On Chooser-Picker positional games

Two new versions of the so-called Maker-Breaker Positional Games are defined by József Beck. In these variants Picker takes unselected pair of elements and Chooser keeps one of these elements and gives back the other to Picker. In the Picker-Chooser version Picker is Maker and Chooser is Breaker, while the roles are swapped in the Chooser-Picker version. It seems that both the Picker-Chooser and Chooser-Picker versions are not worse for Picker than the original Maker-Breaker versions. Here we give winning conditions for Picker in some Chooser-Picker games that extend the results of Beck.