1. =? (1) follows by Theorem 2, as non-image-dictatorship implies nondictatorship. To prove the reverse implication (2) =? (1) assume f is a unanimous, strategy-proof and non-dictatorial choice rule from L(P N , A) to A and undom(P i ) = undom(P j ) for all agents i and j in N. It is sufficient to prove that there are two distinct alternatives a and b with undom(P i ) = {a;The implication (2 a) =? (1) follows by Example 1 and the implication (2 b)