  We introduce and study Maker/Breaker-type positional games on random graphs.
Our main concern is to determine the threshold probability $p_{F}$ for the
existence of Maker's strategy to claim a member of $F$ in the unbiased game
played on the edges of random graph $G(n,p)$, for various target families $F$
of winning sets. More generally, for each probability above this threshold we
study the smallest bias $b$ such that Maker wins the $(1\:b)$ biased game. We
investigate these functions for a number of basic games, like the connectivity
game, the perfect matching game, the clique game and the Hamiltonian cycle
game.
