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Axiomatic characterizations of the family of Weighted priority values

Author

Listed:
  • Sylvain Ferrières

    (GATE Lyon Saint-Étienne - Groupe d'Analyse et de Théorie Economique Lyon - Saint-Etienne - UL2 - Université Lumière - Lyon 2 - UJM - Université Jean Monnet - Saint-Étienne - EM - EMLyon Business School - CNRS - Centre National de la Recherche Scientifique)

  • Adriana Navarro-Ramos

    (GATE Lyon Saint-Étienne - Groupe d'Analyse et de Théorie Economique Lyon - Saint-Etienne - UL2 - Université Lumière - Lyon 2 - UJM - Université Jean Monnet - Saint-Étienne - EM - EMLyon Business School - CNRS - Centre National de la Recherche Scientifique)

  • Philippe Solal

    (GATE Lyon Saint-Étienne - Groupe d'Analyse et de Théorie Economique Lyon - Saint-Etienne - UL2 - Université Lumière - Lyon 2 - UJM - Université Jean Monnet - Saint-Étienne - EM - EMLyon Business School - CNRS - Centre National de la Recherche Scientifique)

  • Sylvain Béal

    (CRESE - Centre de REcherches sur les Stratégies Economiques (UR 3190) - UFC - Université de Franche-Comté - UBFC - Université Bourgogne Franche-Comté [COMUE])

Abstract
We introduce a new family of values for TU-games with a priority structure, which both containsthe Priority value recently introduced by B´eal et al. (2022) and the Weighted Shapley values (Kalaiand Samet, 1987). Each value of this family is called a Weighted priority value and is constructedas follows. A strictly positive weight is associated with each agent and the agents are partiallyordered according to a binary relation. An agent is a priority agent with respect to a coalitionif it is maximal in this coalition with respect to the partial order. A Weighted priority valuedistributes the dividend of each coalition among the priority agents of this coalition in proportionto their weights. We provide an axiomatic characterization of the family of the Weighted Shapleyvalues without the additivity axiom. To this end, we borrow the Priority agent out axiom fromB´eal et al. (2022), which is used to axiomatize the Priority value. We also reuse, in our domain,the principle of Superweak differential marginality introduced by Casajus (2018) to axiomatizethe Positively weighted Shapley values (Shapley, 1953). We add a new axiom of Independence ofnull agent position which indicates that the position of a null agent in the partial order does notaffect the payoff of the other agents. Together with Efficiency, the above axioms characterize theWeighted Shapley values. We show that this axiomatic characterization holds on the subdomainwhere the partial order is structured by levels. This entails an alternative characterization of theWeighted Shapley values. Two alternative characterizations are obtained by replacing our principleof Superweak differential marginality by Additivity and invoking other axioms.

Suggested Citation

  • Sylvain Ferrières & Adriana Navarro-Ramos & Philippe Solal & Sylvain Béal, 2023. "Axiomatic characterizations of the family of Weighted priority values," Post-Print hal-04053363, HAL.
    • Handle: RePEc:hal:journl:hal-04053363
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    References listed on IDEAS

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    1. René Brink & Gerard Laan & Valeri Vasil’ev, 2014. "Constrained core solutions for totally positive games with ordered players," International Journal of Game Theory, Springer;Game Theory Society, vol. 43(2), pages 351-368, May.
    2. René van den Brink, 2002. "An axiomatization of the Shapley value using a fairness property," International Journal of Game Theory, Springer;Game Theory Society, vol. 30(3), pages 309-319.
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    5. Casajus, André, 2018. "Symmetry, mutual dependence, and the weighted Shapley values," Journal of Economic Theory, Elsevier, vol. 178(C), pages 105-123.
    6. Chun, Youngsub, 1991. "On the Symmetric and Weighted Shapley Values," International Journal of Game Theory, Springer;Game Theory Society, vol. 20(2), pages 183-190.
    7. Sylvain Béal & Sylvain Ferrières & Philippe Solal, 2022. "The priority value for cooperative games with a priority structure," International Journal of Game Theory, Springer;Game Theory Society, vol. 51(2), pages 431-450, June.
    8. André Casajus, 2011. "Differential marginality, van den Brink fairness, and the Shapley value," Theory and Decision, Springer, vol. 71(2), pages 163-174, August.
    9. Casajus, André & Huettner, Frank, 2013. "Null players, solidarity, and the egalitarian Shapley values," Journal of Mathematical Economics, Elsevier, vol. 49(1), pages 58-61.
    10. René Brink & Gerard Laan & Valeri Vasil’ev, 2007. "Component efficient solutions in line-graph games with applications," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 33(2), pages 349-364, November.
    11. van den Brink, Rene & Gilles, Robert P., 1996. "Axiomatizations of the Conjunctive Permission Value for Games with Permission Structures," Games and Economic Behavior, Elsevier, vol. 12(1), pages 113-126, January.
    12. Casajus, André, 2021. "Weakly balanced contributions and the weighted Shapley values," Journal of Mathematical Economics, Elsevier, vol. 94(C).
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