Given a social choice function $F$, a subset $B\subset A$ of the candidates and a coalition $S\subset N$ of the voters, $\beta$-effectiveness of $S$ for $B$ is equivalent to $N\setminus S$ not being $\beta$-effective for $A\setminus B$.
However, for a social choice correspondence it is not true - the fact that $N\setminus S$ is not $\beta$-effective for $A\setminus B$ does not, in general, lead to $S$ being $\beta$-effective for $B$.
Could anyone give an example that shows why it is so? What is special about social choice functions that gives this symmetry?