedman, Eric & Moulin, Herve, 1999. "Three Methods to Share Joint Costs or Surplus," Journal of Economic Theory, Elsevier, vol. 87(2), pages 275-312, August.
    8. Martin Shubik, 1962. "Incentives, Decentralized Control, the Assignment of Joint Costs and Internal Pricing," Management Science, INFORMS, vol. 8(3), pages 325-343, April.
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    Cited by:

    1. Koster, M., 2005. "Sharing Variable Returns of Cooperation," CeNDEF Working Papers 05-06, Universiteit van Amsterdam, Center for Nonlinear Dynamics in Economics and Finance.

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    cost sharing;

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