Reallocation with Priorities

We consider a reallocation problem with priorities where each agent is initially endowed with a house and is willing to exchange it but each house has a priority ordering over the agents of the market. In this setting, it is well known that there is no individually rational and stable mechanism. As a result, the literature has introduced a modified stability notion called µ0-stability. In contrast to college admission problems, in which priorities are present but there is no initial endowment, we show that the modified Deferred Acceptance mechanism identified in the literature is not the only individually rational, strategy-proof and µ0-stable mechanism. By introducing a new axiom called the independence of irrelevant agents and using the standard axiom of unanimity, we show that the modified Deferred Acceptance mechanism is the unique mechanism that is individually rational, strategy-proof, µ0-stable, unanimous and independent of irrelevant agents.