Zermelo's theorem (game theory)
Encyclopedia
In game theory
, Zermelo’s theorem, named after Ernst Zermelo
, says that in any finite two-person game of perfect information in which the players move alternatively and in which chance does not affect the decision making process, if the game cannot end in a draw, then one of the two players must have a winning strategy.
More formally, every finite extensive-form game exhibiting full information has a Nash equilibrium
that is discoverable by backward induction
. If every payoff is unique, for every player, this backward induction solution is unique.
Zermelo's paper, published in 1913, was originally published only in German. Ulrich Schwalbe and Paul Walker faithfully translated Zermelo's paper into English in 1997 and published the translation in the appendix to Zermelo and the Early History of Game Theory. 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. 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".
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
, Zermelo’s theorem, named after Ernst Zermelo
Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo was a German mathematician, whose work has major implications for the foundations of mathematics and hence on philosophy. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem.-Life:He graduated...
, says that in any finite two-person game of perfect information in which the players move alternatively and in which chance does not affect the decision making process, if the game cannot end in a draw, then one of the two players must have a winning strategy.
More formally, every finite extensive-form game exhibiting full information has a Nash equilibrium
Nash equilibrium
In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...
that is discoverable by backward induction
Backward induction
Backward induction is the process of reasoning backwards in time, from the end of a problem or situation, to determine a sequence of optimal actions. It proceeds by first considering the last time a decision might be made and choosing what to do in any situation at that time. Using this...
. If every payoff is unique, for every player, this backward induction solution is unique.
Zermelo's paper, published in 1913, was originally published only in German. Ulrich Schwalbe and Paul Walker faithfully translated Zermelo's paper into English in 1997 and published the translation in the appendix to Zermelo and the Early History of Game Theory. 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. 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".