Social Choice and Welfare 55:243-253 (2020)

Wesley H. Holliday
University of California, Berkeley
Mikayla Kelley
Stanford University
In Arrovian social choice theory assuming the independence of irrelevant alternatives, Murakami (1968) proved two theorems about complete and transitive collective choice rules that satisfy 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 (1994) is a generalization of Murakami’s dichotomy theorem and that Wilson’s (1972) impossibility theorem 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 (2006), restores Murakami’s dictator-or-inverse-dictator result.
Keywords social choice without Pareto  non-imposition   strict non-imposition  citizens’ sovereignty  Wilson’s Theorem  incomplete social preference  regularity  minimal comparability  asusuke Murakam
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Social Choice and the Arrow Conditions.Allan F. Gibbard - 2014 - Economics and Philosophy 30 (3):269-284.
Arrow's Decisive Coalitions.Wesley H. Holliday & Eric Pacuit - 2020 - Social Choice and Welfare 54:463–505.

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