The conditional Shapley-Shubik measure for ternary voting games

Ternary voting games (TVGs) model situations where a voter has three options, which can be thought of as yes, no, and abstention. This paper presents ϕ˜, an extension of the Shapley-Shubik power measure to ternary voting games. ϕ˜ measures a voter's power as the probability that the voter will be pivotal given that they do not abstain. This contrasts with other extensions of the Shapley-Shubik measure to TVGs, which measure power as the probability that a player's vote is pivotal no matter what that vote is. Desirable properties of power measures in SVGs are extended to TVGs and ϕ˜ is shown to satisfy these properties.