Characterizing finite posets whose probabilistic powerdomain are RB-domains

We classify the finite posets whose probabilistic powerdomain is an RB-domain. For a finite nonempty poset $P$, let $\Vone(P)$ be the probability powerdomain of $P$, which is the probability simplex ordered by the stochastic order. We prove that $\Vone(P)$ is an RB-domain if and only if $P$ has a least element and the undirected Hasse graph of $P$ is a tree. Consequently, the probabilistic powerdomain does not preserve RB-domains; the four-point diamond gives a finite counterexample. The proof separates two obstructions. First, if $P$ has no least element, then the face of probability measures supported on the minimal points must be fixed pointwise by every deflation below the identity. Secondly, once a least element exists, the Hasse graph is connected, and a cycle in it makes the local stochastic cone non-simplicial. A Euclidean finite-step cone argument then rules out the finite-valued monotone approximations supplied by the RB property.