Winning fast in fair biased Maker-Breaker games

We study the (a:a) Maker-Breaker games played on the edge set of the complete graph on n vertices. In the following four games – perfect matching game, Hamiltonicity game, star factor game and path factor game, our goal is to determine the least number of moves which Maker needs in order to win these games. Moreover, for all games except for the star factor game, we show how Red can win in the strong version of these games.