Generalizations of bold play in red and black.
type: Item
instance of: scholarly article
title: Generalizations of bold play in red and black.
zbMATH Open document ID: 1047.60091
DOI: 10.1016/S0304-4149(00)00069-7
author: Marcus Pendergrass
author: Kyle T. Siegrist
published in: Stochastic Processes and their Applications
publication date: 2004-09-22T00:00:00Z
review text: In the game of red and black the player starts with an initial fortune \(x\) (normalized to lie in \([0,1]\)), and then bets on a sequence of Bernoulli trials, until he either reaches the target (1), or is ruined (0). A famous result is that in the subfair case, an optimal strategy is ``bold play'' whereby at each trial the player bets \(\min (x,1-x)\), where \(x\) is the fortune. In addition to optimality, the bold strategy has a number of interesting properties. In particular before the game ends, the fortune process is deterministic, following the map \(x\mapsto 2x\) mod 1. The purpose of this paper is to study relevant Markov chains in a more general setting that preserves some of these properties. A general class of partially deterministic chains is introduced. These chains follow a deterministic map before entry into a given set of states \(D\). Results for the distribution of the hitting time and place are obtained. Further, chains on the sequence space \(\{0,1,\dots ,k-1\}^\infty \) are studied. When \(k=2\) and the sequences are interpreted as the binary coordinates of the player's fortune, the chain corresponds to standard bold play in red and black. For \(k>2\) the models obey similar properties but, unfortunately, the chains on the sequence space do not seem to have a natural gambling interpretation. Finaly, Markov chains that naturally generalize bold play with \(k\) players are studied.
Mathematics Subject Classification ID: 60K10
Mathematics Subject Classification ID: 60J05
zbMATH DE Number: 2102380
zbMATH Keywords: red and black
zbMATH Keywords: bold play
zbMATH Keywords: Markov chain
zbMATH Keywords: hitting time
reviewed by: Martin Janžura
MaRDI profile type: Publication
cites work: Q3794956
cites work: Q5343895
cites work: A construction of markov processes by piecing out
cites work: Discounted and rapid subfair red-and-black
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cites work: Dynamic programming and gambling models
cites work: The Bold Strategy in Presence of House Limit
full work available at URL: s0304-4149(00)00069-7
OpenAlex ID: W2093374704
Wikidata QID: Q127374553
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description: scientific article; zbMATH DE number 2102380