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 in the Euclidean plane, a rule assigns to each profile of reported dips a point of the polytope. A point <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$x$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>x</mi> </math> </EquationSource> </InlineEquation> of the polytope is called single-best if there is a point <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$y$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>y</mi> </math> </EquationSource> </InlineEquation> of the polytope such that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$x$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>x</mi> </math> </EquationSource> </InlineEquation> is the unique point of the polytope at maximal distance from <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$y$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>y</mi> </math> </EquationSource> </InlineEquation>. 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. Copyright Springer-Verlag Berlin Heidelberg 2014
