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DETERMINACY AND INDETERMINACY OF GAMES PLAYED ON COMPLETE METRIC SPACES

Part of: Game theory

Published online by Cambridge University Press:  12 May 2014

LIOR FISHMAN ,
TUE LY  and
DAVID SIMMONS
LIOR FISHMAN
Affiliation:
University of North Texas, Department of Mathematics, 1155 Union Circle #311430, Denton, TX 76203-5017, USA email [email protected]
TUE LY
Affiliation:
Brandeis University, Department of Mathematics, 415 South Street, Waltham, MA 02454-9110, USA email [email protected]
DAVID SIMMONS*
Affiliation:
Ohio State University, Department of Mathematics, 231 W. 18th Avenue, Columbus, OH 43210-1174, USA email [email protected]
*
✉[email protected]
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Abstract

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Schmidt’s game is a powerful tool for studying properties of certain sets which arise in Diophantine approximation theory, number theory and dynamics. Recently, many new results have been proven using this game. In this paper we address determinacy and indeterminacy questions regarding Schmidt’s game and its variations, as well as more general games played on complete metric spaces (for example, fractals). We show that, except for certain exceptional cases, these games are undetermined on certain sets. Judging by the vast numbers of papers utilising these games, we believe that the results in this paper will be of interest to a large audience of number theorists as well as set theorists and logicians.

Keywords

determinacy of gamesBorel determinacyGale–Stewart gamesSchmidt’s game

MSC classification

Secondary: 91A44: Games involving topology or set theory 91A05: 2-person games 91A18: Games in extensive form 91A50: Discrete-time games
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 2 , October 2014 , pp. 339 - 351
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

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