  We initiate the study of a new variant of the Maker-Breaker positional game,
which we call multistage game. Given a hypergraph
$\mathcal{H}=(\mathcal{X},\mathcal{F})$ and a bias $b \ge 1$, the $(1:b)$
multistage Maker-Breaker game on $\mathcal{H}$ is played in several stages as
follows. Each stage is played as a usual $(1:b)$ Maker-Breaker game, until all
the elements of the board get claimed by one of the players, with the first
stage being played on $\mathcal{H}$. In every subsequent stage, the game is
played on the board reduced to the elements that Maker claimed in the previous
stage, and with the winning sets reduced to those fully contained in the new
board. The game proceeds until no winning sets remain, and the goal of Maker is
to prolong the duration of the game for as many stages as possible. In this
paper we estimate the maximum duration of the $(1:b)$ multistage Maker-Breaker
game, for biases $b$ subpolynomial in $n$, for some standard graph games played
on the edge set of $K_n$: the connectivity game, the Hamilton cycle game, the
non-$k$-colorability game, the pancyclicity game and the $H$-game. While the
first three games exhibit a probabilistic intuition, it turns out that the last
two games fail to do so.
