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For object reallocation problems, if preferences are strict but otherwiseunrestricted, the Top Trading Cycles rule (TTC) is the leading rule: It is theonly rule satisfying efficiency, individual rationality, andstrategy-proofness. However, on the subdomain of single-peaked preferences,Bade (2019) defines a new rule, the "crawler", which also satisfies these threeproperties. (i) The crawler selects an allocation by "visiting" agents in aspecific order. A natural "dual" rule can be defined by proceeding in thereverse order. Our first theorem states that the crawler and its dual areactually the same. (ii) Single-peakedness of a preference profile may in facthold for more than one order and its reverse. Our second theorem states thatthe crawler is invariant to the choice of the order. (iii) For objectallocation problems (as opposed to reallocation problems), we define aprobabilistic version of the crawler by choosing an endowment profile at randomaccording to a uniform distribution, and applying the original definition. Ourthird theorem states that this rule is the same as the "random priority rule".

The Crawler: Three Equivalence Results for Object (Re)allocation Problems when Preferences Are Single-peaked