Strategy-proof and Pareto efficient allocation of indivisible goods: general impossibility domains

We consider the problem of allocating several homogeneous indivisible goods when monetary transfers among agents are possible. We study the possibility of constructing strategy-proof and Pareto efficient mechanisms on restricted domains of agents' valuation profiles. We show that there is no strategy-proof and Pareto efficient mechanism under the weak domain condition that all agents' sets of possible valuations share at least four common valuations satisfying a certain inequality. Moreover, we prove that this impossibility result is robust to any affine transformation of domains, and we examine this impossibility result when the set of agent's possible valuations consists of finite integers.