Secretary Problem

A Secretary Problem is a Famous Problem that ...

• See: Natural Logarithm, Optimal Stopping, Applied Probability, Statistics, Decision Theory, Stopping Rule, Selection Algorithm, e (Mathematical Constant).

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References

2017

• (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/secretary_problem Retrieved:2017-2-14.
  • The secretary problem is a famous problem that uses the optimal stopping theory. The problem has been studied extensively in the fields of applied probability, statistics, and decision theory. It is also known as the marriage problem, the sultan's dowry problem, the fussy suitor problem, the googol game, and the best choice problem.

    The basic form of the problem is the following: imagine an administrator who wants to hire the best secretary out of [math]\displaystyle{ n }[/math] rankable applicants for a position. The applicants are interviewed one by one in random order. A decision about each particular applicant is to be made immediately after the interview. Once rejected, an applicant cannot be recalled. During the interview, the administrator can rank the applicant among all applicants interviewed so far, but is unaware of the quality of yet unseen applicants. The question is about the optimal strategy (stopping rule) to maximize the probability of selecting the best applicant. If the decision can be deferred to the end, this can be solved by the simple maximum selection algorithm of tracking the running maximum (and who achieved it), and selecting the overall maximum at the end. The difficulty is that the decision must be made immediately.

    The problem has an elegant solution. The optimal stopping rule prescribes always rejecting the first [math]\displaystyle{ n/e }[/math] applicants after the interview (where e is the base of the natural logarithm) and then stopping at the first applicant who is better than every applicant interviewed so far (or continuing to the last applicant if this never occurs). Sometimes this strategy is called the [math]\displaystyle{ 1/e }[/math] stopping rule, because the probability of stopping at the best applicant with this strategy is about [math]\displaystyle{ 1/e }[/math] already for moderate values of [math]\displaystyle{ n }[/math] . One reason why the secretary problem has received so much attention is that the optimal policy for the problem (the stopping rule) is simple and selects the single best candidate about 37% of the time, irrespective of whether there are 100 or 100 million applicants. In fact, for any value of [math]\displaystyle{ n }[/math] the probability of selecting the best candidate when using the optimal policy is at least [math]\displaystyle{ 1/e }[/math] .

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