Optimal selection problems based on exchangeable trials

Abstract

We consider optimal stopping problems with loss function q depending on the rank of the stopped random variable. Samuels asked whether there exists an exchangeable sequence of random variables $X_1, \dots, X_n$ without ties for which the observation of the values of the $X_i$'s gives no advantage in comparison with the observation of just the relative ranks of the variables. We call distributions of the sequences with this property q-noninformative and derive necessary and sufficient conditions for this property. Extending an impossibility result of B. Hill, we show that, for any $n > 1$, there are certain losses q for which q-noninformative distributions do not exist. Special attention is given to the classical problem of minimizing the expected rank: for n even we construct explicitly universal randomized stopping rules which are strictly better than the rank rules for any exchangeable sequence.