Is the game tree of any impartial game well-founded? Why or why not?

Question

I'm trying to understand why the game tree of any impartial game (even transfinite, like transfinite Nim) must be well-founded under the "is reachable from" relation (say, if the position $G'$ is reachable from $G$ after some moves, then $G'\mathcal R G$). Why must every non-empty set of positions of an impartial game have some minimal elements? Why can't there be infinite descending chains of game positions in such a non-empty set?

Answer 1

The Wikipedia article on impartial games states that there must be a finite number of positions, so an infinite descending chain of different positions is not possible. It does not specify that there be no cycles, so one could have a game that returned to a position seen earlier. The Wikipedia article on the Sprague-Grundy theorem does specify that all impartial games must end, so they are well-founded. In the early definition of a game in Winning Ways they say that games must end, but admit that some of the games they consider do not meet all the rules. In fact, they discuss in passing the game On, which has just one move from the starting position back to the starting position.

You need to look carefully at the definition you are using of an impartial game to see if they must be well founded or not.