New characterizations of strategy-proofness under single-peakedness

We provide novel representations of strategy-proof voting rules applicable when voters have uni-dimensional single-peaked preferences. In particular, we introduce a 'grading curve' representation which is particularly useful when introducing variable electorates. Our analysis recovers, links and unifies existing results in the literature, and provides new characterizations when strategy-proofness is combined with other desirable properties such as ordinality, participation, consistency, and proportionality. Finally, the new representations are used to compute the strategy-proof methods that maximize the ex-ante social welfare for the L 2-norm and a uniform prior. The resulting strategy-proof welfare maximizer is the linear median (or 'uniform median'), that we also characterize as the unique proportional strategy-proof voting rule.