Rectangular Polyomino Set Weak (1,2)-achievement Games

In a polyomino set (1,2)-achievement game the maker and the breaker alternately mark one and two previously unmarked cells respectively. The maker's goal is to mark a set of cells congruent to one of a given set of polyominoes. The breaker tries to prevent the maker from achieving his goal. The teams of polyominoes for which the maker has a winning strategy is determined up to size 4. In set achievement games, it is natural to study infinitely large polyominoes. This enables the construction of super winners that characterize all winning teams up to a certain size.