Mechanism design for one-facility location game with obnoxious effects on a line

The classic obnoxious facility game [4], [11] is a special case of our problem when d1=0 and d2=1. In this work, we first study the hardness of approximate mechanism design on this generalized problem, which states that our problem cannot admit any deterministic strategy-proof mechanism with bounded approximation ratio if d1≥12. Then we limit the thresholds to some ranges, both deterministic and randomized strategy-proof mechanisms are studied, and the approximation ratios vary with the specific values of d1 and d2.