Abstract
Consider a wager that is more complicated than simply winning or losing the amount of the bet. For example, a pass line bet with double odds is such a wager, as is a bet on video poker using a specified drawing strategy. We are concerned with the probability that, in an independent sequence of identical wagers of this type, the gambler loses L or more betting units (i.e., the gambler is “ruined”) before he wins W or more betting units. Using an idea of Markov, Feller established upper and lower bounds on the probability of ruin, bounds that are often very close to each other. However, his formulation depends on finding a positive nontrivial root of the equation φ (ρ )=1, where φ is the probability generating function for the wager in question. Here we give simpler bounds, which rely on the first few moments of the specified wager, thereby making such gambler's ruin probabilities more easily computable.
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References
R. M. Canjar, “Gambler's ruin revisited: The effects of skew and large jackpots,” Preprint, University of Detroit Mercy, 2000.
H. Dunbar and J. B., “Risk of ruin for video poker and other skewed-up games,” Blackjack Forum vol. 19 (Fall) pp. 21-27, 1999.
W. Feller, An Introduction to Probability Theory and Its Applications, vol. 1 John Wiley & Sons: New York, 1950. 3rd edition, 1968.
P. A. Griffin, The Theory of Blackjack, GBC Press: Las Vegas, 2nd Edition, 1981.
S. Jensen, “Optimal drawing strategy for Deuces Wild video poker,” Research report, University of Utah, 2001.
A. S. Kozek, “A rule of thumb (not only) for gamblers,” Stochastic Process. Appl. vol. 55 pp. 169-181, 1995.
A. A. Markov, Wahrscheinlichkeitsrechnung, B. G. Teubner: Leipzig, 1912.
P. Sileo, “The evaluation of blackjack games using a combined expectation and risk measure,” in Gambling and Commercial Gaming: Essays in Business, Economics, Philosophy and Science (W. R. Eadington and J. A. Cornelius, eds), Institute for the Study of Gambling and Commercial Gaming, University of Nevada, Reno, 1992.
J. V. Uspensky, Introduction to Mathematical Probability, McGraw-Hill: New York, 1937.
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Ethier, S.N., Khoshnevisan, D. Bounds on Gambler's Ruin Probabilities in Terms of Moments. Methodology and Computing in Applied Probability 4, 55–68 (2002). https://doi.org/10.1023/A:1015705430513
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DOI: https://doi.org/10.1023/A:1015705430513