Author:
Hsiau Shoou-Ren,Yao Yi-Ching
Abstract
Abstract
In the classical gambler’s ruin problem, the gambler plays an adversary with initial capitals z and
$a-z$
, respectively, where
$a>0$
and
$0< z < a$
are integers. At each round, the gambler wins or loses a dollar with probabilities p and
$1-p$
. The game continues until one of the two players is ruined. For even a and
$0<z\leq {a}/{2}$
, the family of distributions of the duration (total number of rounds) of the game indexed by
$p \in [0,{\frac{1}{2}}]$
is shown to have monotone (increasing) likelihood ratio, while for
${a}/{2} \leq z<a$
, the family of distributions of the duration indexed by
$p \in [{\frac{1}{2}}, 1]$
has monotone (decreasing) likelihood ratio. In particular, for
$z={a}/{2}$
, in terms of the likelihood ratio order, the distribution of the duration is maximized over
$p \in [0,1]$
by
$p={\frac{1}{2}}$
. The case of odd a is also considered in terms of the usual stochastic order. Furthermore, as a limit, the first exit time of Brownian motion is briefly discussed.
Publisher
Cambridge University Press (CUP)
Subject
Statistics, Probability and Uncertainty,General Mathematics,Statistics and Probability