Percolation and best-choice problem for powers of paths

2017 ◽  
Vol 54 (2) ◽  
pp. 343-362
Author(s):  
Fabricio Siqueira Benevides ◽  
Małgorzata Sulkowska
Keyword(s):  
Optimal Algorithm ◽  
Random Order ◽  
Upper And Lower Bounds ◽  
Directed Path ◽  
Choice Problem ◽  
Percolation Process ◽  
Best Choice Problem ◽  
Probability Of Success ◽  
Maximal Vertex ◽  
A Site

AbstractThe vertices of thekth power of a directed path withnvertices are exposed one by one to a selector in some random order. At any time the selector can see the graph induced by the vertices that have already appeared. The selector's aim is to choose online the maximal vertex (i.e. the vertex with no outgoing edges). We give upper and lower bounds for the asymptotic behaviour ofpn,kn1/(k+1), wherepn,kis the probability of success under the optimal algorithm. In order to derive the upper bound, we consider a model in which the selector obtains some extra information about the edges that have already appeared. We give the exact asymptotics of the probability of success under the optimal algorithm in this case. In order to derive the lower bound, we analyse a site percolation process on a sequence of thekth powers of a directed path withnvertices.

A best-choice problem with multiple selectors

2000 ◽  
Vol 37 (3) ◽  
pp. 718-735 ◽  
Author(s):  
Hagit Glickman
Keyword(s):  
Random Order ◽  
Point Of View ◽  
Full Description ◽  
Dimensional Vector ◽  
Choice Problem ◽  
Random Vectors ◽  
Relative Rank ◽  
Best Choice Problem ◽  
Optimal Value ◽  
Bounded Below

Consider a situation where a known number, n, of objects appear sequentially in a random order. At each stage, the present object is presented to d ≥ 2 different selectors, who must jointly decide whether to select or reject it, irrevocably. Exactly one object must be chosen. The observation at stage j is a d-dimensional vector R(j) = (R1(j),…, Rd(j)), where Ri(j) is the relative rank of the jth object, by the criterion of the ith selector. The decision whether to stop or not at time j is based on the d-dimensional random vectors R(1),…, R(j). The criteria according to which each selector ranks the objects can either be dependent or independent. Although the goal of each selector is to maximize the probability of choosing the best object from his/her point of view, all d selectors must cooperate and chose the same object. The objective studied here is the maximization of the minimum over the d individual probabilities of choosing the best object. We exhibit the structure of the optimal rule. For independent criteria we give a full description of the rule and show that the optimal value tends to d-d/(d-1), as n → ∞. Furthermore, we show that as n → ∞, the liminf of the values under negatively associated criteria is bounded below by d-d/(d-1).

A best-choice problem with multiple selectors

2000 ◽  
Vol 37 (03) ◽  
pp. 718-735 ◽  
Author(s):  
Hagit Glickman
Keyword(s):  
Random Order ◽  
Point Of View ◽  
Full Description ◽  
Dimensional Vector ◽  
Choice Problem ◽  
Random Vectors ◽  
Relative Rank ◽  
Best Choice Problem ◽  
Optimal Value ◽  
Bounded Below

Consider a situation where a known number, n, of objects appear sequentially in a random order. At each stage, the present object is presented to d ≥ 2 different selectors, who must jointly decide whether to select or reject it, irrevocably. Exactly one object must be chosen. The observation at stage j is a d-dimensional vector R (j) = (R 1(j),…, R d (j)), where R i (j) is the relative rank of the jth object, by the criterion of the ith selector. The decision whether to stop or not at time j is based on the d-dimensional random vectors R (1),…, R (j). The criteria according to which each selector ranks the objects can either be dependent or independent. Although the goal of each selector is to maximize the probability of choosing the best object from his/her point of view, all d selectors must cooperate and chose the same object. The objective studied here is the maximization of the minimum over the d individual probabilities of choosing the best object. We exhibit the structure of the optimal rule. For independent criteria we give a full description of the rule and show that the optimal value tends to d -d/(d-1), as n → ∞. Furthermore, we show that as n → ∞, the liminf of the values under negatively associated criteria is bounded below by d -d/(d-1).

scholarly journals Dynamic Threshold Strategy for Universal Best Choice Problem

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Jakub Kozik
Keyword(s):  
Partially Ordered Sets ◽  
Dynamic Threshold ◽  
Ordered Sets ◽  
Threshold Strategy ◽  
Choice Problem ◽  
Best Choice Problem ◽  
Partial Analysis ◽  
Probability Of Success ◽  
New Strategy ◽  
International Audience

International audience We propose a new strategy for universal best choice problem for partially ordered sets. We present its partial analysis which is sufficient to prove that the probability of success with this strategy is asymptotically strictly greater than 1/4, which is the value of the best universal strategy known so far.

scholarly journals From Directed Path to Linear Order---The Best Choice Problem for Powers of Directed Path

2015 ◽  
Vol 29 (1) ◽  
pp. 500-513 ◽  
Author(s):  
Andrzej Grzesik ◽  
Michał Morayne ◽  
Małgorzata Sulkowska
Keyword(s):  
Linear Order ◽  
Directed Path ◽  
Choice Problem ◽  
Best Choice 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

Urn sampling distributions giving alternate correspondences between two optimal stopping problems

2016 ◽  
Vol 48 (3) ◽  
pp. 726-743 ◽  
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.

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.

The Best Choice Problem for a Random Number of Objects

1973 ◽  
Vol 17 (4) ◽  
pp. 657-668 ◽  
Author(s):  
E. L. Presman ◽  
I. M. Sonin
Keyword(s):  
Random Number ◽  
Choice Problem ◽  
Best Choice Problem
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