We prove a general possibility result for collective decision problems where individual allocations are one-dimensional, preferences are single-peaked (strictly convex), and feasible allocation profiles cover a closed convex set. Special cases include the celebrated median voter theorem (Black...

A private commodity is divided among agents with single peaked preferences over their share. A rationing method elicits individual peaks (demands); if the commodity is overdemanded (resp. underdemanded), no agent receives more (resp. less) than his peak. A fixed path rationing method allocates...