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Inheritance of convexity for partition restricted games

Author

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  • Alexandre Skoda

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

Abstract

A correspondence P associates to every subset A ⊆ N a partition P ( A ) of A and to every game ( N , v ) , the P -restricted game ( N , v ¯ ) defined by v ¯ ( A ) = ∑ F ∈ P ( A ) v ( F ) for all A ⊆ N . We give necessary and sufficient conditions on P to have inheritance of convexity from ( N , v ) to ( N , v ¯ ) . The main condition is a cyclic intersecting sequence free condition. As a consequence, we only need to verify inheritance of convexity for unanimity games and for the small class of extremal convex games ( N , v S ) (for any 0̸ ≠ S ⊆ N ) defined for any A ⊆ N by v S ( A ) = | A ∩ S | − 1 if A ∩ S ≠ 0̸ , and v S ( A ) = 0 otherwise. In particular, when ( N , v ¯ ) corresponds to Myerson's network-restricted game, inheritance of convexity can be verified by this way. For the P min correspondence ( P min ( A ) is built by deleting edges of minimum weight in the subgraph G A of a weighted communication graph G ), we show that inheritance of convexity for unanimity games already implies inheritance of convexity

Suggested Citation

  • Alexandre Skoda, 2017. "Inheritance of convexity for partition restricted games," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01487381, HAL.
    • Handle: RePEc:hal:cesptp:halshs-01487381
    DOI: 10.1016/j.disopt.2017.01.004
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    Citations

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    Cited by:

    1. Alexandre Skoda, 2017. "Convexity of Graph-Restricted Games Induced by Minimum Partitions," Documents de travail du Centre d'Economie de la Sorbonne 17049, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    2. Alexandre Skoda, 2017. "Convexity of Graph-Restricted Games Induced by Minimum Partitions," Post-Print halshs-01659804, HAL.
    3. Alexandre Skoda, 2020. "Inheritance of Convexity for the P˜min-Restricted Game," Post-Print halshs-02967120, HAL.
    4. Alexandre Skoda, 2017. "Convexity of Graph-Restricted Games Induced by Minimum Partitions," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01659804, HAL.
    5. Alexandre Skoda, 2020. "Inheritance of Convexity for the P˜min-Restricted Game," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-02967120, HAL.
    6. A. Skoda, 2021. "Inheritance of convexity for the $$\mathcal {P}_{\min }$$ P min -restricted game," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 93(1), pages 1-32, February.
    7. Alexandre Skoda, 2019. "Convexity of graph-restricted games induced by minimum partitions," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01617023, HAL.
    8. Alexandre Skoda, 2020. "Inheritance of Convexity for the P˜min-Restricted Game," Documents de travail du Centre d'Economie de la Sorbonne 20020, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    9. Alexandre Skoda, 2019. "Convexity of graph-restricted games induced by minimum partitions," Post-Print halshs-01617023, HAL.

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