Generalizing the secretary problem

The secretary problem refers to a certain class of optimal stopping problems based on relative ranks. To allow a more realistic formulation of the problem, this paper considers an arbitrary loss function. A finite and an infinite problem are defined and the optimal solutions are obtained. The solution for the infinite problem is given by a differential equation while the finite problem is given by a difference equation. Under general conditions, the finite problem tends to the infinite problem. An example involving the secretary problem with interview cost is considered and illustrates the usefulness of the present paper.

Urn sampling distributions giving alternate correspondences between two optimal stopping problems

Advances in Applied Probability ◽

10.1017/apr.2016.25 ◽

2016 ◽

Vol 48 (3) ◽

pp. 726-743 ◽

Cited By ~ 1

Author(s):

Mitsushi Tamaki

Keyword(s):

Optimal Stopping ◽

Random Variable ◽

Planning Horizon ◽

Information Model ◽

Secretary Problem ◽

Choice Problem ◽

Sampling Distributions ◽

Best Choice Problem ◽

Optimal Stopping Problems ◽

Bounded Random Variable

Abstract The best-choice problem and the duration problem, known as versions of the secretary problem, are concerned with choosing an object from those that appear sequentially. Let (B,p) denote the best-choice problem and (D,p) the duration problem when the total number N of objects is a bounded random variable with prior p=(p1, p2,...,pn) for a known upper bound n. Gnedin (2005) discovered the correspondence relation between these two quite different optimal stopping problems. That is, for any given prior p, there exists another prior q such that (D,p) is equivalent to (B,q). In this paper, motivated by his discovery, we attempt to find the alternate correspondence {p(m),m≥0}, i.e. an infinite sequence of priors such that (D,p(m-1)) is equivalent to (B,p(m)) for all m≥1, starting with p(0)=(0,...,0,1). To be more precise, the duration problem is distinguished into (D1,p) or (D2,p), referred to as model 1 or model 2, depending on whether the planning horizon is N or n. The aforementioned problem is model 1. For model 2 as well, we can find the similar alternate correspondence {p[m],m≥ 0}. We treat both the no-information model and the full-information model and examine the limiting behaviors of their optimal rules and optimal values related to the alternate correspondences as n→∞. A generalization of the no-information model is given. It is worth mentioning that the alternate correspondences for model 1 and model 2 are respectively related to the urn sampling models without replacement and with replacement.

Optimal stopping and dynamic allocation

Advances in Applied Probability ◽

10.2307/1427104 ◽

1987 ◽

Vol 19 (4) ◽

pp. 829-853 ◽

Cited By ~ 12

Author(s):

Fu Chang ◽

Tze Leung Lai

Keyword(s):

Wiener Process ◽

Asymptotic Expansions ◽

Optimal Stopping ◽

Exponential Family ◽

Optimal Solutions ◽

Bandit Problem ◽

Dynamic Allocation ◽

Asymptotically Optimal ◽

Simple Index ◽

Optimal Stopping Problems

A class of optimal stopping problems for the Wiener process is studied herein, and asymptotic expansions for the optimal stopping boundaries are derived. These results lead to a simple index-type class of asymptotically optimal solutions to the classical discounted multi-armed bandit problem: given a discount factor 0<β <1 and k populations with densities from an exponential family, how should x1, x2,… be sampled sequentially from these populations to maximize the expected value of Ʃ∞1 βi−1xi, in ignorance of the parameters of the densities?

Optimal stopping and dynamic allocation

Advances in Applied Probability ◽

10.1017/s0001867800017456 ◽

1987 ◽

Vol 19 (04) ◽

pp. 829-853 ◽

Cited By ~ 1

Author(s):

Fu Chang ◽

Tze Leung Lai

Keyword(s):

Wiener Process ◽

Asymptotic Expansions ◽

Optimal Stopping ◽

Exponential Family ◽

Optimal Solutions ◽

Bandit Problem ◽

Dynamic Allocation ◽

Asymptotically Optimal ◽

Simple Index ◽

Optimal Stopping Problems

A class of optimal stopping problems for the Wiener process is studied herein, and asymptotic expansions for the optimal stopping boundaries are derived. These results lead to a simple index-type class of asymptotically optimal solutions to the classical discounted multi-armed bandit problem: given a discount factor 0<β <1 and k populations with densities from an exponential family, how should x 1, x 2,… be sampled sequentially from these populations to maximize the expected value of Ʃ∞ 1 β i−1 x i , in ignorance of the parameters of the densities?

