On the location of public bads: strategy-proofness under two-dimensional single-dipped preferences

In a model with finitely many agents who have single-dipped Euclidean preferences on a polytope inthe Euclidean plane, a rule assigns to each profile of reported dips a point of the polytope. Asingle-best point is a point which is the unique point at maximal distance from some other pointof the polytope. It is proved that any strategy-proof and Pareto optimal rule is a dictatorshipunless the polytope has exactly two single-best points or it has exactly four single-best pointswhich form the vertices of a rectangle. In the latter cases strategy-proof and Pareto optimalrules can be obtained by committee voting (simple games) between the single-best alternatives.This framework models situations where public bads such as garbage dumping grounds or nuclearplants have to be located within a confined region.