A stochastic game of optimal stopping and order selection
The authors study the unrestricted version of the stochastic two person zero-sum game introduced by \textit{J. P. Gilbert} and \textit{F. Mosteller} [J. Am. Stat. Assoc. 61, 35-73 (1966)] and \textit{Y. S. Chow}, \textit{S. Moriguti}, \textit{H. Robbins} and \textit{S. M. Samuels} [Israel J. Math. 2 (1964), 81-90 (1965; Zbl 0149.144)] in which player 2 compiles \(Y_ 1, Y_ 2, \ldots, Y_ n\) by rearranging the values of an i.i.d. sequence with continuous distribution known to both players. In the unrestricted game, where \({\mathcal J}\) is the class of all exchangeable distributions and player 1 aims to maximize the probability of stopping at the largest observation, the class of all nonanticipating stopping rules \(T\) is considered. Using a convexity idea, \textit{T. P. Hill} and \textit{D. P. Kennedy} [see Ann. Appl. Probab. 2, No. 2, 503-517 (1992; Zbl 0758.60041)] constructed minimax distributions for certain payoff functions and introduced a sequence of r.vs., called Bernoulli pyramid, that generates the process of relative ranks with an independence property. The authors main construction of the traveller's process yields a sequence of observations with the same rank properties as that of the Bernoulli pyramid, leading in some cases to a solution analogous to the solution of Hill and Kennedy. They also characterize explicitly those payoff functions that admit a solution via Bernoulli pyramid or traveller's process.
scientific article; zbMATH DE number 780994