  This note considers a variation of the full-information secretary problem
where the random variables to be observed are independent and identically
distributed. Consider $X_1,\dots,X_n$ to be an independent sequence of random
variables, let $M_n:=\max\{X_1,\dots,X_n\}$, and the objective is to select the
maximum of the sequence. What is the maximum probability of "stopping at the
maximum"? That is, what is the stopping time $\tau$ adapted to $X_1,...,X_n$
that maximizes $P(X_{\tau}=M_n)$? This problem was examined by Gilbert and
Mosteller \cite{GilMost} when in addition the common distribution is
continuous. The optimal win probability in this case is denoted by
$v_{n,max}^*$. What if it is desired to "stop within a proportion of the
maximum"? That is, for $0<\alpha<1$, what is the stopping rule $\tau$ that
maximizes $P(X_{\tau} \geq \alpha M_n)$? In this note both problems are treated
as games, it is proven that for any continuous random variable $X$, if $\tau^*$
is the optimal stopping rule then $P(X_{\tau^*} \geq \alpha M_n)\geq
v_{n,max}^*$, and this lower bound is sharp. Some examples and another
interesting result are presented.
