[math-fun] Double moves

Has anyone studied variants of standard combinatorial games (such as Nim) in which each player makes two standard moves in succession on each turn instead of just one? (This is the way my 8-year-old son likes to play tic-tac-toe against me: he puts down two X's, then I put down two O's, then he puts down two X's and wins. Fair, right?) Note that this form of two-player game can be seen as a four-player game in which players #1 and #2 collaborate and players #3 and #4 collaborate. Jim Propp