On the full information best-choice problem

1996 ◽  
Vol 33 (3) ◽  
pp. 678-687 ◽  
Author(s):  
Alexander V. Gnedin
Keyword(s):  
Poisson Process ◽  
Optimal Stopping ◽  
Infinite Sequence ◽  
Full Information ◽  
Optimal Stopping Problem ◽  
Choice Problem ◽  
Best Choice Problem ◽  
Finite Problem

We introduce the optimal stopping problem of an infinite sequence of records associated with a planar Poisson process. This problem serves as a limiting form of the classical full information best-choice problem. A link between the finite problem and its limiting form is established via embedding n i.i.d. observations into the planar process.

On the full information best-choice problem

1996 ◽  
Vol 33 (03) ◽  
pp. 678-687 ◽  
Author(s):  
Alexander V. Gnedin
Keyword(s):  
Poisson Process ◽  
Optimal Stopping ◽  
Infinite Sequence ◽  
Full Information ◽  
Optimal Stopping Problem ◽  
Choice Problem ◽  
Best Choice Problem ◽  
Finite Problem

We introduce the optimal stopping problem of an infinite sequence of records associated with a planar Poisson process. This problem serves as a limiting form of the classical full information best-choice problem. A link between the finite problem and its limiting form is established via embedding n i.i.d. observations into the planar process.

Why do these quite different best-choice problems have the same solutions?

2004 ◽  
Vol 36 (2) ◽  
pp. 398-416 ◽  
Author(s):  
Stephen M. Samuels
Keyword(s):  
Poisson Process ◽  
Process Model ◽  
Partial Information ◽  
Stopping Rule ◽  
Good Explanation ◽  
Full Information ◽  
Choice Problem ◽  
Best Choice Problem ◽  
Current Observation ◽  
Information Problem

The full-information best-choice problem, as posed by Gilbert and Mosteller in 1966, asks us to find a stopping rule which maximizes the probability of selecting the largest of a sequence of n i.i.d. standard uniform random variables. Porosiński, in 1987, replaced a fixed n by a random N, uniform on {1,2,…,n} and independent of the observations. A partial-information problem, imbedded in a 1980 paper of Petruccelli, keeps n fixed but allows us to observe only the sequence of ranges (max - min), as well as whether or not the current observation is largest so far. Recently, Porosiński compared the solutions to his and Petruccelli's problems and found that the two problems have identical optimal rules as well as risks that are asymptotically equal. His discovery prompts the question: why? This paper gives a good explanation of the equivalence of the optimal rules. But even under the lens of a planar Poisson process model, it leaves the equivalence of the asymptotic risks as somewhat of a mystery. Meanwhile, two other problems have been shown to have the same limiting risks: the full-information problem with the (suboptimal) Porosiński-Petruccelli stopping rule, and the full-information ‘duration of holding the best’ problem of Ferguson, Hardwick and Tamaki, which turns out to be nothing but the Porosiński problem in disguise.

scholarly journals Sum the Multiplicative Odds to One and Stop

2010 ◽  
Vol 47 (03) ◽  
pp. 761-777 ◽  
Author(s):  
Mitsushi Tamaki
Keyword(s):  
Poisson Process ◽  
Optimal Stopping ◽  
Process Approach ◽  
Stopping Rule ◽  
Secretary Problem ◽  
Full Information ◽  
Bernoulli Trials ◽  
Optimal Stopping Problem ◽  
Asymptotic Results ◽  
Optimal Stopping Rule

We consider the optimal stopping problem of maximizing 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 such that 1 ≤ m < n. The optimal stopping rule of this problem has a nice interpretation, that is, it stops on the first success for which the sum of the m-fold multiplicative odds of success for the future trials is less than or equal to 1. This result can be viewed as a generalization of Bruss' (2000) odds theorem. Application will be made to the secretary problem. For more generality, we extend the problem in several directions in the same manner that Ferguson (2008) used to extend the odds theorem. We apply this extended result to the full-information analogue of the secretary problem, and derive the optimal stopping rule and the probability of win explicitly. The asymptotic results, as n tends to ∞, are also obtained via the planar Poisson process approach.

scholarly journals Optimal Stopping with Rank-Dependent Loss

2007 ◽  
Vol 44 (4) ◽  
pp. 996-1011 ◽  
Author(s):  
Alexander V. Gnedin
Keyword(s):  
Poisson Process ◽  
Optimal Stopping ◽  
Nondecreasing Function ◽  
Stopping Rule ◽  
Choice Problem ◽  
History Dependence ◽  
Optimal Rule ◽  
Best Choice Problem ◽  
Complete History ◽  
Independent And Identically Distributed

For τ, a stopping rule adapted to a sequence of n independent and identically distributed observations, we define the loss to be E[q(Rτ)], where Rj is the rank of the jth observation and q is a nondecreasing function of the rank. This setting covers both the best-choice problem, with q(r) = 1(r > 1), and Robbins' problem, with q(r) = r. As n tends to ∞, the stopping problem acquires a limiting form which is associated with the planar Poisson process. Inspecting the limit we establish bounds on the stopping value and reveal qualitative features of the optimal rule. In particular, we show that the complete history dependence persists in the limit; thus answering a question asked by Bruss (2005) in the context of Robbins' problem.

