Relatives and friends of an end-stage renal disease patient who offer to donate a kidney are often found to be incompatible with their intended recipients. Kidney paired donation matches one patient and his incompatible donor with another patient and donor in the same situation for an organ exchange. Let patient- donor pairs be the vertices of an undirected graph G, with an edge connecting any two reciprocally compatible vertices. A matching in G is a feasible set of paired donations. We describe various optimization problems on kidney paired donation graphs G and the merits of each in clinical transplantation. Because some matches are geographically undesirable, and the expected lifespan of a transplanted kidney depends on the immunologic concordance of donor and recipient, we weight the edges of G and seek a maximum edge-weight matching. Unfortunately, such matchings might not have the maximum cardinality; there is a risk of an unpredictable trade-off between the quality and quantity of paired donations. We propose an edge-weighting of G which guarantees that every matching with maximum weight also has maximum cardinality, and also maximizes the number of transplants for an exceptional subset of recipients, while reducing travel and favoring immunologic concordance.