PSPACE-completeness of the Weighted Poset Game

Poset game, which includes some famous games. e.g., Nim and Chomp as sub-games, is an important two-player impartial combinatorial game. The rule of the game is as follows: For a given poset (partial ordered set), each player intern chooses an element and the selected element and it’s descendants (elements succeeding it) are all removed from the poset. A player who choose the last element is the winner. On the complexity of poset game, although it is clearly in PSPACE, it have not known whether it is in P or NP-hard. Recently a weighted poset game, which is a generalization of poset game, have been presented and it was found that some sub-games of it can be solved in polynomial-time. The complexity of this game is also open. This paper shows that weighted poset game is PSPACE-complete even if the weights are restricted in {1,−1}, the dag, which represents the poset, is bipartite, and the length of each path in the dag is at most two.