The Order of Binary Experiments under Endogenous Stopping

Abstract: We study the comparison of binary statistical experiments in large samples where information is acquired sequentially and the number of observations can depend on realized evidence. We introduce two orders. Stopping dominance compares experiments by their ability to reproduce the outcomes of arbitrary stopping policies, while decision dominance compares their value in finite decision problems with costly observations. Our main result shows that the two orders coincide and are characterized by coordinatewise dominance of the two directed Kullback--Leibler divergences. Moreover, strict dominance implies eventual strict value dominance in every decision problem in which learning the state can change the optimal action. The key result behind this characterization is an exact simulation theorem: any binary target experiment can be generated from repeated observations of a source experiment with expected sample sizes attaining the two KL lower bounds up to an additive constant that depends only on the source.