The nucleolus of large majority games

•A bound for the distance between weights and the nucleolus.•Coincidence of the nucleolus and weights for non-homogeneous games.•A limit theorem for the nucleolus, similar to Penrose's limit theorem.•A sufficient criterion for a positive nucleolus for non-null players.

Members of a shareholder meeting or legislative committee have greater or smaller voting power than meets the eye if the nucleolus of the induced majority game differs from the voting weight distribution. We establish a new sufficient condition for the weight and power distributions to be equal, and we characterize the limit behavior of the nucleolus in case all relative weights become small.