  We present an algebraic account of the Wasserstein distances $W_p$ on
complete metric spaces, for $p \geq 1$. This is part of a program of a
quantitative algebraic theory of effects in programming languages. In
particular, we give axioms, parametric in $p$, for algebras over metric spaces
equipped with probabilistic choice operations. The axioms say that the
operations form a barycentric algebra and that the metric satisfies a property
typical of the Wasserstein distance $W_p$. We show that the free complete such
algebra over a complete metric space is that of the Radon probability measures
with finite moments of order $p$, equipped with the Wasserstein distance as
metric and with the usual binary convex sums as operations.
