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This paper considers a multiple stopping time problem for a Markov chainobserved in noise, where a decision maker chooses at most L stopping times tomaximize a cumulative objective. We formulate the problem as a PartiallyObserved Markov Decision Process (POMDP) and derive structural results for theoptimal multiple stopping policy. The main results are as follows: i) Theoptimal multiple stopping policy is shown to be characterized by thresholdcurves in the unit simplex of Bayesian Posteriors. ii) The stopping setsl(defined by the threshold curves) are shown to exhibit a nested structure. iii)The optimal cumulative reward is shown to be monotone with respect to thecopositive ordering of the transition matrix. iv) A stochastic gradientalgorithm is provided for estimating linear threshold policies by exploitingthe structural results. These linear threshold policies approximate thethreshold curves, and share the monotone structure of the optimal multiplestopping policy. As an illustrative example, we apply the multiple stoppingframework to interactively schedule advertisements in live online social media.It is shown that advertisement scheduling using multiple stopping performssignificantly better than currently used methods.

Multiple Stopping Time POMDPs: Structural Results & Application in Interactive Advertising in Social Media