Conditions for Transitive and Quasi-Transitive Majority Decisions

Abstract

This paper has two objectives. First, we prove certain general theorems which reveal the close relationship between two sets of results in the recent discussion of the conditions (formulated in terms of restricted preferences) for consistent majority decisions. The first set of these results is concerned with sufficient (and sometimes with necessary) conditions for transitive majority decisions over a triple of alternatives (a) for the case where individual weak preference relations are orderings2 and the number of individuals concerned over the triple (that is, not indifferent between all the three alternatives) is odd (Arrow, 1963; Black, 1958; Inada, 1964, 1969; Majumdar, 1969; Sen, 1966; Vickrey, 1960; Ward, 1965) and (b) for the case where individual weak preference relations are reflexive, connected and quasi-transitive3 but not necessarily transitive (from now on such a weak preference relation will be called a Q-type weak preference relation), and the number of concerned individuals having transitive preferences is odd (Fishburn, 1970a; Batra and Pattanaik, 1972). The second set of these results is concerned with sufficient (and sometimes with necessary) conditions for quasi-transitive majority decisions for the case where individual weak preference relations are orderings (Pattanaik, 1968; Sen, 1969; Sen and Pattanaik, 1969) and also for the case where individual weak preference relations are of Q-type (Fishburn, 1970a, b; Inada, 1970; Pattanaik, 1970). We prove three theorems which show the tight relation that exists between conditions established by results of the former type and those established by results of the latter type. Our second objective is to use the theorems mentioned above together with a theorem (on quasi-transitive majority decisions with Q-type individual preferences) proved by Inada (1970) and Fishburn (1970b) to complete a line of enquiry initiated by Fishburn (1970a). Fishburn established sufficient conditions for transitive majority decisions when individual preferences are of Q-type and the number of concerned individuals with transitive preferences is odd. (Strictly speaking, he assumed that the total number of concerned individuals is