Why aren't all games impartial games?

Question

We can take any partisan game, like Chess, and make it into an impartial game like so:

Take the game tree of moves for both players and make each game state a spot on the board. Now we start the game with a piece starting at the "beginning" state. The rules are the same for both players, you may move the piece forward to any available space, and a player loses when they can no longer move.

This seems impartial to me. Both players can move on the board in the exact same way. The only difference is who goes first. So what is wrong with this example? What is the precise definition of what "symmetrical play" means?

Answer 1

You have succeeded in producing an impartial game $I$ with the following property:

Player $1$ has a winning strategy in $I$ iff player White has a winning strategy in Chess; similarly, player $2$ has a winning strategy in $I$ iff player Black has a winning strategy in Chess. Moreover, these bi-implications are witnessed by very simple "strategy translation" functions.

However, in combinatorial game theory equivalence of games is a finer notion than simply "same person should win." For example, the one-red-edge and the two-red-edges Hackenbush boards are each won by player Red, but they are not equivalent games. Your $I$ is not in fact equivalent to Chess in any deeper sense: for example, letting $+$ denote the usual sum of games we have by a general "mirroring" argument that $I+I$ is null but (assuming Chess is a win for White) Chess $+$ Chess is a win for White.

So basically, what we're seeing here is just further evidence that we need to be careful in introducing a notion of "equivalence" for games.