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 and the division of a non disposable commodity by the uniform rationing rule.
We construct a canonical peak-only rule equalizing in the leximin sense individual gains from an arbitrary benchmark allocation: it is efficient, group-strategyproof, fair, and (for most problems) continuous. These properties leave room for many other rules, except for symmetric non disposable division problems.