I've been reading about combinatorial games, on an article by Brilliant: https://brilliant.org/wiki/combinatorial-games-winning-positions/#chomp-and-strategy-stealing.

The article makes 4 statements about combinatorial games, which I am confused about:

1. Due to the deterministic nature of the game, one can show via backward induction that every position can be uniquely characterized as a winning or a losing position.

2. The empty game (the game where there are no moves to be made) is a losing position.

3. A position is a winning position if at least one of the positions that can be obtained from this position by a single move is a losing position.

4. A position is a losing position if every position that can be obtained from this position by a single move is a winning position.

What I am confused about is the following:

1. I don't understand why every position can be uniquely characterized as a winning or a losing position. Can't there be positions that don't bring a player close to winning, but don't cause them to lose either?

2. I still don't fully understand what an empty game is. Is it a game where you can't make any moves (like when you are checkmated in chess)? But then why is an empty game an automatic loss? In a misere game, it would be a win, because your opponent is the last to move!

3. I don't understand why the fact that "one of the positions that can be obtained from this position by a single move is a losing position" is sufficient to show something is a winning position. Because that means the position will only cause the opponent to lose if he or she chooses to move to the worst possible position - otherwise you're opponent won't be any closer to losing then before! So how can that be considered a winning position?

• "one of the positions that can be obtained from this position by a single move is a losing position" -- they probably mean, you can move in such a way that the opponent is in a losing position. – Peter Franek Jun 9 '20 at 6:18