Why do these quite different best-choice problems have the same solutions?

2004 ◽  
Vol 36 (02) ◽  
pp. 398-416 ◽  
Author(s):  
Stephen M. Samuels
Keyword(s):  
Poisson Process ◽  
Process Model ◽  
Partial Information ◽  
Stopping Rule ◽  
Good Explanation ◽  
Full Information ◽  
Choice Problem ◽  
Best Choice Problem ◽  
Current Observation ◽  
Information Problem

The full-information best-choice problem, as posed by Gilbert and Mosteller in 1966, asks us to find a stopping rule which maximizes the probability of selecting the largest of a sequence of n i.i.d. standard uniform random variables. Porosiński, in 1987, replaced a fixed n by a random N, uniform on {1,2,…,n} and independent of the observations. A partial-information problem, imbedded in a 1980 paper of Petruccelli, keeps n fixed but allows us to observe only the sequence of ranges (max - min), as well as whether or not the current observation is largest so far. Recently, Porosiński compared the solutions to his and Petruccelli's problems and found that the two problems have identical optimal rules as well as risks that are asymptotically equal. His discovery prompts the question: why? This paper gives a good explanation of the equivalence of the optimal rules. But even under the lens of a planar Poisson process model, it leaves the equivalence of the asymptotic risks as somewhat of a mystery. Meanwhile, two other problems have been shown to have the same limiting risks: the full-information problem with the (suboptimal) Porosiński-Petruccelli stopping rule, and the full-information ‘duration of holding the best’ problem of Ferguson, Hardwick and Tamaki, which turns out to be nothing but the Porosiński problem in disguise.

scholarly journals Sum the Multiplicative Odds to One and Stop

2010 ◽  
Vol 47 (3) ◽  
pp. 761-777 ◽  
Author(s):  
Mitsushi Tamaki
Keyword(s):  
Poisson Process ◽  
Optimal Stopping ◽  
Process Approach ◽  
Stopping Rule ◽  
Secretary Problem ◽  
Full Information ◽  
Bernoulli Trials ◽  
Optimal Stopping Problem ◽  
Asymptotic Results ◽  
Optimal Stopping Rule

We consider the optimal stopping problem of maximizing 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 such that 1 ≤ m < n. The optimal stopping rule of this problem has a nice interpretation, that is, it stops on the first success for which the sum of the m-fold multiplicative odds of success for the future trials is less than or equal to 1. This result can be viewed as a generalization of Bruss' (2000) odds theorem. Application will be made to the secretary problem. For more generality, we extend the problem in several directions in the same manner that Ferguson (2008) used to extend the odds theorem. We apply this extended result to the full-information analogue of the secretary problem, and derive the optimal stopping rule and the probability of win explicitly. The asymptotic results, as n tends to ∞, are also obtained via the planar Poisson process approach.

scholarly journals Optimal Stopping with Rank-Dependent Loss

2007 ◽  
Vol 44 (04) ◽  
pp. 996-1011 ◽  
Author(s):  
Alexander V. Gnedin
Keyword(s):  
Poisson Process ◽  
Optimal Stopping ◽  
Nondecreasing Function ◽  
Stopping Rule ◽  
Choice Problem ◽  
History Dependence ◽  
Optimal Rule ◽  
Best Choice Problem ◽  
Complete History ◽  
Independent And Identically Distributed

For τ, a stopping rule adapted to a sequence ofnindependent and identically distributed observations, we define the loss to be E[q(Rτ)], whereRjis the rank of thejth observation andqis a nondecreasing function of the rank. This setting covers both the best-choice problem, withq(r) =1(r&gt; 1), and Robbins' problem, withq(r) =r. Asntends to ∞, the stopping problem acquires a limiting form which is associated with the planar Poisson process. Inspecting the limit we establish bounds on the stopping value and reveal qualitative features of the optimal rule. In particular, we show that the complete history dependence persists in the limit; thus answering a question asked by Bruss (2005) in the context of Robbins' problem.

The Full-Information Best Choice Problem with Two Choices

1992 ◽  
pp. 278-281 ◽  
Author(s):  
Zdzislaw Porosinski
Keyword(s):  
Full Information ◽  
Choice Problem ◽  
Best Choice Problem
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