Coordinate-wise Median: Not Bad, Not Bad, Pretty Good
We consider the facility location problem in two dimensions. In particular, we consider a setting where agents have Euclidean preferences, defined by their ideal points, for a facility to be located in ℝ^2. For the utilitarian objective and an odd number of agents, we show that the coordinate-wise median mechanism (CM) has a worst-case approximation ratio (WAR) of √(2)√(n^2+1)/n+1. Further, we show that CM has the lowest WAR for this objective in the class of strategyproof, anonymous, continuous mechanism. For the p-norm social welfare objective, we find that the WAR for CM is bounded above by 2^3/2-2/p for p≥ 2. Since it follows from previous results in one-dimension that any deterministic strategyproof mechanism must have WAR at least 2^1-1/p, our upper bound guarantees that the CM mechanism is very close to being the best deterministic strategyproof mechanism for p≥ 2.
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