A characterization of social choice correspondences that implement the core of simple games

In a coalitional voting game, an alternative is said to be in the core when no majority is willing to replace it with another alternative. A social choice correspondence is a mapping which associates any profile of voters' preferences with a non-empty subset of the set of alternatives, which is understood as the set of selected outcomes. We characterize the social choice correspondences which always select the core of a simple game for any profile of preferences.