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Two-armed bandits with a goal, II. Dependent arms

Published online by Cambridge University Press:  01 July 2016

Donald A. Berry  and
Bert Fristedt
Donald A. Berry*
Affiliation:
University of Minnesota
Bert Fristedt*
Affiliation:
University of Minnesota
*
Postal address: ∗ School of Statistics, University of Minnesota, 270 Vincent Hall, 206 Church St S.E., Minneapolis, MN 55455, U.S.A.
Postal address: ∗∗ School of Mathematics, University of Minnesota, 270 Vincent Hall, 206 Church St S.E., Minneapolis, MN 55455, U.S.A.
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Abstract

One of two random variables, X and Y, can be selected at each of a possibly infinite number of stages. Depending on the outcome, one's fortune is either increased or decreased by 1. The probability of increase may not be known for either X or Y. The objective is to increase one's fortune to G before it decreases to g, for some integral g and G; either may be infinite.

In Part I (Berry and Fristedt (1980)), the distribution of X is unknown and that of Y is known. In the current part, it is known that either X or Y has probability α of increasing the current fortune by 1 and the other has probability β of increasing the fortune by 1, where α and β are known, but which goes with X is not known. We show that optimal strategies exist in general and find all optimal schemes when α = 0 and when α + β = 1. In both cases myopic strategies are shown to be optimal. A counterexample is used to show that myopic strategies, while intuitively very appealing, are not optimal for general (α, β).

Keywords

ACHIEVING A GOALTWO-ARMED BANDITSHOW TO GAMBLE IF YOU MUSTGAMBLER'S RUIN: SEQUENTIAL DECISIONS: BAYESIAN DECISION MAKING: SEQUENTIAL MEDICAL TREATMENTSSTOCHASTIC CONTROLOPTIMAL DYNAMIC DESIGNSMYOPIC STRATEGIES
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 4 , December 1980 , pp. 958 - 971
Copyright
Copyright © Applied Probability Trust 1980 

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Footnotes

∗

This author's research sponsored by the NSF under Grant No. MCS 78-02694.

∗∗

This author's research sponsored by the NSF under Grant No. MCS 78-01168 A01.

References

Berry, D. A. (1972) A Bernoulli two-armed bandit. Ann. Math. Statist. 43, 871–897.CrossRefGoogle Scholar
Berry, D. A. and Fristedt, B. (1980) Two-armed bandits with a goal, I. One arm known. Adv. Appl. Prob. 12, 775–798.Google Scholar
Degroot, M. H. (1970) Optimal Statistical Decisions. McGraw-Hill, New York.Google Scholar
Fabius, J. and Van Zwet, W. R. (1970) Some remarks on the two-armed bandit. Ann. Math. Statist. 41, 1906–1916.Google Scholar
Feldman, D. (1962) Contributions to the ‘two-armed bandit’ problem. Ann. Math. Statist. 33, 847–856.CrossRefGoogle Scholar
Kelly, T. A. (1974) A note on the Bernoulli two-armed bandit. Ann. Statist. 2, 1056–1062.Google Scholar