I n a stochastic interdiction game a proliferator aims to minimize the expected duration of a nuclear weapons development project, and an interdictor endeavors to maximize the project duration by delaying some of the project tasks. We formulate static and dynamic versions of the interdictor's decision problem where the interdiction plan is either precommitted or adapts to new information revealed over time, respectively. The static model gives rise to a stochastic program, whereas the dynamic model is formalized as a multiple optimal stopping problem in continuous time and with decision-dependent information. Under a memoryless probabilistic model for the task durations, we prove that the static model reduces to a mixed-integer linear program, whereas the dynamic model reduces to a finite Markov decision process in discrete time that can be solved via efficient value iteration. We then generalize the dynamic model to account for uncertainty in the outcomes of the interdiction actions. We also discuss a crashing game where the proliferator can use limited resources to expedite tasks so as to counterbalance the interdictor's efforts. The resulting problem can be formulated as a robust Markov decision process.
We propose a tightening sequence of optimistic approximations to the Gittins index in “Optimistic Gittins Indices.” We show that the use of these approximations in concert with the use of an increasing discount factor appears to offer a compelling alternative to state-of-the-art index schemes proposed for the Bayesian multiarmed bandit problem. We prove that the use of these optimistic indices constitutes a regret optimal algorithm. Perhaps more interestingly, the use of even the loosest of these approximations appears to offer substantial performance improvements over state-of-the-art alternatives while incurring little to no additional computational overhead relative to the simplest of these alternatives.
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