Abstract
The Shapley–Shubik index is a specialization of the Shapley value and is widely applied to evaluate the power distribution in committees drawing binary decisions. It was generalized to decisions with more than two levels of approval both in the input and the output. The corresponding games are called (j, k) simple games. Here we present a new axiomatization for the Shapley–Shubik index for (j, k) simple games as well as for a continuous variant, which may be considered as the limit case.
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Notes
In our definition of a (j, k) simple game we will deviate from Freixas and Zwicker (2003) by numbering the input and output levels starting from 0 instead of 1 and assuming that lower numbers correspond to a lower level of approval.
Note that we slightly deviate from the original definition of a (j, k) simple game in Freixas and Zwicker (2003), see Footnote 1. With this, we ensure that (2, 2) simple games are in one-to-one correspondence to simple games encoding ‘‘no’’ as 0 and ‘‘yes’’ as 1.
Using the notation introduced at the beginning of Sect. 2, we have \(v(x_{-i},y_i)=v(x_{N\backslash \{i\}},y_i)= v(x_1,\ldots ,x_{i-1},y_i,x_{i+1},\ldots ,x_n)\).
The definition of unanimity games has already been given in the second paragraph of Sect. 2.
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Acknowledgements
Hilaire Touyem benefits from a financial support of the CETIC (Centre d’Excellence Africain en Technologies de l’Information et de la Communication) Project of the University of Yaounde I. We would like to thank the anonymous reviewers for their suggestions and comments.
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Kurz, S., Moyouwou, I. & Touyem, H. Axiomatizations for the Shapley–Shubik power index for games with several levels of approval in the input and output. Soc Choice Welf 56, 569–594 (2021). https://doi.org/10.1007/s00355-020-01296-6
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DOI: https://doi.org/10.1007/s00355-020-01296-6