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

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## Abstract

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 \(x\) of the polytope is called single-best if there is a point \(y\) of the polytope such that \(x\) is the unique point of the polytope at maximal distance from \(y\). 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.

## Keywords

Single-dipped preferences Pareto optimality Strategy-proofness Committee voting## JEL Classification

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