Source
Blackwell (1953), LIT-781, Theorems 1 to 8, as read in NOTE-595. The finite-outcome case is THEORY-717, and the case of a simulated experiment whose outcome space is standard Borel is THEORY-156, which this entry extends. Both are Active.
The claim
For experiments with finitely many states and decision problems with bounded losses, α does at least as well as β in every problem exactly when a Markov kernel turns α’s observation into one distributed as β’s under every state, whatever the outcome spaces. The paper reaches the general case by reducing both orders to a comparison of standard measures on the simplex (Theorems 1 and 2, quoted from Blackwell 1951), and proving in Theorem 8 that the convex order there gives a mean-preserving transformation.
THEORY-156’s note of 2026-10-29 re-derives the converse without the 1951 results. Its last step composes a martingale coupling with the conditional law of β’s outcome given its posterior, which is a kernel when β’s outcome space is standard Borel. Without that regularity the record has only the paper’s word, resting on a theorem it holds as quoted.
What would settle it
The promote_when, carried over unchanged: a second, independent proof of the arbitrary-outcome case read and checked, or the 1953 proof checked together with the 1951 proofs of the two results it quotes. QUESTION-321 asks the same thing as a question, and adds that a counterexample, an outcome space that is not standard Borel where no Markov kernel exists, would answer it the other way.