Fast winning strategies in positional games

For the unbiased Maker-Breaker game, played on the hypergraph H , let τ M ( H ) be the smallest integer t such that Maker can win the game within t moves (if the game is a Breakerʹs win, then set τ M ( H ) = ∞ ). Similarly, for the unbiased Avoider-Enforcer game played on H , let τ E ( H ) be the smallest integer t such that Enforcer can win the game within t moves (if the game is an Avoiderʹs win, then set τ M ( E ) = ∞ ). We investigate τ M and τ E and determine their value for various positional games.