Given a graph G and a set X ⊆ V(G), the relative Wiener index of X in G is defined as
WX (G) = {u,v}∈X
2 dG(u, v) . The graphs G (of even order) in which for every partition
V(G) = V1 +V2 of the vertex set V(G) such that |V1| = |V2| we haveWV1 (G) = WV2 (G) are
called equal opportunity graphs. In this note we prove that a graph G of even order is an
equal opportunity graph if and only if it is a distance-balanced graph. The latter graphs are
known by several characteristic properties, for instance, they are precisely the graphs G in
which all vertices u ∈ V(G) have the same total distance DG(u) = v∈V(G) dG(u, v). Some
related problems are posed along the way, and the so-called Wiener game is introduced.