General restrictions of Wythoff-like games

A.S. Fraenkel and Y. Tanny ([13], 2012) introduced a class of Wythoff-like games. Given an integer function f, the game Wyt(f) is itself a generalization of famous Wythoff's game. In the current paper, four new models of a restricted version of Wythoff-like games, Odd–Odd Wyt(f), Odd–Even Wyt(f), Even–Odd Wyt(f) and Even–Even Wyt(f), are investigated. Under normal play convention, all P-positions of these four models are given for any homogeneous or inhomogeneous polynomial f. For Even–Even Wyt(f), the structure of P-positions is given by recursive characterization in terms of the mex function. For other models, the structures of P  -positions are of algebraic form, which allows us to decide in polynomial time whether or not a given game position (a,b)(a,b) is P.