In a model with finitely many agents who have single-dipped Euclidean preferences on a disc in the Euclidean plane, a rule assigns to each profile of reported dips a point of the disc. It is proved that any strategy-proof and Pareto optimal rule is dictatorial. This framework models situations...

We characterize a generalization of the linear index of individual deprivation. Our class allows for an income increase of a higher-income individual to have a stronger impact the closer they occur to the income of the person under consideration.