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Secretaries with Advice

Paul Duetting
Proceedings of the 22nd ACM Conference on Economics and Computation (EC'21) (2021), pp. 409-429
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The secretary problem is probably the purest model of decision making under uncertainty. In this paper we ask which advice can we give the algorithm to improve its success probability? We propose a general model that unifies a broad range of problems: from the classic secretary problem with no advice, to the variant where the quality of a secretary is drawn from a known distribution and the algorithm learns each candidate’s quality quantile on arrival, to more modern ML-based versions of advice where a binary classifier gives us noisy advice about whether or not the current secretary is the best on the market. Our main technique is a factor revealing LP that captures all of the problems above. We use this LP formulation to gain structural insight into the optimal policy and present two case studies: a re-derivation of the classic known distributions result with tools from linear programming and a tight analysis of the noisy binary advice model.