The Shapley–Shubik Power Index for Multi-Dimensional Rules with Multiple Approval Levels in the Input

In this paper, we introduce multi-dimensional rules with multiple approval levels in the input, which generalize both the multi-dimensional rules introduced by Courtin, S. and Laruelle, A. [2020] Multi-dimensional rules, Math. Soc. Sci. 103, 1–7 and the games with multiple approval levels in the input studied by Freixas, J. and Zwicker, W. S [2003] Weighted voting, abstention, and multiple levels of approval, Soc. Choice Welfare 21, 399–431. The decision process of such a rule is modeled as follows: (i) there are several individuals; (ii) there are several dimensions; (iii) each individual expresses a level of approval on each dimension; (iv) a decision process maps each input profile to a final decision (“accept” or “reject”).