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.
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References
Bak, J. (2001). The anxious gambler's ruin. Math. Mag. 74, 182--193.
Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, New York.
Chen, R. (1976). Subfair discounted red-and-black game with a house limit. J. Appl. Prob. 13, 608--613.
Mathematical Reviews (MathSciNet):
MR418232
Chen, R. (1977). Subfair primitive casino with a discount factor. Z. Wahrscheinlichkeitsth. 39, 167--174.
Mathematical Reviews (MathSciNet):
MR451401
Chen, R. (1978). Subfair `red-and-black' in the presence of inflation. Z. Wahrscheinlichkeitsth. 42, 293--301.
Mathematical Reviews (MathSciNet):
MR494481
Chen, R. and Zame, A. (1979). On discounted subfair primitive casino. Z. Wahrscheinlichkeitsth. 49, 257--266.
Mathematical Reviews (MathSciNet):
MR547827
Dubins, L. (1998). Discrete red-and-black with fortune-dependent win probabilities. Prob. Eng. Inf. Sci. 12, 417--424.
Dubins, L. and Savage, L. J. (1976). Inequalities for Stochastic Processes: How to Gamble if You Must. Dover, New York.
Mathematical Reviews (MathSciNet):
MR410875
Durrett, R. (1996). Probability: Theory and Examples, 2nd edn. Duxbury Press, Belmont, CA.
Fristedt, B. and Gray, L. (1997). A Modern Approach to Probability Theory. Birkhäuser, Boston, MA.
Heath, D., Pruitt, W. and Sudderth, W. (1972). Subfair red-and-black with a limit. Proc. Amer. Math. Soc. 35, 555--560.
Mathematical Reviews (MathSciNet):
MR309574
Klugman, S. (1977). Discounted and rapid subfair red-and-black. Ann. Statist. 5, 734--745.
Mathematical Reviews (MathSciNet):
MR438478
Maitra, A. and Sudderth, W. (1996). Discrete Gambling and Stochastic Games. Springer, New York.
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.
Mathematical Reviews (MathSciNet):
MR812818
Ross, S. (1974). Dynamic programming and gambling models. Adv. Appl. Prob. 6, 593--606.
Mathematical Reviews (MathSciNet):
MR347381
Ruth, K. (1999). Favorable red and black on the integers with a minimum bet. J. Appl. Prob. 36, 837--851.
Secchi, P. (1997). Two-person red-and-black stochastic games. J. Appl. Prob. 34, 107--126.
Wilkins, J. E. (1972). The bold strategy in presence of a house limit. Proc. Amer. Math. Soc. 32, 567--570.
Mathematical Reviews (MathSciNet):
MR292182
