Rational-strategy conjecture for squarefree lattice games

A lattice game has a set of legal moves and a set of positions, classified as P-positions or N-positions. A rational strategy is the finite algebraic description of the P-positions used to decide whether a position is a P-position and to find a legal move from an N-position to a P-position.

Rational-strategy conjecture. Every squarefree lattice game possesses a rational strategy.

A rational strategy would yield algorithms for classifying positions and finding winning moves, with polynomial running time when the strategy is short. The paper notes that it is unknown whether rational strategies exist for general squarefree games, including Dawson's Chess.