Abstract. We introduce a new type of positional games, played on a vertex set of a graph. Given a graph [Formula: see text], two players claim vertices of [Formula: see text], where the outcome of the game is determined by the subgraphs of [Formula: see text] induced by the vertices claimed by each player (or by one of them). We study classical positional games such as Maker-Breaker, Avoider-Enforcer, Waiter-Client, and Client-Waiter games, in both their biased and unbiased versions, where the board of the game is the vertex set of the binomial random graph [Formula: see text]. Under these settings, we consider those games where the target sets are the vertex sets of all graphs containing a copy of a fixed graph [Formula: see text], called [Formula: see text]-games. We focus on the cases in which [Formula: see text] is a clique, a cycle, or a forest, and for Waiter-Client and monotone Avoider-Enforcer games our results apply to any fixed graph [Formula: see text]. We show that, similarly to the edge version of [Formula: see text]-games, there is a strong connection between the threshold probability for these games and the one for the corresponding vertex Ramsey property (that is, the property that every [Formula: see text]-vertex-coloring of [Formula: see text] spans a monochromatic copy of [Formula: see text]). Another similarity to the edge version of these games we demonstrate is that the games in which [Formula: see text] is a triangle or a forest present a different behavior compared to the general case.