Abstract : We consider problems of locating a public bad, such as a nuclearplant or a garbage dumping ground, in some specific area like a city or a region. Each individual (agent) is assumed to consider one location as the worst one, his `dip', and to prefer locations at maximal (Euclidean) distance from this dip. The question is to find a Pareto optimal location for the public bad, based on the reported preferences (dips) of the agents. Formally, we consider Pareto optimal and strategy-proof social choice functions defined on the domain of single-dipped preference profiles, where the dips are assumed to be in a compact subset of the plane. Pareto optimality requires a social choice function not to select an alternative if there exists another alternative which is better for someone and not worse for anyone. Strategy-proofness prevents agents from misrepresenting their preferences to get better results for themselves.
Our main results so far are as follows. If the set of alternatives has a regular polygon or a circle as its boundary, then a Pareto optimal and strategy-proof social choice function must be dictatorial unless the polygon is a square. Extension of the study to more general polytopes or convex sets is in progress.

Keywords: Single-dipped Preferences, Strategy-proofness, Plane.