Game of Chess

Preliminaries of Chess

Finite Game

Von Neumann’s Theorem

Theorem. In every two-player, finite game with perfect information in which the set of outcomes is $\mathcal O=\{\text{I wins, II wins, Draw}\}$, one and only one of the following three alternatives must hold:

• Player I has a winning strategy.
• Player II has a winning strategy.
• Each of the two players has a strategy that guarantees at least a draw.

The proof of the theorem is referred to [1]

The proof is based on the structure of the game tree

Remark. Mycielski [1992] proved the theorem for the case of infinite games with perfect information, and the set of outcomes $\mathcal O$ is finite. The proof is based on transfinite induction.

Zermelo’s Theorem

Theorem. In chess, one and only one of the following must be true:

• White has a winnning strategy.
• Black has a winning strategy.
• Each of the two players has a strategy that guarantees at least a draw.

It is only an existence theorem, and we still do not know which of the three possibilities is true.

Proof 1

As the game of chess is finite, there must be an upper bound on the number of moves in a game, set as $2K$

Q.E.D.