PARETO-OPTIMALITY, ANONYMITY, AND STRATEGY-PROOFNESS IN LOCATION-PROBLEMS

Generalized location problems with n agents are considered, who each report a point in m-dimensional Euclidean space. A solution assigns a compromise point to these n points, and the individual utilities for this compromise point are equal to the negatives of the Euclidean distances to the individual positions. For m = 2 and n odd, it is shown that a solution is Pareto optimal, anonymous, and strategy-proof if, and only if, it is obtained by taking the coordinatewise median with respect to a pair of orthogonal axes. Further, for all other situations with m greater-than-or-equal-to 2, such a solution does not exist. A few results concerning other solution properties, as well as different utility functions, are discussed.