Fast strategies in Maker-Breaker games played on random boards

In this paper we analyze classical Maker-Breaker games played on the edge set of a sparse random board $G\sim \gnp$. We consider the Hamiltonicity game, the perfect matching game and the $k$-connectivity game. We prove that for $p(n)\geq \text{polylog}(n)/n$, the board $G\sim \gnp$ is typically such that Maker can win these games asymptotically as fast as possible, i.e. within $n+o(n)$, $n/2+o(n)$ and $kn/2+o(n)$ moves respectively.