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.
On
There are different approaches depending on your context and goals. If the game can go on forever, then it's fairly common to consider that as a third "drawn" state.
If the game can't go on forever, or you don't want to do that, but the rules would normally declare certain end states are "draws". Then you might split the game up into copies where "draws are counted as a win for player X". For example, if you could show that Chess-where-draws-are-counted-as-a-win-for-White is a win for the White, then you'd know that White could at least force a draw. And if you knew additionally Chess-where-draws-are-counted-as-a-win-for-Black is a win for Black, then you'd know that Chess is drawn under perfect play.
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.