On Two-Level Poset Games

We consider the complexity of determining the winner of a finite, two-level poset game. This is a natural question, as it has been shown recently that determining the winner of a finite, three-level poset game is PSPACE-complete. We give a simple formula allowing one to compute the status of a type of two-level poset game that we call parity-uniform. This class includes significantly more easily solvable two-level games than was known previously. We also establish general equivalences between various two-level games. These equivalences imply that for any n, only finitely many two-level posets with n minimal elements need be considered, and a similar result holds for two-level posets with n maximal elements.