Hostname: page-component-84b7d79bbc-tsvsl Total loading time: 0 Render date: 2024-07-30T04:52:53.362Z Has data issue: false hasContentIssue false
  • English
  • Français

Explicit Optimal Strategy for the Vardi Casino with Limited Playing Time

Part of: Stochastic processes Game theory

Published online by Cambridge University Press:  14 July 2016

Yi-Ching Yao
Yi-Ching Yao*
Affiliation:
Academia Sinica and National Chengchi University
*
∗Postal address: Institute of Statistical Science, Academia Sinica, Taipei, 115, Taiwan, ROC. Email address: yao@stat.sinica.edu.tw
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Vardi casino with parameter 0 < c < 1 consists of infinitely many tables indexed by their odds, each of which returns the same (negative) expected winnings -c per dollar. A gambler seeks to maximize the probability of reaching a fixed fortune by gambling repeatedly with suitably chosen stakes and tables (odds). The optimal strategy is derived explicitly subject to the constraint that the gambler is allowed to play only a given finite number of times. Some properties of the optimal strategy are also discussed.

Keywords

Gambling theorybold playprimitive casinofinite horizon

MSC classification

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 91A60: Probabilistic games; gambling
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 3 , September 2007 , pp. 831 - 837
Copyright
Copyright © Applied Probability Trust 2007 

References

[1] Chen, M.-R. and Hsiau, S.-R. (2006). Two-person red-and-black games with bet-dependent win probability functions. J. Appl. Prob. 43, 905–915.Google Scholar
[2] Chen, R. W., Shepp, L. A. and Zame, A. (2004). A bold strategy is not always optimal in the presence of inflation. J. Appl. Prob. 41, 587–592.Google Scholar
[3] Chen, R. W., Shepp, L. A., Yao, Y.-C. and Zhang, C.-H. (2005). On optimality of bold play for primitive casinos in the presence of inflation. J. Appl. Prob. 42, 121–137.Google Scholar
[4] Dubins, L. E. and Savage, L. J. (1976). Inequalities for Stochastic Processes (How to Gamble if You Must). Dover, New York.Google Scholar
[5] Grigorescu, I., Chen, R. and Shepp, L. (2007). Optimal strategy for the Vardi casino with interest payments. J. Appl. Prob. 44, 199–211.Google Scholar
[6] Schweinsberg, J. (2005). Improving on bold play when the gambler is restricted. J. Appl. Prob. 42, 321–333.Google Scholar
[7] Shepp, L. A. (2006). Bold play and the optimal policy for Vardi's casino. In Random Walk, Sequential Analysis and Related Topics, eds Hsiung, A. C., Ying, Z. and Zhang, C.-H., World Scientific, Singapore, pp. 150–156.Google Scholar
[8] Yao, Y.-C. (2007). On optimality of bold play for discounted Dubins–Savage gambling problems with limited playing times. J. Appl. Prob. 44, 212–225.Google Scholar