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

## Abstract

We introduce a parameterized measure of partial ownership, the \(\alpha \)*-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 \(\alpha \)*-endowment lower bound* (Theorem 2); for efficiency, partial ownership must in fact be complete. We also characterize the rules meeting each \(\alpha \)*-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.

## Keywords

Object reallocation Top trading cycles \(\alpha \)-endowment## JEL Classification

D63 D70## Notes

### Acknowledgements

We gratefully acknowledge comments and suggestions from Jens Gudmundsson, Vikram Manjunath, Thayer Morrill, William Thomson, and two anonymous referees, as well as seminar participants at Lund University, Helsinki Center of Economic Research, the 2015 Workshop on Game Theory and Social Choice in Budapest, and GAMES 2016.

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