Zermelo's Theorem
In finite two-player zero-sum games of perfect information without chance and draws allowed, either one player has a winning strategy or both can force a draw.
Definition
For finite two-player zero-sum games of perfect information with only two outcomes (win/lose), exactly one player has a winning strategy.
With draws allowed, either one side wins or both can secure at least a draw.
Intuition
Determinacy of perfect-information games — someone is guaranteed a good outcome.
Historically the founding result of game theory (Zermelo 1913).
Worked example
Chess: either White wins, Black wins, or both can force a draw — the game is determined, though we do not know which case.
Nim and other impartial games are fully solved (Sprague-Grundy theorem).
The math
Proved by backward induction on the finite game tree.
Generalized to infinite games by Martin's Borel Determinacy Theorem.
Where it is used
Conceptual foundation for game-playing AI and combinatorial game theory.
Basis for alpha-beta pruning and minimax search.
More in Game theory
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