Bart-Moe games, JumbleG and discrepancy

Let A and B be hypergraphs with a common vertex set V. In a (p, q, A ∪ B) Bart-Moe game, the players take turns selecting previously unclaimed vertices of V. The game ends when every vertex has been claimed by one of the players. The first player, called Bart (to denote his role as Breaker and Avoider together), selects p vertices per move and the second player, called Moe (to denote his role as Maker or Enforcer), selects q vertices per move. Bart wins the game iff he has at least one vertex in every hyperedge B ∈ B and no complete hyperedge A ∈ A. We prove a sufficient condition for Bart to win the (p, 1) game, for every positive integer p. We then apply this criterion to two different games in which the first player's aim is to build a pseudo-random graph of density frac(p, p + 1), and to a discrepancy game.