July  2017, 4(3): 191-194. doi: 10.3934/jdg.2017011

On Zermelo's theorem

1. 

Department of Economics, University of Iowa, Iowa City, IA 52242-1994, USA

2. 

Department of Economics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK

* Corresponding author

Received  February 2017 Revised  February 2017 Published  April 2017

A famous result in game theory known as Zermelo's theorem says that ''in chess either White can force a win, or Black can force a win, or both sides can force at least a draw". The present paper extends this result to the class of all finite-stage two-player games of complete information with alternating moves. It is shown that in any such game either the first player has a winning strategy, or the second player has a winning strategy, or both have unbeatable strategies.

Citation: Rabah Amir, Igor V. Evstigneev. On Zermelo's theorem. Journal of Dynamics and Games, 2017, 4 (3) : 191-194. doi: 10.3934/jdg.2017011
References:
[1]

R. AmirI. V. Evstigneev and K. R. Schenk-Hoppé, Asset market games of survival: A synthesis of evolutionary and dynamic games, Annals of Finance, 9 (2013), 121-144.  doi: 10.1007/s10436-012-0210-5.

[2]

R. J. Aumann, Lectures on Game Theory, Westview, Boulder, 1989.

[3]

E. Borel, La théorie du jeu et les équations intégrales á noyau symétrique, Comptes Rendus de l'Académie des Sciences, 173 (1921), 1304-1308.  doi: 10.2307/1906946.

[4]

C. L. Bouton, Nim, a game with a complete mathematical theory, Annals of Mathematics, 3 (1901/02), 35-39. doi: 10.2307/1967631.

[5]

A. Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4190-4.

[6]

Yu. Khomskii, Intensive Course on Infinite Games, Sofia University, 2010. Available from: https://www.math.uni-hamburg.de/home/khomskii/infinitegames2010/InfiniteGamesSofia.pdf.

[7]

F. Kojima, Stability and instability of the unbeatable strategy in dynamic processes, International Journal of Economic Theory, 2 (2006), 41-53.  doi: 10.1111/j.1365-2966.2006.0023.x.

[8]

U. Schwalbe and P. Walker, Zermelo and the early history of game theory, Games and Economic Behavior, 34 (2001), 123-137.  doi: 10.1006/game.2000.0794.

[9]

E. Zermelo, Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels, in Proceedings of the Fifth International Congress of Mathematicians (Cambridge 1912), (eds. E. W. Hobson and A. E. H. Love), Cambridge University Press, Cambridge, 2 (1913), 501-504.

show all references

References:
[1]

R. AmirI. V. Evstigneev and K. R. Schenk-Hoppé, Asset market games of survival: A synthesis of evolutionary and dynamic games, Annals of Finance, 9 (2013), 121-144.  doi: 10.1007/s10436-012-0210-5.

[2]

R. J. Aumann, Lectures on Game Theory, Westview, Boulder, 1989.

[3]

E. Borel, La théorie du jeu et les équations intégrales á noyau symétrique, Comptes Rendus de l'Académie des Sciences, 173 (1921), 1304-1308.  doi: 10.2307/1906946.

[4]

C. L. Bouton, Nim, a game with a complete mathematical theory, Annals of Mathematics, 3 (1901/02), 35-39. doi: 10.2307/1967631.

[5]

A. Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4190-4.

[6]

Yu. Khomskii, Intensive Course on Infinite Games, Sofia University, 2010. Available from: https://www.math.uni-hamburg.de/home/khomskii/infinitegames2010/InfiniteGamesSofia.pdf.

[7]

F. Kojima, Stability and instability of the unbeatable strategy in dynamic processes, International Journal of Economic Theory, 2 (2006), 41-53.  doi: 10.1111/j.1365-2966.2006.0023.x.

[8]

U. Schwalbe and P. Walker, Zermelo and the early history of game theory, Games and Economic Behavior, 34 (2001), 123-137.  doi: 10.1006/game.2000.0794.

[9]

E. Zermelo, Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels, in Proceedings of the Fifth International Congress of Mathematicians (Cambridge 1912), (eds. E. W. Hobson and A. E. H. Love), Cambridge University Press, Cambridge, 2 (1913), 501-504.

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