The Secretary Problem and Its Extensions: A Review

Freeman, P. R.

doi:10.2307/1402748

Cited by 406 publications

(250 citation statements)

References 41 publications

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“…One might suspect that this is due to the fact that there is no penalty (except the possibility of missing the maximum value) in the secretary problem for waiting, but surprisingly, similar results hold for variants of the secretary problem where waiting is punished by discounting the values or by setting an explicit cost for each period that no one is hired [4,7]. The main message of this paper is to show that a long exploration phase is indeed an artifact of the assumption that the values are coming from a worst-case distribution.…”

Section: Introductionmentioning

confidence: 85%

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The Secretary Problem with a Hazard Rate Condition

Mahdian

McAfee

Pennock

2008

Lecture Notes in Computer Science

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Abstract. In the classical secretary problem, the objective is to select the candidate of maximum value among a set of n candidates arriving one by one. The value of the candidates come from an unknown distribution and is revealed at the time the candidate arrives, at which point an irrevocable decision on whether to select the candidate must be made. The well-known solution to this problem, due to Dynkin, waits for n/e steps to set an "aspiration level" equal to the maximum value of the candidates seen, and then accepts the first candidate whose value exceeds this level. This guarantees a probability of at least 1/e of selecting the maximum value candidate, and there are distributions for which this is essentially the best possible. One feature of this algorithm that seems at odds with reality is that it prescribes a long waiting period before selecting a candidate. In this paper, we show that if a standard hazard rate condition is imposed on the distribution of values, the waiting period falls from n/e to n/ log(n), meaning that it is enough to observe a diminishingly small sample to set the optimal aspiration level. This result is tight, as both the hazard condition and the optimal sampling period bind exactly for the exponential distribution.

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Section: Introductionmentioning

confidence: 85%

“…We call such a strategy an "aspiration strategy". Many variants of the secretary problem and this solution, and their applications to problems such as online mechanism design for auctions are studied in the literature; see, for example, the survey papers [1,4].…”

Section: Introductionmentioning

confidence: 99%

The Secretary Problem with a Hazard Rate Condition

Mahdian

McAfee

Pennock

2008

Lecture Notes in Computer Science

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“…These techniques have been applied to several disciplines, including probability and statistics (e.g. Freeman, 1983) and economics and finance (e.g. Myneni, 1992), among others.…”

Section: The Methodological Backgroundmentioning

confidence: 99%

Optimal Duration of Magazine Promotions

2005

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The planning of promotions and other marketing events frequently requires manufacturers to make decisions about the optimal duration of these activities. Yet manufacturers often lack the support tools for decision making. We assume that customer decisions at the aggregated level follow a state-dependent Markov process. On the basis of the expected economic return associated with dynamic response to stimuli, we determine the ideal length of marketing events using dynamic programming optimization and apply the model to a complex promotion event. Results suggest that this methodology could help managers in the publishing industry to plan the optimal duration of promotion events.

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“…In the secretary problem, sometimes called the marriage problem and reviewed by Freeman (1983) and Ferguson (1989), each opportunity only has one characteristic whose probability distribution is not known. The objective is to maximise the probability of choosing the opportunity with the maximum characteristic value.…”

Section: Literature Reviewmentioning

confidence: 99%

Lending decisions with limits on capital available: The polygamous marriage problem

Thomas

Huang

2016

European Journal of Operational Research

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In order to stimulate or subdue the economy, banking regulators have sought to impose caps or floors on individual bank's lending to certain types of borrowers. This paper shows that the resultant decision problem for a bank of which potential borrower to accept is a variant of the marriage/secretary problem where one can accept several applicants. The paper solves the decision problem using dynamic programming. We give results on the form of the optimal lending problem and counter examples to further "reasonable" conjectures which do not hold in the general case. By solving numerical examples we show the potential loss of profit and the inconsistency in the lending decision that are caused by introducing floors and caps on lending. The paper also describes some other situations where the same decision occurs.

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The Secretary Problem and Its Extensions: A Review

Cited by 406 publications

References 41 publications

The Secretary Problem with a Hazard Rate Condition

The Secretary Problem with a Hazard Rate Condition

Optimal Duration of Magazine Promotions

Lending decisions with limits on capital available: The polygamous marriage problem

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