Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T16:17:11.655Z Has data issue: false hasContentIssue false
  • English
  • Français

Improving on bold play when the gambler is restricted

Part of: Stochastic processes Game theory

Published online by Cambridge University Press:  14 July 2016

Jason Schweinsberg
Jason Schweinsberg*
Affiliation:
University of California, San Diego
*
∗Postal address: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0112, USA. Email address: jschwein@math.ucsd.edu
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.

Suppose that a gambler starts with a fortune in (0,1) and wishes to attain a fortune of 1 by making a sequence of bets. Assume that whenever the gambler stakes an amount s, the gambler's fortune increases by s with probability w and decreases by s with probability 1 − w, where w < ½. Dubins and Savage showed that the optimal strategy, which they called ‘bold play’, is always to bet min{f, 1 − f}, where f is the gambler's current fortune. Here we consider the problem in which the gambler may stake no more than ℓ at one time. We show that the bold strategy of always betting min{ℓ, f, 1 − f} is not optimal if ℓ is irrational, extending a result of Heath, Pruitt, and Sudderth.

Keywords

Bold playred-and-blackgamblingsupermartingale

MSC classification

Primary: 91A60: Probabilistic games; gambling
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G42: Martingales with discrete parameter
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 2 , June 2005 , pp. 321 - 333
Copyright
© Applied Probability Trust 2005 

References

Bak, J. (2001). The anxious gambler's ruin. Math. Mag. 74, 182–193.Google Scholar
Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, New York.Google Scholar
Chen, R. (1976). Subfair discounted red-and-black game with a house limit. J. Appl. Prob. 13, 608–613.CrossRefGoogle Scholar
Chen, R. (1977). Subfair primitive casino with a discount factor. Z. Wahrscheinlichkeitsth. 39, 167–174.Google Scholar
Chen, R. (1978). Subfair ‘red-and-black’ in the presence of inflation. Z. Wahrscheinlichkeitsth. 42, 293–301.Google Scholar
Chen, R. and Zame, A. (1979). On discounted subfair primitive casino. Z. Wahrscheinlichkeitsth. 49, 257–266.CrossRefGoogle Scholar
Dubins, L. (1998). Discrete red-and-black with fortune-dependent win probabilities. Prob. Eng. Inf. Sci. 12, 417–424.Google Scholar
Dubins, L. and Savage, L. J. (1976). Inequalities for Stochastic Processes: How to Gamble if You Must. Dover, New York.Google Scholar
Durrett, R. (1996). Probability: Theory and Examples, 2nd edn. Duxbury Press, Belmont, CA.Google Scholar
Fristedt, B. and Gray, L. (1997). A Modern Approach to Probability Theory. Birkhäuser, Boston, MA.Google Scholar
Heath, D., Pruitt, W. and Sudderth, W. (1972). Subfair red-and-black with a limit. Proc. Amer. Math. Soc. 35, 555–560.Google Scholar
Klugman, S. (1977). Discounted and rapid subfair red-and-black. Ann. Statist. 5, 734–745.Google Scholar
Maitra, A. and Sudderth, W. (1996). Discrete Gambling and Stochastic Games. Springer, New York.CrossRefGoogle Scholar
Pestien, V. and Sudderth, W. (1985). Continuous-time red and black: how to control a diffusion to a goal. Math. Operat. Res. 10, 599–611.Google Scholar
Ross, S. (1974). Dynamic programming and gambling models. Adv. Appl. Prob. 6, 593–606.CrossRefGoogle Scholar
Ruth, K. (1999). Favorable red and black on the integers with a minimum bet. J. Appl. Prob. 36, 837–851.Google Scholar
Secchi, P. (1997). Two-person red-and-black stochastic games. J. Appl. Prob. 34, 107–126.Google Scholar
Wilkins, J. E. (1972). The bold strategy in presence of a house limit. Proc. Amer. Math. Soc. 32, 567–570.Google Scholar