Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-26T07:02:00.304Z Has data issue: false hasContentIssue false
  • English
  • Français

Further insight into the structure of bold and timid policies

Published online by Cambridge University Press:  01 July 2016

Pravin K. Johri  and
Michael N. Katehakis
Pravin K. Johri*
Affiliation:
State University of New York at Stony Brook
Michael N. Katehakis*
Affiliation:
State University of New York at Stony Brook
*
∗Postal address: Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA.
∗Postal address: Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

A gambler repeatedly plays a game until either he becomes broke or his fortune becomes equal to or exceeds a target amount. The gambler is allowed to make multiple bets, i.e. stake integral amounts on different alternatives of the game, and more than one bet may win simultaneously. The objective is to determine a strategy that maximizes the probability of attaining the target amount. When all bets have the same gain and alternatives of betting exist such that the relevant plays are feasible, the following results are obtained. For unfavorable games, a bold type of policy is shown to be optimal. A timed type of policy is shown to be best within a restricted class of policies for favorable games. In general, optimal policies contain multiple bets. Based on a numerical example, this is established for roulette also.

Keywords

DYNAMIC PROGRAMMING
Type
Research Article
Information
Advances in Applied Probability , Volume 17 , Issue 2 , June 1985 , pp. 298 - 307
Copyright
Copyright © Applied Probability Trust 1985 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Breiman, L. (1961) Optimal gambling systems for favorable games. Proc. 4th Berkeley Symp. Math. Statist. Prob. 1, 65–78.Google Scholar
Derman, C. (1970) Finite State Markovian Decision Processes. Academic Press, New York.Google Scholar
Dubins, L. (1968) A simpler proof of Smith’s roulette theorem. Ann. Math. Statist. 39, 390–393.Google Scholar
Dubins, L. and Savage, L. (1976) Inequalities for Stochastic Processes. How to Gamble if you Must. Dover Publications, New York.Google Scholar
Freedman, D. (1967) Timid play is optimal. Ann. Math. Statist. 38, 1281–1284.Google Scholar
Molenaar, W. and Van Der Velde, E. A. (1967) How to survive a fixed number of fair bets. Ann. Math. Statist. 38, 1278–1281.Google Scholar
Ross, S. M. (1974) Dynamic programming and gambling models. Adv. Appl. Prob. 6, 593–606.Google Scholar
Scarne, J. (1961) Scarne’s Complete Guide to Gambling. Simon and Schuster, New York.Google Scholar
Smith, G. J. (1967) Optimal strategy at roulette. Z. Wahrscheinlichkeitsth. 8, 91–100.Google Scholar