Does the Multisecretary Problem Always Have Bounded Regret?

12/16/2019

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by   Robert L. Bray, et al.

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Arlotto and Gurvich (2019) showed that the regret in the multisecretary problem is bounded in the number of job openings, n, and the number of applicants, k, provided that the applicant valuations are drawn from a distribution with finite support. I show that this result does not hold when applicant valuations are drawn from a standard uniform distribution. In this case, the regret is between log(n)/16 - 1/4 and log(n+1) / 8, when k = n/2 and n >= 16. I establish these bounds by decomposing the regret into a sum of expected myopic regrets. This decomposition also yields a shorter proof of Arlotto and Gurvich's original result.