When is it best to follow the leader?

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.