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

  • Patrick Harless
  • William PhanEmail author
Research Article


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.


Object reallocation Top trading cycles \(\alpha \)-endowment 

JEL Classification

D63 D70 



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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of ArizonaTucsonUSA
  2. 2.North Carolina State UniversityRaleighUSA

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