Some online Maker-Breaker games

We consider some Maker-Breaker games of the following flavor. We have some set $V$ of items for purchase. Maker's goal is to purchase some member of a given family $\cH$ of subsets of $V$ as cheaply as possible and Breaker's goal is to make the purchase as expensive as possible. Each player has a pointer and during a player's turn their pointer moves through the items in the order of the permutation until the player decides to take one. We mostly focus on the case where the permutation is random and unknown to the players (it is revealed by the players as their pointers move).