An appropriate way to extend the Banzhaf index for multiple levels of approval

This is a post-peer-review, pre-copyedit version of an article published in Group decision and negotiation. The final authenticated version is available online at: http://dx.doi.org/10.1007/s10726-020-09718-7. 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 intu-itive and recognized extension of the index fails to preserve the desirability rela-tion 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 sim-ple 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 exten-sions of well-established notions in a wider context requires a careful analysis. Dif-ferent 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 situa-tion 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. This research was partially supported by funds from: the Spanish Ministry of Economy and Competitiveness (MINECO) and from the European Union (FEDER funds) under grant MTM2015-66818-P (MINECO/FEDER), and the Spanish Ministry of Science and Innovation grant PID2019-I04987GB-I00.