Bettina Klaus () (Harvard Business School, Negotiation, Organizations & Markets Unit) Lars Ehlers () (Departement de Sciences Economiques and CIREQ, Universite de Montreal, Montreal, Quebec)
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Ehlers and Klaus (2003) study so-called house allocation problems and claim to characterize all rules satisfying efficiency, independence of irrelevant objects, and resource-monotonicity on two preference domains (Ehlers and Klaus, 2003, Theorem 1). They explicitly prove Theorem 1 for preference domain R0 which requires that the null object is always the worst object and mention that the corresponding proofs for the larger domain R of unrestricted preferences "are completely analogous." Quesada (2009) in a recent working paper claims to have found a counterexample that shows that Theorem 1 is not correct on the unrestricted domain R. In Lemma 1, we prove that Quesada's (2009) example in not a counterexample to Ehlers and Klaus (2003, Theorem 1). However, in Example 1 and Lemma 2, we demonstrate how to adjust Quesada's (2009) original idea to indeed establish a counterexample to Ehlers and Klaus (2003, Theorem 1) on the general domain R. Quesada (2009) also proposes a way of correcting the result on the general domain R by strengthening independence of irrelevant objects in two ways: in addition to requiring that the chosen allocation should depend only on preferences over the set of available objects (which always includes the null object), he adds two situations in which the allocation should also be invariant when preferences over the null object change. We here demonstrate that it is sufficient to require only one of Quesada's (2009) additional independence requirements to reestablish the result of Theorem 1 on the general domain R. Finally, while Quesada (2009) essentially replicates the original proofs of Ehlers and Klaus (2003) using his stronger independence condition, here we offer a short proof that uses the established result of Theorem 1 for the restricted domain R0.
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