Weighted games of best choice
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The game of best choice (also known as the secretary problem) is a model for sequential decision making with a long history and many variations. The classical setup assumes that the sequence of candidate rankings are uniformly distributed. Given a statistic on permutations, one can generalize the uniform distribution on the symmetric group by weighting each permutation according to an exponential function in the statistic. We play the game of best choice on the Ewens and Mallows distributions that are obtained in this way from the number of left-to-right maxima and number of inversions in the permutation, respectively. For each of these, we give the optimal strategy and probability of winning. We also introduce a general class of permutation statistics that always produces games of best choice whose optimal strategy is positional. Specializing this result produces a new proof of a foundational result from the literature on the secretary problem.
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