Referenda, Quorum and the Abstention Paradox∗

We analyze single binary-choice voting rules and identify the presence of the No-Show paradox in this simple setting, as a consequence of specific turnout or quorum conditions that are included in actual rules. Since these conditions are meant to ensure a representative outcome, we try to formalize this concern and reach our main result: no voting rule can ensure representation if abstention is possible, unless specific restrictive assumptions are made on the preference domain of abstainers. We then focus on the main referendum systems and show that appropriate restrictions do make them compatible with representation. The main purpose of our paper is, however, to provide a tool for referendum design. Unawareness of the fact that defining a representative voting system necessarily entails restrictions on the preference domain of non-voters may lead to the imposition of restrictions that do not satisfy any deliberate criteria and generate such problems as the No-Show paradox. We therefore recommend instead that a conscious choice be made on how abstention is to be interpreted and that this choice be used as a tool to derive the corresponding referendum rule. JEL Classification: D71 ∗The idea for this paper started with some jocose but insightful notes written by José João Marques da Silva at the time of the first referendum held in Portugal (1998). With-out being aware of the ”no-show paradox”, he had the right intuition on the pathologies of the Portuguese referendum. Jose ́ João passed away in August 2000. ISEG lost a bright, interested, clever and a friendly scholar. May we dedicate this paper to his memory