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.

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.

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 that player 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)

Game cases

Zermelo’s Theorem can be applied to all finite-stage two-player games with complete information and alternating moves.

The game must satisfy the following criteria:

• there are two players in the game; the game is of perfect information;
• the board game is finite;
• the two players can take alternate turns;
• and there is no chance element present.

Zero-sum game

Zero-sum game: involves two sides, where the result is an advantage for one side and a loss for the other

Some example:

• Cutting a cake: where taking a more significant piece reduces the amount of cake available for others as much as it increases the amount available for that taker, is a zero-sum game if all participants value each unit of cake equally.
• futures contracts and options

In contrast, non-zero-sum describes a situation in which the interacting parties’ aggregate gains and losses can be less than or more than zero.

References

• https://en.wikipedia.org/wiki/Zermelo%27s_theorem_(game_theory)