We consider domains with a natural property called top-circularity. We show that if such a domain satisfies either the maximal conflict property or the weak conflict property, then it is dictatorial. We obtain the result in Sato :331–342, 2010) as a corollary. Furthermore, it follows from our results that the union of a single-peaked domain and a single-dipped domain is dictatorial.

In this paper, we consider choice functions that are unanimous, anonymous, symmetric, and group strategy-proof and consider domains that are single-peaked on some tree. We prove the following three results in this setting. First, there exists a unanimous, anonymous, symmetric, and group strategy-proof choice function on a path-connected domain if and only if the domain is single-peaked on a tree and the number of agents is odd. Second, a choice function is unanimous, anonymous, symmetric, and group strategy-proof on a single-peaked (...) domain on a tree if and only if it is the pairwise majority rule and the number of agents is odd. Third, there exists a unanimous, anonymous, symmetric, and strategy-proof choice function on a strongly path-connected domain if and only if the domain is single-peaked on a tree and the number of agents is odd. As a corollary of these results, we obtain that there exists no unanimous, anonymous, symmetric, and group strategy-proof choice function on a path-connected domain if the number of agents is even. (shrink)