Fast winning strategies in positional games

For the unbiased Maker-Breaker game, played on the hypergraph H, let τM(H) be the smallest integer t such that Maker can win the game within t moves (if the game is a Breaker's win, then set τM(H)=∞). Similarly, for the unbiased Avoider-Enforcer game played on H, let τE(H) be the smallest integer t such that Enforcer can win the game within t moves (if the game is an Avoider's win, then set τM(E)=∞). We investigate τM and τE and determine their value for various positional games.