The power of the largest player

Decisions in a shareholder meeting or a legislative committee are often modeled as a weighted game. Influence of a member is then measured by a power index. A large variety of different indices has been introduced in the literature. This paper analyzes how power indices differ with respect to the largest possible power of a non-dictatorial player. It turns out that the considered set of power indices can be partitioned into two classes. This may serve as another indication which index to use in a given application.