We study the possibility of strategy-proof and efficient mechanisms in pure exchange economies. In his remarkable paper, Zhou (1991) establishes an elegant impossibility result: there is no strategy-proof, efficient, and nondictatorial mechanism in the two-agent case. He conjectures that there is no strategy-proof, efficient, and "non-inversely-dictatorial" mechanism in the case of three or more agents. However, we discover some counterexamples to his conjecture in the case of four or more agents. We present a new interesting open question: Is there any strategy-proof, efficient, and "non-alternately-dictatorial" mechanism?
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