A stochastic game of optimal stopping and order selection
type: Item
instance of: scholarly article
title: A stochastic game of optimal stopping and order selection
zbMATH Open document ID: 0824.62079
author: Ulrich Krengel
author: Alexander Gnedin
published in: The Annals of Applied Probability
publication date: 1995-08-10T00:00:00Z
review text: The authors study the unrestricted version of the stochastic two person zero-sum game introduced by \textit{J. P. Gilbert} and \textit{F. Mosteller} [J. Am. Stat. Assoc. 61, 35-73 (1966)] and \textit{Y. S. Chow}, \textit{S. Moriguti}, \textit{H. Robbins} and \textit{S. M. Samuels} [Israel J. Math. 2 (1964), 81-90 (1965; Zbl 0149.144)] in which player 2 compiles \(Y_ 1, Y_ 2, \ldots, Y_ n\) by rearranging the values of an i.i.d. sequence with continuous distribution known to both players. In the unrestricted game, where \({\mathcal J}\) is the class of all exchangeable distributions and player 1 aims to maximize the probability of stopping at the largest observation, the class of all nonanticipating stopping rules \(T\) is considered. Using a convexity idea, \textit{T. P. Hill} and \textit{D. P. Kennedy} [see Ann. Appl. Probab. 2, No. 2, 503-517 (1992; Zbl 0758.60041)] constructed minimax distributions for certain payoff functions and introduced a sequence of r.vs., called Bernoulli pyramid, that generates the process of relative ranks with an independence property. The authors main construction of the traveller's process yields a sequence of observations with the same rank properties as that of the Bernoulli pyramid, leading in some cases to a solution analogous to the solution of Hill and Kennedy. They also characterize explicitly those payoff functions that admit a solution via Bernoulli pyramid or traveller's process.
Mathematics Subject Classification ID: 62L15
Mathematics Subject Classification ID: 91A15
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 91A05
zbMATH DE Number: 780994
zbMATH Keywords: best choice problem
zbMATH Keywords: minimax strategy
zbMATH Keywords: stochastic two person zero-sum game
zbMATH Keywords: exchangeable distributions
zbMATH Keywords: nonanticipating stopping rules
zbMATH Keywords: Bernoulli pyramid
zbMATH Keywords: relative ranks
zbMATH Keywords: traveller's process
reviewed by: Adhir K. Basu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1177004842
OpenAlex ID: W2069913401
DOI: 10.1214/AOAP/1177004842
Recommended article: Q4217683
Recommended article: Q4035175
Recommended article: Q4231948
Recommended article: Q3807023
Recommended article: A turn-based game related to the last-success-problem
Recommended article: On a randomized strategy in Neveu's stopping problem
Recommended article: Sequential games with random priority
Recommended article: A Nonzero-Sum Extension of Dynkin's Stopping Problem
description: scientific article; zbMATH DE number 780994