Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-06-03T19:58:35.854Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-person red-and-black stochastic games

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Piercesare Secchi
Piercesare Secchi*
Affiliation:
Università di Pavia
*
∗Postal address: Dipartimento di Economia Politica e Metodi Quantitativi, Università di Pavia, Via San Felice, 5, 27100 Pavia, Italy.
Get access Rights & Permissions [Opens in a new window]

Abstract

We define a leavable stochastic game which is a possible two-person generalization of the classical red-and-black gambling problem. We show that there are three basic possibilities for a two-person red-and-black game which, by analogy with gambling theory, we call the subfair, the fair and the superfair cases. A suitable generalization of what in gambling theory is called bold play is proved to be a uniformly ε-optimal stationary strategy for player I in the fair and the subfair cases whereas a generalization of timid play is shown to be ε-optimal for player I in the superfair possibility.

Keywords

STOCHASTIC GAMESLEAVABLE GAMESRED-AND-BLACKGAMBLING

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 34 , Issue 1 , March 1997 , pp. 107 - 126
Copyright
Copyright © Applied Probability Trust 1997 

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

Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
Dubins, L. E. and Savage, L. J. (1976) Inequalities for Stochastic Processes. How to Gamble if you Must. Dover, New York.Google Scholar
Luce, R. D. and Raiffa, H. (1989) Games and Decisions. Dover, New York.Google Scholar
Maitra, A. and Sudderth, W. (1992) An operator solution of stochastic games. Israel J. Math. 78, 33–19.Google Scholar
Maitra, A. and Sudderth, W. (1996) Discrete Gambling and Stochastic Games. Springer, New York.Google Scholar
Milnor, J. and Shapley, L. S. (1957) On games of survival. In Contributions to the Theory of Games. ed. Dresher, M., Tucker, A. W. and Wolfe, P. Princeton University Press, Princeton, NJ.Google Scholar
Orkin, M. (1972) Recursive matrix games. J. Appl. Prob. 9, 813–820.Google Scholar
Ross, S. M. (1974) Dynamic programming and gambling models. Adv. Appl. Prob. 6, 593–606.Google Scholar
Secchi, P. (1995) Problems in two-person, zero-sum stochastic games. PhD thesis. University of Minnesota, Minneapolis.Google Scholar
Taylor, A. E. (1985) General Theory of Functions and Integration. Dover, New York.Google Scholar
Von Neumann, J. and Morgenstern, O. (1947) Theory of Games and Economic Behavior. 2nd edn. Princeton University Press, Princeton, NJ.Google Scholar