This work deals with the evaluation of decision power in multi-dimensional rules. Courtin and Laruelle (2020) introduced a decision process that specifies the collective acceptance or rejection of a proposal with several dimensions. The decision process is modeled as follows: (i) There are several individuals. (ii) There are several dimensions. (iii) Each of the individuals expresses a binary choice ("Yes"or "No") on each dimension. (iv) A decision process maps each choice to a final binary decision ("Yes"or "No"). We extend and characterize six well-known power indices within this context: the Shapley-Shubik index (Shapley and Shubik, 1954), the Banzhaf index (Banzhaf, 1965), the Public good index (Holler, 1982), the Null individual free index (Alonso-Meijide et al., 2011), the Shift index (Alonso-Meijide and Freixas, 2010) and the Deegan-Packel index (Deegan and Packel, 1978). (c) 2021 Elsevier B.V. All rights reserved.