Object reallocation problems under single-peaked preferences: two characterizations of the crawler

In object reallocation problems, if preferences are strict but otherwise unrestricted, TTC is the leading rule: It is the only rule satisfying efficiency, the endowment lower bound, and strategy-proofness. However, on the subdomain of single-peaked preferences, Bade (J Econ Theory 180:81–99, 2019) defines a new rule, the “crawler”, which also satisfies these properties, and in fact enjoys a stronger strategic property. We identify additional interesting properties that the crawler satisfies, and provide two characterizations of this rule. The first characterization is based on the endowment lower bound and two invariance properties, “adjacent-endowment-swapping invariance” and “separability”. The second characterization is based on the endowment lower bound, strategy-proofness, adjacent-endowment-swapping invariance, and another invariance property, “non-bossiness”.