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