Let $G = (V,E)$ be a connected graph. A probability measure $μ$ on $V$ is called "balanced" if it has the following property: if $T_μ(v)$ denotes the "earth mover's" cost of transporting all the mass of $μ$ from all over the graph to the vertex $v$, then $T_μ$ attains its global maximum at each point in the support of $μ$. We prove a decomposition result that characterizes balanced measures as convex combinations of suitable "extremal" balanced measures that we call "basic." An upper bound on the number of basic balanced measures on $G$ follows, and an example shows that this estimate is essentially sharp.