Ternary voting games

We define ternary voting games (TVGs), a generalization of simple voting games (SVGs). In a play of an SVG each voter has just two options: voting ‘yes’ or ‘no’. In a TVG a third option is added: abstention. Every SVG can be regarded as a (somewhat degenerate) TVG; but the converse is false. We define appropriate generalizations of the Shapley–Shubik and Banzhaf indices for TVGs. We define also the responsiveness (or degree of democratic participation) of a TVG and determine, for each n, the most responsive TVGs with n voters. We show that these maximally responsive TVGs are more responsive than the corresponding SVGs.