The Annals of Applied Probability

A Stochastic Game of Optimal Stopping and Order Selection

Alexander V. Gnedin and Ulrich Krengel

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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

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

First available in Project Euclid: 19 April 2007

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

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


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.

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