Adjoining to (s,t)(s,t)-Wythoff’s game its PP-positions as moves

We define a class of new games obtained from (s,t)(s,t)-Wythoff’s game by adjoining to it an appropriate subset of its PP-positions as additional moves. The preservation of all PP-positions of the new games are analyzed. Our results show that for any integer s≥1s≥1, there exist infinite pairs of (s,t)(s,t) such that all PP-positions of the new games do not depend on the choice of additional moves.