Influence relation in two-output multichoice voting games

Abstract

The influence relation, defined within the set of simple games, is identified as a preorder. Additionally, it is proved to be a subpreorder of the preorders induced by the Shapley–Shubik and Banzhaf–Coleman indices. When this relation extends to voting games with abstention, detailed in Tchantcho et al. (2008), and further to multichoice voting games as in Pongou et al. (2014), it is shown that these extensions aren't always preorders. Even when they are, they don't necessarily align with the preorders induced by the extended Banzhaf–Coleman and Shapley–Shubik power indices in Freixas (2005a) and Freixas (2005b). In this paper, we introduce extensions for two-output multichoice voting games that are both preorders and subpreorders of the Banzhaf–Coleman power index defined in Freixas (2005b). Further, we characterize the two-output multichoice voting games for which one of these new power theories agrees with the generalized Banzhaf–Coleman and Shapley–Shubik power indices in Freixas (2005a) and Freixas (2005b) respectively.