The secretary problem with multiple job vacancies and batch candidate arrivals

In this note we present a bound of the optimal maximum probability for the multiplicative odds theorem of optimal stopping theory. We deal with an optimal stopping problem that maximizes the probability of stopping on any of the last m successes of a sequence of independent Bernoulli trials of length N, where m and N are predetermined integers satisfying 1 ≤ m < N. This problem is an extension of Bruss' (2000) odds problem. In a previous work, Tamaki (2010) derived an optimal stopping rule. We present a lower bound of the optimal probability. Interestingly, our lower bound is attained using a variation of the well-known secretary problem, which is a special case of the odds problem.

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Prophet Inequalities for Independent and Identically Distributed Random Variables from an Unknown Distribution

Mathematics of Operations Research ◽

10.1287/moor.2021.1167 ◽

2021 ◽

Author(s):

José Correa ◽

Paul Dütting ◽

Felix Fischer ◽

Kevin Schewior

Keyword(s):

Stopping Time ◽

Random Variables ◽

Optimal Solution ◽

Random Order ◽

Secretary Problem ◽

Maximum Value ◽

Optimal Stopping Theory ◽

Long Time ◽

Object Of Study ◽

Independent And Identically Distributed

A central object of study in optimal stopping theory is the single-choice prophet inequality for independent and identically distributed random variables: given a sequence of random variables [Formula: see text] drawn independently from the same distribution, the goal is to choose a stopping time τ such that for the maximum value of α and for all distributions, [Formula: see text]. What makes this problem challenging is that the decision whether [Formula: see text] may only depend on the values of the random variables [Formula: see text] and on the distribution F. For a long time, the best known bound for the problem had been [Formula: see text], but recently a tight bound of [Formula: see text] was obtained. The case where F is unknown, such that the decision whether [Formula: see text] may depend only on the values of the random variables [Formula: see text], is equally well motivated but has received much less attention. A straightforward guarantee for this case of [Formula: see text] can be derived from the well-known optimal solution to the secretary problem, where an arbitrary set of values arrive in random order and the goal is to maximize the probability of selecting the largest value. We show that this bound is in fact tight. We then investigate the case where the stopping time may additionally depend on a limited number of samples from F, and we show that, even with o(n) samples, [Formula: see text]. On the other hand, n samples allow for a significant improvement, whereas [Formula: see text] samples are equivalent to knowledge of the distribution: specifically, with n samples, [Formula: see text] and [Formula: see text], and with [Formula: see text] samples, [Formula: see text] for any [Formula: see text].

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The Secretary Problem with a Hazard Rate Condition

Lecture Notes in Computer Science - Internet and Network Economics ◽

10.1007/978-3-540-92185-1_76 ◽

2008 ◽

pp. 708-715 ◽

Cited By ~ 6

Author(s):

Mohammad Mahdian ◽

Randolph Preston McAfee ◽

David Pennock

Keyword(s):

Hazard Rate ◽

Secretary Problem ◽

Rate Condition

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Subjective optimality in finite sequential decision-making

PLoS Computational Biology ◽

10.1371/journal.pcbi.1009633 ◽

2021 ◽

Vol 17 (12) ◽

pp. e1009633

Author(s):

Yeonju Sin ◽

HeeYoung Seon ◽

Yun Kyoung Shin ◽

Oh-Sang Kwon ◽

Dongil Chung

Keyword(s):

Decision Making ◽

Time Window ◽

Programming Model ◽

Secretary Problem ◽

Optimal Decision ◽

Sequential Decision Making ◽

Sequential Decision ◽

Model Based ◽

Subjective Valuation ◽

Optimal Decision Making

Many decisions in life are sequential and constrained by a time window. Although mathematically derived optimal solutions exist, it has been reported that humans often deviate from making optimal choices. Here, we used a secretary problem, a classic example of finite sequential decision-making, and investigated the mechanisms underlying individuals’ suboptimal choices. Across three independent experiments, we found that a dynamic programming model comprising subjective value function explains individuals’ deviations from optimality and predicts the choice behaviors under fewer and more opportunities. We further identified that pupil dilation reflected the levels of decision difficulty and subsequent choices to accept or reject the stimulus at each opportunity. The value sensitivity, a model-based estimate that characterizes each individual’s subjective valuation, correlated with the extent to which individuals’ physiological responses tracked stimuli information. Our results provide model-based and physiological evidence for subjective valuation in finite sequential decision-making, rediscovering human suboptimality in subjectively optimal decision-making processes.

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[Who Solved the Secretary Problem?]: Comment

Statistical Science ◽

10.1214/ss/1177012496 ◽

1989 ◽

Vol 4 (3) ◽

pp. 292-293 ◽

Cited By ~ 1

Author(s):

Minoru Sakaguchi

Keyword(s):

Secretary Problem

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Opportunity Costs in the Game of Best Choice

The Electronic Journal of Combinatorics ◽

10.37236/8322 ◽

2019 ◽

Vol 26 (1) ◽

Cited By ~ 1

Author(s):

Madeline Crews ◽

Brant Jones ◽

Kaitlyn Myers ◽

Laura Taalman ◽

Michael Urbanski ◽

...

Keyword(s):

Decision Making ◽

Classical Case ◽

Secretary Problem ◽

Sequential Decision Making ◽

Opportunity Costs ◽

Sequential Decision ◽

Probability Of Success ◽

Uniform Model ◽

Weighted Model ◽

Over Time

The game of best choice, also known as the secretary problem, is a model for sequential decision making with many variations in the literature. Notably, the classical setup assumes that the sequence of candidate rankings is uniformly distributed over time and that there is no expense associated with the candidate interviews. Here, we weight each ranking permutation according to the position of the best candidate in order to model costs incurred from conducting interviews with candidates that are ultimately not hired. We compare our weighted model with the classical (uniform) model via a limiting process. It turns out that imposing even infinitesimal costs on the interviews results in a probability of success that is about 28%, as opposed to 1/e (about 37%) in the classical case.

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A generalized secretary problem

Sequential Analysis ◽

10.1080/07474946.2016.1165521 ◽

2016 ◽

Vol 35 (2) ◽

pp. 145-157

Author(s):

Abba Krieger ◽

Ester Samuel-Cahn

Keyword(s):

Secretary Problem

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Probability maximizing approach to a secretary problem by random change-point of the distribution law of the observed process

Journal of Applied Probability ◽

10.2307/3213668 ◽

1984 ◽

Vol 21 (1) ◽

pp. 98-107 ◽

Cited By ~ 1

Author(s):

Minoru Yoshida

Keyword(s):

Distribution Function ◽

Stopping Time ◽

Random Variables ◽

Secretary Problem ◽

Independent Random Variables ◽

Distribution Law ◽

Common Distribution ◽

Common Distribution Function ◽

Random Moment ◽

The Moment

Before some random moment θ, independent identically distributed random variables x1, · ··, xθ–1 with common distribution function μ (dx) appear consecutively. After the moment θ, independent random variables xθ, xθ+1, · ·· have another common distribution function f (x)μ (dx). Our information about θ can be constructed only by successively observed values of the x's.In this paper we find an optimal stopping policy by which we can maximize the probability that the quantity associated with the stopping time is the largest of all θ + m – 1 quantities for a given integer m.

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