Measurable, nonleavable gambling problems
type: Item
instance of: scholarly article
title: Measurable, nonleavable gambling problems
zbMATH Open document ID: 0694.60038
DOI: 10.1007/BF02764945
author: Lester E. Dubins
author: A. Maitra
author: Roger A. Purves
author: William D. Sudderth
published in: Israel Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: The paper deals with the \textit{L. E. Dubins} and \textit{L. J. Savage} [Inequalities for stochastic processes. How to gamble if you must. (1976; Zbl 0359.60002)] gambling problem \({\mathcal E}=(X,\Gamma,u)\), where X is a set, u: \(X\to {\mathbb{R}}\), while \(\Gamma\) maps X into sets of finitely additive measures on X; \({\mathcal E}\) is called analytic if X is a Borel set, the graph of \(\Gamma\) is an analytic set (in \(X\times Prob(X))\) while u is bounded and upper analytic (i.e. \(\{x| u(x)>\alpha \}\) is analytic for every real \(\alpha)\); u(\(\sigma\),t) denotes the expected utility under the choice of strategy \(\sigma =(\sigma_ 0,\sigma_ 1,...)\) (which means \(\sigma_ n(x_ 1,...,x_ n)\in \Gamma (x_ n)\) and stopping time t. The authors deal with the functions \[ V(x):=\sup \{\limsup_ tu(\sigma,t)| \quad \sigma \text{ is a strategy available at }x\} \] and \(V_ M(x)\) defined as above but with the choice of strategies restricted to universally measurable functions. The main theorem of the paper, saying that \(V=V_ M\) and being upper analytic whenever \({\mathcal E}\) is analytic, generalizes a result of \textit{R. E. Strauch} [Trans. Am. Math. Soc. 126, 64-72 (1967; Zbl 0166.157)]. The proof is based on a theorem of \textit{Y. N. Moschovakis} [Descriptive set theory. (1980; Zbl 0433.03025), Theorem 7.C.8]. The authors also give alternative characterizations of V.
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 60G35
zbMATH DE Number: 4137110
zbMATH Keywords: analytic functions
zbMATH Keywords: gambling problem
zbMATH Keywords: finitely additive measures
zbMATH Keywords: stopping time
reviewed by: Andrzej Wieczorek
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4137110