On the ordinal equivalence of the Jonhston, Banzhaf and Shapley–Shubik power indices for voting games with abstention

Abstract

The aim of this paper is twofold. We extend the well known Johnston power index usually defined for simple voting games, to voting games with abstention and we provide a full characterization of this extension. On the other hand, we conduct an ordinal comparison of three power indices: the Shapley–Shubik, Banzhaf and newly defined Johnston power indices. We provide a huge class of voting games with abstention in which these three power indices are ordinally equivalent. This is clearly a generalization of the work by Freixas et al. (Eur J Oper Res 216:367–375, 2012) and a twofold extension of Parker (Games Econ Behav 75:867–881, 2012) in the sense that, the ordinal equivalence emerges for three power indices (not just for the Shapley–Shubik and the Banzhaf indices), and it holds for a class of games strictly larger than the class of I-complete (3,2) games namely semi I-complete (3,2) games.