In the game of nim, what's the winning strategy if one or both players can skip a step?

Question

Consider a game of nim with $3$ heaps, the winning strategy is for a player to leave always an even total number of $1$'s, $2$'s, and $4$'s. (source: http://en.wikipedia.org/wiki/Nim)

How would the winning strategy change if one or both players are allowed to skip a step once at any time?

Answer 1

If both players have the option, not at all. If I am in a winning position in the original game, I will make the move I would make in the original game. If my opponent passes, I pass. We are now back in the basic game.

If only one player has the option, that player wins. If he is winning, he never passes and just wins. If he is losing, he passes and becomes the winning player.

This is just a start of some of them material in Winning Ways

Answer 2

If one player is allowed to skip, he can turn a losing position to a winning position, hence he always wins.

If both are allowed to skip once, the first to skip would play with an opponent who can skip and thus win. Hence it would be a bad idea to skip and the classical Nim strategies apply.