In a house allocation with existing tenants model using the TTC mechanism, consider the incentive of an agent to misreport his/her preferences. Can it ever be that misreporting the true preferences by some agent a induces an outcome such that : i) a does not change his/her assignment (i.e. a is assigned to the same house as when truthfully reporting): and ii) some other agent a' is assigned to a different house? If the answer is yes, provide an example, otherwise prove that it is not possible.
If condition i is: a gets better off . The the answer should be not possible, right? Because a TTC mechanism matching is strategic proof. How about in this question?