Generalizing the Secretary Problem with Rank-dependent Rejection Probability

When sequential compliance method is used for Weibull distributions, the shape parameter is usually considered to be fixed. However, because of the life of products are determined by many factors, the shape parameter is variational in practice, that is to say, the shape parameter in the criterions is different from that in the practice. In this paper, the changes of acceptance and rejection probability are researched by the influence of shape parameter changes. Finally, by means of simulation test, changes on the shape parameter affecting on the probability of acceptance and rejection are quantitatively analyzed. As a result, the larger the gap on the shape parameter in the criterions and in the practice is, the larger the gap on the producer’s risk and the consumer’s risk.

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A Note on a Lower Bound for the Multiplicative Odds Theorem of Optimal Stopping

Journal of Applied Probability ◽

10.1017/s0021900200011748 ◽

2014 ◽

Vol 51 (03) ◽

pp. 885-889 ◽

Cited By ~ 1

Author(s):

Tomomi Matsui ◽

Katsunori Ano

Keyword(s):

Lower Bound ◽

Optimal Stopping ◽

Secretary Problem ◽

Bernoulli Trials ◽

Optimal Stopping Problem ◽

Maximum Probability ◽

Optimal Stopping Rule ◽

Optimal Probability ◽

Optimal Stopping Theory ◽

Special Case

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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Rejection Sampling Revisit: How to Choose Parameters in Lattice-Based Signature

Mathematical Problems in Engineering ◽

10.1155/2021/9948618 ◽

2021 ◽

Vol 2021 ◽

pp. 1-12

Author(s):

Zhongxiang Zheng ◽

Anyu Wang ◽

Lingyue Qin

Keyword(s):

Uniform Distribution ◽

Sampling Theorem ◽

New Method ◽

Security Level ◽

Rejection Sampling ◽

Forgery Attack ◽

Rejection Probability ◽

Standardization Process ◽

Security Levels ◽

Security Bound

Rejection sampling technology is a core tool in the design of lattice-based signatures with ‘Fiat–Shamir with Aborts’ structure, and it is related to signing efficiency and signature, size as well as security. In the rejection sampling theorem proposed by Lyubashevsky, the masking vector of rejection sampling is chosen from discrete Gaussian distribution. However, in practical designs, the masking vector is more likely to be chosen from bounded uniform distribution due to better efficiency and simpler implementation. Besides, as one of the third-round candidate signatures in the NIST postquantum cryptography standardization process, the 3rd round version of CRYSTALS-Dilithium has proposed a new method to decrease the rejection probability in order to achieve better efficiency and smaller signature size by decreasing the number of nonzero coefficients of the challenge polynomial according to the security levels. However, it is seen that small entropies in this new method may lead to higher risk of forgery attack compared with former schemes proposed in its 2nd version. Thus, in this paper, we first analyze the complexity of forgery attack for small entropies and then introduce a new method to decrease the rejection probability without loss of security including the security against forgery attack. This method is achieved by introducing a new rejection sampling theorem with tighter bound by utilizing Rényi divergence where masking vector follows uniform distribution. By observing large gaps between the security claim and actual security bound in CRYSTALS-Dilithium, we propose two series of adapted parameters for CRYSTALS-Dilithium. The first set can improve the efficiency of the signing process in CRYSTALS-Dilithium by factors of 61.7 % and 41.7 % , according to the security levels, and ensure the security against known attacks, including forgery attack. And, the second set can reduce the signature size by a factor of 14.09 % with small improvements in efficiency at the same security level.

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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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