Usually, a topological game runs for countably many innings. Players can make some choices and then, after $\omega$ innings, the winner is declared. Several games are of the form first player gives a collection, second player chooses some elements from the collection -- this is done once per inning. Classically, the variations for the second player are ``choose one'' or ``choose finitely many''. These variations have different outcomes (e.g.~Rothberger versus Menger games). Here we will present some differences that appear when we go even more specific: what kind of differences can appear when we allow second player to pick two choices? Or three? If we let second player to choose finitely many, but in the end, the number of choices per inning is bounded -- does it make any difference?

This is a joint work with M. Duzi. This work was supported by FAPESP (2017/09252-3).