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Inheritance of Convexity for Partition Restricted Games

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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,vP) defined by vP(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,vP). 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,vS) (for any Ø ≠ S ⊆ N) defined for any A ⊆ N by vS(A) = |A ∩ S | − 1 if |A ∩ S | ≥ 1, and vs(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 Pmin correspondence (Pmin(A) is built by deleting edges of minimum weight in the subgraph GA of a weighted communication graph G, we show that inheritance of convexity for unanimity games already implies inheritance of convexity. Assuming only inheritance of superadditivity, we also compute the Shapley value of the restricted game (N,vP) for an arbitrary correspondence P.

Suggested Citation

  • Alexandre Skoda, 2016. "Inheritance of Convexity for Partition Restricted Games," Post-Print halshs-01318105, HAL.
    • Handle: RePEc:hal:journl:halshs-01318105
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-01318105
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    References listed on IDEAS

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    1. Michel Grabisch, 2013. "The core of games on ordered structures and graphs," Annals of Operations Research, Springer, vol. 204(1), pages 33-64, April.
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    8. Alexandre Skoda, 2016. "Complexity of inheritance of F-convexity for restricted games induced by minimum partitions," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01382502, HAL.
    9. E. Algaba & J.M. Bilbao & J.J. López, 2001. "A unified approach to restricted games," Theory and Decision, Springer, vol. 50(4), pages 333-345, June.
    10. Alexandre Skoda, 2016. "Complexity of inheritance of F-convexity for restricted games induced by minimum partitions," Post-Print halshs-01382502, HAL.
    11. Alexandre Skoda, 2016. "Complexity of inheritance of F-convexity for restricted games induced by minimum partitions," Documents de travail du Centre d'Economie de la Sorbonne 16055, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    12. Alexandre Skoda, 2016. "Convexity of Network Restricted Games Induced by Minimum Partitions," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-01305005, HAL.
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