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Cost sharing solutions defined by non-negative eigenvectors


  • Subiza, Begoña

    () (Universidad de Alicante, Departamento de Métodos Cuantitativos y Teoría Económica)

  • Silva-Reus, José Ángel

    () (Instituto Interuniversitario de Desarrollo Social y Paz)

  • Peris, Josep E.

    () (Universidad de Alicante, Departamento de Métodos Cuantitativos y Teoría Económica)


We consider a cost sharing problem, where each individual is identi ed by a characteristic (a positive real number) ci: The two main positions on how to share a common cost M are the Egalitarian and the Proportional solutions. These solutions can be obtained as the Perron's eigenvectors (right and left, respectively) of a characteristics matrix A, with rk(A) = 1; de fined from the individuals' characteristics ci: Then, the right Perron's eigenvector associated to any Levinger's transformation L(α) = α A'+(1-α )A; α є [0;1] ; provides a dif ferent solution to the cost sharing problem (from the egalitarian one, for α = 0; to the proportional one, for α = 1). We are interested in analyzing the properties of the components of these Perron's eigenvectors as (non linear) functions of the parameter α that de fines the convex combination of matrices A and A': These components de fine the solution of the cost sharing problem, that could be understood as a compromise between the egalitarian and proportional approaches. We prove that when the associated positive eigenvector is normalized, its components have a monotone behaviour in the unit interval [0;1]: Additional properties and a way of selecting a particular compromise solution are provided.

Suggested Citation

  • Subiza, Begoña & Silva-Reus, José Ángel & Peris, Josep E., 2013. "Cost sharing solutions defined by non-negative eigenvectors," QM&ET Working Papers 13-6, University of Alicante, D. Quantitative Methods and Economic Theory.
  • Handle: RePEc:ris:qmetal:2013_006

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    References listed on IDEAS

    1. Fiestras-Janeiro, M.G. & García-Jurado, I. & Meca, A. & Mosquera, M.A., 2011. "Cooperative game theory and inventory management," European Journal of Operational Research, Elsevier, vol. 210(3), pages 459-466, May.
    2. Peris, Josep E. & Jiménez-Gómez, José M., 2012. "A Proportional Approach to Bankruptcy Problems with a guaranteed minimum," QM&ET Working Papers 12-7, University of Alicante, D. Quantitative Methods and Economic Theory.
    3. Giménez-Gómez, José-Manuel & Peris, Josep E., 2014. "A proportional approach to claims problems with a guaranteed minimum," European Journal of Operational Research, Elsevier, vol. 232(1), pages 109-116.
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      • Meca-Martinez, A. & Timmer, J.B. & Garcia-Jurado, I. & Borm, P.E.M., 1999. "Inventory Games," Discussion Paper 1999-53, Tilburg University, Center for Economic Research.
      • Meca, A. & Timmer, J.B. & Garcia-Jurado, I. & Borm, P.E.M., 2004. "Inventory games," Other publications TiSEM 49368f2d-02fc-49c9-9d74-8, Tilburg University, School of Economics and Management.
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    6. Karsten, Frank & Basten, Rob J.I., 2014. "Pooling of spare parts between multiple users: How to share the benefits?," European Journal of Operational Research, Elsevier, vol. 233(1), pages 94-104.
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    More about this item


    Cost Sharing; Egalitarian; Proportional; Perrons Eigenvector; Compromise Solution;

    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
    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations

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