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 (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

Social Choice Theory and Theories of Democracy

本稿では，アローの不可能性定理を中核とする社会的選択理論と民主主義について論じる．まず，社会的選択理論の枠組みと不可能性定理を概観し，その特徴を意味と構造というそれぞれの観点から説明する．そして，制度としての民主主義と討議としての民主主義をそれらの観点から検討する．さらに，討議の不安定的な性質について説明する.This paper argues the relationship between Arrow’s impossibility theorem and　democracy. First, we overview the framework of social choice theory and the Arrow　theorem. Second, we discuss democracy as an institution anddemocracy as deliberation.　Third, the instability of deliberation is argued.特集　社会科学における「善」と「正義