## Bernoulli

- Bernoulli
- Volume 8, Number 1 (2002), 39-52.

### Prophet inequalities for optimal stopping rules with probabilistic recall

David Assaf and Ester Samuel-Cahn

#### Abstract

Let *X*_{i}, *i = 1, ..., n*, be independent random variables, and consider an optimal stopping problem where an observation not chosen in the past is still available *i* steps later with some probability *p*_{i}, *1 ≥ p*_{1}* ≥ ... ≥ p*_{n -1}* ≥ 0*. Only one object may be chosen. After formulating the general solution to this optimal stopping problem, we consider `prophet inequalities' for this situation. Let *V*_{\bf p}* (X*_{1}*, ..., X*_{n}*)* be the optimal value to the statistician. We show that for all non-trivial, non-negative *X*_{i} and all *n ≥ 2*, the `ratio prophet inequality' *\rm E[ \max (X*_{1}*, ..., X*_{n}*)] < (2 - p*_{n -1}* ) V*_{\bf p}* (X*_{1}*, ..., X*_{n}*)* holds, and *2 - p*_{n -1} is the `best constant'. This generalizes the classical prophet inequality with no recall, in which the best constant is 2. For any *0 ≤ X*_{i}* ≤ 1*, the `difference prophet inequality' *\rm E[\max (X*_{1}*, ..., X*_{n}*)] - V*_{\bf p}* (X*_{1}*, ..., X*_{n}*) ≤ (1- p*_{n-1}*) [ 1 - (1 - p*_{n - 1}*)*^{1/2}* ]*^{2}* / p*^{2}_{n-1} holds. Prophet regions are also discussed.

#### Article information

**Source**

Bernoulli, Volume 8, Number 1 (2002), 39-52.

**Dates**

First available in Project Euclid: 10 March 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1078951088

**Mathematical Reviews number (MathSciNet)**

MR2002m:60074

**Zentralblatt MATH identifier**

1006.60036

**Keywords**

backward solicitation optimal stopping probabilistic recall prophet inequalities, prophet region recall

#### Citation

Assaf, David; Samuel-Cahn, Ester. Prophet inequalities for optimal stopping rules with probabilistic recall. Bernoulli 8 (2002), no. 1, 39--52. https://projecteuclid.org/euclid.bj/1078951088