In a model with finitely many agents who have single-dipped Euclidean preferences on a polytope in the Euclidean plane, a rule assigns to each profile of reported dips a point of the polytope. A point of the polytope is called single-best if there is a point of the polytope such that is the unique point of the polytope at maximal distance from . It is proved that if the polytope does not have either exactly two single-best points or exactly four single-best points which form the vertices of a rectangle, then any Pareto optimal and strategy-proof rule is dictatorial. If the polytope has exactly two single-best points, then there are non-dictatorial strategy-proof and Pareto optimal rules, which can be described by committee voting (simple games) between the two single-best points. This also holds if there are exactly four single-best points which form the vertices of a rectangle, but in that case, we limit ourselves to describing an example of such a rule. The framework under consideration models situations where public bads such as garbage dumping grounds or nuclear plants have to be located within a confined region.
Öztürk, M., Peters, H. J. M., & Storcken, A. J. A. (2014). On the location of public bads: strategy-proofness under two-dimensional single-dipped preferences. Economic Theory, 56(1), 83-108. https://doi.org/10.1007/s00199-013-0785-8