  We provide novel simple representations of strategy-proof voting rules when
voters have uni-dimensional single-peaked preferences (as well as
multi-dimensional separable preferences). The analysis recovers, links and
unifies existing results in the literature such as Moulin's classic
characterization in terms of phantom voters and Barber\`a, Gul and Stacchetti's
in terms of winning coalitions ("generalized median voter schemes"). First, we
compare the computational properties of the various representations and show
that the grading curve representation is superior in terms of computational
complexity. Moreover, the new approach allows us to obtain new
characterizations when strategy-proofness is combined with other desirable
properties such as anonymity, responsiveness, ordinality, participation,
consistency, or proportionality. In the anonymous case, two methods are single
out: the -- well know -- ordinal median and the -- most recent -- linear
median.
