Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-29T18:34:02.906Z Has data issue: false hasContentIssue false
  • English
  • Français

The dependence of betting strategies on the probability of winning

Published online by Cambridge University Press:  14 July 2016

Eduardo J. Subelman
Eduardo J. Subelman*
Affiliation:
University of California, Los Angeles
*
∗Postal address: Department of System Science, School of Engineering and Applied Science, University of California, Los Angeles, CA 90024, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the optimal wagers to be made by a gambler facing cointossing games who desires to maximize the expected value of the utility of his final fortune in a fixed number n of plays. In the case of fixed probability of a win, the optimal bet is shown to be increasing in the probability. In the case of unknown probability of a win, the wager is shown to be monotone in the prior distribution under the monotone likelihood ratio ordering of these distributions.

Keywords

GAMBLING THEORYBETTING STRATEGIESMONOTONE POLICIESADAPTIVE BETTINGBAYESIAN DECISIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 4 , December 1979 , pp. 855 - 866
Copyright
Copyright © Applied Probability Trust 1979 

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.)

Footnotes

Partially supported by the U.S. Army Research Office under Grant DAAG29–77–0040 with the University of California.

References

[1] Bellman, R. and Kalaba, R. (1957) On the role of dynamic programming in statistical communication theory. IRE Trans. Inf. Theory IT–3, 197–203.CrossRefGoogle Scholar
[2] Breiman, L. (1965) Optimal gambling systems for favorable games. Proc. 4th Berkeley Symp. Math. Statist. Prob. 1, 65–78.Google Scholar
[3] Dubins, L. and Savage, L. (1965) How to Gamble if You Must. McGraw-Hill, New York.Google Scholar
[4] Ferguson, T. (1965) Betting systems which minimize the probability of ruin. SIAM J. 13, 795–818.Google Scholar
[5] Freedman, D. (1967) Timid play is optimal. Ann. Math. Statist. 38, 1281–1284.Google Scholar
[6] Kelly, J. L. Jr. (1956) A new interpretation of information rate. Bell System Tech. J. 35, 917–926.Google Scholar
[7] Molenaar, W. and Van Der Velde, E. (1967) How to survive a fixed number of fair bets. Ann. Math. Statist. 38, 1278–1281.Google Scholar
[8] Pasternack, B. (1974) Optimal gambling and investment systems under discounting and disbursement, ORC 74–1, Operations Research Center, University of California, Berkeley.Google Scholar
[9] Ross, S. M. (1970) Applied Probability Models with Optimization Applications Holden-Day, San Francisco.Google Scholar
[10] Ross, S. M. (1974) Dynamic programming and gambling models. Adv. Appl. Prob. 6, 593–606.Google Scholar
[11] Subelman, E. J. (1979) Optimal betting strategies for favourable games. Naval Res. Logist. Quart. 26, 355–363.Google Scholar
[12] Topkis, D. M. (1978) Minimizing a submodular function on a lattice. Operat. Res. 26, 305–321.CrossRefGoogle Scholar