Variable-population voting rules

Let X be a set of social alternatives, and let V be a set of `votes' or `signals'. (We do not assume any structure on X or V). A `variable population voting rule' F takes any number of anonymous votes drawn from V as input, and produces a nonempty subset of X as output. The rule F satisfies `reinforcement' if, whenever two disjoint sets of voters independently select some subset Y of X, the union of these two sets will also select Y. We show that F satisfies reinforcement if and only if F is a `balance rule'. If F satisfies a form of neutrality, then F is satisfies reinforcement if and only if F is a scoring rule (with scores taking values in an abstract linearly ordered abelian group R); this generalizes a result of Myerson (1995). We also discuss the sense in which the balance or scoring representation of F is unique. Finally, we provide a characterization of two scoring rules: `formally utilitarian' voting and `range voting'. a