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When is it best to follow the leader?

Author

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  • Ernst, Philip A.
  • Rogers, L.C.G.
  • Zhou, Quan

Abstract

An object is hidden in one of N boxes. Initially, the probability that it is in box i is πi(0). You then search in continuous time, observing box Jt at time t, and receiving a signal as you observe: if the box you are observing does not contain the object, your signal is a Brownian motion, but if it does contain the object your signal is a Brownian motion with positive drift μ. It is straightforward to derive the evolution of the posterior distribution π(t) for the location of the object. If T denotes the first time that one of the πj(t) reaches a desired threshold 1−ε, then the goal is to find a search policy (Jt)t≥0 which minimizes the mean of T. This problem was studied by Posner and Rumsey (1966) and by Zigangirov (1966), who derive an expression for the mean time of a conjectured optimal policy, which we call follow the leader (FTL); at all times, observe the box with the highest posterior probability. Posner and Rumsey assert without proof that this is optimal, and Zigangirov offers a proof that if the prior distribution is uniform then FTL is optimal. In this paper, we show that if the prior is not uniform, then FTL is not always optimal; for uniform prior, the question remains open.

Suggested Citation

  • Ernst, Philip A. & Rogers, L.C.G. & Zhou, Quan, 2020. "When is it best to follow the leader?," Stochastic Processes and their Applications, Elsevier, vol. 130(6), pages 3394-3407.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:6:p:3394-3407
    DOI: 10.1016/j.spa.2019.09.017
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    References listed on IDEAS

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    1. Vladimir Dragalin, 1996. "A simple and effective scanning rule for a multi-channel system," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 43(1), pages 165-182, December.
    2. Klimko, E. M. & Yackel, James, 1975. "Optimal search strategies for Wienér processes," Stochastic Processes and their Applications, Elsevier, vol. 3(1), pages 19-33, January.
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