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The multi-armed bandit, with constraints

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  • Eric Denardo
  • Eugene Feinberg
  • Uriel Rothblum

Abstract

Presented in this paper is a self-contained analysis of a Markov decision problem that is known as the multi-armed bandit. The analysis covers the cases of linear and exponential utility functions. The optimal policy is shown to have a simple and easily-implemented form. Procedures for computing such a policy are presented, as are procedures for computing the expected utility that it earns, given any starting state. For the case of linear utility, constraints that link the bandits are introduced, and the constrained optimization problem is solved via column generation. The methodology is novel in several respects, which include the use of elementary row operations to simplify arguments. Copyright Springer Science+Business Media New York 2013

Suggested Citation

  • Eric Denardo & Eugene Feinberg & Uriel Rothblum, 2013. "The multi-armed bandit, with constraints," Annals of Operations Research, Springer, vol. 208(1), pages 37-62, September.
    • Handle: RePEc:spr:annopr:v:208:y:2013:i:1:p:37-62:10.1007/s10479-012-1250-y
    DOI: 10.1007/s10479-012-1250-y
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    References listed on IDEAS

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    1. Eugene A. Feinberg & Uriel G. Rothblum, 2012. "Splitting Randomized Stationary Policies in Total-Reward Markov Decision Processes," Mathematics of Operations Research, INFORMS, vol. 37(1), pages 129-153, February.
    2. Eric V. Denardo & Haechurl Park & Uriel G. Rothblum, 2007. "Risk-Sensitive and Risk-Neutral Multiarmed Bandits," Mathematics of Operations Research, INFORMS, vol. 32(2), pages 374-394, May.
    3. Schlag, Karl H., 1998. "Why Imitate, and If So, How?, : A Boundedly Rational Approach to Multi-armed Bandits," Journal of Economic Theory, Elsevier, vol. 78(1), pages 130-156, January.
    4. Sonin, Isaac M., 2008. "A generalized Gittins index for a Markov chain and its recursive calculation," Statistics & Probability Letters, Elsevier, vol. 78(12), pages 1526-1533, September.
    5. Michael N. Katehakis & Arthur F. Veinott, 1987. "The Multi-Armed Bandit Problem: Decomposition and Computation," Mathematics of Operations Research, INFORMS, vol. 12(2), pages 262-268, May.
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    Cited by:

    1. Esther Frostig & Gideon Weiss, 2016. "Four proofs of Gittins’ multiarmed bandit theorem," Annals of Operations Research, Springer, vol. 241(1), pages 127-165, June.
    2. Boxiao Chen & Xiuli Chao & Hyun-Soo Ahn, 2019. "Coordinating Pricing and Inventory Replenishment with Nonparametric Demand Learning," Operations Research, INFORMS, vol. 67(4), pages 1035-1052, July.
    3. Nicole Bäuerle & Ulrich Rieder, 2014. "More Risk-Sensitive Markov Decision Processes," Mathematics of Operations Research, INFORMS, vol. 39(1), pages 105-120, February.
    4. Hyeong Soo Chang, 2020. "An asymptotically optimal strategy for constrained multi-armed bandit problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(3), pages 545-557, June.
    5. Malekipirbazari, Milad & Çavuş, Özlem, 2024. "Index policy for multiarmed bandit problem with dynamic risk measures," European Journal of Operational Research, Elsevier, vol. 312(2), pages 627-640.
    6. Felipe Caro & Aparupa Das Gupta, 2022. "Robust control of the multi-armed bandit problem," Annals of Operations Research, Springer, vol. 317(2), pages 461-480, October.

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