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Capital process and optimality properties of a Bayesian Skeptic in coin-tossing games

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  • Masayuki Kumon
  • Akimichi Takemura
  • Kei Takeuchi
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    Abstract

    We study capital process behavior in the fair-coin game and biased-coin games in the framework of the game-theoretic probability of Shafer and Vovk (2001). We show that if Skeptic uses a Bayesian strategy with a beta prior, the capital process is lucidly expressed in terms of the past average of Reality's moves. From this it is proved that the Skeptic's Bayesian strategy weakly forces the strong law of large numbers (SLLN) with the convergence rate of O(\sqrt{\log n/n})$ and if Reality violates SLLN then the exponential growth rate of the capital process is very accurately described in terms of the Kullback divergence between the average of Reality's moves when she violates SLLN and the average when she observes SLLN. We also investigate optimality properties associated with Bayesian strategy.

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    File URL: http://arxiv.org/pdf/math/0510662
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number math/0510662.

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    Date of creation: Oct 2005
    Date of revision: Sep 2008
    Publication status: Published in Stochastic Analysis and Applications,26:6,1161-1180,2008
    Handle: RePEc:arx:papers:math/0510662

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    Web page: http://arxiv.org/

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    Cited by:
    1. Vladimir Vovk, 2009. "Continuous-time trading and the emergence of probability," Papers 0904.4364, arXiv.org, revised Aug 2010.
    2. Kumon, Masayuki & Takemura, Akimichi & Takeuchi, Kei, 2011. "Sequential optimizing strategy in multi-dimensional bounded forecasting games," Stochastic Processes and their Applications, Elsevier, vol. 121(1), pages 155-183, January.
    3. Vladimir Vovk, 2012. "Continuous-time trading and the emergence of probability," Finance and Stochastics, Springer, vol. 16(4), pages 561-609, October.
    4. Shin-ichiro Takazawa, 2012. "Exponential inequalities and the law of the iterated logarithm in the unbounded forecasting game," Annals of the Institute of Statistical Mathematics, Springer, vol. 64(3), pages 615-632, June.
    5. Kei Takeuchi & Masayuki Kumon & Akimichi Takemura, 2008. "Multistep Bayesian strategy in coin-tossing games and its application to asset trading games in continuous time," Papers 0802.4311, arXiv.org, revised Mar 2008.
    6. Kei Takeuchi & Akimichi Takemura & Masayuki Kumon, 2011. "New Procedures for Testing Whether Stock Price Processes are Martingales," Computational Economics, Society for Computational Economics, vol. 37(1), pages 67-88, January.
    7. Kei Takeuchi & Akimichi Takemura & Masayuki Kumon, 2009. "New procedures for testing whether stock price processes are martingales," Papers 0907.3273, arXiv.org, revised Feb 2010.
    8. Masayuki Kumon & Jing Li & Akimichi Takemura & Kei Takeuchi, 2012. "Bayesian logistic betting strategy against probability forecasting," Papers 1204.3496, arXiv.org.

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