Zermelo's theorem (game theory)

In game theory, Zermelo's theorem is a theorem about finite two-person games of perfect information in which the players move alternately and in which chance does not affect the decision making process. It says that if the game cannot end in a draw, then one of the two players must have a winning strategy (i.e. force a win). An alternate statement is that for a game meeting all of these conditions except the condition that a draw is not possible, then either the first-player can force a win, or the second-player can force a win, or both players can force a draw.[1] The theorem is named after Ernst Zermelo.

For Zermelo's theorem in set theory, see well-ordering theorem.

Conclusions of Zermelo's theorem

Zermelo's work shows that in two-person zero-sum games with perfect information, if a player is in a winning position, then they can always force a win no matter what strategy the other player may employ. Furthermore, and as a consequence, if a player is in a winning position, it will never require more moves than there are positions in the game (with a position defined as position of pieces as well as the player next to move).[1]

Publication history

Zermelo's original paper describing the theorem, Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels, was published in German in 1913. Ulrich Schwalbe and Paul Walker translated Zermelo's paper into English in 1997 and published the translation in the appendix to Zermelo and the Early History of Game Theory.[1]

Details

Zermelo considers the class of two-person games without chance, where players have strictly opposing interests and where only a finite number of positions are possible. Although in the game only finitely many positions are possible, Zermelo allows infinite sequences of moves since he does not consider stopping rules. Thus, he allows for the possibility of infinite games. Then he addresses two problems:

  1. What does it mean for a player to be in a 'winning' position and is it possible to define this in an objective mathematical manner?
  2. If the player is in a winning position, can the number of moves needed to force the win be determined?

To answer the first question, Zermelo states that a necessary and sufficient condition is the nonemptyness of a certain set, containing all possible sequences of moves such that a player wins independently of how the other player plays. But should this set be empty, the best a player could achieve would be a draw. So he defines another set containing all possible sequences of moves such that a player can postpone her loss for an infinite number of moves, which implies a draw. This set may also be empty, i. e., the player can avoid her loss for only finitely many moves if her opponent plays correctly. But this is equivalent to the opponent being able to force a win. This is the basis for all modern versions of Zermelo's theorem.

About the second question, Zermelo claimed that it will never take more moves than there are positions in the game. His proof is a proof by contradiction: Assume that a player can win in a number of moves larger than the number of positions. Of course, at least one winning position must have appeared twice. So the player could have played at the first occurrence in the same way as he does at the second and thus could have won in fewer moves than there are positions.

Example

When applied to chess, Zermelo's Theorem states "either White can force a win, or Black can force a win, or both sides can force at least a draw".[2][3]

Notes
  1. Schwalbe, Ulrich; Walker, Paul. "Zermelo and the Early History of Game Theory" (PDF). Cite journal requires |journal= (help)
  2. MacQuarrie, John. "Mathematics and Chess".
  3. Aumann, R. J. (1989). Lectures on Game Theory. Boulder, CO: Westview Press. p. 1.
  • Original Paper(in German)
  • Ulrich Schwalbe, Paul Walker, Zermelo and the Early History of Game Theory, Games and Economic Behavior, Volume 34, 2001, 123-137, online
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