On the complexity of Chooser-Picker positional games

TL;DR

This paper proves that the Picker-Chooser and Chooser-Picker positional games are NP-hard, highlighting their computational complexity and similarities to traditional Maker-Breaker games, and explores pairing strategies in Maker-Breaker contexts.

Contribution

It establishes the NP-hardness of Picker-Chooser and Chooser-Picker games and analyzes pairing strategies, connecting these findings to the game "Snaky."

Findings

Both games are NP-hard.

Picker-Chooser and Chooser-Picker behave similarly to Maker-Breaker games.

Pairing strategies are effective in Maker-Breaker games.

Abstract

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."