## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 5, Number 1 (1995), 310-321.

### A Stochastic Game of Optimal Stopping and Order Selection

Alexander V. Gnedin and Ulrich Krengel

#### Abstract

We study the following two-person zero-sum game. $n$ random numbers are drawn independently from a continuous distribution known to both players. Player 2 observes all the numbers and selects an order to present them to the opponent. Player 1 learns the numbers sequentially as they are presented and may stop learning whenever he/she pleases. If the stop occurred at the number that is the $k$th largest among all $n$ numbers, Player 1 pays the amount $q(k)$ to Player 2, where $q(1) \leq \cdots \leq q(n)$ is a given payoff function. Player 1 aims to minimize the expected payoff; Player 2 aims to maximize it. We find an explicit solution of the game for a wide class of payoff functions including those $q$'s typically considered in the context of best choice problems.

#### Article information

**Source**

Ann. Appl. Probab., Volume 5, Number 1 (1995), 310-321.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177004842

**Digital Object Identifier**

doi:10.1214/aoap/1177004842

**Mathematical Reviews number (MathSciNet)**

MR1325055

**Zentralblatt MATH identifier**

0824.62079

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

**Keywords**

Optimal stopping rank order selection arrangement best-choice problem minimax strategy

#### Citation

Gnedin, Alexander V.; Krengel, Ulrich. A Stochastic Game of Optimal Stopping and Order Selection. Ann. Appl. Probab. 5 (1995), no. 1, 310--321. doi:10.1214/aoap/1177004842. https://projecteuclid.org/euclid.aoap/1177004842