Evaluating Optimal Stopping Problems Under Multivariate Settings and Model Uncertainty

SSRN Electronic Journal ◽

10.2139/ssrn.3549891 ◽

2020 ◽

Author(s):

Ankush Agarwal ◽

Christian-Oliver Ewald ◽

Yihan Zou

Keyword(s):

Optimal Stopping ◽

Model Uncertainty ◽

Optimal Stopping Problems

Intensity-Based Framework and Penalty Formulation of Optimal Stopping Problems

SSRN Electronic Journal ◽

10.2139/ssrn.886880 ◽

2006 ◽

Cited By ~ 1

Author(s):

Yue Kuen Kwok ◽

Min Dai ◽

Hong You

Keyword(s):

Optimal Stopping ◽

Penalty Formulation ◽

Optimal Stopping Problems

A quantization algorithm for solving multidimensional discrete-time optimal stopping problems

Bernoulli ◽

10.3150/bj/1072215199 ◽

2003 ◽

Vol 9 (6) ◽

pp. 1003-1049 ◽

Cited By ~ 109

Author(s):

Vlad Bally ◽

Gilles Pagès

Keyword(s):

Discrete Time ◽

Optimal Stopping ◽

Time Optimal ◽

Optimal Stopping Problems

p-adic confluence of q-difference equations

Compositio Mathematica ◽

10.1112/s0010437x07003454 ◽

2008 ◽

Vol 144 (4) ◽

pp. 867-919 ◽

Cited By ~ 6

Author(s):

Andrea Pulita

Keyword(s):

Differential Equation ◽

Difference Equation ◽

Differential Equations ◽

Difference Equations ◽

Root Of Unity

AbstractWe develop the theory of p-adic confluence of q-difference equations. The main result is the fact that, in the p-adic framework, a function is a (Taylor) solution of a differential equation if and only if it is a solution of a q-difference equation. This fact implies an equivalence, called confluence, between the category of differential equations and those of q-difference equations. We develop this theory by introducing a category of sheaves on the disk D−(1,1), for which the stalk at 1 is a differential equation, the stalk at q isa q-difference equation if q is not a root of unity, and the stalk at a root of unity ξ is a mixed object, formed by a differential equation and an action of σξ.

On wald-type optimal stopping for Brownian motion

Journal of Applied Probability ◽

10.2307/3215175 ◽

1997 ◽

Vol 34 (1) ◽

pp. 66-73 ◽

Cited By ~ 4

Author(s):

S. E. Graversen ◽

G. Peškir

Keyword(s):

Brownian Motion ◽

Optimal Stopping ◽

Stopping Time ◽

Maximal Inequality ◽

Stopping Times ◽

Standard Brownian Motion ◽

Square Root ◽

Real Analysis ◽

Optimal Stopping Problems ◽

Wald's Identity

The solution is presented to all optimal stopping problems of the form supτE(G(|Β τ |) – cτ), where is standard Brownian motion and the supremum is taken over all stopping times τ for B with finite expectation, while the map G : ℝ+ → ℝ satisfies for some being given and fixed. The optimal stopping time is shown to be the hitting time by the reflecting Brownian motion of the set of all (approximate) maximum points of the map . The method of proof relies upon Wald's identity for Brownian motion and simple real analysis arguments. A simple proof of the Dubins–Jacka–Schwarz–Shepp–Shiryaev (square root of two) maximal inequality for randomly stopped Brownian motion is given as an application.

Optimal Stopping of the Maximum Process

Journal of Applied Probability ◽

10.1017/s0021900200011694 ◽

2014 ◽

Vol 51 (03) ◽

pp. 818-836 ◽

Cited By ~ 1

Author(s):

Luis H. R. Alvarez ◽

Pekka Matomäki

Keyword(s):

Brownian Motion ◽

Free Boundary ◽

Optimal Stopping ◽

Original Problem ◽

Alternative Route ◽

Linear Diffusion ◽

Maximum Process ◽

Running Maximum ◽

Parameterized Family ◽

Optimal Stopping Problems

We consider a class of optimal stopping problems involving both the running maximum as well as the prevailing state of a linear diffusion. Instead of tackling the problem directly via the standard free boundary approach, we take an alternative route and present a parameterized family of standard stopping problems of the underlying diffusion. We apply this family to delineate circumstances under which the original problem admits a unique, well-defined solution. We then develop a discretized approach resulting in a numerical algorithm for solving the considered class of stopping problems. We illustrate the use of the algorithm in both a geometric Brownian motion and a mean reverting diffusion setting.