A Stochastic Game of Optimal Stopping and Order Selection

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.