Let $G$ be an impartial combinatorial game. I claim that there is a game $G'$ such that $G$ (without terminal positions; see below) under the misère play rule is equivalent to $G'$ under the normal play rule. I wonder if this construction is already known and written down somewhere in the literature (so that I can cite it)? Any reference would be appreciated. Here is the construction of $G'$:
- The positions of $G'$ are the non-terminal positions of $G$.
- A move in $G'$ is a move in $G$ which does not lead to a terminal position of $G$.
- Hence, the terminal positions of $G'$ are those non-terminal positions of $G$ which only move to terminal positions of $G$, i.e. which have nimber $1$ under the normal play rule.
Then I think that $G$ wins/loses under the misère play rule iff $G'$ wins/loses under the normal play rule. Basically this just captures the following philosophy behind misère games: Try to avoid the terminal positions!
I am interested in this because of the following: I have read at various places that there are no nimbers for misère games. However, with the replacement of $G$ by $G'$ above, we can just define the nimber of a non-terminal position of $G$ to be the nimber of the corresponding position of $G'$ under the normal play rule. Terminal positions are boring, they get no nimber. As soon as we know them, we can discard them from the analysis of our misère game $G$. What's wrong with that? This has been applied in the game of noetherian rings.