We study matching markets with aligned preferences and establish a connection between common design objectives---stability, efficiency, and fairness---and the theory of optimal transport. Optimal transport gives new insights into the structural properties of matchings obtained from pursuing these objectives, and into the trade-offs between different objectives. Matching markets with aligned preferences provide a tractable stylized model capturing supply-demand imbalances in a range of settings such as partnership formation, school choice, organ donor exchange, and markets with transferable utility where bargaining over transfers happens after a match is formed.

We study two-sided matching as introduced in Gale and Shapley [1962]. In contrast to much of the literature, we ignore stability and characterize all mechanisms which are group strategy-proof, efficient and treat the two sides symmetrically in the sense that the mechanism is invariant with respect to a reflection between the two sides. We find that all group strategy-proof, efficient, and "gender-neutral" mechanisms can be implemented by an algorithm which operates in a sequence of rounds. In each round, two agents are selected, one from each side and their matching is determined before moving on to the next round. We refer to these agents as the "royals." The royals are given their most-preferred available matches whenever possible. However, when their preferences conflict one of two "regimes" is used. The royals are either "matched-by-default" or "unmatched-by-default." In the former case, either of the royals can unilaterally force the other to match with them while in the latter case, they may only match together if both agree. In either case, if this pair of agents is not matched together, each gets their top choices among the set of remaining agents, excluding one another. We call the set of all such mechanisms "royalty mechanisms."