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On Gittins' index theorem in continuous time

Author

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  • Bank, Peter
  • Küchler, Christian

Abstract

We give a new and comparably short proof of Gittins' index theorem for dynamic allocation problems of the multi-armed bandit type in continuous time under minimal assumptions. This proof gives a complete characterization of optimal allocation strategies as those policies which follow the current leader among the Gittins indices while ensuring that a Gittins index is at an all-time low whenever the associated project is not worked on exclusively. The main tool is a representation property of Gittins index processes which allows us to show that these processes can be chosen to be pathwise lower semi-continuous from the right and quasi-lower semi-continuous from the left. Both regularity properties turn out to be crucial for our characterization and the construction of optimal allocation policies.

Suggested Citation

  • Bank, Peter & Küchler, Christian, 2007. "On Gittins' index theorem in continuous time," Stochastic Processes and their Applications, Elsevier, vol. 117(9), pages 1357-1371, September.
  • Handle: RePEc:eee:spapps:v:117:y:2007:i:9:p:1357-1371
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    Cited by:

    1. Samuel N. Cohen & Tanut Treetanthiploet, 2019. "Gittins' theorem under uncertainty," Papers 1907.05689, arXiv.org, revised Jun 2021.
    2. Diana M. Negoescu & Kostas Bimpikis & Margaret L. Brandeau & Dan A. Iancu, 2018. "Dynamic Learning of Patient Response Types: An Application to Treating Chronic Diseases," Management Science, INFORMS, vol. 64(8), pages 3469-3488, August.
    3. Li, Xiao & Li, Yuqiang & Wu, Xianyi, 2023. "Empirical Gittins index strategies with ε-explorations for multi-armed bandit problems," Computational Statistics & Data Analysis, Elsevier, vol. 180(C).

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