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Large deviations and fairness for a betting game with a constant ratio of capital

Published online by Cambridge University Press:  23 August 2024

Toshio Nakata
Toshio Nakata*
Affiliation:
Department of Mathematics, University of Teacher Education Fukuoka, Munakata, Fukuoka, 811-4192, Japan e-mail: nakata@fukuoka-edu.ac.jp
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This Article is a follow-up to a recent Gazette Article about a probabilistic betting game studied by Abdin et al. [1]. We examine the speed of convergence of the probability needed to investigate this game by giving concrete examples, using the large deviation, which is a valuable tool for estimating probabilities of repeated trials (see [2], [3, Chapter 6], [4, Section 5.11]). Moreover, to get a deep understanding of the game, we study fairness when it is repeated infinite times. Let us call it fairness in the sense of infinity, whose exact definition will be given in the final section.

Type
Articles
Information
The Mathematical Gazette , Volume 108 , Issue 572 , July 2024 , pp. 237 - 247
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Copyright
© The Authors, 2024 Published by Cambridge University Press on behalf of The Mathematical Association

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