Condorcet Consistency and the strong no show paradoxes

We identify the maximal voting correspondence which is Condorcet Consistent and satisfies two participation conditions, namely the Top Property and the Bottom Property thereby extending a result in Perez (2001). The former participation condition says that if an alternative is in the chosen set at a profile of rankings and a ranking is added with that alternative on top, then it remains to be a member of the chosen set. The latter says that if an alternative is not in the chosen set at a profile of rankings and a ranking is added with that alternative at bottom, then the alternative is again not in the chosen set. In particular, voting functions (single-valued voting correspondences) with these three properties select from this maximal correspondence, and we demonstrate several ways in which this can or cannot be done.