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Games on lattices, multichoice games and the shapley value: a new approach

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  • Michel Grabisch
  • Fabien Lange
Abstract
Multichoice games, as well as many other recent attempts to generalize the notion of classical cooperative game, can be casted into the framework of lattices. We propose a general definition for games on lattices, together with an interpretation. Several definitions of the Shapley value of a multichoice games have already been given, among them the original one due to Hsiao and Raghavan, and the one given by Faigle and Kern. We propose a new approach together with its axiomatization, more in the spirit of the original axiomatization of Shapley, and avoiding a high computational complexity. Copyright Springer-Verlag 2007

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  • Michel Grabisch & Fabien Lange, 2007. "Games on lattices, multichoice games and the shapley value: a new approach," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 65(1), pages 153-167, February.
    • Handle: RePEc:spr:mathme:v:65:y:2007:i:1:p:153-167
    DOI: 10.1007/s00186-006-0109-x
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    1. Rodica Branzei & Dinko Dimitrov & Stef Tijs, 2008. "Models in Cooperative Game Theory," Springer Books, Springer, edition 0, number 978-3-540-77954-4, February.
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    4. Tijs, S.H. & Brânzei, R. & Ishihara, S. & Muto, S., 2004. "On cores and stable sets for fuzzy games," Other publications TiSEM 66dd20be-cb4b-4b6d-937e-0, Tilburg University, School of Economics and Management.
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    Cited by:

    1. J.M. Alonso Meijide & M. à lvarez-Mozos & M.G. Fiestras-Janeiro & A. Jiménez-Losada, 2024. "The Partition Lattice Value for Global Cooperative Games," UB School of Economics Working Papers 2024/473, University of Barcelona School of Economics.
    2. Michel Grabisch & Agnieszka Rusinowska, 2010. "A model of influence in a social network," Theory and Decision, Springer, vol. 69(1), pages 69-96, July.
    3. David Lowing & Kevin Techer, 2022. "Marginalism, egalitarianism and efficiency in multi-choice games," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 59(4), pages 815-861, November.
    4. Michael Jones & Jennifer Wilson, 2013. "Two-step coalition values for multichoice games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(1), pages 65-99, February.
    5. Lange, Fabien & Grabisch, Michel, 2009. "Values on regular games under Kirchhoff's laws," Mathematical Social Sciences, Elsevier, vol. 58(3), pages 322-340, November.
    6. GRABISCH, Michel & LABREUCHE, Christophe & RIDAOUI, Mustapha, 2019. "On importance indices in multicriteria decision making," European Journal of Operational Research, Elsevier, vol. 277(1), pages 269-283.
    7. C. Manuel & E. González-Arangüena & R. Brink, 2013. "Players indifferent to cooperate and characterizations of the Shapley value," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(1), pages 1-14, February.
    8. Faigle, U. & Grabisch, M. & Heyne, M., 2010. "Monge extensions of cooperation and communication structures," European Journal of Operational Research, Elsevier, vol. 206(1), pages 104-110, October.
    9. Michel Grabisch & Lijue Xie, 2011. "The restricted core of games on distributive lattices: how to share benefits in a hierarchy," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 73(2), pages 189-208, April.
    10. Béal, Sylvain & Rémila, Eric & Solal, Philippe, 2010. "Rooted-tree solutions for tree games," European Journal of Operational Research, Elsevier, vol. 203(2), pages 404-408, June.
    11. Sébastien Courtin & Zéphirin Nganmeni & Bertrand Tchantcho, 2016. "The Shapley–Shubik power index for dichotomous multi-type games," Theory and Decision, Springer, vol. 81(3), pages 413-426, September.
    12. S. Béal & A. Lardon & E. Rémila & P. Solal, 2012. "The average tree solution for multi-choice forest games," Annals of Operations Research, Springer, vol. 196(1), pages 27-51, July.
    13. Mustapha Ridaoui & Michel Grabisch & Christophe Labreuche, 2017. "Axiomatization of an importance index for Generalized Additive Independence models," Documents de travail du Centre d'Economie de la Sorbonne 17048, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    14. David Lowing & Kevin Techer, 2021. "Marginalism, Egalitarianism and E ciency in Multi-Choice Games," Working Papers halshs-03334056, HAL.
    15. David Lowing, 2023. "Allocation rules for multi-choice games with a permission tree structure," Annals of Operations Research, Springer, vol. 320(1), pages 261-291, January.
    16. Michel Grabisch, 2011. "Ensuring the boundedness of the core of games with restricted cooperation," Annals of Operations Research, Springer, vol. 191(1), pages 137-154, November.
    17. Michel Grabisch & Lijue Xie, 2008. "The core of games on distributive lattices: how to share benefits in a hierarchy," Post-Print halshs-00344802, HAL.
    18. R. Branzei & N. Llorca & J. Sánchez-Soriano & S. Tijs, 2014. "A constrained egalitarian solution for convex multi-choice games," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(3), pages 860-874, October.
    19. Michael Jones & Jennifer Wilson, 2010. "Multilinear extensions and values for multichoice games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 72(1), pages 145-169, August.
    20. Courtin, Sébastien & Nganmeni, Zéphirin & Tchantcho, Bertrand, 2017. "Dichotomous multi-type games with a coalition structure," Mathematical Social Sciences, Elsevier, vol. 86(C), pages 9-17.
    21. Sébastien Courtin & Zéphirin Nganmeni & Bertrand Tchantcho, 2017. "Dichotomous multi-type games with a coalition structure," Post-Print halshs-01545772, HAL.
    22. Yu-Hsien Liao, 2012. "Converse consistent enlargements of the unit-level-core of the multi-choice games," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 20(4), pages 743-753, December.
    23. Branzei, R. & Tijs, S. & Zarzuelo, J., 2009. "Convex multi-choice games: Characterizations and monotonic allocation schemes," European Journal of Operational Research, Elsevier, vol. 198(2), pages 571-575, October.

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