Characterization of group-strategyproof mechanisms for facility location in strictly convex space

We characterize the class of group-strategyproof mechanisms for the single facility location game in any unconstrained strictly convex space. A mechanism is group-strategyproof,if no group of agents can misreport so that all its members are strictlybetter off. A strictly convex space is a normed vector space where |x+y|<2 holds for any pair of different unit vectors x ≠ y, e.g., any Lp space with p∈ (1,∞). We show that any deterministic, unanimous, group-strategyproof mechanism must be dictatorial, and that any randomized, unanimous, translation-invariant, group-strategyproof mechanism must be 2-dictatorial.Here a randomized mechanism is 2-dictatorial if the lottery output of the mechanism must be distributed on the line segment between two dictators' inputs. A mechanism is translation-invariant if the output of the mechanism follows the same translation of the input. Our characterization directly implies that any (randomized) translation-invariant approximation algorithm satisfying the group-strategyproofness property has a lower bound of 2-approximation for maximum cost (whenever n ≥ 3), and n/2 - 1 for social cost. We also find an algorithm that 2-approximates the maximum cost and n/2-approximates the social cost, proving the bounds to be (almost) tight.