  We study the biased $(1:b)$ Maker--Breaker positional games, played on the
edge set of the complete graph on $n$ vertices, $K_n$. Given Breaker's bias
$b$, possibly depending on $n$, we determine the bounds for the minimal number
of moves, depending on $b$, in which Maker can win in each of the two standard
graph games, the Perfect Matching game and the Hamilton Cycle game.
