Finding the Second-Best Candidate under the Mallows Model
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The well-known secretary problem in sequential analysis and optimal stopping theory asks one to maximize the probability of finding the optimal candidate in a sequentially examined list under the constraint that accept/reject decisions are made in real-time. The problem has received significant interest in the mathematics community and is related to practical questions arising in online search, data streaming, daily purchase modeling and multi-arm bandit mechanisms. A version of the problem is the so-called postdoc problem, for which the question of interest is to devise a strategy that identifies the second-best candidate with highest possible probability of success. We solve the postdoc problem for the untraditional setting where the candidates are not presented uniformly at random but rather according to permutations drawn from the Mallows distribution. The Mallows distribution represents the counterpart of a Gaussian distribution for rankings. To address the problem, we extend the work in [1] for the secretary setting and introduce a number of new proof techniques. Our results reveal that the optimal stopping criteria depends on the choice of the Mallows model parameter t: For t>1, the optimal strategy is to reject the first k'(t) candidates and then accept the next left-to-right (abbreviated as l-r) second-best candidate. This coincides with the optimal strategy for the classical postdoc problem derived in [3,2]. For 0<t<=1/2, the optimal strategy is to reject the first k"(t) candidates and then accept the next l-r best candidate; if no selection is made before the last candidate, then the last candidate is accepted. The most interesting strategy arises for 1/2<t<1, the optimal strategy is to reject the first k_1(t) candidates and then accept the next l-r maximum, or reject the first k_2(t)>=k_1(t) candidates and then accept the next l-r second maximum, whichever comes first.
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