On the complexity of Chooser-Picker positional games

Two new versions of the so-called Maker-Breaker Positional Games are defined by J\'ozsef Beck in [{\em Combinatorica} {\bf 22}(2) (2002) 169--216]. He defines two players, Picker and Chooser. In each round, Picker takes a pair of elements not already selected and Chooser keeps one and returns the other to Picker. In the Picker-Chooser version Picker plays as Maker and Chooser plays as Breaker, while the roles are swapped in the Chooser-Picker version. The outcome of these games is sometimes very similar to that of the traditional Maker-Breaker games. Here we show that both Picker-Chooser and Chooser-Picker games are NP-hard, which gives support to the paradigm that the games behave similarly while being quite different in definition. We also investigate the pairing strategies for Maker-Breaker games, and apply these results to the game called "Snaky."