Unbiased Canonical Set-Valued Oracles Via Lattice Theory

An oracle that tells you the probability of some future event can change that very probability because you act on the answer. We argue that this performativity is OK as people consult oracles to be informed, and hence moved, by the answer. We worry about instead that asking for a self-consistent answer, one that still holds once it has been announced, may leave the oracle with several answers to pick from, and whichever rule it uses to pick is a lever it could learn to pull. We propose to take away that choice: The oracle reports a credal set instead of a point estimate, which lifts the oracle's reaction function to an isotone operator on a lattice. We make the oracle report that operator's least fixed point, which exists because of Knaster and Tarski. As that answer is fixed by a rule laid down in advance, nothing is left to choose by the oracle. We show that solution exists, is self-consistent, is never empty, can be computed by iterating from below, and equals the ordinary point estimate if the question is not performative after all. For simple queries about probabilities, we propose to restrict answers to intervals and show that, under a mild monotonicity assumption, the answer is simply the interval from the no-information baseline to the self-fulfilling equilibrium one would end up at by iteratively querying a point oracle until the answer is self-consistent. As our proposal is purely order-theoretic, it carries over unchanged to arbitrary random variables and to bounded continuous statistics of their law. Finally we show that a fixed finite family of polytopes suffices to approximate every answer uniformly, which turns the construction into a terminating computation. We close by placing the construction inside the Scientist AI programme, where it offers a choice-free criterion for a step that programme currently hands to audited human judgement.