Baltic Way 2019

Let $S$ be a finite set with $n\ge 2$ elements. Two players, $A$ and $B$, alternately choose nonempty proper subsets of $S$, where

(1) it is not allowed to choose a set that contains a set previously chosen by any player,

(2) it is not allowed to choose a set that is contained in any previously chosen set,

(3) it is not allowed to choose a set whose union with any previously chosen set is $S$.

$A$ begins. The player who first cannot choose a set anymore loses. Which player has a winning strategy?