Abstract-We consider problems in which a maximizer and a minimizer agent take actions in turn, such as games or optimal control with uncertainty modeled as an opponent. We extend the ideas of optimistic optimization to this setting, obtaining a search algorithm that has been previously considered as the best-first search variant of the B* method. We provide a novel analysis of the algorithm relying on a certain structure for the values of action sequences, under which earlier actions are more important than later ones. An asymptotic branching factor is defined as a measure of problem complexity, and it is used to characterize the relationship between computation invested and near-optimality. In particular, when action importance decreases exponentially, convergence rates are obtained. Throughout, examples illustrate analytical concepts such as the branching factor. In an empirical study, we compare the optimistic best-first algorithm with two classical game tree search methods, and apply it to a challenging HIV infection control problem.

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