  For object reallocation problems, if preferences are strict but otherwise
unrestricted, the Top Trading Cycles rule (TTC) is the leading rule: It is the
only rule satisfying efficiency, individual rationality, and
strategy-proofness. However, on the subdomain of single-peaked preferences,
Bade (2019) defines a new rule, the "crawler", which also satisfies these three
properties. (i) The crawler selects an allocation by "visiting" agents in a
specific order. A natural "dual" rule can be defined by proceeding in the
reverse order. Our first theorem states that the crawler and its dual are
actually the same. (ii) Single-peakedness of a preference profile may in fact
hold for more than one order and its reverse. Our second theorem states that
the crawler is invariant to the choice of the order. (iii) For object
allocation problems (as opposed to reallocation problems), we define a
probabilistic version of the crawler by choosing an endowment profile at random
according to a uniform distribution, and applying the original definition. Our
third theorem states that this rule is the same as the "random priority rule".
