A game with three players
type: Item
instance of: scholarly article
title: A game with three players
zbMATH Open document ID: 0664.60049
DOI: 10.1016/0167-7152(88)90089-2
author: Dennis Sandell
published in: Statistics \& Probability Letters
publication date: 1988-01-01T00:00:00Z
review text: This game consists of a sequence of rounds. In a round three players \(A_ 1\), \(A_ 2\), \(A_ 3\) toss a symmetric coin. If one player gets a result different from those of the others, he receives one dollar from each. If they all obtain the same result they toss their coins again. The initial amount of money of \(A_ i\) is \(a_ i\) dollars, \(i=1,2,3\). If T is the number of rounds until one of the players is ruined, it is proved that \[ E(T)=a_ 1a_ 2a_ 3/(a-2),\quad a=a_ 1+a_ 2+a_ 3. \] This result is obtained by a martingale method. More precisely, if \(Z_ i(n)\) is the capital of player \(A_ i\) after n rounds, \(i=1,2,3\), and \[ S_ n=Z_ 1(n)Z_ 2(n)Z_ 3(n)+n(a-2),\quad \sigma_ n=\sigma (Z_ i(k),\quad i=1,2,3,\quad 0\leq k\leq n) \] the author proves that \((S_ n,\sigma_ n)\) is a martingale, and that the optional sampling theorem holds for \((S_ n)\) and T, i.e. we have \(E(S_ T)=E(S_ 0).\) Since \(S_ 0=a_ 1a_ 2a_ 3\) and \(S_ T=T(a-2)\), the main result is deduced easily.
Mathematics Subject Classification ID: 60G40
zbMATH DE Number: 4086665
zbMATH Keywords: stopping time
zbMATH Keywords: martingale method
zbMATH Keywords: optional sampling theorem
reviewed by: J.-P. Lepeltier
MaRDI profile type: Publication
full work available at URL: 0167-7152(88)90089-2
OpenAlex ID: W2037138968
cites work: Q3956129
cites work: Q3848153
description: scientific article; zbMATH DE number 4086665