A Maker-Breaker game is a game with two players, the Maker and the Breaker, played on a finite set $X$ and a family $\Cal{E}$ of subsets of $X$. The two players move alternately to claim an element of $X$ in each round without replacement where the Breaker moves first. The Maker wins if she claims all elements of a set from $\Cal{E}$, and the Breaker wins otherwise. Therefore the elements of $\Cal{E}$ are called winning sets. A biased $(1, b)$ Maker@-Breaker game allows the Breaker to claim $b$ elements in each move. The authors study two typical Maker-Breaker games on the edge set of the complete graph $K_n$, namely the Perfect Matching game and the Hamilton Cycle game, with a winning set being a perfect matching and a Hamilton cycle, respectively. Attention is paid not only to what value of $b$ the Maker can win, but also how fast she can win. \par The paper consists of four theorems and their proofs. The first three theorems, one for the Perfect Matching game and two for the Hamilton Cycle game, provide conditions on $b$ and upper bounds of the minimal number of moves needed for the Maker to win, for large enough $n$. For large enough $b$ and $n$, the Breaker can postpone Maker's win, however, for at least a certain number of moves in both games. The results are summarized in the last theorem. With the help of an existing result from [M. Krivelevich, J. Amer. Math. Soc. {\bf 24} (2011), no.~1, 125--131; MR2726601], lower and upper bounds for the minimal number of moves that the Maker needs to win in both games are obtained.