For finite binary experiments under the paper’s technical assumptions, two state-specific information scores are enough to determine whether one experiment dominates another when observations stop according to the evidence collected. The scores are directed Kullback–Leibler (KL) divergences, a pair of measures describing how strongly observations distinguish the two possible states, with one calculated in each direction. Within this setting, stopping dominance and decision dominance—being at least as valuable across finite decision problems—are equivalent to having at least as much KL information in both directions.
Nothing in the paper comes from participants or an empirical dataset. It studies finite, full-support, identified binary source and target experiments. Its proposed simulator draws source signals sequentially, decides when to stop from the history it has observed, and adds randomization that does not depend on the true state, allowing it to reproduce the target experiment exactly.
How the conversion works
That exact-conversion result comes with information bounds. Every exact simulator must use an expected number of source observations above a lower bound in each state direction, tied to the target’s directed KL information. The paper constructs one whose expected counts satisfy upper bounds in both states up to an additive term depending only on the source experiment.
The construction uses continuous path revelation: it turns each discrete jump in a likelihood ratio—a running measure of which state the observed history favors—into a continuous path that can be stopped at one of two boundaries.
That ordering also has an implication for finite decision problems in which observations cost something. If one experiment strictly decision-dominates another, the first has weakly higher value when the observation cost is sufficiently small; if no action is optimal in both states, the advantage is strict for all sufficiently small costs.
Adaptive stopping changes the rate
For producing repeated copies of a target experiment from a source, the asymptotic observation rate is the target-to-source directed-KL ratio for each state direction. The overall rate is the larger of the two, so the more demanding state direction sets the pace as the number of target copies grows.
A constructed pair sharpens the contrast with fixed-length sampling. The two experiments have equal directed KL divergences and endogenous conversion rates of 1 in both directions. For every m, an exact simulator’s expected statewise sample count lies between m and m+4, uniformly over the construction’s parameter, while the corresponding fixed-length conversion rates diverge as that parameter tends to zero.
At very small observation costs, a binary-testing calculation gives a leading risk term of c log(1/c), weighted by the prior probabilities and the inverses of the two directed KL divergences. The expression is asymptotic, with a relative remainder that vanishes as c tends to zero.
A result with clear boundaries
The work is a version 1 arXiv preprint rather than an empirical study. It is limited to finite, full-support, identified binary experiments and finite decision problems with small observation costs. For experiments with more than two states, pairwise KL dominance is presented as necessary, but whether it is sufficient remains unresolved.
Paper data and sources
Original title: The Order of Binary Experiments under Endogenous Stopping
Authors: Zihao Li
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text