Consistency and dynamic approach of indexes

The purpose of this article is to study two indexes, the marginal index and the Banzhaf–Coleman index. For each of these two indexes, there is a corresponding reduced game that can be used to characterize it. In addition, we consider the efficient extensions of two indexes. In comparison to each characterization of two indexes, we establish a similar characterization for each extension of two indexes through an identical approach. Finally, for each of two efficient indexes, we propose a dynamic process leading to that corresponding efficient index, starting from an arbitrary efficient payoff vector.