On nonoptimality of bold play for subfair red-and-black with a rational-valued house limit

May Ru Chen, Shoou Ren Hsiau, Pei Shou Chung, Y. A.O. Yi-Ching

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Abstract

In the subfair red-and-black gambling problem, a gambler can stake any amount in his possession, winning an amount equal to the stake with probability w and losing the stake with probability 1 - w, where 0 < w < 1/2. The gambler seeks to maximize the probability of reaching a fixed fortune (to be normalized to unity) by gambling repeatedly with suitably chosen stakes. In their classic work, Dubins and Savage (1965), (1976) showed that it is optimal to play boldly. When there is a house limit of ℓ (0 ℓ t < 1/2), so that the gambler can stake no more than &ill, Wilkins (1972) showed that bold play remains optimal provided that 1/ℓ is an integer. On the other hand, building on an earlier surprising result of Heath, Pruitt and Sudderth (1972), Schweinsberg (2005) recently showed that, for all irrational 0 < ℓ < 1/2 and all 0 < w < 1/2, bold play is not optimal for some initial fortune. The purpose of the present paper is to present several results supporting the conjecture that, for all rational ℓ with 1/ℓ not an integer and all 0 < w < 1/2, bold play is not optimal for some initial fortune. While most of these results are based on Schweinsberg's method, in a special case where his method is shown to be inapplicable, we argue that the conjecture can be verified with the help of symbolic-computation software.

Original languageEnglish
Pages (from-to)1024-1038
Number of pages15
JournalJournal of Applied Probability
Volume45
Issue number4
DOIs
Publication statusPublished - 2008 Dec 1

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All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Mathematics(all)
  • Statistics, Probability and Uncertainty

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