The rational-strategy conjecture for lattice games

A lattice game is a game played on a set of lattice points in a polyhedron, with a finite compatible rule set determining its winning and losing positions. For a lattice game, let $\mathcal{P}$ denote the set of P-positions, and let its generating function be the corresponding formal power series. A rational strategy is a rational generating function for the set of P-positions.

Rational-strategy conjecture. Every lattice game possesses a rational strategy; equivalently, the generating function for its P-positions is a ratio of polynomials with integer coefficients.

A rational strategy gives algorithms for recognizing P-positions and N-positions and for computing a legal move to a P-position from any N-position; the paper notes that examples and heuristic arguments motivate the conjecture, but does not establish it in general.