Characterization and equilibrium of bichromatic max-sum matchings

We study maximum-sum red-blue matchings and matching equilibrium for finite planar point sets. For a red-blue perfect matching $M = \{(a_i,b_i) : 1 \le i \le n\}$, we define the gain of a directed red cycle as the change in total weight produced by cyclically shifting the corresponding blue partners. We prove that $M$ is maximum-sum if and only if every directed red cycle has nonpositive gain, and we derive a geometric sufficient condition for optimality from cyclic intersections of distance-difference regions. We then characterize balanced matchings, in which all red-blue perfect matchings have the same total weight. Equilibrium is shown to be equivalent to vanishing cycle gains, to an additive form of the distance matrix, and to a common level-set condition for distance-difference functions. In the squared Euclidean case this yields an orthogonality classification, while in the Euclidean case it yields a hyperbolic level-set description and a collinear-separation classification in the nondegenerate setting.