A rule of thumb (not only) for gamblers

Andrzej S. Kozek

doi:10.1016/0304-4149(95)91546-D

A rule of thumb (not only) for gamblers

Andrzej S. Kozek^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

5 Citations (Scopus)

Abstract

Let prize X in a game be a random variable with a cumulative distribution function F, E[X] ≠ 0, and Var(X) < ∞. In a Gambler's Ruin Problem we consider the probability P_F(A, B) of accumulating fortune A before losing the initial fortune B. Suppose our Gambler is to choose between different strategies with the same expected values and different variances. P_F(A, B) is known to depend in general on the whole cumulative distribution function F of X. In this paper we derive an approximation which implies the following rule called A Rule of Thumb (not only) for Gamblers: if E(X) < 0 then the strategy with the greater variance is superior, while in case E[X] > 0 the strategy with the smaller variance is more favorable to the Gambler. We include some examples of applications of The Rule. Moreover we derive a general solution in the Roulette case and use it to show good behavior of The Rule explicitly.

Original language	English
Pages (from-to)	169-181
Number of pages	13
Journal	Stochastic Processes and their Applications
Volume	55
Issue number	1
DOIs	https://doi.org/10.1016/0304-4149(95)91546-D
Publication status	Published - 1995

Keywords

Gambling
Martingale
Random walk
Ruin
Stopping times

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10.1016/0304-4149(95)91546-D

Cite this

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AB - Let prize X in a game be a random variable with a cumulative distribution function F, E[X] ≠ 0, and Var(X) < ∞. In a Gambler's Ruin Problem we consider the probability PF(A, B) of accumulating fortune A before losing the initial fortune B. Suppose our Gambler is to choose between different strategies with the same expected values and different variances. PF(A, B) is known to depend in general on the whole cumulative distribution function F of X. In this paper we derive an approximation which implies the following rule called A Rule of Thumb (not only) for Gamblers: if E(X) < 0 then the strategy with the greater variance is superior, while in case E[X] > 0 the strategy with the smaller variance is more favorable to the Gambler. We include some examples of applications of The Rule. Moreover we derive a general solution in the Roulette case and use it to show good behavior of The Rule explicitly.

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KW - Stopping times

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A rule of thumb (not only) for gamblers

Abstract

Keywords

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