Shapley-Scarf Markets with Objective Indifferences

In many object allocation problems, some of the objects may effectively be indistinguishable from each other, such as with dorm rooms or school seats. In such cases, it is reasonable to assume that agents are indifferent between identical copies of the same object. We call this setting ``objective indifferences.'' Top trading cycles (TTC) with fixed tie-breaking has been suggested and used in practice to deal with indifferences in object allocation problems. Under general indifferences, TTC with fixed tie-breaking is not Pareto efficient nor group strategy-proof. Furthermore, it may not select the core, even when it exists. Under objective indifferences, agents are always and only indifferent between copies of the same object. In this setting, TTC with fixed tie-breaking maintains Pareto efficiency, group strategy-proofness, and core selection. In fact, we present domain characterization results which together show that objective indifferences is the most general setting where TTC with fixed tie-breaking maintains these important properties.