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Decomposition of interaction indices: alternative interpretations of cardinal-probabilistic interaction indices

Author

Listed:
  • Sébastien Courtin

    (IDEES - Identité et Différenciation de l’Espace, de l’Environnement et des Sociétés - UNICAEN - Université de Caen Normandie - NU - Normandie Université - ULH - Université Le Havre Normandie - NU - Normandie Université - UNIROUEN - Université de Rouen Normandie - NU - Normandie Université - CNRS - Centre National de la Recherche Scientifique - IRIHS - Institut de Recherche Interdisciplinaire Homme et Société - UNIROUEN - Université de Rouen Normandie - NU - Normandie Université, UNICAEN - Université de Caen Normandie - NU - Normandie Université)

  • Rodrigue Tido Takeng

    (CREM - Centre de recherche en économie et management - UNICAEN - Université de Caen Normandie - NU - Normandie Université - UR - Université de Rennes - CNRS - Centre National de la Recherche Scientifique)

  • Frédéric Chantreuil

    (CEMOI - Centre d'Économie et de Management de l'Océan Indien - UR - Université de La Réunion)

Abstract

In cooperative game theory, the concept of interaction index is an extension of the concept of value, considering interaction among players. In this paper we focus on cardinal-probabilistic interaction indices which are generalizations of the class of semivalues. We provide two types of decompositions. With the first one, a cardinal-probabilistic interaction index for a given coalition equals the difference between its external interaction index (or co-Möbius transfom) and a weighted sum of the individual impact of the remaining players on the interaction index of the considered coalition. The second decomposition, based on the notion of the "decomposer", splits an interaction index into a direct part, the decomposer, which measures the interaction in the coalition considered, and an indirect part, which indicates how all remaining players individually affect the interaction of the coalition considered. We propose alternative characterization of the cardinal-probabilistic interaction indices. We then propose an illustration with a well-known example in Multicriteria Aid for Decisions.

Suggested Citation

  • Sébastien Courtin & Rodrigue Tido Takeng & Frédéric Chantreuil, 2020. "Decomposition of interaction indices: alternative interpretations of cardinal-probabilistic interaction indices ," Working Papers hal-02952516, HAL.
    • Handle: RePEc:hal:wpaper:hal-02952516
    DOI: 10.1007/s11238-023-09970-8
    Note: View the original document on HAL open archive server: https://hal.science/hal-02952516
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    References listed on IDEAS

    as
    1. Mayag, Brice & Bouyssou, Denis, 2020. "Necessary and possible interaction between criteria in a 2-additive Choquet integral model," European Journal of Operational Research, Elsevier, vol. 283(1), pages 308-320.
    2. Fujimoto, Katsushige & Kojadinovic, Ivan & Marichal, Jean-Luc, 2006. "Axiomatic characterizations of probabilistic and cardinal-probabilistic interaction indices," Games and Economic Behavior, Elsevier, vol. 55(1), pages 72-99, April.
    3. Brice Mayag & Michel Grabisch & Christophe Labreuche, 2011. "A representation of preferences by the Choquet integral with respect to a 2-additive capacity," Theory and Decision, Springer, vol. 71(3), pages 297-324, September.
    4. Michel Grabisch & Christophe Labreuche, 2010. "A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid," Annals of Operations Research, Springer, vol. 175(1), pages 247-286, March.
    5. Marc Roubens & Michel Grabisch, 1999. "An axiomatic approach to the concept of interaction among players in cooperative games," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(4), pages 547-565.
    6. Pradeep Dubey & Abraham Neyman & Robert James Weber, 1981. "Value Theory Without Efficiency," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 122-128, February.
    7. Guillermo Owen, 1972. "Multilinear Extensions of Games," Management Science, INFORMS, vol. 18(5-Part-2), pages 64-79, January.
    8. Frédéric Chantreuil & Sébastien Courtin & Kevin Fourrey & Isabelle Lebon, 2019. "A note on the decomposability of inequality measures," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 53(2), pages 283-298, August.
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    Cited by:

    1. Rodrigue Tido Takeng & Arnold Cedrick Soh Voutsa & Kévin Fourrey, 2023. "Decompositions of inequality measures from the perspective of the Shapley–Owen value," Theory and Decision, Springer, vol. 94(2), pages 299-331, February.

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    More about this item

    Keywords

    Game theory; Multicriteria Aid for Decisions; Cardinal-probabilistic interaction indices; External interaction index JEL Codes: C71;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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