The Shapley–Shubik power index for games with several levels of approval in the input and output

Voting systems with several levels of approval in the input and output are considered in this paper. That means games with n=2 players, j=2 ordered qualitative alternatives in the input level and k=2 possible ordered quantitative alternatives in the output. We introduce the Shapley–Shubik power index notion when passing from ordinary simple games or ternary voting games with abstention to this wider class of voting systems. The pivotal role of players is analysed by means of several examples and an axiomatization in the spirit of Shapley and Dubey is given for the proposed power index.