Distributed Facility Location Games with Candidate Locations

We study the distributed facility location games with candidate locations, where agents on a line are partitioned into groups. Both desirable and obnoxious facility location settings are discussed. In distributed location problems, distortion can serve as a standard for quantifying performance, measuring the degree of difference between the actual location plan and the ideal location plan. For the desirable setting, under the max of sum cost objective, we give a strategyproof distributed mechanism with $5$-distortion, and prove that no strategyproof mechanism can have a distortion better than $\sqrt{2}+1$. Under the sum of max cost objective, we give a strategyproof distributed mechanism with $5$-distortion, and prove that no strategyproof mechanism can have a distortion better than $\frac{\sqrt{5}+1}{2}$. Under the max of max cost, we get a strategyproof distributed mechanism with $3$-distortion, and prove that no strategyproof mechanism can have a distortion better than $\frac{\sqrt{5}+1}{2}$. For the obnoxious setting, under three social objectives, we present that there is no strategyproof mechanism with bounded distortion in the case of discrete candidate locations, and no group strategyproof mechanism with bounded distortion in the case of continuous candidate locations.