The three perenial goals of prior-free mechanism design, efficiency, incentive compatibility (strategyproofness) and fairness (horizontal equity) are known to be incompatible for abstract voting (Gibbard-Satterthwaite (1974)), the distribution of private goods (Hurwicz (1972) Serizawa (2002)), the provision of public goods (Green La¤ont (1977) Serizawa (1999)), and more. Two notable “one-dimensional” exceptions are voting with single-peaked preferences (Black (1958), Dummett and Farquharson (1961), Pattanaik (1974)) and dividing a non disposable commodity with convex private preferences (Sprumont (1991), Ching (1994)).
We show that these three goals are still compatible in any collective decision problem where each agent cares about a one-dimensional parameter and preferences are convex, provided the range of allocation pro…les is an arbitrary convex and closed set. We construct a mechanism equalizing bene…ts from a benchmark allocation, that is efficient, (strongly) groupstrategyproof, peak-only, continuous, and fair: it respects the symmetries of the problem and rules out envy between symmetric agents. Applications of our result include: the division of shares in a joint venture under quota and other linear constraints; the coordination of workloads between workers and managers, or between several teams in a production chain when inputs are substitutes or complementary; and the distribution of shipping loads under bilateral constraints. In contrast to the negative message of the seminal results, in the one-dimensional contexts we expect an embarrassment of riches, leaving much ‡exibility to the mechanism designer.