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A conjecture on the Feldman bandit problem

Part of:Stochastic processesSequential methods

Published online by Cambridge University Press:  28 March 2018

Maher Nouiehed and
Sheldon M. Ross
Maher Nouiehed*
Affiliation:
University of Southern California
Sheldon M. Ross*
Affiliation:
University of Southern California
*
* Postal address: Department of Industrial and Systems Engineering, University of Southern California, Los Angeles, CA 90089, USA.
* Postal address: Department of Industrial and Systems Engineering, University of Southern California, Los Angeles, CA 90089, USA.
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Abstract

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We consider the Bernoulli bandit problem where one of the arms has win probability α and the others β, with the identity of the α arm specified by initial probabilities. With u = max(α, β), v = min(α, β), call an arm with win probability u a good arm. Whereas it is known that the strategy of always playing the arm with the largest probability of being a good arm maximizes the expected number of wins in the first n games for all n, we conjecture that it also stochastically maximizes the number of wins. That is, we conjecture that this strategy maximizes the probability of at least k wins in the first n games for all k, n. The conjecture is proven when k = 1, and k = n, and when there are only two arms and k = n - 1.

Keywords

Multi-armed banditstochastically maximizing

MSC classification

Primary:60G40: Stopping times; optimal stopping problems; gambling theory
Secondary:62L05: Sequential design

Information

Type
Short Communications
Information
Journal of Applied Probability , Volume 55 , Issue 1 , March 2018, pp. 318 - 324
Copyright
Copyright © Applied Probability Trust 2018 

References

[1]Feldman, D. (1962). Contributions to the "two-armed bandit" problem. Ann. Math. Statist. 33, 847–856. Google Scholar
[2]Rodman, L. (1978). On the many-armed bandit problem. Ann. Prob. 6, 491–498. CrossRefGoogle Scholar
[3]Presman, É. L. and Sonin, I. N. (1990). Sequential Control With Incomplete Information: The Bayesian Approach to Multi-Armed Bandit Problems. Academic Press, San Diego, CA. Google Scholar