The concept of equivalent games in Sprague-Grundy theorem

Question

I'm trying to understand the concept of equivalent games in the context of Sprague Grundy theory. Suppose we have an impartial game $G_1$ that is equivalent to another impartial game $G_2$, that is, $G_1 \approx G_2$, and also suppose that it is possible to move to some position $G_1'$ from $G_1$ in a single move. Then does there always exist a position $G_2'$ that can be reached from $G_2$ in a single move, such that $G_1' \approx G_2'$?

Answer 1

No. Get $G_1$ be the Nim game with two piles of size $1$, and $G_2$ be the Nim game where you have already lost, i.e., where there are no piles.

$G_1$ and $G_2$ both have nim-value $0$, but there are moves from position $G_1$ while $G_2$ has no available moves.

$G_1$ and $G_2$ both having nim-value $0$ only means they can't move to another position with nim-value $0$; they could move to many other positions, possibly different ones from each other.

For a more involved example, let $G_1$ be the game $\{8,9\}$ and $G_2$ the game $\{2,3\}$; both have nim-value $1$, but $G_1$ can move to a position with nim-value $8$ while $G_2$ cannot.

You could define an equivalence of games as you describe, at least for games whose positions are guaranteed to go on for finite time (and which are bounded for each position, as opposed to something like Chomp on a $1\times\omega$ board). You can define "strong equivalence" by saying that two games are equivalent if they both have no moves available, or if the set of positions they can move to can be put in a bijection with each pair being strong-equivalent.

However, as the name suggests, this would be a very hard condition to satisfy, and wouldn't let you reduce impartial games very usefully in most cases. (It would also take longer to check.)