## The Annals of Probability

### Optimal Stopping in an Urn

#### Abstract

An urn contains $N$ objects, labelled with the integers $1, \cdots, N$. One object is removed at a time, without replacement. If after $n$ draws the largest number which has been observed is $m_n$, and the process is terminated, we receive a payoff $f(n, m_n)$. For payoff functions $f$ in a certain class, the optimal time to stop is with draw $$\tau_f = \inf\{n \geqslant 0: m_n - n \geqslant j_n\}$$ where the $j_n$ are computable from a simple algorithm, which permits also exact computation of the value $$V_f = E\{f(\tau_f, m_{\tau_f})\}.$$ We also study the behavior of $V_f$ when $N$ is large in special cases.

#### Article information

Source
Ann. Probab., Volume 8, Number 3 (1980), 451-464.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176994720

Digital Object Identifier
doi:10.1214/aop/1176994720

Mathematical Reviews number (MathSciNet)
MR573286

Zentralblatt MATH identifier
0434.60046

JSTOR