Zermelo’s Theorem, when applied to chess, states:
“either white can force a win, or black can force a win, or both sides can force at least a draw ”
I do not get this. How can it be proven?
And why do people lose in chess then, if they can force a draw at least?
The theorem is about perfect players.
Assume you were able to enumerate and evaluate all the gazillion chess positions. Then the following can happen:
- You find out that you can force a win as white, no matter how well black plays
- You find out that even with your best play, you cannot force a win as white if black plays well enough
- You find out that you can force a win as black, no matter how well white plays
- You find out that even with your best play, you cannot force a win as black, if white plays well enough.
In fact, with your complete knowledge, you will find out that exactly one of 1 and 2 holds (but we imperfect beings don’t know which): either you can or you can’t force a win.
Similarly exactly one of 3 and 4 must hold.
Of course, 1 and 3 cannot hold simultaneously. That leaves us with three exhaustive and mutually exclusive options in total:
- A perfect white player can force a win
- A perfect black player can force a win
- None of the above. That is: if only they play well enough, both white and black can avoid losing. In other words, both can force at least a draw (though they may win against a stupid opponent).
Real life people (and computers) lose because the premise of the above is not fulfilled for them: nobody has perfect knowledge of all chess positions.
We do have perfect knowledge about simplified games, for example chess endgames with very few pieces. The perfect knowledge about these even led to a change in chess rules regarding the 50-move-rule, because endgames were discovered that could be won, but might require more than 50 moves. More precisely (thanks to Voo‘s comment): The 50 moves rule was changed so that if you could show you’d win in N moves with N≤100 it would still be allowed (and players had to agree on which positions that extension was allowed and god knows what else) and then abolished again later on. In §9.3 of the current FIDE rules the 50 moves rule is once again strictly enforced.
EDIT:(inspired by vaxquis‘ comment) The above considerations are not specific to chess, they apply to a large class of games, the important properties of these games being that there are no elements of secrecy or probability (e.g. Poker fails on both these criteria because the cards are dealt at random and only you know your own hand):
- TicTacToe is so simple that one can enumerate all possibiliteis by hand to show that no side can force a win
- Who can force a win in Nim (a draw is not possible by the rules) depends on the specific variant and the starting position, but the analysis is not too complicated and may be done in introductory courses – or even by puzzlers
- That Nine men’s morris is a draw was found by a lot of brute-forcing (plus smart ideas)
- In 2007, it ws found that Checkers is a draw, also by the use of massive computer power
- It is a “two-liner” to show that the game of Hex can be won by the first player (“strategy stealing”). Nevertheless, explicit winning strategies are known only for small boards (again, by throwing CPU power at the problem)
Wikipedia has a nice overview listing more games that are more or less solved.