combinatorial-game-theorydiscrete mathematics
I've been reading about combinatorial games, specifically about positions in such games can be classified as either winning or losing positions. However, what I'm not sure about now is how I can represent draws using this: situations where neither player wins or loses. Do I use a winning position or losing position to represent a position where a draw has occurred? Why is such a representation correct?
Thanks in advance.
The misère Grundy number is defined almost exactly as you have defined the "nimber for misère games": terminal positions get misère Grundy number $1$, and all others get the mex (minimal excludant) of the options' misère Grundy numbers, as usual (so that positions that can only move to terminal positions have misère Grundy number $\mathrm{mex}(\{1\})=0$). This can be found in Meghan Rose Allen's MSc thesis at http://miseregames.org/docs/meghan.pdf , as well as in On Numbers and Games, and perhaps Winning Ways for your Mathematical Plays as well.
Why then, have you found the claim that there are "no nimbers for misère games"? Because unlike in the normal play case, these misère Grundy numbers don't tell you nearly enough information for how to play sums with all sorts of games under misère play. They don't even tell you enough to play misère Nim! Two heaps of size $2$ (call the position $G$) has misère Grundy number $0$, and so does a single heap of size $1$. But $G+G$ is a $\mathcal{P}$ position, whereas $1+1$ is a $\mathcal{N}$ position under misère play.
You also asked for a reference for something like your $G'$ construction. I think the closest standard object to what you are looking for is the "mate" of $G$, denoted by $G^-$. It is defined by: $$G^{-}=\begin{cases} \left\{ \emptyset\right\} & \text{ if }G\cong\emptyset\text{;}\\ \left\{ \left(G'\right)^{-}:G'\in G\right\} & \text{ otherwise}. \end{cases}$$
For references, you can find this as definition V.3.1 of Aaron N. Siegel's "Combinatorial Game Theory". It can also be found in "On Numbers and Games" (check the index for "mate").
If I understand correctly (you're not asking about perfect play),
seems to be a draw.
I got that from here.
Best Answer
The analysis on the Brilliant page only works with games in which a draw is not possible. In general, you'd need to represent drawn positions as their own type of position; a drawn position can move to another drawn position (because both players can force a draw), or move to a winning position (if the person who just plays messes up and moves to a position where the other can win). It can't move to a losing position, because otherwise it'd be possible to win from the current state, and hence would be a winning state, not a drawn state.