An Appropriate Way to Extend the Banzhaf Index for Multiple Levels of Approval

The Banzhaf power index for games admits several extensions if the players have more than two ordered voting options. In this paper we prove that the most intuitive and recognized extension of the index fails to preserve the desirability relation for games with more than three ordered input levels of approval, a failure that undermines the index to be a good measure of power. This leads us to think of an alternative to the Banzhaf index for several input levels of approval. We propose a candidate for which it is proved that: (1) coincides with the Banzhaf index for simple games, (2) it is proportional to its known extension for three levels of approval, and (3) preserves the desirability relation regardless of the number of input levels of approval. This new index is based on measuring the total capacity the player has to alter the outcome. In addition, it can be expressed through a very appropriate mathematical formulation that greatly facilitates its computation. Defining extensions of well-established notions in a wider context requires a careful analysis. Different extensions can provide complementary nuances and, when this occurs, none of them can be considered to be ‘the’ extension. As shown in this paper, this situation applies when trying to extend the Banzhaf power index from simple games to the broader context of games with several ordered input levels of approval.