Rulesets for Beatty games

We describe a ruleset for a 2-pile subtraction game with P-positions ({(lfloor alpha n rfloor ,lfloor beta n rfloor ) : n in mathbb Z_{ge 0} }) for any irrational (1< alpha < 2), and (beta ) such that (1/alpha +1/beta = 1). We determine the (alpha )’s for which the game can be represented as a finite modification of t-Wythoff (Holladay, Math Mag 41:7–13, 1968; Fraenkel, Am Math Mon 89(6):353–361, 1982) and describe this modification.