A simple proof of Sen's possibility theorem on majority decisions

Abstract

Condorcet’s voting paradox shows that pairwise majority voting may lead to cyclical majority preferences. In a famous paper, Sen (1966) identified a general condition on a profile of individual preference orderings, called triplewise value-restriction, which is sufficient for the avoidance of such cycles. This note aims to make Sen’s result easily accessible. We provide an elementary proof of Sen's possibility theorem and a simple reformulation of Sen’s condition. We discuss how Sen’s condition is logically related to a number of precursors. Finally, we state a necessary and sufficient condition for the avoidance of cycles, and suggest that, although there is still some logical space between that condition and Sen’s sufficient condition, Sen’s condition cannot be further generalized in an appealing way.