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."

The Gibbard-Satterthwaite Impossibility Theorem [Gibbard, 1973, Satterthwaite, 1975] holds that dictatorship is the only Pareto optimal and strategyproof social choice function on the full domain of preferences. Much of the work in mechanism design aims at getting around this impossibility theorem. Three grand success stories stand out. On the domains of single-peaked preferences, of object assignment, and of quasilinear preferences, there are appealing Pareto optimal and strategyproof social choice functions. We investigate whether these success stories are robust to strengthening strategyproofness to obvious strategyproofness, a stronger incentive property that was recently introduced by Li [2015] and has since garnered considerable attention.