Before revision:  We extend  and at the same time simplify this result. Moreover we show that the difficulties they highlighted relaxing the assumption on the cardinality of the range, disappear under a proper assumption on the domain.
After revision:  We extend and at the same time simplify it. Our analysis points out the role of individual strategy-proofness in deriving the preference reversal property, giving back to the latter its original individual nature (cfr. Eliaz, 2004). Moreover, we show that the difficulties they highlighted relaxing the assumption on the cardinality of the range, disappear under a proper assumption on the domain.