The voter who wasn't there: Referenda, representation and abstention

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 formalize this concern and reach our main result: no voting rule can ensure representation if abstention is possible, unless 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: rather than imposing arbitrary restrictions on the preference domain of non-voters, we recommend instead that a conscious choice be made on how abstention is to be interpreted and that this choice be used to derive the corresponding referendum rule.