On endowments and indivisibility: partial ownership in the Shapley–Scarf model

We introduce a parameterized measure of partial ownership, the α -endowment lower bound , appropriate to probabilistic allocation. Strikingly, among all convex combinations of efficient and group strategy-proof rules, only Gale’s Top Trading Cycles is sd efficient and meets a positive α -endowment lower bound (Theorem 2 ); for efficiency, partial ownership must in fact be complete. We also characterize the rules meeting each α -endowment lower bound (Theorem 1 ). For each bound, the family is a semilattice ordered by strength of ownership rights. It includes rules where agents’ partial ownership lower bounds are met exactly, rules conferring stronger ownership rights, and the full endowments of TTC. This illustrates the trade-off between sd efficiency and flexible choice of ownership rights.