A note on Murakami’s theorems and incomplete social choice without the Pareto principle

In Arrovian social choice theory assuming the independence of irrelevant alternatives, Murakami (Logic and social choice, Dover Publications, New York, 1968) proved two theorems about complete and transitive collective choice rules satisfying strict non-imposition (citizens’ sovereignty), one being a dichotomy theorem about Paretian or anti-Paretian rules and the other a dictator-or-inverse-dictator impossibility theorem without the Pareto principle. It has been claimed in the later literature that a theorem of Malawski and Zhou (Soc Choice Welf 11(2):103–107, 1994) is a generalization of Murakami’s dichotomy theorem and that Wilson’s impossibility theorem (J Econ Theory 5(3):478–486, 1972) is stronger than Murakami’s impossibility theorem, both by virtue of replacing Murakami’s assumption of strict non-imposition with the assumptions of non-imposition and non-nullness. In this note, we first point out that these claims are incorrect: non-imposition and non-nullness are together equivalent to strict non-imposition for all transitive collective choice rules. We then generalize Murakami’s dichotomy and impossibility theorems to the setting of incomplete social preference. We prove that if one drops completeness from Murakami’s assumptions, his remaining assumptions imply (i) that a collective choice rule is either Paretian, anti-Paretian, or dis-Paretian (unanimous individual preference implies noncomparability) and (ii) that adding proposed constraints on noncomparability, such as the regularity axiom of Eliaz and Ok (Games Econ Behav 56:61–86, 2006), restores Murakami’s dictator-or-inverse-dictator result.