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Description:
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 1948, Dummett and Farquharson 1961) and the division of a nondisposable commodity by the uniform rationing rule (Sprumont 1991). We construct a canonical peak-only rule that equalizes, 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 nondisposable division problems.