Abstract
Any power index defines a total preorder in a simple game and, thus, induces a hierarchy among its players. The desirability relation, which is also a preorder, induces the same hierarchy as the Banzhaf and the Shapley indices on linear games, i.e., games in which the desirability relation is total. The desirability relation is a sub–preorder of another preorder, the weak desirability relation, and the class of weakly linear games, i.e., games for which the weak desirability relation is total, is larger than the class of linear games. The weak desirability relation induces the same hierarchy as the Banzhaf and the Shapley indices on weakly linear games. In this paper, we define a chain of preorders between the desirability and the weak desirability preorders. From them we obtain new classes of totally preordered games between linear and weakly linear games.
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Notes
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Linear games are also called complete, ordered or directed games in the literature, see Taylor and Zwicker [16] for references on these names.
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Acknowledgements
This research was partially supported by grant MTM2015–66818-P(MINECO/FEDER) from the Spanish Ministry of Economy and Competitiveness (MINECO) and from the European Union (FEDER funds).
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Freixas, J., Pons, M. (2017). Preorders in Simple Games. In: Mercik, J. (eds) Transactions on Computational Collective Intelligence XXVII. Lecture Notes in Computer Science(), vol 10480. Springer, Cham. https://doi.org/10.1007/978-3-319-70647-4_5
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