Strongly maximal matchings in infinite weighted graphs

Given an assignment of weights w to the edges of an infinite graph G, a matching M in G is called strongly w-maximal if for any matching N there holds ∑{w(e) | e ∈ N \ M} ≤ ∑{w(e) | e ∈ M \ N}. We prove that if w assumes only finitely many values all of which are rational then G has a strongly w-maximal matching. This result is best possible in the sense that if we allow irrational values or infinitely many values then there need not be a strongly w-maximal matching.